Euclidean tilings by convex regular polygons: Difference between revisions

Revision as of 07:25, 10 October 2022

Revision as of 21:42, 10 December 2022

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== ''k''-uniform tilings ==

{| class=wikitable align=right width=300

|+ 3-uniform tiling #57 of 61 colored

|- align=center valign=top

|[[File:3-uniform 57.svg|150px]]<br/>by sides, yellow triangles, red squares (by polygons)

|[[File:3-uniform n57.svg|150px]]<br/>by 4-isohedral positions, 3 shaded colors of triangles (by orbits)

|}

Such periodic tilings may be classified by the number of [[Group action (mathematics)#Orbits and stabilizers|orbits]] of vertices, edges and tiles. If there are {{mvar|k}} orbits of vertices, a tiling is known as {{mvar|k}}-uniform or {{mvar|k}}-isogonal; if there are {{mvar|t}} orbits of tiles, as {{mvar|t}}-isohedral; if there are {{mvar|e}} orbits of edges, as {{mvar|e}}-isotoxal.

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Finally, if the number of types of vertices is the same as the uniformity (''m'' = ''k'' below), then the tiling is said to be ''[[oeis:A068600|Krotenheerdt]]''. In general, the uniformity is greater than or equal to the number of types of vertices (''m'' ≥ ''k''), as different types of vertices necessarily have different orbits, but not vice versa. Setting ''m'' = ''n'' = ''k'', there are 11 such tilings for ''n'' = 1; 20 such tilings for ''n'' = 2; 39 such tilings for ''n'' = 3; 33 such tilings for ''n'' = 4; 15 such tilings for ''n'' = 5; 10 such tilings for ''n'' = 6; and 7 such tilings for ''n'' = 7.

Below is an example of a 3-unifom tiling:

{| class=wikitable align=rightleft width=300

|+ Colored 3-uniform tiling #57 of 61 colored

|- align=center valign=top

|[[File:3-uniform 57.svg|150px]]<br/>by sides, yellow triangles, red squares (by polygons)

|[[File:3-uniform n57.svg|150px]]<br/>by 4-isohedral positions, 3 shaded colors of triangles (by orbits)

|}

{{Clr}}

{| class="wikitable" align=left style="margin: auto; text-align:center;"

|+ ''k''-uniform, ''m''-Archimedean tiling counts<ref>{{Cite web|url=http://probabilitysports.com/tilings.html|title=n-Uniform Tilings|website=probabilitysports.com|access-date=2019-06-21}}</ref>

!colspan=2 rowspan=2 | !! colspan="16" |''m''-Archimedean

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|-

|}

{{Clr}}

=== 2-uniform tilings ===