Talk:Trivial group

What are the smallest abstractly asymmetrical polytopes? (By reasonable metrics such as element count, facet count, etc.) Vel (talk) 04:23, 12 January 2023 (UTC)
 * You can't have asymmetric polytopes of rank < 3. There is a abstract polyhedron with 4 faces that has a trivial automorphism group. Faces are 1 digon, 1+1 triangles, and 1 square, for 24 flags. You can't have an abstract polyhedron with 1 face, and any polyhedron with 2 faces is symmetric. 3 is the minimum, but I suspect that it's impossible because any polyhedron seems to have a mirror symmetry (an involution). If you are looking to minimize flags, this 4 sided polyhedron seems pretty good. If you want to minimize facets, 3 seems like it could be achieved in 4D. Sycamore916 (talk) 04:43, 12 January 2023 (UTC)
 * Quick proof that the described polytope is the smallest asymmetric abstract polytope of rank 3 by flag count:
 * Lemma 1. No asymmetric abstract polyhedron (AsAP) has less than 4 faces:
 * Omitted for brevity.
 * Lemma 2. An AsAP with an n-gonal face has a minimum of 4n+8 flags
 * A polyhedron with $e$ edges has $4e$ flags. This means an AsAP with an n-gonal face must have n edges to make the n-gon, and any additional minimum of 2 extra edges for other faces due to lemma 1. This means a minimum of n+2 edges and 4n+8 flags.
 * QED
 * As a corollary no AsAP with n-gonal faces where n > 4, has 24 or fewer flags. And any AsAP with fewer than 24 flags has only 2-gon and 3-gon faces.
 * Lemma 3. An AsAP with only 2-gon and 3-gon faces must have more than 24 flags.
 * A polyhedron must have an even number of faces with odd edges. Since 2-gons are even, there must be an even number of 3-gons faces. Each 3-gon face of a polyhedron has 6 flags. This means the number of triangular faces must be 2 (12 flag minimum) or 4 (24 flag minimum). In the case of 4 triangular faces, there must be no 2-gons.  Both possible ways to construct a polyhedron with 4 triangular faces are face-transitive. Thus there must be exactly two 3-gons and up to three 2-gons. There are 6 possible polytopes to consider 1 with a single 2-gon, 2 with two 2-gons and 3 with three 3-gons. All of them are transitive on their triangular faces.
 * QED
 * Corollary: No non-trivial asymmetric abstract polytope has fewer than 24 flags.
 * The minimum number of flags for a a rank-$n$ polytope is $$2^n$$. This would be the ditope of ditopes. However a non-trivial asymmetric abstract polytope must have at least 3 facets, thus the minimum for non-trivial asymmetric polytopes is $$3\times 2^{n-1}$$. This gives 24 for rank 4 and greater than 24 for higher ranks.
 * Sycamore916 (talk) 20:38, 12 March 2023 (UTC)
 * Sycamore916 (talk) 20:38, 12 March 2023 (UTC)