geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

of identical 60°

rhombi In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

on the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

. Each rhombus has two 60° and two 120°

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

s; rhombi with this shape are sometimes also called

diamonds Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, b ...

. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.

Properties

The rhombille tiling can be seen as a subdivision of a

hexagonal tiling In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathemat ...

with each

hexagon In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

divided into three

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:. This is the dual tiling of the

trihexagonal tiling In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2. ...

kagome lattice In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2. ...

. As the dual to a

uniform tiling In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the f ...

, it is one of eleven possible

Laves tiling This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dua ...

s, and in the

face configuration In geometry, a vertex configurationCrystallography ...

for

monohedral tiling A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

s it is denoted .6.3.6 It is also one of 56 possible

isohedral tiling In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

s by quadrilaterals, and one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling. It is possible to embed the rhombille tiling into a subset of a three-dimensional

integer lattice In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid l ...

, 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 A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences ...

between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite

unit distance graph In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graph ...

and

partial cube In graph theory, a partial cube is a graph that is isometric to a subgraph of a hypercube. In other words, a partial cube can be identified with a subgraph of a hypercube in such a way that the distance between any two vertices in the partial cu ...

Artistic and decorative applications

The rhombille tiling can be interpreted as an

isometric projection Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is an axonometric projection in which the three coordinate axes appear equally foreshortened and ...

view of a set of cubes in two different ways, forming a

reversible figure Ambiguous images or reversible figures are visual forms that create ambiguity by exploiting graphical similarities and other properties of visual system interpretation between two or more distinct image forms. These are famous for inducing the ...

related to the

Necker Cube The Necker cube is an optical illusion that was first published as a Rhomboid in 1832 by Swiss crystallographer Louis Albert Necker. It is a simple wire-frame, two dimensional drawing of a cube with no visual cues as to its orientation, so it c ...

. In this context it is known as the "reversible cubes" illusion. In the

M. C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in t ...

artworks ''

Metamorphosis I ''Metamorphosis I'' is a woodcut print by the Dutch artist M. C. Escher which was first printed in May, 1937. This piece measures and is printed on two sheets. The concept of this work is to morph one image into a tessellated pattern, then gradu ...

'', ''

Metamorphosis II ''Metamorphosis II'' is a woodcut print by the Dutch artist M. C. Escher. It was created between November, 1939 and March, 1940. The print measures and was printed from 20 blocks on 3 combined sheets. ''Metamorphosis II'' is a long, horizontal ...

'', and ''

Metamorphosis III ''Metamorphosis III'' is a woodcut print by the Dutch artist M. C. Escher created during 1967 and 1968. Measuring , this is Escher's largest print. It was printed on thirty-three blocks on six combined sheets and mounted on canvas. This print was ...

'' Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms. 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. 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. The rhombille tiling is also used as a design for

parquetry Parquet (; French for "a small compartment") is a geometric mosaic of wood pieces used for decorative effect in flooring. Parquet patterns are often entirely geometrical and angular—squares, triangles, lozenges—but may contain curves. T ...

and for floor or wall tiling, sometimes with variations in the shapes of its rhombi. It appears in ancient Greek floor

mosaic A mosaic is a pattern or image made of small regular or irregular pieces of colored stone, glass or ceramic, held in place by plaster/mortar, and covering a surface. Mosaics are often used as floor and wall decoration, and were particularly pop ...

s from

Delos The island of Delos (; el, Δήλος ; Attic: , Doric: ), near Mykonos, near the centre of the Cyclades archipelago, is one of the most important mythological, historical, and archaeological sites in Greece. The excavations in the island are ...

and from Italian floor tilings from the 11th century,. although the tiles with this pattern in

Siena Cathedral Siena Cathedral ( it, Duomo di Siena) is a medieval church in Siena, Italy, dedicated from its earliest days as a Roman Catholic Marian church, and now dedicated to the Assumption of Mary. It was the episcopal seat of the Diocese of Siena, and ...

are of a more recent vintage. In

quilting Quilting is the term given to the process of joining a minimum of three layers of fabric together either through stitching manually using a needle and thread, or mechanically with a sewing machine or specialised longarm quilting system. A ...

, it has been known since the 1850s as the "tumbling blocks" pattern, referring to the visual dissonance caused by its doubled three-dimensional interpretation... This is a mystery novel, but it also include
a brief description of the tumbling blocks quilt pattern
in its front matter. As a quilting pattern it also has many other names including cubework, heavenly stairs, and Pandora's box. It has been suggested that the tumbling blocks quilt pattern was used as a signal in the

Underground Railroad The Underground Railroad was a network of clandestine routes and safe houses established in the United States during the early- to mid-19th century. It was used by enslaved African Americans primarily to escape into free states and Canada. T ...

: when slaves saw it hung on a fence, they were to box up their belongings and escape. ''See

Quilts of the Underground Railroad Quilts of the Underground Railroad describes a controversial belief that quilts were used to communicate information to African slaves about how to escape to freedom via the Underground Railroad. It has been disputed by a number of historians. Bo ...

.'' 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

English heraldry English heraldry is the form of coats of arms and other heraldic bearings and insignia used in England. It lies within the so-called Gallo-British tradition. Coats of arms in England are regulated and granted to individuals by the English kings ...

– in the Geal/e arms.

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 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 ''Q*bert'' (also known as ''Qbert'') is an arcade video game developed and published for the North American market by Gottlieb in 1982. It is a 2D action game with puzzle elements that uses isometric graphics to create a pseudo-3D effect. The ...

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 A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...

via a simulation of

billiard ball computer A billiard-ball computer, a type of conservative logic circuit, is an idealized model of a reversible mechanical computer based on Newtonian dynamics, proposed in 1982 by Edward Fredkin and Tommaso Toffoli. Instead of using electronic signals li ...

s. In

condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...

, 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 The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...

s and related systems of

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

interactions in

diatomic Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Ot ...

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...

s, and it has also been studied in

percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...

Related polyhedra and tilings

The rhombille tiling is the dual of the

. 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 A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of the plane by non-overlapping polygons or other shapes, and ''aperiodic'' means that shifting any tiling with these shapes by any finite distance, without r ...

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 projection Axonometric projection is a type of orthographic projection used for creating a pictorial drawing of an object, where the object is rotated around one or more of its axes to reveal multiple sides.Gary R. Bertoline et al. (2002) ''Technical Graphi ...

s of three dimensional cubic steps. There are only eight

edge tessellation In geometry, an edge tessellation is a partition of the plane into non-overlapping polygons (a tessellation) with the property that the reflection of any of these polygons across any of its edges is another polygon in the tessellation. All of the r ...

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..

Examples

File:P3 dual.png, The rhombille tiling overlaid on its dual, the

. File:Delos cubic floor mosaic.jpg, Rhombille tiling floor mosaic in

File:Siena Duomo rhombille floor tiles.jpg, Rhombille tiling pattern on the floor of

File:Rhombille Tiling in Tallinn EST.jpg, Rhombille tiling in cruise terminal in

Tallinn, Estonia Tallinn () is the most populous and capital city of Estonia. Situated on a bay in north Estonia, on the shore of the Gulf of Finland of the Baltic Sea, Tallinn has a population of 437,811 (as of 2022) and administratively lies in the Harju ' ...

Properties

Artistic and decorative applications

Other applications

Related polyhedra and tilings

Examples

See also

References

Further reading