Tiling on the plane

The follows are notes on various tilings on the plane.




Regular Pentagon tiling examples

Written by Paul Bourke
December 2001





Truchet tiles in 2D and 3D

Written by Paul Bourke
June 2006

Named after Sebastien Truchet (1657 - 1729) who studied tiling of squares with a diagonal orientation.

2 Dimensions

Single cell

POVRay source, tile on the x-y plane
#declare RADIUS = 0.1;
#declare oneunit = intersection {
   box { <0,0,-RADIUS-1>, <1,1,RADIUS+1> }
   union {
      torus {
         0.5, RADIUS
         rotate <90,0,0>
      }
      torus {
         0.5, RADIUS
         rotate <90,0,0>
         translate <1,1,0>
      }
   }
   translate <-0.5,-0.5,0>
}  

2x2 tilings

Each unit is randomly rotated by 0 or 90 degrees or flipped vertically or horizontally, there are only two unit states.

5x5 tilings

Larger tiling [POVRay source]

3 Dimensions

Single cell

POVRay source
#declare RADIUS = 0.1;
#declare oneunit = intersection {
   box { <0,0,0>, <1,1,1> }
   union {
      torus {
         1/2, RADIUS
         rotate <90,0,0>
         translate <0,1,1/2>
      }
      torus {
         1/2, RADIUS
         rotate <0,0,90>
         translate <1/2,0,1>
      }
      torus {
         1/2, RADIUS
         translate <1,1/2,0>
      }
   }
   translate <-1/2,-1/2,-1/2>
}

2x2 tilings

Each unit is randomly rotated by integer multiples of 90 degrees about each axis and randomly mirrored about each Cartesian axis plane (xy, xz, yz).

3x3x3 tilings

Larger tiling [POVRay source]




Contribution by Robert McGrego


Contribution by Robert McGrego





Block tiling

Written by Paul Bourke
December 2006


2D tiling ... actually an orthographic projection of a 3D model


Perspective from the same position as above


Perspective projection from another position

POVRay scene file: blocktile.pov





Weave

Written by Paul Bourke
Inspired by work by Quincy Kim

December 2006


Single tiling unit (Unit square in red).


Perspective view of single tiling unit.


Regular tiling.


Randomized tiling.


Perspective view of the randomized tiling.

POVRay scene file: weave.pov for the single unit.
Scene file for a random tiling of the plane: weave.pov.




Hexagonal Lattice

Written by Paul Bourke
December 1997, Updated February 2004

C libraries: hexlib.h, and hexlib.c

Of the three 2 dimensional shapes (equilateral triangle, rectangle, and hexagon) that can be used to tile the plane without holes, the hexagon is the most complex and has many interesting properties. In what follows, an indexing system will be described for a hexagonal tiling called a Spiral Honeycomb Mosaic (SHM). A SHM consists of groups of 7n hexagons (n > 0) called super-hexagons. It uses a base 7 numbering system for the hexagonal mesh, this is illustrated below for n = 1, 2, and 3.

Software

An application has been developed to explore operations in SHM space. It is based upon X-Windows and OpenGL and is currently available for Mac OS-X and Linux (by request). Note that before running the Mac OS-X version check that X-Windows is running. Download: macosx.tar.gz

Interface

The interface is straightforward, to find out about any command line options type "hex -h". The left mouse button pans the image, the middle button rolls, and the right button presents a list of menu options. In order to position/rotate/scale the hexagon to the image see the image mapping dialog box. To experiment with SHM operations see the transformation dialog box.

>hex -h
Usage: hex [options]
Options
     -h      this text (help)
     -i s    load input TGA file
     -v      verbose mode
Key commands
    -,+      zoom in/out
      r      reset
      w      write TGA image of window contents
   1..6      set resolution
    ESC      quit
Mouse buttons
   left      translate
 middle      rotate
  right      menus

Addition and Multiplication

The two basic arithmetic operations can be defined for the SMH, addition and multiplication. These operations act on the addresses of the SHM and result in translation (addition) and rotation/scaling (multiplication) when applied to images represented with the "pixels" of the SHM. The following two C snippets (HexAdd() and HexMul()) implement addition and multiplication. An important characteristics of the operations defined in this way is they are bijective, that is, it is a one to one mapping of each address. Every address (hexagonal pixel) maps uniquely to another address so that no information is lost.

Examples

Notes - Spirals

The curve through powers of 7 (10hex) is an equiangular spiral described by

r = a exp(b theta)

Where r is the radius and theta the angle to the x axis. For example, for the spiral through 1hex, 10hex, 100hex, 1000hex, ... the parameters a and b are

a = sqrt(3)
b = loge(sqrt(7)) / arctan(sqrt(3) / 2) = 1.3632084.

Since the angle in the above case is taken from the negative y (imaginary) axis, the curve would be traced by

x = - r sin(theta)
y = - r cos(theta)

The angle between each successive multiples of 10hex is arctan(sqrt(3) / 2) = 40.893395o, the ratio of two successive radii is sqrt(7) = 2.6457513.

Online SHM Calculator

Base Conversion


Base 7
Base 10
Addition/multiplication


Base 7
Base 10
Add
Multiply
Basis

Base 7
Base 10
Other Functions


Base 7
Base 10
Basis
Inverse
Square root(s)
Magnitude
Cartesian


Base 7
Base 10

References

Alexander, D. and Sheridan, P.
Proceedings Australian Neuroscience Society (1995)
Algebraic Geometric Model of the Receptive Fields Properties of the Macaque Striate Cortex.

Sheridan, P.
PhD Thesis, University of Technology, Sydney (1996)
Spiral Architecture for Machine Vision

Sheridan, P., Alexander, D.M.
Proceedings of Vision, Recognition, Action: Boston. (1997)
Invariant transformations on a space-variant hexagonal grid

Sheridan, P., Hintz, T., Alexander, D.
Image and Vision Computing, 18 (2000)
Pseudo-invariant image transformations on a hexagonal lattice




Non Periodic Tiling of the Plane

eg: Penrose tiles, Danzer tiles, Chair tiles, Trilobite tile, Pinwheel tile.
Written by Paul Bourke
October 1995

Tiling a plane with single geometric shapes is an old exercise in geometry. The platonic solids were shapes that tile 3D space using regular polyhedra, in 2 dimensions examples of regular polygons that tile (without gaps or overlap) the plane are the equilateral triangle, square, and hexagon.

Periodic tilings

There are many examples of general polygons (not regular) that tile the plane, but in the most common cases they form periodic tiling. A periodic tiling is one where it is possible to make a parallelogram (generally larger than the tiles) that can be repeated to produce the same tiling. An example of the more common periodic tiling is shown below for two tiles (blue and yellow), the red boundary illustrates one possible region that can be replicated to create the same effect as the tiles. Note that in general the period forming shape need not be a rectangle but any parallelogram. Note also that there are generally a large, if not infinite, number of periodic parallelograms for a particular periodic tiling.

Periodic and nonperiodic

An example of a single tile that can tile periodically and nonperiodically is a decagon (ten sided) with a "bite" taken out of it. Tiles that can form both periodic and nonperiodic tilings are often referred to as "versatile" tiles.

An example of this tiled periodically is given below. Note that for a tiling not to have holes all the angles at a vertex must sum to 360 degrees. In this case each angle is either 72, 144, or 216 degrees. A number of combinations of these numbers can sum to 360.

The periodic tiling as shown above have the decagons orientated in multiples of 180 degrees. If the decagons are rotated in multiples of 36 degrees (10 different orientations) the tilings can become nonperiodic as shown below.

Some shapes that can be combined to form larger versions of themselves have an obvious nonperiodic tiling, for example the following is called the sphinx. A periodic tiling with the same tiles is obvious.

Nonperiodic only

How about purely non periodic tiling, that is, a set of polygons that tile the plane but never form a periodic tiling. For quite a while it was assumed that any set of tiles that formed a non periodic tiling could also be used to form a periodic tiling. In 1966 Robert Berger showed that it was not possible to determine whether a particular set of tiles would tile non periodically!

This following set of six nonperiodic tilings were "discovered" by Roger Penrose around 1973. It is illustrative to print and cut a few sets of these tiles out and "experience" how they tile the plane.

The following six tiles are attributed to Richard Robinson who in 1971 reduced a 104 aperiodic tile set by Robert Berger (they were a reduction from a set of originally 20426) to just six tiles.

Around 1973/74 Roger Penrose found a set of two tiles that only tile non periodically. While these come in a number of forms the most common one is illustrated below. Note that phi indicated below is the golden ratio of (1 + sqrt(5))/2.

The start of the tiling of these two tiles is illustrated below. The two tiles are often referred to as the kite and dart. The rhombus would of course tile periodically, the triangular pieces force a select combination of allowed connections.

There are a number of elegant properties of this tiling. For example, the ratio of kits to darts in an infinite tiling is the golden ratio!

Another two tile non periodic tiling by Roger Penrose

Floor of the Chemistry building, University of Western Australia.




Danzer Aperiodic Tiling

Attributed to L Danzer

Written by Paul Bourke
July 2002

Three pieces

The three tiles making up the so called "Danzer" aperiodic tile are shown on the right. The rules for coincident edges are shown as edge direction markers, these must match before two pieces can be put together along a shared edge.

Angles
Mapping - First iteration

This tiling was initially thought interesting because it was one of the first found with a seven fold symmetry, something that for many years earlier was considered unlikely. Each of the tiles can be constructed from the three pieces, the first iteration of this is shown on the right. All three tiles are required in each case.

Second iteration

The second iteration (reduction scaling applied) is shown on the right. Note that with correct tiling no edges are bisected by other edges.

Third iteration, piece 1
Third iteration, piece 2
Third iteration, piece 3
Fourth iteration, piece 3




Chair Aperiodic Tiling

Written by Paul Bourke
June 2000

Seed

The so called "chair tile" is formed by starting with the seed tile as shown on the right.

Mapping - first iteration

The first iteration is formed by replicating the seed 4 times, one orientated the same way, one rotated 90 degrees clockwise and the other rotated 90 degrees anticlockwise.

Second iteration

Subsequent iterations are formed by replicating and applying the rotations and translations to the last iteration. The size of the tile increases by a factor of two along each axis.

Third iteration

Fifth iteration

Coloured

The tiles can be coloured in order to bring out some features such as the diagonal structure. The tile below is the fifth iteration using the coloured arrangement on the right.




Trilobite - Cross aperiodic tiling

Written by Paul Bourke
June 2002

Trilobite

This tiling consists of two tiles called the trilobite and the cross. The tiling is only possible as an aperiodic tile, that is, there are no periodic tilings of infinite extent on the plane using these two tiles and the joining rule.

Cross

The second tile, the cross, got its name because it and the trilobite is often drawn rotated by 45 degree to those shown here.

Constraint

The tips of two tiles can only be connected if the black and white strips align. It is this constraint that forces the only infinite tiling to be aperiodic.

Stage 1

Four trilobites can be connected as follows without the need for any cross tiles. At each iteration the total bounds of the tiling increases by a factor of two.

Stage 2

The next stage of the tiling can be constructed as in the chair tile. The result shown on the right is made up of 4 copies of the earlier stage, 4 cross elements are required to fill in the gaps. Note that one of the copies is only translated, one other is rotated clockwise by 90 degrees and the other is rotated anticlockwise by 90 degrees. The same applies for the filling cross tiles.

Stage 3

The next stage uses 4 copies of the last stage with the same rotational and translation relationships as in the last iteration. The cross files required to fill in the holes are again shown in blue below.

Stage 4

And finally the fourth iteration scaled down by a factor of 2.




Pinwheel Aperiodic Tiling

Featured on the Melbourne Federation Square Buildings

Attributed to Charles Radin

Written by Paul Bourke
June 2002

An iterative procedure for creating the pinwheel tiling is given below. This infinite tiling of the plane is known to be aperiodic, that is, there is no parallelogram shaped region that can be repeated to form the same tiling.

  

Seed

The pinwheel tiling can be constructed by starting with a right angle triangle with side lengths of 1, 2 and square root of 5.



Mapping - first iteration

Five of these triangles can be rotated, mirrored, and translated to form a larger similar triangle as shown in red on the right. The rotation angles are one of the angles of the triangle, that is, atan(0.5), pi/2-atan(0.5), or pi/2 degrees.

Second iteration

This process is repeated, on each iteration the global triangle increases in size by the square root of 5. Five copies of the last iteration are packed into the new triangle. The second iteration is shown on the right.

Third iteration

From the right and top down the transformation for each triangle is as follows (leaving out the final translation).

  • Vertical reflection
    Clockwise rotation by sqrt(0.5)

  • Clockwise rotation by pi + sqrt(0.5)

  • Rotation by sqrt(0.5)

  • Vertical reflection
    Clockwise rotation by sqrt(0.5)

  • Vertical reflection
    Clockwise rotation by pi/2 + sqrt(0.5)

Fourth iteration

Locally one can pick out what seems to be regular structures but of course they don't form a larger regular tiling. For example the groups of 8 triangles that share a single vertex can be readily identified, it is these "pinwheel" shapes that give the tiling its name.

The properties of this tiling were discovered and named by Charles Radin of the University of Texas in 1991 and first published in 1994 in the Annals of Mathematics number 139, page 661-702.

Fifth iteration

In this case the tile has been scaled down (click on the image for a larger version) and another triangle is used to form a rectangular region.

A periodic tiling is one where it is possible to make a parallelogram (generally larger than the tiles) that can be repeated to produce the same tiling. Continuing this iterative process results in an infinite tiling on the plane that is never periodic.

Note that forming a larger tiling by repeating this rectangular shape obviously isn't appropriate, the tiling would then no longer be aperiodic!

[Click on the image on the right for a higher resolution version]
Coloured (fourth iteration)

The lack of periodicity can be highlighted by colouring each triangle as shown below. The human visual system and brain are good at picking out patterns, and while there often seem to be small scale patterns they don't persist at larger scales.

Example - Federation Square, Melbourne, Australia

POVRay examples
scene1.pov
scene2.pov




Contribution by Dave Smith

March 2015

The basic tiling unit is shown below, the constraint is that two semi circles cannot be placed adjacent to each other. All angles are integer multiples of 36 degrees, similarly rotations of the tiling unit are also multiples of 36 degrees. All edge lengths are equal.

A legal tiling might be as follows. Question: can this tile the plane indefinitely?

Witch hat

The basic tiling unit is shown below, the internal angles are integer multiples (as shown) of 30 degrees. All edge lengths are equal.

90 degree tile

60 degree tile

Mixtures