Abstract
The icosahedron and the dodecahedron have the same graph structures as their algebraic conjugates, the great dodecahedron and the great stellated dodecahedron. All four polyhedra are equilateral and have planar faces—thus “EP”—and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and “pathological” because of intersecting faces. Approaching the problem analytically, we sought alternate EP-embeddings for Platonic and Archimedean solids. We prove that the number of equations—E edge length equations (enforcing equilaterality) and 2 E − 3 F face (torsion) equations (enforcing planarity)—and of variables ( 3 V − 6 ) are equal. Therefore, solutions of the equations up to equivalence generally leave no degrees of freedom. As a result, in general there is a finite (but very large) number of solutions. Unfortunately, even with state-of-the-art computer algebra, the resulting systems of equations are generally too complicated to completely solve within reasonable time. We therefore added an additional constraint, symmetry, specifically requiring solutions to display (at least) tetrahedral symmetry. We found 77 non-classical embeddings, seven without intersecting faces—two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron.
Highlights
Mathematicians have long been fascinated by the symmetry and regularity of certain three-dimensional solids
(Again, it has edge length 2.) If we replace every occurrence of φ by φin these coordinates, we obtain an algebraic conjugate embedding that corresponds to the great stellated dodecahedron (Figure 2e,f)
We have shown how to set up the system of equations that need to be solved to find all EP-embeddings of a polyhedron, i.e., embeddings in three-dimensional Euclidian space with planar faces and edges of equal length
Summary
Mathematicians have long been fascinated by the symmetry and regularity of certain three-dimensional solids. Permitting more than one type of face, but still requiring identical vertices, regular faces, full symmetry and convexity, the Greeks and later Johannes Kepler found the 13 semi-regular (or “Archimedean”) solids [1]. With identical vertices, these polyhedra are all described as “uniform”. We are able to find all equiplanar solutions for many (but not all) of the icosahedral, octahedral and tetrahedral, regular and semi-regular solids reduced to geometric Td , Th and T symmetry These new methods provide the foundation for exploring the new vein much further for still lower symmetry, for larger and for non-uniform solids, with numerical methods. We provide a website with interactive three-dimensional models of the results of Sections 6 and 7 at URL http://caagt.ugent.be/ep-embeddings/
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