View source for Rhombille tiling

{{Short description|Tiling of the plane with 60° rhombi}}
{{Redirect|Tumbling blocks||Jacob's ladder (toy)}}
{{Infobox face-uniform tiling
| Name=Rhombille tiling
|  Image_File=1-uniform 7 dual.svg
|  Type=[[Laves tiling]]
|  Cox={{CDD|node|3|node_f1|6|node}}<BR>{{CDD|node_h1|3|node|6|node_f1}}
|  Face_List=[[Rhombus|60°–120° rhombus]]
|  Symmetry_Group=p6m, [6,3], *632<BR>p3m1, [3<sup>[3]</sup>], *333
|  Rotation_Group  = p6, [6,3]<sup>+</sup>, (632)<BR>p3, [3<sup>[3]</sup>]<sup>+</sup>, (333)
|  Face_Type=V3.6.3.6[[File:Tiling face 3-6-3-6.svg|40px|right]]
|  Dual=[[Trihexagonal tiling]]
|  Property_List=[[edge-transitive]], [[face-transitive]]
}}

In [[geometry]], the '''rhombille tiling''',<ref>{{citation
 | last1 = Conway | first1 = John | author1-link = John Horton Conway
 | last2 = Burgiel | first2 = Heidi
 | last3 = Goodman-Strauss | first3 = Chaim | author3-link = Chaim Goodman-Strauss
 | contribution = Chapter 21: Naming Archimedean and Catalan polyhedra and tilings
 | isbn = 978-1-56881-220-5
 | page = 288
 | publisher = AK Peters
 | title = The Symmetries of Things | title-link = The Symmetries of Things
 | year = 2008}}.</ref> also known as '''tumbling blocks''',<ref name="tumble1"/> '''reversible cubes''', or the '''dice lattice''', is a [[tessellation]] of identical 60° [[rhombus|rhombi]] on the [[Euclidean plane]]. Each rhombus has two 60° and two 120° [[angle]]s; rhombi with this shape are sometimes also called [[Polyiamond|diamonds]]. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.

==Properties==
[[File:Rhombus_(Rhombille).svg|frame|Rhombus unit for the rhombille tiling dual to unit-length trihexagonal tiling.|left]]
[[File:Compound of two hexagonal tiling.svg|left|thumb|180px|Two [[hexagonal tiling]]s with red and blue edges within rhombille tiling]]
[[File:1-uniform_7_dual_edgecolor4.svg|left|thumb|180px|Four hexagonal tilings with red, green, blue, and magenta edges within the rhombille tiling<ref>Richard K. Guy & Robert E. Woodrow, ''The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History'', 1996, p.79, Figure 10</ref>]]
The rhombille tiling can be seen as a subdivision of a [[hexagonal tiling]] with each [[hexagon]] divided into three [[rhombi]] meeting at the center point of the hexagon. This subdivision represents a [[regular compound tiling]]. It can also be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi.

The diagonals of each rhomb are in the ratio 1:{{radic|3}}.
This is the [[Dual polyhedron|dual tiling]] of the [[trihexagonal tiling]] or [[kagome lattice]]. As the dual to a [[uniform tiling]], it is one of eleven possible [[Laves tiling]]s, and in the [[face configuration]] for [[monohedral tiling]]s it is denoted [3.6.3.6].<ref>{{citation
 | last1 = Grünbaum
 | first1 = Branko
 | author1-link = Branko Grünbaum
 | last2 = Shephard
 | first2 = G. C.
 | author2-link = Geoffrey Colin Shephard
 | isbn = 0-7167-1193-1
 | location = New York
 | publisher = W. H. Freeman
 | title = Tilings and Patterns
 | year = 1987
 | url-access = registration
 | url = https://archive.org/details/isbn_0716711931
 }}. Section 2.7, Tilings with regular vertices, pp. 95–98.</ref>

It is also one of 56 possible [[isohedral tiling]]s by quadrilaterals,<ref>{{harvtxt|Grünbaum|Shephard|1987}}, Figure 9.1.2, Tiling ''P''<sub>4</sub>-42, p.&nbsp;477.</ref> and one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.<ref name="Edge tessellations and stamp foldin">{{citation
 | last1 = Kirby | first1 = Matthew
 | last2 = Umble | first2 = Ronald
 | arxiv = 0908.3257
 | doi = 10.4169/math.mag.84.4.283
 | issue = 4
 | journal = Mathematics Magazine
 | mr = 2843659
 | pages = 283–289
 | title = Edge tessellations and stamp folding puzzles
 | volume = 84
 | year = 2011}}.</ref>

It is possible to embed the rhombille tiling into a subset of a three-dimensional [[integer lattice]], consisting of the points (''x'',''y'',''z'') with |''x''&nbsp;+&nbsp;''y''&nbsp;+&nbsp;''z''|&nbsp;≤&nbsp;1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, and more strongly such that the number of edges in the shortest path between any two vertices of the tiling is the same as the [[Manhattan distance]] between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite [[unit distance graph]] and [[partial cube]].<ref>{{citation
 | last1 = Deza | first1 = Michel | author1-link = Michel Deza
 | last2 = Grishukhin | first2 = Viatcheslav
 | last3 = Shtogrin | first3 = Mikhail
 | doi = 10.1142/9781860945489
 | isbn = 1-86094-421-3
 | location = London
 | mr = 2051396
 | page = 150
 | publisher = Imperial College Press
 | title = Scale-isometric polytopal graphs in hypercubes and cubic lattices: Polytopes in hypercubes and <math>\mathbb{Z}_n</math>
 | url = https://books.google.com/books?id=i1WLG6NeDCkC&pg=PA150
 | year = 2004}}.</ref>

==Artistic and decorative applications==
The rhombille tiling can be interpreted as an [[isometric projection]] view of a set of cubes in two different ways, forming a [[reversible figure]] related to the [[Necker cube]]. In this context it is known as the "reversible cubes" illusion.<ref>{{citation|title=Human psychology|first=Howard Crosby|last=Warren|publisher=Houghton Mifflin|year=1919|page=262|url=https://books.google.com/books?id=ByxVAAAAMAAJ&pg=PA262}}.</ref>

In the [[M. C. Escher]] artworks ''[[Metamorphosis I]]'', ''[[Metamorphosis II]]'', and ''[[Metamorphosis III]]'' Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms.<ref>{{citation
 | last = Kaplan | first = Craig S.
 | contribution = Metamorphosis in Escher's art
 | pages = 39–46
 | title = Bridges 2008: Mathematical Connections in Art, Music and Science
 | url = http://www.cgl.uwaterloo.ca/~csk/papers/kaplan_bridges2008.pdf
 | year = 2008}}.</ref> In another of his works, ''Cycle'' (1938), Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements (drawn isometrically) and an upstairs patio tiled with the rhombille tiling. A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so.<ref>{{citation|title=M.C. Escher, the Graphic Work|first=Maurits Cornelis|last=Escher|author-link=M. C. Escher|publisher=[[Taschen]]|year=2001|isbn=9783822858646|url=https://books.google.com/books?id=lq-EZiRdPswC&pg=PR1|pages=29–30}}.</ref> These works involve only a single three-dimensional interpretation of the tiling, but in ''[[Convex and Concave]]'' Escher experiments with reversible figures more generally, and includes a depiction of the reversible cubes illusion on a flag within the scene.<ref>{{citation
 | last = De May | first = Jos
 | editor1-last = Schattschneider | editor1-first = D. | editor1-link = Doris Schattschneider
 | editor2-last = Emmer | editor2-first = M.
 | contribution = Painting after M. C. Escher
 | pages = 130–141
 | publisher = Springer
 | title = M. C. Escher's Legacy: A Centennial Celebration
 | year = 2003}}.</ref>

The rhombille tiling is also used as a design for [[parquetry]]<ref>{{citation|title=Treasure Chests: The Legacy of Extraordinary Boxes|first1=Lon|last1=Schleining|first2=Randy|last2=O'Rourke|publisher=Taunton Press|year=2003|isbn=9781561586516|contribution=Tricking the eyes with tumbling blocks|page=58|url=https://books.google.com/books?id=xA91sJ1LNQIC&pg=PA58}}.</ref> and for floor or wall tiling, sometimes with variations in the shapes of its rhombi.<ref>[http://mathtourist.blogspot.com/2011/02/tessellation-tango.html Tessellation Tango], The Mathematical Tourist, Drexel University, retrieved 2012-05-23.</ref> It appears in ancient Greek floor [[mosaic]]s from [[Delos]]<ref>{{citation|title=Mosaics of the Greek and Roman World|first=Katherine M. D.|last=Dunbabin|publisher=Cambridge University Press|year=1999|isbn=9780521002301|page=32|url=https://books.google.com/books?id=U7Uu_Dq8oY4C&pg=PA32}}.</ref> and from Italian floor tilings from the 11th century,<ref name="tumble3">{{citation|title=Quilt of Joy: Stories of Hope from the Patchwork Life|first=Mary|last=Tatem|publisher=Revell|year=2010|isbn=9780800733643|contribution=Tumbling Blocks|page=115|url=https://books.google.com/books?id=BXxuCv8InhUC&pg=PA115}}.</ref> although the tiles with this pattern in [[Siena Cathedral]] are of a more recent vintage.<ref>{{citation|title=Italian ceramic art|first=Henry|last=Wallis|publisher=Bernard Quaritch|year=1902|page=xxv|url=https://books.google.com/books?id=YQQ4AAAAMAAJ&pg=PR25}}.</ref> In [[quilting]], it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation.<ref name="tumble1">{{citation|title=Tumbling Blocks: New Quilts from an Old Favorite|first=Barbara|last=Smith|publisher=Collector Books|year=2002|isbn=9781574327892}}.</ref><ref name="tumble3"/><ref name="tumble2">{{citation|title=Tumbling Blocks|series=Benni Harper Mysteries|first=Earlene|last=Fowler|publisher=Penguin|year=2008|isbn=9780425221235|author-link=Earlene Fowler}}. This is a mystery novel, but it also includes [https://books.google.com/books?id=An1vm22_2GcC&pg=PT10 a brief description of the tumbling blocks quilt pattern] in its front matter.</ref> As a quilting pattern it also has many other names including cubework, heavenly stairs, and Pandora's box.<ref name="tumble2"/> It has been suggested that the tumbling blocks quilt pattern was used as a signal in the [[Underground Railroad]]: when slaves saw it hung on a fence, they were to box up their belongings and escape. ''See [[Quilts of the Underground Railroad]].''<ref>{{citation|title=Hidden in Plain View: A Secret Story of Quilts and the Underground Railroad|first1=Jacqueline L.|last1=Tobin|first2=Raymond G.|last2=Dobard|publisher=Random House Digital, Inc.|year=2000|isbn=9780385497671|page=[https://archive.org/details/hiddeninplainvie0000tobi/page/81 81]|url=https://archive.org/details/hiddeninplainvie0000tobi/page/81}}.</ref> In these decorative applications, the rhombi may appear in multiple colors, but are typically given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality.
There is a single known instance of implicit rhombille and [[trihexagonal tiling]] in [[English heraldry]] – in the Geal/e arms.<ref>[http://www.pleiade.org/col_geal/aux_armes-symbolism.html Aux armes: symbolism], Symbolism in arms, Pleiade, retrieved 2013-04-17.</ref>

==Other applications==
The rhombille tiling may be viewed as the result of overlaying two different hexagonal tilings, translated so that some of the vertices of one tiling land at the centers of the hexagons of the other tiling. Thus, it can be used to define [[Block cellular automaton|block cellular automata]] in which the cells of the automaton are the rhombi of a rhombille tiling and the blocks in alternating steps of the automaton are the hexagons of the two overlaid hexagonal tilings. In this context, it is called the "Q*bert neighborhood", after the video game [[Q*bert]] which featured an isometric view of a pyramid of cubes as its playing field.
The Q*bert neighborhood may be used to support [[Turing completeness|universal computation]] via a simulation of [[billiard ball computer]]s.<ref>[http://cell-auto.com/neighbourhood/qbert/index.html The Q*Bert neighbourhood], Tim Tyler, accessed 2012-05-23.</ref>

In [[condensed matter physics]], the rhombille tiling is known as the '''dice lattice''', '''diced lattice''', or '''dual kagome lattice'''. It is one of several repeating structures used to investigate [[Ising model]]s and related systems of [[Spin (physics)|spin]] interactions in [[Diatomic molecule|diatomic]] [[crystal]]s,<ref>{{citation
 | last = Fisher | first = Michael E. | author-link = Michael Fisher
 | doi = 10.1103/PhysRev.113.969
 | issue = 4
 | journal = Physical Review
 | pages = 969–981
 | title = Transformations of Ising models
 | volume = 113
 | year = 1959| bibcode = 1959PhRv..113..969F}}.</ref> and it has also been studied in [[percolation theory]].<ref>{{citation
 | last1 = Yonezawa | first1 = Fumiko
 | last2 = Sakamoto | first2 = Shoichi
 | last3 = Hori | first3 = Motoo
 | doi = 10.1103/PhysRevB.40.636
 | issue = 1
 | journal = Phys. Rev. B
 | pages = 636–649
 | title = Percolation in two-dimensional lattices. I. A technique for the estimation of thresholds
 | volume = 40
 | year = 1989| bibcode = 1989PhRvB..40..636Y}}.</ref>

==Related polyhedra and tilings==
{{multiple image
|image1=Rhombic tiling tilted1.svg
|image2=Skew_rhombille.png
|footer=Combinatorially equivalent tilings by parallelograms
}}
The rhombille tiling is the dual of the [[trihexagonal tiling]].
It is one of many different ways of tiling the plane by congruent rhombi.
Others include a [[Square tiling#Topologically equivalent tilings|diagonally flattened variation of the square tiling]] (with translational symmetry on all four sides of the rhombi), the tiling used by the [[Miura-ori]] folding pattern (alternating between translational and reflectional symmetry), and the [[Penrose tiling]] which uses two kinds of rhombi with 36° and 72° acute angles [[aperiodic tiling|aperiodically]].
When more than one type of rhombus is allowed, additional tilings are possible, including some that are topologically equivalent to the rhombille tiling but with lower symmetry.

Tilings combinatorially equivalent to the rhombille tiling can also be realized by parallelograms, and interpreted as [[axonometric projection]]s of three dimensional cubic steps.

There are only eight [[edge tessellation]]s, tilings of the plane with the property that reflecting any tile across any one of its edges produces another tile; one of them is the rhombille tiling.<ref name="Edge tessellations and stamp foldin"/>

== Examples ==
<gallery widths="240" heights="160">
File:P3 dual.svg|The rhombille tiling overlaid on its dual, the [[trihexagonal tiling]].
File:Delos cubic floor mosaic.jpg|Rhombille tiling [[floor mosaic]] in [[Delos]]
File:Siena Duomo rhombille floor tiles.jpg|Rhombille tiling pattern on the floor of [[Siena Cathedral]]
File:Rhombille Tiling in Tallinn EST.jpg|Rhombille tiling in cruise terminal in [[Tallinn, Estonia]]
</gallery>

== See also==
{{Commons category}}
* [[Tiling by regular polygons]]

==References==
{{reflist|35em}}

==Further reading==
* Keith Critchlow, ''Order in Space: A design source book'', 1970, pp.&nbsp;77–76, pattern 1

{{Tessellation}}

[[Category:Euclidean tilings]]
[[Category:Isohedral tilings]]
[[Category:Isotoxal tilings]]
[[Category:Quasiregular polyhedra]]