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Rhombille tiling

March 16, 2024

"Tumbling blocks" redirects here. For other uses, see Jacob's ladder (toy).

In geometry, the rhombille tiling,[1] also known as tumbling blocks,[2] reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.

Rhombille tiling
TypeLaves tiling
Faces60°–120° rhombus
Coxeter diagram
Symmetry groupp6m, [6,3], *632
p3m1, [3[3]], *333
Rotation groupp6, [6,3]+, (632)
p3, [3[3]]+, (333)
Dual polyhedronTrihexagonal tiling
Face configurationV3.6.3.6
Propertiesedge-transitive, face-transitive

Properties edit

 
Rhombus unit for the rhombille tiling dual to unit-length trihexagonal tiling.
 
Two hexagonal tilings with red and blue edges within rhombille tiling
 
Four hexagonal tilings with red, green, blue, and magenta edges within the rhombille tiling[3]

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:√3. This is the dual tiling of the trihexagonal tiling or kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, and in the face configuration for monohedral tilings it is denoted [3.6.3.6].[4]

It is also one of 56 possible isohedral tilings by quadrilaterals,[5] and one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.[6]

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 + y + z| ≤ 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.[7]

Artistic and decorative applications edit

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.[8]

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.[9] 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.[10] 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.[11]

The rhombille tiling is also used as a design for parquetry[12] and for floor or wall tiling, sometimes with variations in the shapes of its rhombi.[13] It appears in ancient Greek floor mosaics from Delos[14] and from Italian floor tilings from the 11th century,[15] although the tiles with this pattern in Siena Cathedral are of a more recent vintage.[16] 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.[2][15][17] As a quilting pattern it also has many other names including cubework, heavenly stairs, and Pandora's box.[17] 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.[18] 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.[19]

Other applications edit

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 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 universal computation via a simulation of billiard ball computers.[20]

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 models and related systems of spin interactions in diatomic crystals,[21] and it has also been studied in percolation theory.[22]

Related polyhedra and tilings edit

 
 
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 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 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 projections of three dimensional cubic steps.

There are only eight edge tessellations, 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.[6]

Examples edit

See also edit

 
Wikimedia Commons has media related to Rhombille tiling.

References edit

  1. ^ Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings", The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5.
  2. ^ a b Smith, Barbara (2002), Tumbling Blocks: New Quilts from an Old Favorite, Collector Books, ISBN 9781574327892.
  3. ^ 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
  4. ^ Grünbaum, Branko; Shephard, G. C. (1987), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1193-1. Section 2.7, Tilings with regular vertices, pp. 95–98.
  5. ^ Grünbaum & Shephard (1987), Figure 9.1.2, Tiling P4-42, p. 477.
  6. ^ a b Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles", Mathematics Magazine, 84 (4): 283–289, arXiv:0908.3257, doi:10.4169/math.mag.84.4.283, MR 2843659.
  7. ^ Deza, Michel; Grishukhin, Viatcheslav; Shtogrin, Mikhail (2004), Scale-isometric polytopal graphs in hypercubes and cubic lattices: Polytopes in hypercubes and  , London: Imperial College Press, p. 150, doi:10.1142/9781860945489, ISBN 1-86094-421-3, MR 2051396.
  8. ^ Warren, Howard Crosby (1919), Human psychology, Houghton Mifflin, p. 262.
  9. ^ Kaplan, Craig S. (2008), "Metamorphosis in Escher's art", Bridges 2008: Mathematical Connections in Art, Music and Science (PDF), pp. 39–46.
  10. ^ Escher, Maurits Cornelis (2001), M.C. Escher, the Graphic Work, Taschen, pp. 29–30, ISBN 9783822858646.
  11. ^ De May, Jos (2003), "Painting after M. C. Escher", in Schattschneider, D.; Emmer, M. (eds.), M. C. Escher's Legacy: A Centennial Celebration, Springer, pp. 130–141.
  12. ^ Schleining, Lon; O'Rourke, Randy (2003), "Tricking the eyes with tumbling blocks", Treasure Chests: The Legacy of Extraordinary Boxes, Taunton Press, p. 58, ISBN 9781561586516.
  13. ^ Tessellation Tango, The Mathematical Tourist, Drexel University, retrieved 2012-05-23.
  14. ^ Dunbabin, Katherine M. D. (1999), Mosaics of the Greek and Roman World, Cambridge University Press, p. 32, ISBN 9780521002301.
  15. ^ a b Tatem, Mary (2010), "Tumbling Blocks", Quilt of Joy: Stories of Hope from the Patchwork Life, Revell, p. 115, ISBN 9780800733643.
  16. ^ Wallis, Henry (1902), Italian ceramic art, Bernard Quaritch, p. xxv.
  17. ^ a b Fowler, Earlene (2008), Tumbling Blocks, Benni Harper Mysteries, Penguin, ISBN 9780425221235. This is a mystery novel, but it also includes a brief description of the tumbling blocks quilt pattern in its front matter.
  18. ^ Tobin, Jacqueline L.; Dobard, Raymond G. (2000), Hidden in Plain View: A Secret Story of Quilts and the Underground Railroad, Random House Digital, Inc., p. 81, ISBN 9780385497671.
  19. ^ Aux armes: symbolism, Symbolism in arms, Pleiade, retrieved 2013-04-17.
  20. ^ The Q*Bert neighbourhood, Tim Tyler, accessed 2012-05-23.
  21. ^ Fisher, Michael E. (1959), "Transformations of Ising models", Physical Review, 113 (4): 969–981, Bibcode:1959PhRv..113..969F, doi:10.1103/PhysRev.113.969.
  22. ^ Yonezawa, Fumiko; Sakamoto, Shoichi; Hori, Motoo (1989), "Percolation in two-dimensional lattices. I. A technique for the estimation of thresholds", Phys. Rev. B, 40 (1): 636–649, Bibcode:1989PhRvB..40..636Y, doi:10.1103/PhysRevB.40.636.

Further reading edit

  • Keith Critchlow, Order in Space: A design source book, 1970, pp. 77–76, pattern 1

March 16, 2024
rhombille, tiling, tumbling, blocks, redirects, here, other, uses, jacob, ladder, geometry, rhombille, tiling, also, known, tumbling, blocks, reversible, cubes, dice, lattice, tessellation, identical, rhombi, euclidean, plane, each, rhombus, angles, rhombi, wi. Tumbling blocks redirects here For other uses see Jacob s ladder toy In geometry the rhombille tiling 1 also known as tumbling blocks 2 reversible cubes or the dice lattice is a tessellation of identical 60 rhombi on the Euclidean plane Each rhombus has two 60 and two 120 angles rhombi with this shape are sometimes also called diamonds Sets of three rhombi meet at their 120 angles and sets of six rhombi meet at their 60 angles Rhombille tilingTypeLaves tilingFaces60 120 rhombusCoxeter diagramSymmetry groupp6m 6 3 632p3m1 3 3 333Rotation groupp6 6 3 632 p3 3 3 333 Dual polyhedronTrihexagonal tilingFace configurationV3 6 3 6Propertiesedge transitive face transitive Contents 1 Properties 2 Artistic and decorative applications 3 Other applications 4 Related polyhedra and tilings 5 Examples 6 See also 7 References 8 Further readingProperties edit nbsp Rhombus unit for the rhombille tiling dual to unit length trihexagonal tiling nbsp Two hexagonal tilings with red and blue edges within rhombille tiling nbsp Four hexagonal tilings with red green blue and magenta edges within the rhombille tiling 3 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 3 This is the dual tiling of the trihexagonal tiling or kagome lattice As the dual to a uniform tiling it is one of eleven possible Laves tilings and in the face configuration for monohedral tilings it is denoted 3 6 3 6 4 It is also one of 56 possible isohedral tilings by quadrilaterals 5 and one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling 6 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 y z 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 7 Artistic and decorative applications editThe 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 8 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 9 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 10 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 11 The rhombille tiling is also used as a design for parquetry 12 and for floor or wall tiling sometimes with variations in the shapes of its rhombi 13 It appears in ancient Greek floor mosaics from Delos 14 and from Italian floor tilings from the 11th century 15 although the tiles with this pattern in Siena Cathedral are of a more recent vintage 16 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 2 15 17 As a quilting pattern it also has many other names including cubework heavenly stairs and Pandora s box 17 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 18 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 19 Other applications editThe 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 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 universal computation via a simulation of billiard ball computers 20 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 models and related systems of spin interactions in diatomic crystals 21 and it has also been studied in percolation theory 22 Related polyhedra and tilings edit nbsp nbsp 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 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 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 projections of three dimensional cubic steps There are only eight edge tessellations 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 6 Examples edit nbsp The rhombille tiling overlaid on its dual the trihexagonal tiling nbsp Rhombille tiling floor mosaic in Delos nbsp Rhombille tiling pattern on the floor of Siena Cathedral nbsp Rhombille tiling in cruise terminal in Tallinn EstoniaSee also edit nbsp Wikimedia Commons has media related to Rhombille tiling Tiling by regular polygonsReferences edit Conway John Burgiel Heidi Goodman Strauss Chaim 2008 Chapter 21 Naming Archimedean and Catalan polyhedra and tilings The Symmetries of Things AK Peters p 288 ISBN 978 1 56881 220 5 a b Smith Barbara 2002 Tumbling Blocks New Quilts from an Old Favorite Collector Books ISBN 9781574327892 Richard K Guy amp Robert E Woodrow The Lighter Side of Mathematics Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History 1996 p 79 Figure 10 Grunbaum Branko Shephard G C 1987 Tilings and Patterns New York W H Freeman ISBN 0 7167 1193 1 Section 2 7 Tilings with regular vertices pp 95 98 Grunbaum amp Shephard 1987 Figure 9 1 2 Tiling P4 42 p 477 a b Kirby Matthew Umble Ronald 2011 Edge tessellations and stamp folding puzzles Mathematics Magazine 84 4 283 289 arXiv 0908 3257 doi 10 4169 math mag 84 4 283 MR 2843659 Deza Michel Grishukhin Viatcheslav Shtogrin Mikhail 2004 Scale isometric polytopal graphs in hypercubes and cubic lattices Polytopes in hypercubes and Z n displaystyle mathbb Z n nbsp London Imperial College Press p 150 doi 10 1142 9781860945489 ISBN 1 86094 421 3 MR 2051396 Warren Howard Crosby 1919 Human psychology Houghton Mifflin p 262 Kaplan Craig S 2008 Metamorphosis in Escher s art Bridges 2008 Mathematical Connections in Art Music and Science PDF pp 39 46 Escher Maurits Cornelis 2001 M C Escher the Graphic Work Taschen pp 29 30 ISBN 9783822858646 De May Jos 2003 Painting after M C Escher in Schattschneider D Emmer M eds M C Escher s Legacy A Centennial Celebration Springer pp 130 141 Schleining Lon O Rourke Randy 2003 Tricking the eyes with tumbling blocks Treasure Chests The Legacy of Extraordinary Boxes Taunton Press p 58 ISBN 9781561586516 Tessellation Tango The Mathematical Tourist Drexel University retrieved 2012 05 23 Dunbabin Katherine M D 1999 Mosaics of the Greek and Roman World Cambridge University Press p 32 ISBN 9780521002301 a b Tatem Mary 2010 Tumbling Blocks Quilt of Joy Stories of Hope from the Patchwork Life Revell p 115 ISBN 9780800733643 Wallis Henry 1902 Italian ceramic art Bernard Quaritch p xxv a b Fowler Earlene 2008 Tumbling Blocks Benni Harper Mysteries Penguin ISBN 9780425221235 This is a mystery novel but it also includes a brief description of the tumbling blocks quilt pattern in its front matter Tobin Jacqueline L Dobard Raymond G 2000 Hidden in Plain View A Secret Story of Quilts and the Underground Railroad Random House Digital Inc p 81 ISBN 9780385497671 Aux armes symbolism Symbolism in arms Pleiade retrieved 2013 04 17 The Q Bert neighbourhood Tim Tyler accessed 2012 05 23 Fisher Michael E 1959 Transformations of Ising models Physical Review 113 4 969 981 Bibcode 1959PhRv 113 969F doi 10 1103 PhysRev 113 969 Yonezawa Fumiko Sakamoto Shoichi Hori Motoo 1989 Percolation in two dimensional lattices I A technique for the estimation of thresholds Phys Rev B 40 1 636 649 Bibcode 1989PhRvB 40 636Y doi 10 1103 PhysRevB 40 636 Further reading editKeith Critchlow Order in Space A design source book 1970 pp 77 76 pattern 1 Retrieved from https en wikipedia org w index php title Rhombille tiling amp oldid 1160190185 Related polyhedra and tilings, wikipedia, wiki, book, books, library,

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