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→Dissected regular polygons: more unsourced cruft |
Renamed "Types of vertices with regular polygons" to "Plane-vertex tilings," moved section below "Archimedean, uniform or semiregular tilings". Rearranged order of list of plane-vertexes, by order of quantity (top-to-bottom) and relevancy to article. Reworded phrasing and redirected links, as well as noted on the last plane-vertex specific tilings as included for completion |
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In order to solve those problems, GomJau-Hogg’s notation <ref>{{cite journal |last1=Gomez-Jauregui |first1=Valentin |last2=Hogg |first2=Harrison|display-authors=etal |title=GomJau-Hogg’s Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0 |journal=Symmetry |date=2021 |volume=13 |issue=12 |doi=10.3390/sym13122376 |url=https://doi.org/10.3390/sym13122376}}</ref> is a slightly modified version of the research and notation presented in 2012,<ref name="Gomez-Jauregui 2012" /> about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0,<ref>{{cite web |last1=Hogg |first1=Harrison |last2=Gomez-Jauregui |first2=Valentin |title=Antwerp 3.0 |url=https://antwerp.hogg.io/<}}</ref> a free online application, allows for the infinite generation of regular polygon tilings through a set of shape placement stages and iterative rotation and reflection operations, obtained directly from the GomJau-Hogg’s notation.
Three of them can make [[#Regular_tilings|regular tilings]] (6<sup>3</sup>, 4<sup>4</sup>, 3<sup>6</sup>), and eight more can make [[#Archimedean,_uniform_or_semiregular_tilings|semiregular or archimedean tilings]], (3.12.12, 4.6.12, 4.8.8, (3.6)<sup>6</sup>, 3.4.6.4, 3.3.4.3.4, 3.3.3.4.4, 3.3.3.3.6). Four of them can exist in higher [[#k-uniform_tilings|''k''-uniform tilings]] (3.3.4.12, 3.4.3.12, 3.3.6.6, 3.4.4.6), while six can not be used to completely tile the plane by regular polygons with no gaps or overlaps (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, 5.5.10).<ref>Tilings and Patterns, p.58-69</ref>▼
{| class=wikitable▼
|- align=center ▼
!3▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 7 42.svg|50px]]<BR>3.7.42▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 8 24.svg|40px]]<BR>3.8.24▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 9 18.svg|40px]]<BR>3.9.18▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 10 15.svg|40px]]<BR>3.10.15▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 12 12.svg|40px]]<BR>3.12.12▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 5 20.svg|40px]]<BR>4.5.20▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 6 12.svg|40px]]<BR>4.6.12▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 8 8.svg|50px]]<BR>4.8.8▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 5 5 10.svg|40px]]<BR>5.5.10▼
|valign=bottom|[[File:Regular polygons meeting at vertex 3 6 6 6.svg|50px]]<BR>6<sup>3</sup>▼
|- align=center▼
!valign=center|4▼
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 4 12.svg|40px]]<BR>3.3.4.12▼
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 3 12.svg|40px]]<BR>3.4.3.12▼
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 6 6.svg|40px]]<BR>3.3.6.6▼
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 6 3 6.svg|40px]]<BR>(3.6)<sup>6</sup>▼
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 4 6.svg|40px]]<BR>3.4.4.6▼
|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 6 4.svg|40px]]<BR>3.4.6.4▼
|valign=bottom|[[File:Regular polygons meeting at vertex 4 4 4 4 4.svg|40px]]<BR>4<sup>4</sup>▼
|- align=center▼
!5▼
|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 4 3 4.svg|40px]]<BR>3.3.4.3.4▼
|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 4 4.svg|40px]]<BR>3.3.3.4.4▼
|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 3 6.svg|40px]]<BR>3.3.3.3.6▼
!6▼
|valign=bottom|[[File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|40px]]<BR>3<sup>6</sup>▼
|}▼
== Regular tilings ==
Line 103 ⟶ 65:
<br/>
Grünbaum and Shephard distinguish the description of these tilings as ''Archimedean'' as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as ''uniform'' as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.
==Plane-vertex tilings==
There are 17 combinations of regular convex polygons that form 21 types of [[Vertex (geometry)#Of a plane tiling|plane-vertex tilings]] possible.<ref>{{citation|title=The Elements of Plane Practical Geometry, Etc|first=Elmslie William|last=Dallas|publisher=John W. Parker & Son|year=1855|page=134|url=https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134}}</ref> Polygons in these meet at a point with no gap or overlap. Listing by their [[vertex figure]]s, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons.<ref>Tilings and Patterns, Figure 2.1.1, p.60</ref> <ref>Tilings and Patterns, p.58-69</ref>
▲
▲{| class=wikitable
▲|- align=center
▲|valign=bottom|[[File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|40px]]<BR>3<sup>6</sup>
▲|- align=center
▲!5
▲|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 4 3 4.svg|40px]]<BR>3.3.4.3.4
▲|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 4 4.svg|40px]]<BR>3.3.3.4.4
▲|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 3 6.svg|40px]]<BR>3.3.3.3.6
!4
▲|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 4 12.svg|40px]]<BR>3.3.4.12
▲|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 3 12.svg|40px]]<BR>3.4.3.12
▲|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 6 6.svg|40px]]<BR>3.3.6.6
▲|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 6 3 6.svg|40px]]<BR>(3.6)<sup>6</sup>
▲|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 4 6.svg|40px]]<BR>3.4.4.6
▲|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 6 4.svg|40px]]<BR>3.4.6.4
▲|valign=bottom|[[File:Regular polygons meeting at vertex 4 4 4 4 4.svg|40px]]<BR>4<sup>4</sup>
▲|- align=center
▲!6
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 7 42.svg|50px]]<BR>3.7.42
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 8 24.svg|40px]]<BR>3.8.24
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 9 18.svg|40px]]<BR>3.9.18
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 10 15.svg|40px]]<BR>3.10.15
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 12 12.svg|40px]]<BR>3.12.12
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 5 20.svg|40px]]<BR>4.5.20
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 6 12.svg|40px]]<BR>4.6.12
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 8 8.svg|50px]]<BR>4.8.8
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 5 5 10.svg|40px]]<BR>5.5.10
▲|valign=bottom|[[File:Regular polygons meeting at vertex 3 6 6 6.svg|50px]]<BR>6<sup>3</sup>
▲|}
== ''k''-uniform tilings ==
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