Talk:Isotoxal figure

Poll to move to edge-transitive

Edge-transitive is a standard term for polytopes. Edge-uniform was an incorrect term derived from uniform polytopes which are vertex-transitive.

• Yes - Tom Ruen 22:26, 3 February 2007 (UTC)

• Yes and no. I would prefer to move it to Isotoxal because it's shorter, and redirect from Edge-transitive. Steelpillow 13:03, 6 February 2007 (UTC)

Isotoxal polytopes

It looks like a GENERAL rule is that a vertex transitive polytope is also edge-transitive if its vertex figure is vertex transitive! (Since each vertex in the vertex figure represents an edges in the polytope) The dual-rule of polyhedra doesn't quite apply for other dimensional polytopes, but it's sort of dimensionally reversed. (The dual of a vertex-transitive, face-transitive 4-polytope will be cell-transitive, edge-transitive because vertices/cells swap, and edges-faces swap.)

I'll see if I can find any sources to support these assertions! Tom Ruen (talk) 23:18, 2 October 2009 (UTC)

Every dynkin graph, that has only one marked node, has only one kind of edge, is therefore isotoxal. Those that have several marked nodes, are isotoxal, when the kinds of node by themselves, lead to the same polytope differently orientated in the symmetry. One must understand that the nodes in the graph are pedal nodes, þat is, a point where þe mirror bisects a kind of edge.

Þe implication is then that if þere is only one kind of edge, then it is carried by symmetry to every edge, and is therefore isotoxal giving. --Wendy.krieger (talk) 08:31, 7 October 2009 (UTC)

Isotoxal stars in four dimensions

The rule of a single marked node, gives an isotoxal figure, applies even to the star-figures.

In practice, the star groups like V53, 5V5, 53V, V5V, 3V5, and 33V (V=5/2), leads to no case of coincidence, since the faces are invariantly different.

In the remaining cases, where nodes are identical, such as x5o5/2o5x, x5/2o5o5/2x, and o5/2x5x5/2o, these lead to a double-cover, such that the faces coincide in pairs. None the same, the single cover is valid.

The case of o5x5/2x5o leads to a sixwise cover of the 533, but there is a figure o5x5/3x5o, which leads to a double cover of 240 alt-truncated (5/2,5) [verf = 10/3, 10/3, 5] these fall by pairs into 120 of the same face. --Wendy.krieger (talk) 08:38, 7 October 2009 (UTC)

Isotoxal 4-polytopes/honeycombs

I think this is a complete list of regular convex and uniform 4-polytopes/honeycombs that are isotoxal. (And a subset of nonconvex forms from the nonconvex regulars)

Moved list to User:Tomruen/List of isotoxal polychora and honeycombs until I can get some references... Tom Ruen (talk) 07:28, 11 October 2009 (UTC)

Counter-example

An edge-transitive tiling is not necessarily vertex- or face-transitive!

Consider an irregular spherical dihedron, made of a small rhombus in the Northern hemisphere, and a large rhombus on the rest of the sphere, including the Equator and the Southern hemisphere. The dihedron's symmetries are : two perpendicular reflections (across the rhombus' diagonals), the resulting 180deg rotation (around the rhombus' center), and the identity. These four symmetries map any edge to any other, so it is edge-transitive. But a vertex only gets mapped to itself and the opposite vertex, not the adjacent two, so it is not vertex-transitive. And, of course, the two rhombi have different areas, so the dihedron is not face-transitive either. — Preceding unsigned comment added by Mr e man2017 (talk • contribs) 00:09, 18 August 2017 (UTC)

What is the most appropriate notion?

I have written "concave (isotoxal) polyhedra" in the Isotoxal figure article & in the List of isotoxal polyhedra and tilings article, & i wonder if concave is the most appropriate notion to uniform polyhedra & to dual uniform polyhedra:

concave, or non-convex, or star polyhedra? Are there different cases?

In advance, thank you for your answer! :-)

RavBol (talk) 14:28, 4 September 2020 (UTC)