Composite Concave Cupolae as Geometric and Architectural Forms
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2015
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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 (no...t 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.