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Journal for Geometry and Graphics 19 (2015), No. 1, 079--091
Copyright Heldermann Verlag 2015

Composite Concave Cupolae as Geometric and Architectural Forms

Slobodan Misic
Faculty of Civil Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia

Marija Obradovic
Faculty of Civil Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia

Gordana Dukanovic
Faculty of Forestry, University of Belgrade, Serbia

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 alternating 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 remove 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 examine the possibilities of combining the considered polyhedra into unified composite structures.

Keywords: Concave polyhedra, cupola, deltahedron, equilateral triangle.

MSC: 51N05; 51M20, 00A67

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