Timeline for Are there Monohedra with odd numbers of faces?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 5 at 0:34 | comment | added | LSpice | @OscarLanzi's answer referenced above. | |
| Jan 5 at 0:29 | comment | added | Oscar Lanzi | Nicely done! To recover some congruence characteristics, consider the construction of a quadrangular enneahedron which lends itself to a threefold symmetry; see my answer. | |
| Jan 14, 2022 at 12:16 | vote | accept | Nandakumar R | ||
| Feb 27, 2022 at 13:56 | |||||
| Nov 17, 2021 at 13:22 | comment | added | Yaakov Baruch | However in both the Herschel and and the two colorings of the 11-face graph, there are vertices containing all four angles of the kite, which adds up to $2\pi$ and is therefore not possible in a convex polyhedron. So an example, if it exists, will have to come from more complex figures. | |
| Nov 17, 2021 at 11:56 | comment | added | Yaakov Baruch | @DavidEppstein Following up on your remark that all faces need to be kites: this means that lengths come in 2 types (S, "short", and L, "long") and opposite edges must have different types. Both Herschel's graph and the 11-face example here have the property that the all edges can be colored S or L in such a way; the coloring is unique (up to permutations) for Herschel's graph, and comes in 2 flavors for the 11-face. | |
| Oct 17, 2021 at 7:47 | comment | added | David Eppstein | See also en.wikipedia.org/wiki/Herschel_graph for a simpler 9-face example of a polyhedron with an odd number of (non-congruent) quadrilateral faces. | |
| Oct 17, 2021 at 5:35 | comment | added | Nandakumar R | Nice! Guess this is at least a very strong indication that monohedra with odd number of faces exist - maybe the number of faces may be considerably more than 11. And whether simply connectedness and convexity (lack thereof) have anything to do with the question would also hopefully have interesting answers. | |
| Oct 16, 2021 at 21:40 | history | answered | Yaakov Baruch | CC BY-SA 4.0 |