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Face-transitive is a standard term for polytopes. Face-uniform was an incorrect term derived from uniform polytopes which are are vertex-transitive, and their duals which are face-transitive.
Looking behind the recent to-and-fro moves, it seems to me that Isohedral should be more than a redirect. It could be made a disambuguation page, or it could be basically this one, with links to isohedral numbers and the like. My vote is to move this page across. -- Cheers, Steelpillow ( Talk) 19:35, 9 July 2008 (UTC)
If this is to be moved, move it to face-transitive polytope or face-transitive polyhedron or isohedral polytope or isohedral polyhedron or some other appropriate noun phrase, not to an adjective. Michael Hardy ( talk) 20:48, 9 July 2008 (UTC)
Anyway I DEFINITE dislike adding the word "polyhedron" to terms that apply to all polytopes/tilings/honeycombs. Tom Ruen ( talk) 18:12, 12 July 2008 (UTC)
Would that last one be OK by you? -- Cheers, Steelpillow ( Talk) 19:23, 12 July 2008 (UTC)
I am confused by the section k-isohedral. The definitions for polytopes and tilings appear to be very different: The former makes no mention of symmetry orbits and explicitly forbids different shapes. Strangely, it does not refer to any k. The latter defines k as the number of symmetry orbits and explicitly allows different shapes. But why m<k? Apparently confirming my suspicion that the definition for polytopes is simply wrong, the example is a polytope that has 2 or 3 symmetry orbits. (An ambiguity, that, btw, doesn't make it a good choice for an example.) But if the definition for tilings is the correct one for any figures, we run into another problem: The example says it is "k-isohedral but not isohedral", which seems to contradict my understanding that isohedral is the same as 1-isohedral. Is my understanding wrong or is there a special condition that k > 1? Why would that be? — Sebastian 16:24, 19 August 2015 (UTC)
I see r-hedral apparently means r shapes of tiles, monohedral (r=1), dihedral (r=2), trihedral (r=3)... [1] So maybe this could be defined somewhere else, and then monohedral could be moved there. Monohedral tiling directs to Tessellation#Introduction_to_tessellations. Tom Ruen ( talk) 19:09, 19 August 2015 (UTC)
The lede says: " Isohedron redirects here", but it doesn't. This article actually links to Isohedron! (If that statement were correct, this link would create a circular – and useless – self-reference.)
Also, Pentagonal tiling#Reinhardt (1918) links to an old name for this page, Isohedral, which redirects to this article's page, i.e. to Isohedral figure. (According to earlier talk on this discussion page, the expression "Isohedral figure" was a compromise between alternatives such as (IIRC) "Isohedral polytope" and "Isohedral polygon", whilst trying to avoid continuing to use the adjective "Isohedral" for a page title.)
Another difficulty for readers at present is that the discussion of Isohedral figures and Isohedral is disjointed, being split across both of the named pages. Readers would get a clearer picture from a single, well-organised page that combines information from both.
Here's a plan:
Based on those constraints, I suggest that the best title we could use would be Isohedral polytope.
I look forward to a fruitful discussion with interested readers and editors. yoyo ( talk) 14:50, 5 November 2015 (UTC)
Isohedron became a redirect to Isohedral figure (here) on 2018-06-26.
This page started as [Face-uniform] on (2006-08-05); moved to [Face-transitive] on 2007-02-10; moved to [Face-transitive polyhedron] and back to [Face-transitive] on 2008-07-08; moved to [Isohedral figure] on 2008-07-13, where it is today. - A876 ( talk) 23:57, 3 April 2019 (UTC)
Counterexamples might be worth mentioning, if only on this talk page.
The following are not isohedral figures. They have all identical (congruent) faces, but they are not "face-transitive":
These lead to some questions:
Addition?
Subtraction??
A876, those are good questions. I also am looking for a name for the superset you asked for -- polyhedron composed entirely of identical faces (mirror-image faces also allowed).
In addition to the polyhedra you already listed, more congruent-face but not face-transitive polyhedron include:
Perhaps monohedrons ( monohedral figure) are a good name for that superset, "polyhedron composed entirely of identical faces, whether or not they are face-transitive"? (Alas, plesiohedron uses "monohedral" to mean something entirely different -- apparently a kind of polyhedron that may have a variety of irregular faces, but large numbers of identical copies of the entire polyhedron tile space with no gaps).
It's apparently still an open question in 2020 as to how many kinds of monohedral but non-isohedral polyhedrons exist (see "What are the known convex polyhedra with congruent faces?", unanswered as of 2020). -- DavidCary ( talk) 20:47, 4 December 2020 (UTC)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||||||
|
Face-transitive is a standard term for polytopes. Face-uniform was an incorrect term derived from uniform polytopes which are are vertex-transitive, and their duals which are face-transitive.
Looking behind the recent to-and-fro moves, it seems to me that Isohedral should be more than a redirect. It could be made a disambuguation page, or it could be basically this one, with links to isohedral numbers and the like. My vote is to move this page across. -- Cheers, Steelpillow ( Talk) 19:35, 9 July 2008 (UTC)
If this is to be moved, move it to face-transitive polytope or face-transitive polyhedron or isohedral polytope or isohedral polyhedron or some other appropriate noun phrase, not to an adjective. Michael Hardy ( talk) 20:48, 9 July 2008 (UTC)
Anyway I DEFINITE dislike adding the word "polyhedron" to terms that apply to all polytopes/tilings/honeycombs. Tom Ruen ( talk) 18:12, 12 July 2008 (UTC)
Would that last one be OK by you? -- Cheers, Steelpillow ( Talk) 19:23, 12 July 2008 (UTC)
I am confused by the section k-isohedral. The definitions for polytopes and tilings appear to be very different: The former makes no mention of symmetry orbits and explicitly forbids different shapes. Strangely, it does not refer to any k. The latter defines k as the number of symmetry orbits and explicitly allows different shapes. But why m<k? Apparently confirming my suspicion that the definition for polytopes is simply wrong, the example is a polytope that has 2 or 3 symmetry orbits. (An ambiguity, that, btw, doesn't make it a good choice for an example.) But if the definition for tilings is the correct one for any figures, we run into another problem: The example says it is "k-isohedral but not isohedral", which seems to contradict my understanding that isohedral is the same as 1-isohedral. Is my understanding wrong or is there a special condition that k > 1? Why would that be? — Sebastian 16:24, 19 August 2015 (UTC)
I see r-hedral apparently means r shapes of tiles, monohedral (r=1), dihedral (r=2), trihedral (r=3)... [1] So maybe this could be defined somewhere else, and then monohedral could be moved there. Monohedral tiling directs to Tessellation#Introduction_to_tessellations. Tom Ruen ( talk) 19:09, 19 August 2015 (UTC)
The lede says: " Isohedron redirects here", but it doesn't. This article actually links to Isohedron! (If that statement were correct, this link would create a circular – and useless – self-reference.)
Also, Pentagonal tiling#Reinhardt (1918) links to an old name for this page, Isohedral, which redirects to this article's page, i.e. to Isohedral figure. (According to earlier talk on this discussion page, the expression "Isohedral figure" was a compromise between alternatives such as (IIRC) "Isohedral polytope" and "Isohedral polygon", whilst trying to avoid continuing to use the adjective "Isohedral" for a page title.)
Another difficulty for readers at present is that the discussion of Isohedral figures and Isohedral is disjointed, being split across both of the named pages. Readers would get a clearer picture from a single, well-organised page that combines information from both.
Here's a plan:
Based on those constraints, I suggest that the best title we could use would be Isohedral polytope.
I look forward to a fruitful discussion with interested readers and editors. yoyo ( talk) 14:50, 5 November 2015 (UTC)
Isohedron became a redirect to Isohedral figure (here) on 2018-06-26.
This page started as [Face-uniform] on (2006-08-05); moved to [Face-transitive] on 2007-02-10; moved to [Face-transitive polyhedron] and back to [Face-transitive] on 2008-07-08; moved to [Isohedral figure] on 2008-07-13, where it is today. - A876 ( talk) 23:57, 3 April 2019 (UTC)
Counterexamples might be worth mentioning, if only on this talk page.
The following are not isohedral figures. They have all identical (congruent) faces, but they are not "face-transitive":
These lead to some questions:
Addition?
Subtraction??
A876, those are good questions. I also am looking for a name for the superset you asked for -- polyhedron composed entirely of identical faces (mirror-image faces also allowed).
In addition to the polyhedra you already listed, more congruent-face but not face-transitive polyhedron include:
Perhaps monohedrons ( monohedral figure) are a good name for that superset, "polyhedron composed entirely of identical faces, whether or not they are face-transitive"? (Alas, plesiohedron uses "monohedral" to mean something entirely different -- apparently a kind of polyhedron that may have a variety of irregular faces, but large numbers of identical copies of the entire polyhedron tile space with no gaps).
It's apparently still an open question in 2020 as to how many kinds of monohedral but non-isohedral polyhedrons exist (see "What are the known convex polyhedra with congruent faces?", unanswered as of 2020). -- DavidCary ( talk) 20:47, 4 December 2020 (UTC)