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A newly discovered heptahedron named epitahedron. (English) Zbl 1324.51012

Summary: We introduce a newly discovered polytope, the epitahedron, which can be assigned as a 3-dimensional representation of the Penrose Kites and darts tiling (P2), a cell of the 5-dimensional space. The lengths of the edges as well as the volumes of the epitahedra – E- (concave) E+ (convex) and both together, EE (E- : E+ = E+ : EE)– are conform to the golden ratio. ( \(\varphi\equiv\) 1.618). Thus epitahedra are tiling the space in the golden ratio analogous to the irregular Penrose Tiling in the plane, which is a slice of the 5-dimensional space. Two of these 3-dimensional pentagrams inscribed into an icosahedron are creating the boundaries of the dodecahedron by intersection. Their circle decorations on its faces allow to uniquely demonstrate symmetry operations for the visualization of groups. The composition of 12 epitahedra creates naturally a 26 dimensional space with a complex intersecting space configuration in the center. By updown generation the infinite space in the shape of a dodecahedron can be obtained.

MSC:

51M20 Polyhedra and polytopes; regular figures, division of spaces
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