TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2023
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/3355
AB  - The paper shows a way to create a set of tools that allows the construction of different models of spatial deltahedral structures as a result of the game. It is a set of accessories with which participants (school children, students) through play, experimenting with different layouts and combinations of elements of the whole, create spatial forms from uniform elements: sticks of length "a" and ringshaped joint connections. The game uses a STEM learning approach and the "learning by doing" method to provide knowledge about the spatial relationships of elements and static properties of the structure, in the process of playing. The process of forming structures via mentioned connecting elements uses the same principle of connecting supports and nodes as in the real space truss. A similar method is used by some other gaming platforms, such as Geomag, Magna-tiles, Picasso, or Blockaroo. Unlike them, our solution does not use magnets, but prestressing.
PB  - Faculty of Technical Sciences, University of Novi Sad
PB  - Serbian Society for Geometry and Graphics SUGIG
C3  - The 9th International Scientific Conference on Geometry and Graphics ''Mongeometrija 2023''
T1  - Learning while playing - Througie platform for creating models of spatial structures
EP  - 180
SP  - 169
UR  - https://hdl.handle.net/21.15107/rcub_grafar_3355
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2023",
abstract = "The paper shows a way to create a set of tools that allows the construction of different models of spatial deltahedral structures as a result of the game. It is a set of accessories with which participants (school children, students) through play, experimenting with different layouts and combinations of elements of the whole, create spatial forms from uniform elements: sticks of length "a" and ringshaped joint connections. The game uses a STEM learning approach and the "learning by doing" method to provide knowledge about the spatial relationships of elements and static properties of the structure, in the process of playing. The process of forming structures via mentioned connecting elements uses the same principle of connecting supports and nodes as in the real space truss. A similar method is used by some other gaming platforms, such as Geomag, Magna-tiles, Picasso, or Blockaroo. Unlike them, our solution does not use magnets, but prestressing.",
publisher = "Faculty of Technical Sciences, University of Novi Sad, Serbian Society for Geometry and Graphics SUGIG",
journal = "The 9th International Scientific Conference on Geometry and Graphics ''Mongeometrija 2023''",
title = "Learning while playing - Througie platform for creating models of spatial structures",
pages = "180-169",
url = "https://hdl.handle.net/21.15107/rcub_grafar_3355"
}

Obradović, M.,& Mišić, S.. (2023). Learning while playing - Througie platform for creating models of spatial structures. in The 9th International Scientific Conference on Geometry and Graphics ''Mongeometrija 2023''
Faculty of Technical Sciences, University of Novi Sad., 169-180.
https://hdl.handle.net/21.15107/rcub_grafar_3355

Obradović M, Mišić S. Learning while playing - Througie platform for creating models of spatial structures. in The 9th International Scientific Conference on Geometry and Graphics ''Mongeometrija 2023''. 2023;:169-180.
https://hdl.handle.net/21.15107/rcub_grafar_3355 .

Obradović, Marija, Mišić, Slobodan, "Learning while playing - Througie platform for creating models of spatial structures" in The 9th International Scientific Conference on Geometry and Graphics ''Mongeometrija 2023'' (2023):169-180,
https://hdl.handle.net/21.15107/rcub_grafar_3355 .

TY  - CHAP
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2022
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2779
AB  - The paper considers a type of radial pentagon-based tiling consisting of two shapes: triangle and rectangle. The ob tained solution has a spatial interpretation in a 3D arrangement of equilateral triangles and squares dictated by the particular array of concave cupolae of the second sort, minor type (CC-II 5.m). These cupolae are arranged so that their decagonal bases partly overlap, making a pentagonal pattern (similar to the one of the Penrose tiling). Covering the folds between the faces of such a polyhedral structure with polygons, we use exactly equi lateral triangles and squares, thanks to the trigonometric prop erties of CC-II-5.m. Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles will be projected into congruent isosceles triangles corresponding to those obtained by the radial sec tion of a regular pentagon in 5 parts. The squares are project ed into rectangles whose ratio is: a:b = 1:φ/√(1+φ2), where φ 
is the golden ratio. These triangles and rectangles form a ra dial tiling consisting of 5 sectors of the plane, where the pat terns of the established tiles are repeated locally periodically. 
However, with 5-fold rotation of the pattern, the tiling itself is non-periodic. The various tiling solutions that can be obtained in this way may serve as inspiration for the geometric design, 
especially interesting in architecture and applied arts, e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the like
PB  - University of Arts in Belgrade, Faculty of Applied Arts, Kralja Petra 4, 11000 Belgrade, Serbia
T2  - Уметност и наука у примени: искуство и визија: Art and Science Applied: Experience and Vision
T1  - Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation
EP  - 371
SP  - 353
VL  - 2
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2779
ER  - 

@inbook{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2022",
abstract = "The paper considers a type of radial pentagon-based tiling consisting of two shapes: triangle and rectangle. The ob tained solution has a spatial interpretation in a 3D arrangement of equilateral triangles and squares dictated by the particular array of concave cupolae of the second sort, minor type (CC-II 5.m). These cupolae are arranged so that their decagonal bases partly overlap, making a pentagonal pattern (similar to the one of the Penrose tiling). Covering the folds between the faces of such a polyhedral structure with polygons, we use exactly equi lateral triangles and squares, thanks to the trigonometric prop erties of CC-II-5.m. Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles will be projected into congruent isosceles triangles corresponding to those obtained by the radial sec tion of a regular pentagon in 5 parts. The squares are project ed into rectangles whose ratio is: a:b = 1:φ/√(1+φ2), where φ 
is the golden ratio. These triangles and rectangles form a ra dial tiling consisting of 5 sectors of the plane, where the pat terns of the established tiles are repeated locally periodically. 
However, with 5-fold rotation of the pattern, the tiling itself is non-periodic. The various tiling solutions that can be obtained in this way may serve as inspiration for the geometric design, 
especially interesting in architecture and applied arts, e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the like",
publisher = "University of Arts in Belgrade, Faculty of Applied Arts, Kralja Petra 4, 11000 Belgrade, Serbia",
journal = "Уметност и наука у примени: искуство и визија: Art and Science Applied: Experience and Vision",
booktitle = "Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation",
pages = "371-353",
volume = "2",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2779"
}

Obradović, M.,& Mišić, S.. (2022). Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation. in Уметност и наука у примени: искуство и визија: Art and Science Applied: Experience and Vision
University of Arts in Belgrade, Faculty of Applied Arts, Kralja Petra 4, 11000 Belgrade, Serbia., 2, 353-371.
https://hdl.handle.net/21.15107/rcub_grafar_2779

Obradović M, Mišić S. Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation. in Уметност и наука у примени: искуство и визија: Art and Science Applied: Experience and Vision. 2022;2:353-371.
https://hdl.handle.net/21.15107/rcub_grafar_2779 .

Obradović, Marija, Mišić, Slobodan, "Pentagon-Based Radial Tiling with Triangles and Rectangles and Its Spatial Interpretation" in Уметност и наука у примени: искуство и визија: Art and Science Applied: Experience and Vision, 2 (2022):353-371,
https://hdl.handle.net/21.15107/rcub_grafar_2779 .

TY  - CHAP
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2022
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2698
AB  - Finding a possibility to create unique polyhedral surfaces with such specific geometric regularities as the congruence of faces, a high level of symmetry and the ability to describe an infinite polyhedron, served as a starting point for the explorations in this study. Such surfaces can be obtained through a single element, the equilateral triangle, given that they are deltahedral. Here, we will focus on deltahedral rings composed of fragments of the concave antiprisms of the second sort, type major, that we identify as CA-II-n.M. This paper shows not only that it is possible to close a full ring using fragments of selected CA-II-n.M, but also that we can predict the shape of the ring depending on the number of sides of the base-polygon {n} within the chosen CA-II-n.M. The number of solutions obtained for each CA-II-n.M’s representative depends on n and can vary from 1 to 5, out of 8 possible solutions.
PB  - Birkhäuser, Cham, 2022
T2  - Polyhedra and Beyond
T1  - Concave Deltahedral Rings Based on the Geometry of Concave Antiprisms of the Second Sort
EP  - 68
SP  - 53
DO  - 10.1007/978-3-030-99116-6_4
ER  - 

@inbook{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2022",
abstract = "Finding a possibility to create unique polyhedral surfaces with such specific geometric regularities as the congruence of faces, a high level of symmetry and the ability to describe an infinite polyhedron, served as a starting point for the explorations in this study. Such surfaces can be obtained through a single element, the equilateral triangle, given that they are deltahedral. Here, we will focus on deltahedral rings composed of fragments of the concave antiprisms of the second sort, type major, that we identify as CA-II-n.M. This paper shows not only that it is possible to close a full ring using fragments of selected CA-II-n.M, but also that we can predict the shape of the ring depending on the number of sides of the base-polygon {n} within the chosen CA-II-n.M. The number of solutions obtained for each CA-II-n.M’s representative depends on n and can vary from 1 to 5, out of 8 possible solutions.",
publisher = "Birkhäuser, Cham, 2022",
journal = "Polyhedra and Beyond",
booktitle = "Concave Deltahedral Rings Based on the Geometry of Concave Antiprisms of the Second Sort",
pages = "68-53",
doi = "10.1007/978-3-030-99116-6_4"
}

Obradović, M.,& Mišić, S.. (2022). Concave Deltahedral Rings Based on the Geometry of Concave Antiprisms of the Second Sort. in Polyhedra and Beyond
Birkhäuser, Cham, 2022., 53-68.
https://doi.org/10.1007/978-3-030-99116-6_4

Obradović M, Mišić S. Concave Deltahedral Rings Based on the Geometry of Concave Antiprisms of the Second Sort. in Polyhedra and Beyond. 2022;:53-68.
doi:10.1007/978-3-030-99116-6_4 .

Obradović, Marija, Mišić, Slobodan, "Concave Deltahedral Rings Based on the Geometry of Concave Antiprisms of the Second Sort" in Polyhedra and Beyond (2022):53-68,
https://doi.org/10.1007/978-3-030-99116-6_4 . .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2021
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2528
AB  - The paper considers a type of radial pentagon-based tiling consisted of two shapes: triangle
and rectangle. The obtained solution has a spatial interpretation in a 3D arrangement of
equilateral triangles and squares dictated by the particular array of concave cupolae of the
second sort, minor type (CC-II-5.m). These cupolae are arranged so that their decagonal
bases partly overlap, making a pentagonal pattern (similar to one of the Penrose tiling).
Covering the folds between the faces of such a polyhedral structure with polygons,we use
exactly equilateral triangles and squares, thanks to the trigonometric properties of CC-II-5.m.
Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles
will be projected into congruent isosceles triangles corresponding to those obtained by the
radial section of a regular pentagon in 5 parts. The squares are projected into rectangles
whose ratio a:b = 1:φ/√(1+φ2), where φ is the golden ratio. These triangles and rectangles
form a radial tiling consisting of 5 sectors of the plane, where the established tiles’ patterns
are repeated locally periodically. However, with 5-fold rotation of the pattern, the tilling itself
is non-periodic. The various tiling solution that can be obtained in this way may serve as
inspiration for the geometric design, especially interesting in architecture and applied arts,
e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the like
PB  - Faculty of Applied Arts, University of Arts in Belgrade, Serbia
C3  - SmartArt | Књига апстраката| Book of Abstracts
T1  - Pentagon-based radial tiling with triangles and rectangles and its spatial interpretation
SP  - 58
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2528
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2021",
abstract = "The paper considers a type of radial pentagon-based tiling consisted of two shapes: triangle
and rectangle. The obtained solution has a spatial interpretation in a 3D arrangement of
equilateral triangles and squares dictated by the particular array of concave cupolae of the
second sort, minor type (CC-II-5.m). These cupolae are arranged so that their decagonal
bases partly overlap, making a pentagonal pattern (similar to one of the Penrose tiling).
Covering the folds between the faces of such a polyhedral structure with polygons,we use
exactly equilateral triangles and squares, thanks to the trigonometric properties of CC-II-5.m.
Observed in the orthogonal projection onto the plane of the polygonal bases, this 3D “covering” is viewed as a pentagonal-based radial tiling in the Euclidean plane. Equilateral triangles
will be projected into congruent isosceles triangles corresponding to those obtained by the
radial section of a regular pentagon in 5 parts. The squares are projected into rectangles
whose ratio a:b = 1:φ/√(1+φ2), where φ is the golden ratio. These triangles and rectangles
form a radial tiling consisting of 5 sectors of the plane, where the established tiles’ patterns
are repeated locally periodically. However, with 5-fold rotation of the pattern, the tilling itself
is non-periodic. The various tiling solution that can be obtained in this way may serve as
inspiration for the geometric design, especially interesting in architecture and applied arts,
e.g. for rosettes, brise soleils, mosaics, stained glass, fences, partition screens and the like",
publisher = "Faculty of Applied Arts, University of Arts in Belgrade, Serbia",
journal = "SmartArt | Књига апстраката| Book of Abstracts",
title = "Pentagon-based radial tiling with triangles and rectangles and its spatial interpretation",
pages = "58",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2528"
}

Obradović, M.,& Mišić, S.. (2021). Pentagon-based radial tiling with triangles and rectangles and its spatial interpretation. in SmartArt | Књига апстраката| Book of Abstracts
Faculty of Applied Arts, University of Arts in Belgrade, Serbia., 58.
https://hdl.handle.net/21.15107/rcub_grafar_2528

Obradović M, Mišić S. Pentagon-based radial tiling with triangles and rectangles and its spatial interpretation. in SmartArt | Књига апстраката| Book of Abstracts. 2021;:58.
https://hdl.handle.net/21.15107/rcub_grafar_2528 .

Obradović, Marija, Mišić, Slobodan, "Pentagon-based radial tiling with triangles and rectangles and its spatial interpretation" in SmartArt | Књига апстраката| Book of Abstracts (2021):58,
https://hdl.handle.net/21.15107/rcub_grafar_2528 .

TY  - CONF
AU  - Mišić, Slobodan
AU  - Obradović, Marija
AU  - Milakić, Mirjana
PY  - 2020
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2202
AB  - In this paper concave deltahedral surfaces are applied to link the two concepts of geometric rosette design – the polar distribution of the unit element (circular arc) around the center of the contour circle and the rosettes obtained by means of  regular polygons. Forming composite polyhedral structures based on the geometry of concave deltahedral surfaces over a n-sided polygonal base, we have demonstrated one possible method of geometrical generation of three-dimensional rosettes. The concave polyhedral surfaces are lateral surfaces of the concave polyhedrons of the second, fourth and higher sorts, consisting of series of equilateral triangles, grouped into spatial pentahedrons and hexahedrons. Positioned polarly around the central axis of the regular polygon in the polyhedron’s basis and linked by triangles, the spatial pentahedrons and hexahedrons form the deltahedral surface. The sort of the concave polyhedron is determined by the number of equilateral triangle rows in thus obtained polyhedron’s net. In this study, composite polyhedral structures whose surface areas form the three-dimensional rosette are obtained through the combination of concave cupolae of the second sort (CC-II), concave cupolae of the fourth sort (CC-IV), concave antiprisms of the second sort (CA-II) and concave pyramids (CP). By means of elongation, gyro-elongation and augmentation of the listed concave polyhedrons it was possible to generate complex polyhedral structures, which can be used to create three-dimensional rosettes. The parameters of the solids were determined constructively by geometric methods and analytical methods which useiterative numericalprocedures.
PB  - Факултет примењених уметности, Београд, 2020 Faculty of Applied Arts, Belgrade, 2020
C3  - ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIEDFrom Inspiration to Interaction
T1  - Three-Dimensional Rosettes Based on the Geometry of Concave Deltahedral Surfaces
EP  - 422
SP  - 410
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2202
ER  - 

@conference{
author = "Mišić, Slobodan and Obradović, Marija and Milakić, Mirjana",
year = "2020",
abstract = "In this paper concave deltahedral surfaces are applied to link the two concepts of geometric rosette design – the polar distribution of the unit element (circular arc) around the center of the contour circle and the rosettes obtained by means of  regular polygons. Forming composite polyhedral structures based on the geometry of concave deltahedral surfaces over a n-sided polygonal base, we have demonstrated one possible method of geometrical generation of three-dimensional rosettes. The concave polyhedral surfaces are lateral surfaces of the concave polyhedrons of the second, fourth and higher sorts, consisting of series of equilateral triangles, grouped into spatial pentahedrons and hexahedrons. Positioned polarly around the central axis of the regular polygon in the polyhedron’s basis and linked by triangles, the spatial pentahedrons and hexahedrons form the deltahedral surface. The sort of the concave polyhedron is determined by the number of equilateral triangle rows in thus obtained polyhedron’s net. In this study, composite polyhedral structures whose surface areas form the three-dimensional rosette are obtained through the combination of concave cupolae of the second sort (CC-II), concave cupolae of the fourth sort (CC-IV), concave antiprisms of the second sort (CA-II) and concave pyramids (CP). By means of elongation, gyro-elongation and augmentation of the listed concave polyhedrons it was possible to generate complex polyhedral structures, which can be used to create three-dimensional rosettes. The parameters of the solids were determined constructively by geometric methods and analytical methods which useiterative numericalprocedures.",
publisher = "Факултет примењених уметности, Београд, 2020 Faculty of Applied Arts, Belgrade, 2020",
journal = "ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIEDFrom Inspiration to Interaction",
title = "Three-Dimensional Rosettes Based on the Geometry of Concave Deltahedral Surfaces",
pages = "422-410",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2202"
}

Mišić, S., Obradović, M.,& Milakić, M.. (2020). Three-Dimensional Rosettes Based on the Geometry of Concave Deltahedral Surfaces. in ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIEDFrom Inspiration to Interaction
Факултет примењених уметности, Београд, 2020 Faculty of Applied Arts, Belgrade, 2020., 410-422.
https://hdl.handle.net/21.15107/rcub_grafar_2202

Mišić S, Obradović M, Milakić M. Three-Dimensional Rosettes Based on the Geometry of Concave Deltahedral Surfaces. in ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIEDFrom Inspiration to Interaction. 2020;:410-422.
https://hdl.handle.net/21.15107/rcub_grafar_2202 .

Mišić, Slobodan, Obradović, Marija, Milakić, Mirjana, "Three-Dimensional Rosettes Based on the Geometry of Concave Deltahedral Surfaces" in ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIEDFrom Inspiration to Interaction (2020):410-422,
https://hdl.handle.net/21.15107/rcub_grafar_2202 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2020
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2203
AB  - Using the concave polyhedra of the second sort, we are creating spatial structures in the shape of lattice panels to be applied in architecture. The procedure is based on a rectangular (or, less often, a polar) array of identical representatives of the concave polyhedra that include: concave antiprisms, concave cupolae and concave pyramids of the second sort. The selected representative, as a unit cell, can be arrayed so to touch the adjacent cells by vertex, by edge or by face. Thereby, they are  forming 3D lattice, similar to the 2D lattices patterns. We are using a single layer of these structures to form a shape most convenient for architectural usage, which is a shape of a panel. These 3D lattice panels are proposed to be used as brise-soleil, room dividers, fences, etc. The additional layer of visual design when using such a panels is accomplished with the shadows they cast, depending on the time and day of the year. 3D shape emphasizes the play of light and shadow, so these lattice panels can have a significant role as an element of decoration, i.e. architectural ornament.
PB  - Факултет примењених уметности, Београд, 2020 Faculty of Applied Arts, Belgrade, 2020
C3  - ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIED From Inspiration to Interaction
T1  - 3D Lattice Panels Based on the Concave Polyhedra of the Second Sort: Ideas for Architectural Ornaments
EP  - 409
SP  - 394
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2203
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2020",
abstract = "Using the concave polyhedra of the second sort, we are creating spatial structures in the shape of lattice panels to be applied in architecture. The procedure is based on a rectangular (or, less often, a polar) array of identical representatives of the concave polyhedra that include: concave antiprisms, concave cupolae and concave pyramids of the second sort. The selected representative, as a unit cell, can be arrayed so to touch the adjacent cells by vertex, by edge or by face. Thereby, they are  forming 3D lattice, similar to the 2D lattices patterns. We are using a single layer of these structures to form a shape most convenient for architectural usage, which is a shape of a panel. These 3D lattice panels are proposed to be used as brise-soleil, room dividers, fences, etc. The additional layer of visual design when using such a panels is accomplished with the shadows they cast, depending on the time and day of the year. 3D shape emphasizes the play of light and shadow, so these lattice panels can have a significant role as an element of decoration, i.e. architectural ornament.",
publisher = "Факултет примењених уметности, Београд, 2020 Faculty of Applied Arts, Belgrade, 2020",
journal = "ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIED From Inspiration to Interaction",
title = "3D Lattice Panels Based on the Concave Polyhedra of the Second Sort: Ideas for Architectural Ornaments",
pages = "409-394",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2203"
}

Obradović, M.,& Mišić, S.. (2020). 3D Lattice Panels Based on the Concave Polyhedra of the Second Sort: Ideas for Architectural Ornaments. in ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIED From Inspiration to Interaction
Факултет примењених уметности, Београд, 2020 Faculty of Applied Arts, Belgrade, 2020., 394-409.
https://hdl.handle.net/21.15107/rcub_grafar_2203

Obradović M, Mišić S. 3D Lattice Panels Based on the Concave Polyhedra of the Second Sort: Ideas for Architectural Ornaments. in ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIED From Inspiration to Interaction. 2020;:394-409.
https://hdl.handle.net/21.15107/rcub_grafar_2203 .

Obradović, Marija, Mišić, Slobodan, "3D Lattice Panels Based on the Concave Polyhedra of the Second Sort: Ideas for Architectural Ornaments" in ЗБОРНИК РАДОВА: ПРВА МЕЂУНАРОДНА КОНФЕРЕНЦИЈА SMARTART – УМЕТНОСТ И НАУКА У ПРИМЕНИОд инспирације до интеракције / PROCEEDINGS: FIRST INTERNATIONAL CONFERENCE SMARTART – ART AND SCIENCE APPLIED From Inspiration to Interaction (2020):394-409,
https://hdl.handle.net/21.15107/rcub_grafar_2203 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2019
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1981
AB  - Concave polyhedra of the second sort (abbreviated: C-II-n) constitute a group of polyhedra formed over regular n-sided base polygons and having a deltahedral lateral surfaces. This group includes: concave cupolae, concave pyramids and concave antiprisms of the second sort (CC-II-n, CP-II-n and CA-II-n, respectively). The common feature of these solids is that their lateral surfaces consist of a double row of equilateral triangles which can be assembled in two ways, making two different solids’ heights: major (C-II-nM) and minor (C-II-nm). The geometrical regularities and a high level of symmetry that characterizes these polyhedra, makes them suitable for joining and combining, so they can be arrayed infinitely in space, in x, y and z direction forming 3D lattice structures. For some representatives of these solids, the congruity of their lateral deltahedral surfaces occurs, so 3D tessellations are formed. 
In this paper, we focus on a single "layer" of such a structure, a panel-like 3D lattice. It is generated by multiplication of the chosen unit cell – the selected C-II-n representative – along the x-y directions. In the z direction the lateral surfaces form a deltahedral structure which makes the thickness of the panel. 
The method we used is based on the continuous connection of the edges of the two adjacent units, by joining relevant vertices. When we remove the base polygons, the unit cells become hollow, so they can create a honeycombed structure, more desirable for the purpose of application. Then, observed in 2D, by applying symmetry transformations, we form patterns similarly to the formation of wallpaper groups. In this way, we get visually interesting patterns in 2D, which transform into 3D lattice depending on the viewing angle. 
The thickness of the panel can be halved in some cases, so we get a thinner structure with "face" and "back", having different tessellations of polygons appearing on them. As an artistic intervention, these panels can be modified by joining deltahedral surfaces of other C-II-n onto the compatible faces, whereby we add another layer of patterns to the resulting structure. 
3D patterns and lattices are currently experiencing real boom in the design and industry, thanks to the 3D printing capabilities. As for architecture, they can be applied not only as an element of ornamentation, but also as a functional component of the project, especially concerning climate responsive facades.   
Due to the simplicity of the geometry of C-II-n, such 3D structures are feasible and easy to perform in terms of production and assembly. They are achievable not only with 3D printing, but can also be manually assembled or folded like origami, which allows the use of a much wider range of materials.
PB  - Beograd: Fakultet primenjenih umetnosti, Univerzitet umetnosti u Beogradu, Srbija
C3  - Smart Art Knjiga apstrakata / Book of abstracts
T1  - 3D lattice panels based on the concave polyhedra of the second sort: ideas for architectural ornaments
EP  - 88
SP  - 87
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1981
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2019",
abstract = "Concave polyhedra of the second sort (abbreviated: C-II-n) constitute a group of polyhedra formed over regular n-sided base polygons and having a deltahedral lateral surfaces. This group includes: concave cupolae, concave pyramids and concave antiprisms of the second sort (CC-II-n, CP-II-n and CA-II-n, respectively). The common feature of these solids is that their lateral surfaces consist of a double row of equilateral triangles which can be assembled in two ways, making two different solids’ heights: major (C-II-nM) and minor (C-II-nm). The geometrical regularities and a high level of symmetry that characterizes these polyhedra, makes them suitable for joining and combining, so they can be arrayed infinitely in space, in x, y and z direction forming 3D lattice structures. For some representatives of these solids, the congruity of their lateral deltahedral surfaces occurs, so 3D tessellations are formed. 
In this paper, we focus on a single "layer" of such a structure, a panel-like 3D lattice. It is generated by multiplication of the chosen unit cell – the selected C-II-n representative – along the x-y directions. In the z direction the lateral surfaces form a deltahedral structure which makes the thickness of the panel. 
The method we used is based on the continuous connection of the edges of the two adjacent units, by joining relevant vertices. When we remove the base polygons, the unit cells become hollow, so they can create a honeycombed structure, more desirable for the purpose of application. Then, observed in 2D, by applying symmetry transformations, we form patterns similarly to the formation of wallpaper groups. In this way, we get visually interesting patterns in 2D, which transform into 3D lattice depending on the viewing angle. 
The thickness of the panel can be halved in some cases, so we get a thinner structure with "face" and "back", having different tessellations of polygons appearing on them. As an artistic intervention, these panels can be modified by joining deltahedral surfaces of other C-II-n onto the compatible faces, whereby we add another layer of patterns to the resulting structure. 
3D patterns and lattices are currently experiencing real boom in the design and industry, thanks to the 3D printing capabilities. As for architecture, they can be applied not only as an element of ornamentation, but also as a functional component of the project, especially concerning climate responsive facades.   
Due to the simplicity of the geometry of C-II-n, such 3D structures are feasible and easy to perform in terms of production and assembly. They are achievable not only with 3D printing, but can also be manually assembled or folded like origami, which allows the use of a much wider range of materials.",
publisher = "Beograd: Fakultet primenjenih umetnosti, Univerzitet umetnosti u Beogradu, Srbija",
journal = "Smart Art Knjiga apstrakata / Book of abstracts",
title = "3D lattice panels based on the concave polyhedra of the second sort: ideas for architectural ornaments",
pages = "88-87",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1981"
}

Obradović, M.,& Mišić, S.. (2019). 3D lattice panels based on the concave polyhedra of the second sort: ideas for architectural ornaments. in Smart Art Knjiga apstrakata / Book of abstracts
Beograd: Fakultet primenjenih umetnosti, Univerzitet umetnosti u Beogradu, Srbija., 87-88.
https://hdl.handle.net/21.15107/rcub_grafar_1981

Obradović M, Mišić S. 3D lattice panels based on the concave polyhedra of the second sort: ideas for architectural ornaments. in Smart Art Knjiga apstrakata / Book of abstracts. 2019;:87-88.
https://hdl.handle.net/21.15107/rcub_grafar_1981 .

Obradović, Marija, Mišić, Slobodan, "3D lattice panels based on the concave polyhedra of the second sort: ideas for architectural ornaments" in Smart Art Knjiga apstrakata / Book of abstracts (2019):87-88,
https://hdl.handle.net/21.15107/rcub_grafar_1981 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2019
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1982
AB  - We have shown that there is a link between the geometry of the CA II-nM’s  with bases n∈{3, 4, 5} and that of the convex antiprisms with the same bases. An integer number (K) of CA II-nM’s fragments, can be used to form a full multilaterally symmetrical ring of concave deltahedral surfaces, either flower-like (case A) or star-like (case B). The obtained rings can also be termed “of the second sort” (denoted by CDR II-n) as they inherit from the given  CA II-nM the following: a) the linear and angular measurements needed for their graphic and mathematical elaboration, b) two rows of equilateral triangles in the lateral surface, and c) the high level of symmetry. The possible formation of CDR II-n’s  with the highest level of symmetry (i.e. excluding the cases A), and with the number of petals/star-points in which any integer K ≥ 2 can be a subject of further research.
PB  - Porto: Aproged - Associação dos Professores de Geometria e de Desenho
C3  - GEOMETRIAS’19: BOOK OF ABSTRACTS
T1  - Concave deltahedral rings based on the geometry of the concave antiprisms of the second sort
SP  - 85
VL  - 89
DO  - 10.24840/978-989-98926-8-2
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2019",
abstract = "We have shown that there is a link between the geometry of the CA II-nM’s  with bases n∈{3, 4, 5} and that of the convex antiprisms with the same bases. An integer number (K) of CA II-nM’s fragments, can be used to form a full multilaterally symmetrical ring of concave deltahedral surfaces, either flower-like (case A) or star-like (case B). The obtained rings can also be termed “of the second sort” (denoted by CDR II-n) as they inherit from the given  CA II-nM the following: a) the linear and angular measurements needed for their graphic and mathematical elaboration, b) two rows of equilateral triangles in the lateral surface, and c) the high level of symmetry. The possible formation of CDR II-n’s  with the highest level of symmetry (i.e. excluding the cases A), and with the number of petals/star-points in which any integer K ≥ 2 can be a subject of further research.",
publisher = "Porto: Aproged - Associação dos Professores de Geometria e de Desenho",
journal = "GEOMETRIAS’19: BOOK OF ABSTRACTS",
title = "Concave deltahedral rings based on the geometry of the concave antiprisms of the second sort",
pages = "85",
volume = "89",
doi = "10.24840/978-989-98926-8-2"
}

Obradović, M.,& Mišić, S.. (2019). Concave deltahedral rings based on the geometry of the concave antiprisms of the second sort. in GEOMETRIAS’19: BOOK OF ABSTRACTS
Porto: Aproged - Associação dos Professores de Geometria e de Desenho., 89, 85.
https://doi.org/10.24840/978-989-98926-8-2

Obradović M, Mišić S. Concave deltahedral rings based on the geometry of the concave antiprisms of the second sort. in GEOMETRIAS’19: BOOK OF ABSTRACTS. 2019;89:85.
doi:10.24840/978-989-98926-8-2 .

Obradović, Marija, Mišić, Slobodan, "Concave deltahedral rings based on the geometry of the concave antiprisms of the second sort" in GEOMETRIAS’19: BOOK OF ABSTRACTS, 89 (2019):85,
https://doi.org/10.24840/978-989-98926-8-2 . .

TY  - JOUR
AU  - Mišić, Slobodan
AU  - Obradović, Marija
AU  - Đukanović, Gordana
PY  - 2015
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1036
AB  - In this paper, the geometry of concave cupolae has been the starting
point for the generation of composite polyhedral structures, usable as formative
patterns for architectural purposes. Obtained by linking paper folding geometry
with the geometry of polyhedra, concave cupolae are polyhedra that follow the
method of generating cupolae (Johnson’s solids: J3, J4 and J5); but we removed
the convexity criterion and omitted squares in the lateral surface. Instead of alter-
nating triangles and squares there are now two or more paired series of equilateral
triangles. The criterion of face regularity is respected, as well as the criterion of
multiple axial symmetry. The distribution of the triangles is based on strictly
determined and mathematically defined parameters, which allows the creation of
such structures in a way that qualifies them as an autonomous group of polyhedra
— concave cupolae of sorts II, IV, VI (2N). If we want to see these structures as
polyhedral surfaces (not as solids) connecting the concept of the cupola (dome) in
the architectural sense with the geometrical meaning of (concave) cupola, we re-
move the faces of the base polygons. Thus we get a deltahedral structure — a shell
made entirely from equilateral triangles, which is advantageous for the purpose
of prefabrication. Due to the congruence of the major 2n-sided bases of concave
cupolae of sort II with the minor bases of the corresponding concave cupolae of
sort IV, it is possible to combine these polyhedra in composite polyhedra. But
also their elongation with concave antiprisms of sort II or the augmentation with
concave pyramids of sort II could be performed. Based on the foregoing, we exam-
ine the possibilities of combining the considered polyhedra into unified composite
structures.
T2  - Journal for Geometry and Graphics
T1  - Composite Concave Cupolae as Geometric and Architectural Forms
EP  - 91
EP  - 
EP  - 
IS  - 1
SP  - 79
VL  - 19
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1036
ER  - 

@article{
author = "Mišić, Slobodan and Obradović, Marija and Đukanović, Gordana",
year = "2015",
abstract = "In this paper, the geometry of concave cupolae has been the starting
point for the generation of composite polyhedral structures, usable as formative
patterns for architectural purposes. Obtained by linking paper folding geometry
with the geometry of polyhedra, concave cupolae are polyhedra that follow the
method of generating cupolae (Johnson’s solids: J3, J4 and J5); but we removed
the convexity criterion and omitted squares in the lateral surface. Instead of alter-
nating triangles and squares there are now two or more paired series of equilateral
triangles. The criterion of face regularity is respected, as well as the criterion of
multiple axial symmetry. The distribution of the triangles is based on strictly
determined and mathematically defined parameters, which allows the creation of
such structures in a way that qualifies them as an autonomous group of polyhedra
— concave cupolae of sorts II, IV, VI (2N). If we want to see these structures as
polyhedral surfaces (not as solids) connecting the concept of the cupola (dome) in
the architectural sense with the geometrical meaning of (concave) cupola, we re-
move the faces of the base polygons. Thus we get a deltahedral structure — a shell
made entirely from equilateral triangles, which is advantageous for the purpose
of prefabrication. Due to the congruence of the major 2n-sided bases of concave
cupolae of sort II with the minor bases of the corresponding concave cupolae of
sort IV, it is possible to combine these polyhedra in composite polyhedra. But
also their elongation with concave antiprisms of sort II or the augmentation with
concave pyramids of sort II could be performed. Based on the foregoing, we exam-
ine the possibilities of combining the considered polyhedra into unified composite
structures.",
journal = "Journal for Geometry and Graphics",
title = "Composite Concave Cupolae as Geometric and Architectural Forms",
pages = "91---79",
number = "1",
volume = "19",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1036"
}

Mišić, S., Obradović, M.,& Đukanović, G.. (2015). Composite Concave Cupolae as Geometric and Architectural Forms. in Journal for Geometry and Graphics, 19(1), 79-91.
https://hdl.handle.net/21.15107/rcub_grafar_1036

Mišić S, Obradović M, Đukanović G. Composite Concave Cupolae as Geometric and Architectural Forms. in Journal for Geometry and Graphics. 2015;19(1):79-91.
https://hdl.handle.net/21.15107/rcub_grafar_1036 .

Mišić, Slobodan, Obradović, Marija, Đukanović, Gordana, "Composite Concave Cupolae as Geometric and Architectural Forms" in Journal for Geometry and Graphics, 19, no. 1 (2015):79-91,
https://hdl.handle.net/21.15107/rcub_grafar_1036 .

TY  - JOUR
AU  - Obradović, Marija
AU  - Mišić, Slobodan
AU  - Popkonstantinović, Branislav
PY  - 2015
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1930
AB  - Concave pyramids of second sort (CP II) are formed similarly to concave cupolae of second sort
(CC II) with a comparable logic of emergence to the regular-faced convex pyramids. They are formed by enclosing the space over a regular polygonal base by the lateral deltahedral wrapper consisting of equilateral triangles meeting in common apex. The plane net is composed of double row of equilateral triangles, so it is possible to form a regular faced pyramid even with bases greater than pentagonal, whereat the obtained polyhedron will be concave. In this paper we focus on a type of concave pyramids of second sort formed over the basic polygons with an even number of sides, type CP II-B which has half the number of lateral unit cells than the type CP II-A.
PB  - Bucharest, Romania: SORGING–ROMANIAN SOCIETYFORENGINEERINGGRAPHICS
PB  - Brasov, Romania: University ‘Transilvania’ of Brasov
T2  - Journal of Industrial Design and Engineering Graphics (JIDEG)
T1  - Variations of Concave Pyramids of Second Sort with an Even Number of Base Sides
EP  - 50
IS  - 2
SP  - 45
VL  - 10
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1930
ER  - 

@article{
author = "Obradović, Marija and Mišić, Slobodan and Popkonstantinović, Branislav",
year = "2015",
abstract = "Concave pyramids of second sort (CP II) are formed similarly to concave cupolae of second sort
(CC II) with a comparable logic of emergence to the regular-faced convex pyramids. They are formed by enclosing the space over a regular polygonal base by the lateral deltahedral wrapper consisting of equilateral triangles meeting in common apex. The plane net is composed of double row of equilateral triangles, so it is possible to form a regular faced pyramid even with bases greater than pentagonal, whereat the obtained polyhedron will be concave. In this paper we focus on a type of concave pyramids of second sort formed over the basic polygons with an even number of sides, type CP II-B which has half the number of lateral unit cells than the type CP II-A.",
publisher = "Bucharest, Romania: SORGING–ROMANIAN SOCIETYFORENGINEERINGGRAPHICS, Brasov, Romania: University ‘Transilvania’ of Brasov",
journal = "Journal of Industrial Design and Engineering Graphics (JIDEG)",
title = "Variations of Concave Pyramids of Second Sort with an Even Number of Base Sides",
pages = "50-45",
number = "2",
volume = "10",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1930"
}

Obradović, M., Mišić, S.,& Popkonstantinović, B.. (2015). Variations of Concave Pyramids of Second Sort with an Even Number of Base Sides. in Journal of Industrial Design and Engineering Graphics (JIDEG)
Bucharest, Romania: SORGING–ROMANIAN SOCIETYFORENGINEERINGGRAPHICS., 10(2), 45-50.
https://hdl.handle.net/21.15107/rcub_grafar_1930

Obradović M, Mišić S, Popkonstantinović B. Variations of Concave Pyramids of Second Sort with an Even Number of Base Sides. in Journal of Industrial Design and Engineering Graphics (JIDEG). 2015;10(2):45-50.
https://hdl.handle.net/21.15107/rcub_grafar_1930 .

Obradović, Marija, Mišić, Slobodan, Popkonstantinović, Branislav, "Variations of Concave Pyramids of Second Sort with an Even Number of Base Sides" in Journal of Industrial Design and Engineering Graphics (JIDEG), 10, no. 2 (2015):45-50,
https://hdl.handle.net/21.15107/rcub_grafar_1930 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
AU  - Popkonstantinović, Branislav
PY  - 2014
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1252
AB  - Correspondingly to the method of generating the Concave Cupolae of second sort, the Concave Pyramids of second sort have the similar logic of origination, and their counterpart in regular faced convex pyramids (tetrahedron, Johnson's solids J1 and J2). The difference is that instead of onefold series of equilateral triangles in the lateral surface of the solid, there appear twofold series, forming deltahedral lateral surface with a common point, while bases are also regular polygons. This time, instead of the bases from n=3 to n=5, there are the basis from n=6 to n=9. The same lateral surface’s net can be folded and creased in two different ways, which produces the two types of Concave Pyramids of second sort: with a major and with a minor solid height. Combining and joining so obtained solids by the correspondent bases, the concave (ortho) bipyramids of second sort emerge, which then may be elongated, gyroelongated, and conca-elongated, creating a distinctive family of diverse concave polyhedral structures.
PB  - Faculty of Civil Engineering and Architecture ; Serbian Society for Geometry and Graphics SUGIG, Niš ; Beograd
C3  - Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia
T1  - Concave Pyramids of Second Sort -The Occurrence, Types, Variations
EP  - 168
SP  - 157
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1252
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan and Popkonstantinović, Branislav",
year = "2014",
abstract = "Correspondingly to the method of generating the Concave Cupolae of second sort, the Concave Pyramids of second sort have the similar logic of origination, and their counterpart in regular faced convex pyramids (tetrahedron, Johnson's solids J1 and J2). The difference is that instead of onefold series of equilateral triangles in the lateral surface of the solid, there appear twofold series, forming deltahedral lateral surface with a common point, while bases are also regular polygons. This time, instead of the bases from n=3 to n=5, there are the basis from n=6 to n=9. The same lateral surface’s net can be folded and creased in two different ways, which produces the two types of Concave Pyramids of second sort: with a major and with a minor solid height. Combining and joining so obtained solids by the correspondent bases, the concave (ortho) bipyramids of second sort emerge, which then may be elongated, gyroelongated, and conca-elongated, creating a distinctive family of diverse concave polyhedral structures.",
publisher = "Faculty of Civil Engineering and Architecture ; Serbian Society for Geometry and Graphics SUGIG, Niš ; Beograd",
journal = "Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia",
title = "Concave Pyramids of Second Sort -The Occurrence, Types, Variations",
pages = "168-157",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1252"
}

Obradović, M., Mišić, S.,& Popkonstantinović, B.. (2014). Concave Pyramids of Second Sort -The Occurrence, Types, Variations. in Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia
Faculty of Civil Engineering and Architecture ; Serbian Society for Geometry and Graphics SUGIG, Niš ; Beograd., 157-168.
https://hdl.handle.net/21.15107/rcub_grafar_1252

Obradović M, Mišić S, Popkonstantinović B. Concave Pyramids of Second Sort -The Occurrence, Types, Variations. in Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia. 2014;:157-168.
https://hdl.handle.net/21.15107/rcub_grafar_1252 .

Obradović, Marija, Mišić, Slobodan, Popkonstantinović, Branislav, "Concave Pyramids of Second Sort -The Occurrence, Types, Variations" in Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia (2014):157-168,
https://hdl.handle.net/21.15107/rcub_grafar_1252 .

TY  - CONF
AU  - Mišić, Slobodan
AU  - Obradović, Marija
AU  - Popkonstantinović, Branislav
PY  - 2014
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1151
AB  - Concave Cupolae of fourth sort (CC IV) are polyhedra which lateral surface is created by folding a quadruple strip of equilateral triangles, while the bases are regular polygons. The unit cell that forms the deltahedral lateral surface by radial sequenceing around the axis of the solid is composed of two spatial hexahedrals. These unit cells are linked together in its upper zone by spatial quadrihedals, which consist of four equilateral triangles grouped around a common vertex. Over the same polygonal base, there can be formed four diverse Concave Cupolae of IV sort, using different variants of constructive procedure. In this paper, the application of the program for modeling the lateral surface of concave cupolae of IV sort, using the software package MATLAB, allows visual monitoring of the structural transformation of the concave cupolae of IV sort, for different polygonal bases and different variants of constructive procedure.
PB  - Faculty of Civil Engineering and Architecture ; Serbian Society for Geometry and Graphics SUGIG, Niš ; Beograd
C3  - Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia
T1  - The Structural Transformation of Concave Cupolae of Fourth Sort Using Different Variants of Constructive Procedure, 4th International Scientific Conference on Geometry and Graphics
EP  - 156
SP  - 147
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1151
ER  - 

@conference{
author = "Mišić, Slobodan and Obradović, Marija and Popkonstantinović, Branislav",
year = "2014",
abstract = "Concave Cupolae of fourth sort (CC IV) are polyhedra which lateral surface is created by folding a quadruple strip of equilateral triangles, while the bases are regular polygons. The unit cell that forms the deltahedral lateral surface by radial sequenceing around the axis of the solid is composed of two spatial hexahedrals. These unit cells are linked together in its upper zone by spatial quadrihedals, which consist of four equilateral triangles grouped around a common vertex. Over the same polygonal base, there can be formed four diverse Concave Cupolae of IV sort, using different variants of constructive procedure. In this paper, the application of the program for modeling the lateral surface of concave cupolae of IV sort, using the software package MATLAB, allows visual monitoring of the structural transformation of the concave cupolae of IV sort, for different polygonal bases and different variants of constructive procedure.",
publisher = "Faculty of Civil Engineering and Architecture ; Serbian Society for Geometry and Graphics SUGIG, Niš ; Beograd",
journal = "Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia",
title = "The Structural Transformation of Concave Cupolae of Fourth Sort Using Different Variants of Constructive Procedure, 4th International Scientific Conference on Geometry and Graphics",
pages = "156-147",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1151"
}

Mišić, S., Obradović, M.,& Popkonstantinović, B.. (2014). The Structural Transformation of Concave Cupolae of Fourth Sort Using Different Variants of Constructive Procedure, 4th International Scientific Conference on Geometry and Graphics. in Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia
Faculty of Civil Engineering and Architecture ; Serbian Society for Geometry and Graphics SUGIG, Niš ; Beograd., 147-156.
https://hdl.handle.net/21.15107/rcub_grafar_1151

Mišić S, Obradović M, Popkonstantinović B. The Structural Transformation of Concave Cupolae of Fourth Sort Using Different Variants of Constructive Procedure, 4th International Scientific Conference on Geometry and Graphics. in Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia. 2014;:147-156.
https://hdl.handle.net/21.15107/rcub_grafar_1151 .

Mišić, Slobodan, Obradović, Marija, Popkonstantinović, Branislav, "The Structural Transformation of Concave Cupolae of Fourth Sort Using Different Variants of Constructive Procedure, 4th International Scientific Conference on Geometry and Graphics" in Proceedings. Vol. 2 / 4th International Scientific Conference on Geometry and Graphics moNGeometrija 2014, June 20th - 22nd, 2014 Vlasina, Serbia (2014):147-156,
https://hdl.handle.net/21.15107/rcub_grafar_1151 .

TY  - CONF
AU  - Đukanovic, Gordana
AU  - Đorđević, Đorđe
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2014
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1263
PB  - Innsbruck university press, 2014 Universität Innsbruck,1st edition, Innsbruck
C3  - Proceedings of the 16th International Conference on Geometry and Graphics, Innsbruck, August 4-8 2014
T1  - Application of Curves and Surfaces of Higher Orders Obtained by Inversion in the Practice of Architecture
EP  - 53
SP  - 45
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1263
ER  - 

@conference{
author = "Đukanovic, Gordana and Đorđević, Đorđe and Obradović, Marija and Mišić, Slobodan",
year = "2014",
publisher = "Innsbruck university press, 2014 Universität Innsbruck,1st edition, Innsbruck",
journal = "Proceedings of the 16th International Conference on Geometry and Graphics, Innsbruck, August 4-8 2014",
title = "Application of Curves and Surfaces of Higher Orders Obtained by Inversion in the Practice of Architecture",
pages = "53-45",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1263"
}

Đukanovic, G., Đorđević, Đ., Obradović, M.,& Mišić, S.. (2014). Application of Curves and Surfaces of Higher Orders Obtained by Inversion in the Practice of Architecture. in Proceedings of the 16th International Conference on Geometry and Graphics, Innsbruck, August 4-8 2014
Innsbruck university press, 2014 Universität Innsbruck,1st edition, Innsbruck., 45-53.
https://hdl.handle.net/21.15107/rcub_grafar_1263

Đukanovic G, Đorđević Đ, Obradović M, Mišić S. Application of Curves and Surfaces of Higher Orders Obtained by Inversion in the Practice of Architecture. in Proceedings of the 16th International Conference on Geometry and Graphics, Innsbruck, August 4-8 2014. 2014;:45-53.
https://hdl.handle.net/21.15107/rcub_grafar_1263 .

Đukanovic, Gordana, Đorđević, Đorđe, Obradović, Marija, Mišić, Slobodan, "Application of Curves and Surfaces of Higher Orders Obtained by Inversion in the Practice of Architecture" in Proceedings of the 16th International Conference on Geometry and Graphics, Innsbruck, August 4-8 2014 (2014):45-53,
https://hdl.handle.net/21.15107/rcub_grafar_1263 .

TY  - JOUR
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2014
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/618
AB  - There is a widespread opinion in different sources, ranging from popular to scientific, that the project of the Petrovaradin Fortress was conceived under the influence of the most important European military engineer and innovator of the time, Sebastien de Vauban. By examining the historical context as well as by comparing Vauban's geometrical methods for determination of the fortification master line (la ligne magistrale) with Austrian plans and the actual state of the Petrovaradin fortress, especially its Wasserstadt part, we have examined how well-founded this claim is.
PB  - Birkhauser Verlag AG
T2  - Nexus Network Journal
T1  - Are Vauban's Geometrical Principles Applied in the Petrovaradin Fortress?
EP  - 776
IS  - 3
SP  - 751
VL  - 16
DO  - 10.1007/s00004-014-0205-9
ER  - 

@article{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2014",
abstract = "There is a widespread opinion in different sources, ranging from popular to scientific, that the project of the Petrovaradin Fortress was conceived under the influence of the most important European military engineer and innovator of the time, Sebastien de Vauban. By examining the historical context as well as by comparing Vauban's geometrical methods for determination of the fortification master line (la ligne magistrale) with Austrian plans and the actual state of the Petrovaradin fortress, especially its Wasserstadt part, we have examined how well-founded this claim is.",
publisher = "Birkhauser Verlag AG",
journal = "Nexus Network Journal",
title = "Are Vauban's Geometrical Principles Applied in the Petrovaradin Fortress?",
pages = "776-751",
number = "3",
volume = "16",
doi = "10.1007/s00004-014-0205-9"
}

Obradović, M.,& Mišić, S.. (2014). Are Vauban's Geometrical Principles Applied in the Petrovaradin Fortress?. in Nexus Network Journal
Birkhauser Verlag AG., 16(3), 751-776.
https://doi.org/10.1007/s00004-014-0205-9

Obradović M, Mišić S. Are Vauban's Geometrical Principles Applied in the Petrovaradin Fortress?. in Nexus Network Journal. 2014;16(3):751-776.
doi:10.1007/s00004-014-0205-9 .

Obradović, Marija, Mišić, Slobodan, "Are Vauban's Geometrical Principles Applied in the Petrovaradin Fortress?" in Nexus Network Journal, 16, no. 3 (2014):751-776,
https://doi.org/10.1007/s00004-014-0205-9 . .

TY  - CONF
AU  - Mišić, Slobodan
AU  - Obradović, Marija
AU  - Lazović, Goran
AU  - Popkonstantinović, Branislav
PY  - 2013
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1977
AB  - The paper discusses the generation of a specific group of polyhedra, Concave Cupolae of Fourth Sort (CC IV) with regular polygonal bases, using constructive and analytical procedures. Beside determination of the parameters of these polyhedra, the paper deals with their visualization, by the application of graphical software MATLAB. We consider one of the four possible types of forming the lateral surfaces of the Concave Cupolae of fourth sort.
PB  - Timişoara: Editura Politehnica
C3  - Scientific Bulletin of the "Politehnica" University of Timişoara. Transactions on Hydrotechnics
T1  - Generating a Type of Concave Cupolae of Fourth Sort
EP  - 82
IS  - 1
SP  - 79
VL  - 58(72)
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1977
ER  - 

@conference{
author = "Mišić, Slobodan and Obradović, Marija and Lazović, Goran and Popkonstantinović, Branislav",
year = "2013",
abstract = "The paper discusses the generation of a specific group of polyhedra, Concave Cupolae of Fourth Sort (CC IV) with regular polygonal bases, using constructive and analytical procedures. Beside determination of the parameters of these polyhedra, the paper deals with their visualization, by the application of graphical software MATLAB. We consider one of the four possible types of forming the lateral surfaces of the Concave Cupolae of fourth sort.",
publisher = "Timişoara: Editura Politehnica",
journal = "Scientific Bulletin of the "Politehnica" University of Timişoara. Transactions on Hydrotechnics",
title = "Generating a Type of Concave Cupolae of Fourth Sort",
pages = "82-79",
number = "1",
volume = "58(72)",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1977"
}

Mišić, S., Obradović, M., Lazović, G.,& Popkonstantinović, B.. (2013). Generating a Type of Concave Cupolae of Fourth Sort. in Scientific Bulletin of the "Politehnica" University of Timişoara. Transactions on Hydrotechnics
Timişoara: Editura Politehnica., 58(72)(1), 79-82.
https://hdl.handle.net/21.15107/rcub_grafar_1977

Mišić S, Obradović M, Lazović G, Popkonstantinović B. Generating a Type of Concave Cupolae of Fourth Sort. in Scientific Bulletin of the "Politehnica" University of Timişoara. Transactions on Hydrotechnics. 2013;58(72)(1):79-82.
https://hdl.handle.net/21.15107/rcub_grafar_1977 .

Mišić, Slobodan, Obradović, Marija, Lazović, Goran, Popkonstantinović, Branislav, "Generating a Type of Concave Cupolae of Fourth Sort" in Scientific Bulletin of the "Politehnica" University of Timişoara. Transactions on Hydrotechnics, 58(72), no. 1 (2013):79-82,
https://hdl.handle.net/21.15107/rcub_grafar_1977 .

TY  - JOUR
AU  - Obradović, Marija
AU  - Mišić, Slobodan
AU  - Popkonstantinović, Branislav
AU  - Petrović, Maja
AU  - Malešević, Branko
AU  - Obradović, Ratko
PY  - 2013
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/1928
AB  - Concave cupolae of second sort, combined with the other regular faced polyhedra with at least one matching side, provide many possibilities for the formation of various composite polyhedra. The
paper presents research on regular-faced polyhedral structures obtained by joining the bases of square concave cupolae of second sort, with the appropriate sides of Archimedean solid - truncated 
cube, and its application in architecture, based on geometric, structural, and functional analysis.
PB  - Sarajevo: Society for development of teaching and business processes in New net environment in B&H (DRUNPP)
T2  - Technics Technologies Education Management
T1  - Investigation of Concave Cupolae Based Polyhedral Structures and Their Potential Application in Architecture
EP  - 1214
IS  - 3
SP  - 1198
VL  - 8
UR  - https://hdl.handle.net/21.15107/rcub_grafar_1928
ER  - 

@article{
author = "Obradović, Marija and Mišić, Slobodan and Popkonstantinović, Branislav and Petrović, Maja and Malešević, Branko and Obradović, Ratko",
year = "2013",
abstract = "Concave cupolae of second sort, combined with the other regular faced polyhedra with at least one matching side, provide many possibilities for the formation of various composite polyhedra. The
paper presents research on regular-faced polyhedral structures obtained by joining the bases of square concave cupolae of second sort, with the appropriate sides of Archimedean solid - truncated 
cube, and its application in architecture, based on geometric, structural, and functional analysis.",
publisher = "Sarajevo: Society for development of teaching and business processes in New net environment in B&H (DRUNPP)",
journal = "Technics Technologies Education Management",
title = "Investigation of Concave Cupolae Based Polyhedral Structures and Their Potential Application in Architecture",
pages = "1214-1198",
number = "3",
volume = "8",
url = "https://hdl.handle.net/21.15107/rcub_grafar_1928"
}

Obradović, M., Mišić, S., Popkonstantinović, B., Petrović, M., Malešević, B.,& Obradović, R.. (2013). Investigation of Concave Cupolae Based Polyhedral Structures and Their Potential Application in Architecture. in Technics Technologies Education Management
Sarajevo: Society for development of teaching and business processes in New net environment in B&H (DRUNPP)., 8(3), 1198-1214.
https://hdl.handle.net/21.15107/rcub_grafar_1928

Obradović M, Mišić S, Popkonstantinović B, Petrović M, Malešević B, Obradović R. Investigation of Concave Cupolae Based Polyhedral Structures and Their Potential Application in Architecture. in Technics Technologies Education Management. 2013;8(3):1198-1214.
https://hdl.handle.net/21.15107/rcub_grafar_1928 .

Obradović, Marija, Mišić, Slobodan, Popkonstantinović, Branislav, Petrović, Maja, Malešević, Branko, Obradović, Ratko, "Investigation of Concave Cupolae Based Polyhedral Structures and Their Potential Application in Architecture" in Technics Technologies Education Management, 8, no. 3 (2013):1198-1214,
https://hdl.handle.net/21.15107/rcub_grafar_1928 .

TY  - JOUR
AU  - Obradović, Marija
AU  - Popkonstantinović, Branislav
AU  - Mišić, Slobodan
PY  - 2013
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/487
AB  - The paper examines geometrical, static and dynamic properties of the polyhedral structures obtained by folding and creasing the two-rowed segment of equilateral triangular net. Bases of these concave polyhedra are regular, identical polygons in parallel planes, connected by the alternating series of triangles, as in the case of convex antiprisms. There are two ways of folding such a net, and therefore the two types of concave antiprisms of second sort. The paper discusses the methods of obtaining the accurate position of the vertices and other linear parameters of these polyhedra, with the use of mathematical algorithm. Structural analysis of a representative of these polyhedra is presented using the SolidWorks program applications.
AB  - Rad se bavi ispitivanjem geometrijskih, statičkih i dinamičkih osobina jedne poliedarske strukture nastale nabiranjem dvorednog segmenta mreže jednakostraničnih trouglova. Osnove ovih konkavnih poliedara su pravilni, identični poligoni u paralelnim ravnima, povezani nizom naizmeničnih trouglova, kao i u slučaju konveksnih antiprizmi. Postoje dve varijante savijanja ovakve mreže, pa samim tim i dva tipa konkavnih antiprizmi druge vrste (KA II) za svaku posmatranu osnovu od n=5, n=∞. U radu su razmotreni načini dobijanja tačnog položaja temena i drugih linearnih parametara ovih poliedara, uz primenu algoritma za njihovo matematičko izračunavanje. Strukturalna analiza jednog predstavnika ovih poliedara data je korišćenjem aplikacija programa SolidWorks, kako bi se ispitala mogućnost primene ovih oblika u inženjerstvu.
PB  - Univerzitet u Beogradu - Mašinski fakultet, Beograd
T2  - FME Transactions
T1  - On the properties of the concave antiprisms of second sort
T1  - O osobinama konkavnih antiprizmi druge vrste
EP  - 263
IS  - 3
SP  - 256
VL  - 41
UR  - https://hdl.handle.net/21.15107/rcub_grafar_487
ER  - 

@article{
author = "Obradović, Marija and Popkonstantinović, Branislav and Mišić, Slobodan",
year = "2013",
abstract = "The paper examines geometrical, static and dynamic properties of the polyhedral structures obtained by folding and creasing the two-rowed segment of equilateral triangular net. Bases of these concave polyhedra are regular, identical polygons in parallel planes, connected by the alternating series of triangles, as in the case of convex antiprisms. There are two ways of folding such a net, and therefore the two types of concave antiprisms of second sort. The paper discusses the methods of obtaining the accurate position of the vertices and other linear parameters of these polyhedra, with the use of mathematical algorithm. Structural analysis of a representative of these polyhedra is presented using the SolidWorks program applications., Rad se bavi ispitivanjem geometrijskih, statičkih i dinamičkih osobina jedne poliedarske strukture nastale nabiranjem dvorednog segmenta mreže jednakostraničnih trouglova. Osnove ovih konkavnih poliedara su pravilni, identični poligoni u paralelnim ravnima, povezani nizom naizmeničnih trouglova, kao i u slučaju konveksnih antiprizmi. Postoje dve varijante savijanja ovakve mreže, pa samim tim i dva tipa konkavnih antiprizmi druge vrste (KA II) za svaku posmatranu osnovu od n=5, n=∞. U radu su razmotreni načini dobijanja tačnog položaja temena i drugih linearnih parametara ovih poliedara, uz primenu algoritma za njihovo matematičko izračunavanje. Strukturalna analiza jednog predstavnika ovih poliedara data je korišćenjem aplikacija programa SolidWorks, kako bi se ispitala mogućnost primene ovih oblika u inženjerstvu.",
publisher = "Univerzitet u Beogradu - Mašinski fakultet, Beograd",
journal = "FME Transactions",
title = "On the properties of the concave antiprisms of second sort, O osobinama konkavnih antiprizmi druge vrste",
pages = "263-256",
number = "3",
volume = "41",
url = "https://hdl.handle.net/21.15107/rcub_grafar_487"
}

Obradović, M., Popkonstantinović, B.,& Mišić, S.. (2013). On the properties of the concave antiprisms of second sort. in FME Transactions
Univerzitet u Beogradu - Mašinski fakultet, Beograd., 41(3), 256-263.
https://hdl.handle.net/21.15107/rcub_grafar_487

Obradović M, Popkonstantinović B, Mišić S. On the properties of the concave antiprisms of second sort. in FME Transactions. 2013;41(3):256-263.
https://hdl.handle.net/21.15107/rcub_grafar_487 .

Obradović, Marija, Popkonstantinović, Branislav, Mišić, Slobodan, "On the properties of the concave antiprisms of second sort" in FME Transactions, 41, no. 3 (2013):256-263,
https://hdl.handle.net/21.15107/rcub_grafar_487 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
AU  - Petrović, Maja
PY  - 2012
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2031
AB  - Concave cupolae of II sort, combined with the other concave or convex polyhedra with at least one matching side, provide many possibilities for the formation of various composite polyhedra. The paper presents research on regular –faced polyhedral structures obtained by joining bases of some concave cupolae of II sort, with the appropriate sides of Archimedean solids: truncated cube, truncated dodecahedron and great rhombicosidodecahedron.
PB  - Novi Sad: Faculty of Technical Sciences, University of Novi Sad
PB  - Novi Sad: Serbian Society for Geometry and Graphics (SUGIG)
C3  - Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012
T1  - Investigating Composite Polyhedral forms obtained by combining concave cupolae of  II sort with Archimedean Solids
EP  - 123
SP  - 109
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2031
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan and Petrović, Maja",
year = "2012",
abstract = "Concave cupolae of II sort, combined with the other concave or convex polyhedra with at least one matching side, provide many possibilities for the formation of various composite polyhedra. The paper presents research on regular –faced polyhedral structures obtained by joining bases of some concave cupolae of II sort, with the appropriate sides of Archimedean solids: truncated cube, truncated dodecahedron and great rhombicosidodecahedron.",
publisher = "Novi Sad: Faculty of Technical Sciences, University of Novi Sad, Novi Sad: Serbian Society for Geometry and Graphics (SUGIG)",
journal = "Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012",
title = "Investigating Composite Polyhedral forms obtained by combining concave cupolae of  II sort with Archimedean Solids",
pages = "123-109",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2031"
}

Obradović, M., Mišić, S.,& Petrović, M.. (2012). Investigating Composite Polyhedral forms obtained by combining concave cupolae of  II sort with Archimedean Solids. in Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012
Novi Sad: Faculty of Technical Sciences, University of Novi Sad., 109-123.
https://hdl.handle.net/21.15107/rcub_grafar_2031

Obradović M, Mišić S, Petrović M. Investigating Composite Polyhedral forms obtained by combining concave cupolae of  II sort with Archimedean Solids. in Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012. 2012;:109-123.
https://hdl.handle.net/21.15107/rcub_grafar_2031 .

Obradović, Marija, Mišić, Slobodan, Petrović, Maja, "Investigating Composite Polyhedral forms obtained by combining concave cupolae of  II sort with Archimedean Solids" in Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012 (2012):109-123,
https://hdl.handle.net/21.15107/rcub_grafar_2031 .

TY  - CONF
AU  - Obradović, Marija
AU  - Popkonstantinović, Branislav
AU  - Mišić, Slobodan
PY  - 2012
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2033
AB  - The paper discusses determining the positions and spatial coordinates of the vertices of polyhedral structures - concave antiprisms of second sort. These polyhedra originate from folding the double row strip of equilateral triangles, closed by two identical regular polygons, the principle akin to the way of the concave cupolae of second sort formation. The paper considers constructive -geometrical solution, including the method of solving the problem using the mechanisms, by application of SolidWorks software package.
PB  - Novi Sad: Faculty of Technical Sciences, University of Novi Sad
PB  - Novi Sad: Serbian Society for Geometry and Graphics (SUGIG)
C3  - Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012
T1  - Concave Antiprisms of Second Sort with Regular Polygonal Bases
EP  - 143
SP  - 133
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2033
ER  - 

@conference{
author = "Obradović, Marija and Popkonstantinović, Branislav and Mišić, Slobodan",
year = "2012",
abstract = "The paper discusses determining the positions and spatial coordinates of the vertices of polyhedral structures - concave antiprisms of second sort. These polyhedra originate from folding the double row strip of equilateral triangles, closed by two identical regular polygons, the principle akin to the way of the concave cupolae of second sort formation. The paper considers constructive -geometrical solution, including the method of solving the problem using the mechanisms, by application of SolidWorks software package.",
publisher = "Novi Sad: Faculty of Technical Sciences, University of Novi Sad, Novi Sad: Serbian Society for Geometry and Graphics (SUGIG)",
journal = "Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012",
title = "Concave Antiprisms of Second Sort with Regular Polygonal Bases",
pages = "143-133",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2033"
}

Obradović, M., Popkonstantinović, B.,& Mišić, S.. (2012). Concave Antiprisms of Second Sort with Regular Polygonal Bases. in Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012
Novi Sad: Faculty of Technical Sciences, University of Novi Sad., 133-143.
https://hdl.handle.net/21.15107/rcub_grafar_2033

Obradović M, Popkonstantinović B, Mišić S. Concave Antiprisms of Second Sort with Regular Polygonal Bases. in Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012. 2012;:133-143.
https://hdl.handle.net/21.15107/rcub_grafar_2033 .

Obradović, Marija, Popkonstantinović, Branislav, Mišić, Slobodan, "Concave Antiprisms of Second Sort with Regular Polygonal Bases" in Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012 (2012):133-143,
https://hdl.handle.net/21.15107/rcub_grafar_2033 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
AU  - Popkonstantinović, Branislav
AU  - Petrović, Maja
PY  - 2011
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2030
AB  - In recent years, we are witnessing an increasing use of triangular modular building systems, which, especially in representative objects, almost push out the orthogonal system. Considering that deltahedra are assembled of equilateral triangles matching system, we investigated a possible application of such polyhedral forms in geometry of architectural edifices. Deltahedral concave cupola with a regular polygonal base is formed by connecting two concentric regular polygons: n-gon and 2n-gon by multiple series of equilateral triangles. Lateral surface of the cupola is obtained by folding a net of equilateral triangles to a concave polyhedral surface. The paper analyzes the prospect of such surfaces in architecture, not only in the geometry of roof structures, which is imposing as an obvious solution, but also as a structural skeleton of the whole building. The analysis includes several criteria for evaluating the suitability of these forms for practical application: their static properties, atmospheric effects, functional analysis, preferential use of certain constructive systems, and finally, aesthetic properties of these forms.
PB  - Iasi: The Gheorghe Asachi Technical University of Iasi
C3  - Buletinul Institutului Politehnic din Iaşi, Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi - Proceedings of ICEGD Conference, Iasi 2011
T1  - Possibilities of Deltahedral Concave Cupola Form Application in Architecture
EP  - 140
IS  - Fasc. 3
SP  - 123
VL  - Tomul LVII (LXI)
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2030
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan and Popkonstantinović, Branislav and Petrović, Maja",
year = "2011",
abstract = "In recent years, we are witnessing an increasing use of triangular modular building systems, which, especially in representative objects, almost push out the orthogonal system. Considering that deltahedra are assembled of equilateral triangles matching system, we investigated a possible application of such polyhedral forms in geometry of architectural edifices. Deltahedral concave cupola with a regular polygonal base is formed by connecting two concentric regular polygons: n-gon and 2n-gon by multiple series of equilateral triangles. Lateral surface of the cupola is obtained by folding a net of equilateral triangles to a concave polyhedral surface. The paper analyzes the prospect of such surfaces in architecture, not only in the geometry of roof structures, which is imposing as an obvious solution, but also as a structural skeleton of the whole building. The analysis includes several criteria for evaluating the suitability of these forms for practical application: their static properties, atmospheric effects, functional analysis, preferential use of certain constructive systems, and finally, aesthetic properties of these forms.",
publisher = "Iasi: The Gheorghe Asachi Technical University of Iasi",
journal = "Buletinul Institutului Politehnic din Iaşi, Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi - Proceedings of ICEGD Conference, Iasi 2011",
title = "Possibilities of Deltahedral Concave Cupola Form Application in Architecture",
pages = "140-123",
number = "Fasc. 3",
volume = "Tomul LVII (LXI)",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2030"
}

Obradović, M., Mišić, S., Popkonstantinović, B.,& Petrović, M.. (2011). Possibilities of Deltahedral Concave Cupola Form Application in Architecture. in Buletinul Institutului Politehnic din Iaşi, Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi - Proceedings of ICEGD Conference, Iasi 2011
Iasi: The Gheorghe Asachi Technical University of Iasi., Tomul LVII (LXI)(Fasc. 3), 123-140.
https://hdl.handle.net/21.15107/rcub_grafar_2030

Obradović M, Mišić S, Popkonstantinović B, Petrović M. Possibilities of Deltahedral Concave Cupola Form Application in Architecture. in Buletinul Institutului Politehnic din Iaşi, Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi - Proceedings of ICEGD Conference, Iasi 2011. 2011;Tomul LVII (LXI)(Fasc. 3):123-140.
https://hdl.handle.net/21.15107/rcub_grafar_2030 .

Obradović, Marija, Mišić, Slobodan, Popkonstantinović, Branislav, Petrović, Maja, "Possibilities of Deltahedral Concave Cupola Form Application in Architecture" in Buletinul Institutului Politehnic din Iaşi, Publicat de Universitatea Tehnică „Gheorghe Asachi” din Iaşi - Proceedings of ICEGD Conference, Iasi 2011, Tomul LVII (LXI), no. Fasc. 3 (2011):123-140,
https://hdl.handle.net/21.15107/rcub_grafar_2030 .

TY  - CONF
AU  - Obradović, Marija
AU  - Dimitrijević, Magdalena
AU  - Mišić, Slobodan
PY  - 2010
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/3097
AB  - As a result of introducing Bologna process in education, classes of Descriptive geometry were reduced. In order to acquire such a complex matter as Descriptive geometry, it was necessary to provide an additional task as homework. After three years of experience in practicing classic additional supplementary problems, we made an attempt to innovate homework, following an idea to achieve creativity in appliance of Descriptive geometry knowledge in engineering practice. During the training classes, we recognized the students’ problem of connecting abstract apprehensions and principles of Descriptive geometry with basic problems and tasks of engineering practice. Form of seminary paper was chosen as an appropriate one for handling the themes and method units of Descriptive geometry program. Each student, in a group of 30 students, had its own theme to elaborate in text interpretation and drawings, also recognizing
(photos ...images) and presenting examples in real life (buildings or other examples in art, design or industry).
PB  - Faculty of Architecture, University of Belgrade
PB  - Serbian Society for Geometry and Graphics
C3  - Proceedings of the 2nd Int. Sci Conf. moNGeometrija 2010, Belgrade, Serbia
T1  - Seminary paper as an additional task in teaching descriptive geometry
EP  - 446
SP  - 435
UR  - https://hdl.handle.net/21.15107/rcub_grafar_3097
ER  - 

@conference{
author = "Obradović, Marija and Dimitrijević, Magdalena and Mišić, Slobodan",
year = "2010",
abstract = "As a result of introducing Bologna process in education, classes of Descriptive geometry were reduced. In order to acquire such a complex matter as Descriptive geometry, it was necessary to provide an additional task as homework. After three years of experience in practicing classic additional supplementary problems, we made an attempt to innovate homework, following an idea to achieve creativity in appliance of Descriptive geometry knowledge in engineering practice. During the training classes, we recognized the students’ problem of connecting abstract apprehensions and principles of Descriptive geometry with basic problems and tasks of engineering practice. Form of seminary paper was chosen as an appropriate one for handling the themes and method units of Descriptive geometry program. Each student, in a group of 30 students, had its own theme to elaborate in text interpretation and drawings, also recognizing
(photos ...images) and presenting examples in real life (buildings or other examples in art, design or industry).",
publisher = "Faculty of Architecture, University of Belgrade, Serbian Society for Geometry and Graphics",
journal = "Proceedings of the 2nd Int. Sci Conf. moNGeometrija 2010, Belgrade, Serbia",
title = "Seminary paper as an additional task in teaching descriptive geometry",
pages = "446-435",
url = "https://hdl.handle.net/21.15107/rcub_grafar_3097"
}

Obradović, M., Dimitrijević, M.,& Mišić, S.. (2010). Seminary paper as an additional task in teaching descriptive geometry. in Proceedings of the 2nd Int. Sci Conf. moNGeometrija 2010, Belgrade, Serbia
Faculty of Architecture, University of Belgrade., 435-446.
https://hdl.handle.net/21.15107/rcub_grafar_3097

Obradović M, Dimitrijević M, Mišić S. Seminary paper as an additional task in teaching descriptive geometry. in Proceedings of the 2nd Int. Sci Conf. moNGeometrija 2010, Belgrade, Serbia. 2010;:435-446.
https://hdl.handle.net/21.15107/rcub_grafar_3097 .

Obradović, Marija, Dimitrijević, Magdalena, Mišić, Slobodan, "Seminary paper as an additional task in teaching descriptive geometry" in Proceedings of the 2nd Int. Sci Conf. moNGeometrija 2010, Belgrade, Serbia (2010):435-446,
https://hdl.handle.net/21.15107/rcub_grafar_3097 .

TY  - CONF
AU  - Obradović, Marija
AU  - Dimitrijević, Magdalena
AU  - Mišić, Slobodan
PY  - 2010
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2051
AB  - As a result of introducing Bologna process in education, classes
of Descriptive geometry were reduced. In order to acquire such a
complex matter as Descriptive geometry, it was necessary to provide
an additional task as homework. After three years of experience in
practicing classic additional supplementary problems, we made an
attempt to innovate homework, following an idea to achieve
creativity in appliance of Descriptive geometry knowledge in
engineering practice. During the training classes, we recognized the
students’ problem of connecting abstract apprehensions and principles of Descriptive geometry with basic problems and tasks of engineering practice. Form of seminary paper was chosen as an appropriate one for handling the themes and method units of Descriptive geometry program. Each student, in a group of 30 students, had its own theme to elaborate in text interpretation and drawings, also recognizing (photos ...images) and presenting examples in real life (buildings or other examples in art, design or industry).
PB  - Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)
C3  - PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010
T1  - Seminary Paper as an Additional Task in Teaching Descriptive Geometry
EP  - 446
SP  - 435
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2051
ER  - 

@conference{
author = "Obradović, Marija and Dimitrijević, Magdalena and Mišić, Slobodan",
year = "2010",
abstract = "As a result of introducing Bologna process in education, classes
of Descriptive geometry were reduced. In order to acquire such a
complex matter as Descriptive geometry, it was necessary to provide
an additional task as homework. After three years of experience in
practicing classic additional supplementary problems, we made an
attempt to innovate homework, following an idea to achieve
creativity in appliance of Descriptive geometry knowledge in
engineering practice. During the training classes, we recognized the
students’ problem of connecting abstract apprehensions and principles of Descriptive geometry with basic problems and tasks of engineering practice. Form of seminary paper was chosen as an appropriate one for handling the themes and method units of Descriptive geometry program. Each student, in a group of 30 students, had its own theme to elaborate in text interpretation and drawings, also recognizing (photos ...images) and presenting examples in real life (buildings or other examples in art, design or industry).",
publisher = "Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)",
journal = "PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010",
title = "Seminary Paper as an Additional Task in Teaching Descriptive Geometry",
pages = "446-435",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2051"
}

Obradović, M., Dimitrijević, M.,& Mišić, S.. (2010). Seminary Paper as an Additional Task in Teaching Descriptive Geometry. in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010
Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)., 435-446.
https://hdl.handle.net/21.15107/rcub_grafar_2051

Obradović M, Dimitrijević M, Mišić S. Seminary Paper as an Additional Task in Teaching Descriptive Geometry. in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010. 2010;:435-446.
https://hdl.handle.net/21.15107/rcub_grafar_2051 .

Obradović, Marija, Dimitrijević, Magdalena, Mišić, Slobodan, "Seminary Paper as an Additional Task in Teaching Descriptive Geometry" in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010 (2010):435-446,
https://hdl.handle.net/21.15107/rcub_grafar_2051 .

TY  - CONF
AU  - Mišić, Slobodan
AU  - Obradović, Marija
PY  - 2010
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2047
AB  - The cupolae with concave polyhedral surfaces consist of: two regular
polygons, n-gone and 2n-gone in parallel planes, interconnected by an
envelope constituted of series of equilateral triangles. The paper
describes cupolae which originate by corrugating of a fourfold strip of
equilateral triangles, forming thereby the envelope of a cupola. In
this manner, a non-convex polyhedron is emerged. Such a method of corrugating the envelope, allows the solutions for generating cupolae with base polygon which number of sides exceedes n=10, which was the maximal number of sides for cupole with the envelope consisted of twofold strip of equilateral triangles. By analyzing the elements of these polyhedra and by help of their paper models, we find geometric constructions and projection procedures by which it is possible to graphically display the cupolae.
PB  - Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)
C3  - PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010
T1  - Forming the Cupolae With Concave Polyhedral Surfaces by Corrugating a Fourfold Strip of Equilateral Triangles
EP  - 374
SP  - 363
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2047
ER  - 

@conference{
author = "Mišić, Slobodan and Obradović, Marija",
year = "2010",
abstract = "The cupolae with concave polyhedral surfaces consist of: two regular
polygons, n-gone and 2n-gone in parallel planes, interconnected by an
envelope constituted of series of equilateral triangles. The paper
describes cupolae which originate by corrugating of a fourfold strip of
equilateral triangles, forming thereby the envelope of a cupola. In
this manner, a non-convex polyhedron is emerged. Such a method of corrugating the envelope, allows the solutions for generating cupolae with base polygon which number of sides exceedes n=10, which was the maximal number of sides for cupole with the envelope consisted of twofold strip of equilateral triangles. By analyzing the elements of these polyhedra and by help of their paper models, we find geometric constructions and projection procedures by which it is possible to graphically display the cupolae.",
publisher = "Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)",
journal = "PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010",
title = "Forming the Cupolae With Concave Polyhedral Surfaces by Corrugating a Fourfold Strip of Equilateral Triangles",
pages = "374-363",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2047"
}

Mišić, S.,& Obradović, M.. (2010). Forming the Cupolae With Concave Polyhedral Surfaces by Corrugating a Fourfold Strip of Equilateral Triangles. in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010
Belgrade: Faculty of Architecture in Belgrade; Serbian Society for Geometry and Graphics (SUGIG)., 363-374.
https://hdl.handle.net/21.15107/rcub_grafar_2047

Mišić S, Obradović M. Forming the Cupolae With Concave Polyhedral Surfaces by Corrugating a Fourfold Strip of Equilateral Triangles. in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010. 2010;:363-374.
https://hdl.handle.net/21.15107/rcub_grafar_2047 .

Mišić, Slobodan, Obradović, Marija, "Forming the Cupolae With Concave Polyhedral Surfaces by Corrugating a Fourfold Strip of Equilateral Triangles" in PROCEEDINGS | BILTEN of 25th National and 2nd International Scientific Conference moNGeometrija 2010 (2010):363-374,
https://hdl.handle.net/21.15107/rcub_grafar_2047 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2008
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2041
AB  - Cilj istraživanja ovog rada je bio da se iznađu geometrijska rešenja za formiranje prostornih rešetki nad pravilnim poligonalnim osnovama, tako da se postigne prostorna struktura koja ispunjava zahteve rigidnosti datog sklopa, po osnovi geometrijske stabilnosti trougla kao
ravne figure. Takođe, težnja je bila i da se nađe što jednostavnije rešenje, sa upotrebom minimalnog broja različitih štapova u samoj konfiguraciji prostorne rešetke. U tom smislu, moguće je primeniti nekoliko varijanti rasporeda štapova. Najekonomičnije od njih se svodi upravo na geometriju konkavnih kupola druge vrste, buduci da se radi o jednakoivičnim poliedrima kao osnovama za dalju nadogradnju tridimenzionalnih stuktura. Omotač konkavnih kupola druge vrste nastaje nabiranjem ravne mreže, a prevođenjem ivica ovog omotača u sistem štapova, uz dodavanje minimalnog broja različitih tipskih štapova razupirača, dobijamo strukturu kompozitnog poliedra prostorne rešetke.
AB  - The aim of this paper is to find geometric solutions for the formation of space frames over regular polygonal bases, so to achieve a spatial structure that meets the rigidity requirements of a given assembly, based on the geometric stability of a triangle as a flat figure. Also, the aim is to find the simplest possible solution, with the use of a minimum number of different struts in the configuration of the space frame. In that sense, it is possible to apply several variants of strut arrangement. The most material-efficient of them comes down to the geometry of concave cupolae of the second sort, since they are polyhedra of all the edges of the same length, as a basis for further upgrading of three-dimensional structures. The lateral surface of a concave cupola of the second sort is created by folding its triangular planar net, and by transposing the edges of this surface into a system of struts, having the minimum number of different type struts, we obtain the structure of a composite polyhedron in the role of a space frame.
PB  - Niš: Faculty of Architecture and Civil Engineering in Niš; Serbian Society for Geometry and Graphics (SUGIG)
C3  - Proccedings | Zbornik radova (24th national and 1st international scientific conference moNGeometrija 2008)
T1  - Prevođenje konkavnih kupola druge vrste u tridimenzionalne konstruktivne sisteme – prostorne rešetke
EP  - 221
SP  - 209
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2041
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2008",
abstract = "Cilj istraživanja ovog rada je bio da se iznađu geometrijska rešenja za formiranje prostornih rešetki nad pravilnim poligonalnim osnovama, tako da se postigne prostorna struktura koja ispunjava zahteve rigidnosti datog sklopa, po osnovi geometrijske stabilnosti trougla kao
ravne figure. Takođe, težnja je bila i da se nađe što jednostavnije rešenje, sa upotrebom minimalnog broja različitih štapova u samoj konfiguraciji prostorne rešetke. U tom smislu, moguće je primeniti nekoliko varijanti rasporeda štapova. Najekonomičnije od njih se svodi upravo na geometriju konkavnih kupola druge vrste, buduci da se radi o jednakoivičnim poliedrima kao osnovama za dalju nadogradnju tridimenzionalnih stuktura. Omotač konkavnih kupola druge vrste nastaje nabiranjem ravne mreže, a prevođenjem ivica ovog omotača u sistem štapova, uz dodavanje minimalnog broja različitih tipskih štapova razupirača, dobijamo strukturu kompozitnog poliedra prostorne rešetke., The aim of this paper is to find geometric solutions for the formation of space frames over regular polygonal bases, so to achieve a spatial structure that meets the rigidity requirements of a given assembly, based on the geometric stability of a triangle as a flat figure. Also, the aim is to find the simplest possible solution, with the use of a minimum number of different struts in the configuration of the space frame. In that sense, it is possible to apply several variants of strut arrangement. The most material-efficient of them comes down to the geometry of concave cupolae of the second sort, since they are polyhedra of all the edges of the same length, as a basis for further upgrading of three-dimensional structures. The lateral surface of a concave cupola of the second sort is created by folding its triangular planar net, and by transposing the edges of this surface into a system of struts, having the minimum number of different type struts, we obtain the structure of a composite polyhedron in the role of a space frame.",
publisher = "Niš: Faculty of Architecture and Civil Engineering in Niš; Serbian Society for Geometry and Graphics (SUGIG)",
journal = "Proccedings | Zbornik radova (24th national and 1st international scientific conference moNGeometrija 2008)",
title = "Prevođenje konkavnih kupola druge vrste u tridimenzionalne konstruktivne sisteme – prostorne rešetke",
pages = "221-209",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2041"
}

Obradović, M.,& Mišić, S.. (2008). Prevođenje konkavnih kupola druge vrste u tridimenzionalne konstruktivne sisteme – prostorne rešetke. in Proccedings | Zbornik radova (24th national and 1st international scientific conference moNGeometrija 2008)
Niš: Faculty of Architecture and Civil Engineering in Niš; Serbian Society for Geometry and Graphics (SUGIG)., 209-221.
https://hdl.handle.net/21.15107/rcub_grafar_2041

Obradović M, Mišić S. Prevođenje konkavnih kupola druge vrste u tridimenzionalne konstruktivne sisteme – prostorne rešetke. in Proccedings | Zbornik radova (24th national and 1st international scientific conference moNGeometrija 2008). 2008;:209-221.
https://hdl.handle.net/21.15107/rcub_grafar_2041 .

Obradović, Marija, Mišić, Slobodan, "Prevođenje konkavnih kupola druge vrste u tridimenzionalne konstruktivne sisteme – prostorne rešetke" in Proccedings | Zbornik radova (24th national and 1st international scientific conference moNGeometrija 2008) (2008):209-221,
https://hdl.handle.net/21.15107/rcub_grafar_2041 .

TY  - CONF
AU  - Obradović, Marija
AU  - Mišić, Slobodan
PY  - 2008
UR  - https://grafar.grf.bg.ac.rs/handle/123456789/2044
AB  - Cupola is a polyhedron which is consisted of two regular polygons: n-gon and 2n-gon in parallel planes, connected by alternating sequence of squares and equilateral triangles, i.e. Johnson’s solids J3, J4 and J5. However, it is possible to form a polyhedron with the analogously chosen regular n-gon and 2n-gon in parallel planes, in that manner to have a base polygon with n ≥
6, and whose envelope would be formed of series of equilateral triangles, creating a concave
polyhedron, similar to the Johnson’s cupolae, and furthermore to the Johnson’s rotundae. In a lack of an adequate name, we deemed to engender the meaning of the term cupola to a concave
polyhedron that includes regular faces polygons in its geometry, whereat two of them are n-gon and 2n-gon in parallel planes. The method of forming such a cupola is based on wrinkling the net of equilateral triangles, which produce a twofold strip, by folding of which we obtain a deltahedral
envelope surface. Such manner of creating a polyhedron, gives a solution to a problem of creating a regular faced solid which includes even ‘unconstructable’ polygons, as heptagon and nonagon. In this paper, there are described concave regular faced cupolae originated by wrinkling the envelope net consisted of two rows [(2x3+1)n] of equilateral triangles; therefore they are named: the cupolae of second sort. The cupolae originated by using envelope net made of tree rows of equilateral triangles would thus be named: concave regular faced cupolae of third sort, and so on. Concave regular faced cupolae of second sort can have the starting bases from n=4 to n=10. Hendecagon can not be used for the start base polygon (n-gon), because the distance from its double sided counterpart polygon in the parallel plane, would exceed the double value of equilateral triangle’s altitude, the width of the envelope strip. The main parameters of these solids can be found by determining the trajectory of the envelope strip’s elementary cell’s vertices, consisted of six equilateral triangles, which will move around the edge of 2n-gon, behaving as mechanism. The shape of trajectory would show the curve of higher order, therefore there would be two ways to assemble the envelope, so there would exist two possible altitudes of such obtained polyhedrons. There are fourteen solids that would be classified as the concave regular faced cupolae of second sort.
PB  - Dresden: Technische Universität Dresden; International Society for Geometry and Graphics (ISGG)
C3  - Proceedings of the 13th International Conference on Geometry and Graphic - ICGG 2008
T1  - Concave Regular Faced Cupolae of Second Sort
EP  - 10
SP  - 1
SP  - 164 (in Program book)
UR  - https://hdl.handle.net/21.15107/rcub_grafar_2044
ER  - 

@conference{
author = "Obradović, Marija and Mišić, Slobodan",
year = "2008",
abstract = "Cupola is a polyhedron which is consisted of two regular polygons: n-gon and 2n-gon in parallel planes, connected by alternating sequence of squares and equilateral triangles, i.e. Johnson’s solids J3, J4 and J5. However, it is possible to form a polyhedron with the analogously chosen regular n-gon and 2n-gon in parallel planes, in that manner to have a base polygon with n ≥
6, and whose envelope would be formed of series of equilateral triangles, creating a concave
polyhedron, similar to the Johnson’s cupolae, and furthermore to the Johnson’s rotundae. In a lack of an adequate name, we deemed to engender the meaning of the term cupola to a concave
polyhedron that includes regular faces polygons in its geometry, whereat two of them are n-gon and 2n-gon in parallel planes. The method of forming such a cupola is based on wrinkling the net of equilateral triangles, which produce a twofold strip, by folding of which we obtain a deltahedral
envelope surface. Such manner of creating a polyhedron, gives a solution to a problem of creating a regular faced solid which includes even ‘unconstructable’ polygons, as heptagon and nonagon. In this paper, there are described concave regular faced cupolae originated by wrinkling the envelope net consisted of two rows [(2x3+1)n] of equilateral triangles; therefore they are named: the cupolae of second sort. The cupolae originated by using envelope net made of tree rows of equilateral triangles would thus be named: concave regular faced cupolae of third sort, and so on. Concave regular faced cupolae of second sort can have the starting bases from n=4 to n=10. Hendecagon can not be used for the start base polygon (n-gon), because the distance from its double sided counterpart polygon in the parallel plane, would exceed the double value of equilateral triangle’s altitude, the width of the envelope strip. The main parameters of these solids can be found by determining the trajectory of the envelope strip’s elementary cell’s vertices, consisted of six equilateral triangles, which will move around the edge of 2n-gon, behaving as mechanism. The shape of trajectory would show the curve of higher order, therefore there would be two ways to assemble the envelope, so there would exist two possible altitudes of such obtained polyhedrons. There are fourteen solids that would be classified as the concave regular faced cupolae of second sort.",
publisher = "Dresden: Technische Universität Dresden; International Society for Geometry and Graphics (ISGG)",
journal = "Proceedings of the 13th International Conference on Geometry and Graphic - ICGG 2008",
title = "Concave Regular Faced Cupolae of Second Sort",
pages = "10-1-164 (in Program book)",
url = "https://hdl.handle.net/21.15107/rcub_grafar_2044"
}

Obradović, M.,& Mišić, S.. (2008). Concave Regular Faced Cupolae of Second Sort. in Proceedings of the 13th International Conference on Geometry and Graphic - ICGG 2008
Dresden: Technische Universität Dresden; International Society for Geometry and Graphics (ISGG)., 1-10.
https://hdl.handle.net/21.15107/rcub_grafar_2044

Obradović M, Mišić S. Concave Regular Faced Cupolae of Second Sort. in Proceedings of the 13th International Conference on Geometry and Graphic - ICGG 2008. 2008;:1-10.
https://hdl.handle.net/21.15107/rcub_grafar_2044 .

Obradović, Marija, Mišić, Slobodan, "Concave Regular Faced Cupolae of Second Sort" in Proceedings of the 13th International Conference on Geometry and Graphic - ICGG 2008 (2008):1-10,
https://hdl.handle.net/21.15107/rcub_grafar_2044 .