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1 day ago · Platonic solid. In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex.
2 days ago · Learn about the five convex regular polyhedra or Platonic solids, their properties, symmetries, and applications. Explore the proof that there are only five Platonic solids, Euler's formula, and the Schläfli symbols.
4 days ago · These high-symmetry molecules have shapes corresponding to the five platonic solids: tetrahedral, octahedral, cube, dodecahedral and icosahedra. Tetrahedral Point Groups: The highest-fold axis in these point groups is C 3 axis, which is occur in multiples.
1 day ago · There are 4 regular projective polyhedra related to 4 of 5 Platonic solids. The hemi-cube and hemi-octahedron generalize as hemi- n -cubes and hemi- n - orthoplexes to any rank. Regular projective polyhedra
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5 days ago · The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato (c. 424 – 348 BC), Theaetetus (c. 417 BC – 369 BC), Timaeus of Locri (c. 420–380 BC), and Euclid (fl. 300 BC). The Etruscans discovered the regular dodecahedron before 500 BC. [3]
2 days ago · A regular map is an abstract generalization of a Platonic solid. It describes a group, a topological cell decomposition of a 2-manifold of type {p, q} with only p-gons, such that q of them meet at each vertex in a circular manner, and we have maximal combinatorial symmetry, expressed by the flag transitivity of the symmetry group. On the one hand, we have articles on topological surface embeddings of regular maps by F. Razafindrazaka and K. Polthier, C. Séquin, and J. J. van Wijk.On the ...
4 days ago · This ratio completes an elegant triplet of ratios for vertex-to-face dual pairings when the outer Platonic solid is the tetrahedron, octahedron, and icosahedron (i.e., those with triangular faces), specifically 1 : 3, √2, and ϕ: 3, respectively.
