<p>E. Schulte Polytops and Symmetry.</p><p>Regular polytopes.</p><p>Regular polytopes or platonic solids are convex solids (closed) where all the building blocks (vertices, edges, faces, hyperfaces) have the same characteristics. That is, vertices have the same number of neighbours, edges are all the same length, polygons are all the same shape and area, and hyperfaces have the same volume. </p><p>In 2 dimensions there an infinity of regular polytopes. For example, a circle can be approximated by polygons with 2 edges (line), 3 edges (triangle), 4 edges (square), 5 edges, 6 edges indeed any number of edges. In 2 dimensions, every regular polytope is its own dual. </p><p>In 3 dimensions there are just 5, these are typically known as the platonic solids. In 3 dimensions, the tetrahedron is self-dual. The dual of the cube is the octahedron. And the dual of the dodecahedron is the icosahedron. </p><p>Convex polytope P: A convex polytope may be defined as the convex hull of a finite set of points (which are always bounded) in E n -Euclidian space, or as a bounded intersection of a finite set of half-spaces.</p><p>A symbol of the form used to describe regular polygons, polyhedra, and their higher-dimensional counterparts. n=2 polygons {p} (Schlafle - symbol) n=3 platonic solids {p,q}</p><p>Dimension n=4</p><p>Name Symbol Facets Group</p><p>Simplex {3,3,3} 5 120 Dimension n>=5</p><p>Simplex {3,…,3} n+1 (n+1)! Cross-polytope {3,…,3,4} 2n n!2n Cube {4,3,…,3} 2n n!2n</p><p>Symmetry group of { p,q,r} is the Coxeter group with the string diagram. .___p____.___q____.____r___. </p><p>Tesselations and Honeycombs. A regular tiling of polygons (in two dimensions), polyhedra (three dimensions), or polytops ( dimensions) is called a tessellation. Tessellations can be specified using a Schlafli symbol. Honeycombs are n-dimensional polytops.</p><p>Euclidian space. n=2: with triangles, hexagons, squares - {6,3}, {4,4}, {3,6}</p><p> n>=3: with cubes {4,3…,3,4} n=4: with 24 cells {3,4,3,3} with cross-polytops</p><p>Abstract Polytope P of the rank ( or dimension) n</p><p>P ranked partially ordered set i-face elements of rank i ( = -1,0,1,…,n) i=0 vertices ( points) i=1 edges ( lines) i=n-1 facets (hyperplanes)</p><p> Faces F- 1,Fn (of rank -1,n)  Each flag of P contains exactly n+2 faces  P is connected  Intervals of rank 1 are diamonds</p><p>Def: P is regular if and only if G(P) is flag transitive.</p><p>Dimensions n=0, 1, 2 ( points, segments, polygons) appropriately.</p><p>Dimension n=3 P is a regular map ( cell complex, tesselation) on a closed surface.</p><p>Abstract vs. Geometric Theory.</p><p>Use following approach. Given an abstract Polytiope. Find its realiszations in Euclidian Spaces.( Regular, Chiral, etc.) Polyhedra in E 3 Discrete faithful realizations of abstract Polyhedra ( 3-Polytopes) in E 3 .</p><p>Enumeration either finite or infinite(discrete)</p><p> 18 finite polyhedra: 5 platonic, 4 Kepler-Poinsot, 9 Petrials. Group of Petrial: </p><p>R0R2,R1R2  6 planar apairohedra: 3 tesselations, 3 Petrials  24 (discrete) apairohedra</p><p>- 12 reducible (blend) </p><p>- 12 irreducible (pure)</p>