Talk:Trivial group

Smallest abstractly asymmetric polytopes
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 4$n$+8 flags
 * A polyhedron with $e$ edges has f$e$ 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 4$n$+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)


 * Strange trichoron.svg]]
 * I found an asymmetric abstract polychoron with 3 facets. I've drawn up the flag graph on the right. This proves that 3 facets is possible as I suspected. It may be possible to reduce this example in some ways, but I just wanted to get a quick and easy example out, so I can beat the listed record of 4 given in the article. It's not possible to create a 2-facet abstract polytope that is asymmetric, so 3 is optimal.
 * Sycamore916 (talk) 23:57, 12 March 2023 (UTC)


 * I think that Lemma 1 above is incorrect. This abstract polyhedron (Geogebra link) with 7 vertices has an underlying graph which I am 99% sure is asymmetric, which would mean that the resulting polyhedron must be asymmetric too. It still has more than 24 flags but I thought it was worth mentioning. Plasmath (talk) 16:47, 13 March 2023 (UTC)
 * I cannot open the Geogebra link, but here is an image with all abstract polyhedra with 7 vertices and 3 faces. Each is drawn with a mirror line demonstrating a symmetry:
 * 3F7V symmetries.svg
 * More generally these mirror lines should appear for any 3 faced abstract polyhedron since they all take on a similar form, unless I'm missing some class for which this does not hold.
 * Sycamore917 (talk) 19:03, 13 March 2023 (UTC)


 * I should have guessed that Geogebra would make these links private. Here's an image of what is in the Geogebra file:
 * Abstract asymmetric polyhedron with 3 faces.png
 * The 1-skeletons that you show above all have 8 edges, but this polyhedron has 9.
 * Also, may I suggest that you join the Polytope Discord? Almost all discussion of polytopes happens there, and in future discussions we wouldn't be uploading lots of image files to Miraheze that are only used in talk pages.
 * Plasmath (talk) 03:08, 14 March 2023 (UTC)
 * That's a good example! Definitely asymmetric. I was only counting cases where faces border each other once. Sycamore916 (talk) 03:15, 14 March 2023 (UTC)