The Art and Mathematics of Tessellation
Reza Sarhangi
Mathematics Department
Towson University
Towson, MD 21252
E-mail: rsarhangi@towson.edu
Abstract
The purpose of this workshop is to familiarize educators with the art of tessellation and the mathematics behind it.
Moreover, it introduces some useful software that can be integrated in the mathematics curriculum from elementary to
secondary levels.
1. Introduction
Tessellation is the art of periodic patterns of one or more shapes that can be extended across an entire
plane infinitely. It dates back to the early time of civilization – the time that man started using stones and
clay bricks to cover floors and walls and tried to make the designs pleasing by using repeating colors and
shapes. Many civilizations created and integrated tessellation designs in their everyday life, from the
designs in clothing to patterns on the walls of their houses and temples. Sumerians used the geometric
tilings as decorations about 4000 BC.
The word tessellation comes from the Latin word tessella, meaning the small square stone or tile used in
ancient Roman mosaics. The Romans and other Mediterraneans were most concerned with portraying
human beings and natural scenes in intricate mosaics.
Archimedes and other ancient mathematicians investigated properties of regular polygons and
combinations of regular polygons that tessellated the plane.
Some of the most extensive works with mosaic designs were done by Moorish artists. A Moorish dynasty
from North Africa invaded Spain in the eighth century and established a civilization in Andalusia that
lasted until 1492. The Alhambra, a citadel overlooking Granada, Spain, built by Moorish kings in the 12th
and 13th centuries, is the most famous building in the western world for its tessellation designs. The
Moorish tessellation manifested itself in the use of a few shapes and colors of tiles to build complex
geometric designs. Contrary to the extensive use of living beings in the Roman tilings, forbidden by
Moorish religious scholars, the Moorish artists’ works were very abstract and excluded representation of
people, animals or real-world objects.
At the same time, Persian and other Middle Eastern artists created impressive tiling designs that can be
found on many tombs, palaces, mosques, and in carpet designs.
Over the centuries tessellations has appeared in many different media including pottery, wood carvings,
and stained glass. Even though the art of tessellation designs is very old and well developed, the study of
their mathematical properties is recent and many parts of the subject remain unexplored.
Johann Kepler, the great mathematician and astronomer of the seventeenth century studied the tessellating
properties of regular polygons in his book Harmonice Mundi. His studies were not further investigated in
large until the beginning of this century.
Today, the use of computers and sophisticated software have brought this art to the reach of more
audiences, with less background in art, and allows them to be creative. The best known artist of our time
for the application of tessellation designs in his work, M. C. Escher popularized this art in the western
hemisphere. Students in all grade levels, by applying some mathematical transformations in a software
program, such as Tessellation Exploration or TesselMania may create Escher-like designs that would take
several days for an artist to create just a few years ago using pencil drawings.
2. The Monohedral Tessellations of Polygons
Polygons, the closed figures formed by line segments as their sides, underlie most tessellations. By a
closed figure we mean a connected topological area without any hole whose boundary forms a loop which
has no crossings or branches. The simplest polygon, the triangle, can tessellate a plane by itself. By that
we mean if we have congruent copies of any particular triangle, we can cover a surface with them without
any gaps or overlaps. In a technical language, we say the prototile triangle “admits” the monohedral
triangle tessellation. The word monohedral means that every tile in the tessellation is congruent to one
fixed tile. The fixed tile then is the prototile of the tessellation. There are two reasons why triangles can
do a monohedral tessellation. First, the sum of angles of any triangle is . Therefore by an appropriate
arrangement we may cover a plane using exact copies of a particular triangle, as illustrated in Figure1.
Figure 1. The monhedral tessellation of triangle
With the same reasoning as the triangle case, since the sum of angles in any convex or concave
quadrilateral is 2, we can say any quadrilateral tessellates the plane by itself. In general, triangles and
quadrilaterals are the only polygons that can tessellate the plane by themselves.
A branch of convex polygons, regular polygons, is of particular interest in tessellation designs. In a
regular polygon all sides and all angles are congruent. The angle of a regular n-gon is equal to the
quantity (n – 2)/n. This quantity shows that besides the equilateral triangle and square, the only regular
polygon that can tessellate a plane by itself is the regular hexagon. Therefore the first set of tessellations
of our interest, regular tessellations, has three members – the regular 3-, 4-, and 6-gon.
Figure 2. The three regular tessellations
There are interesting problems that can be posed using any of these regular tessellations. For instance, let
us consider a polygon which has been created from a set of seven joined congruent equilateral triangles.
There are exactly twenty-four possible figures for this polygon due to different arrangements of these
triangles, as illustrated in Figure 3. The question is then which one of these tiles can tessellate itself –
which one is the prototile of a monohedral tessellation. It has been shown that only one of these polygons
is not a prototile.
Figure 3. The twenty-four possible figures formed by seven equilateral triangles
A similar problem can be posed for students at the different grade levels by the arrangements of 3, 4, or 5
squares, as in Figure 4 for the case of 5 squares.
Figure 4. The twelve possible figures formed by five squares
3. Tessellations of Two or More Polygons
We would like to study the tessellation of two or more polygons, by using the idea from the previous
section of filling the space around a point, and the idea of the angles of regular polygons.
First, it can be proved that the possible arrangements of regular polygons that fill the space around a point
are limited to 21 cases as shown in Figure 5. These cases start from six equilateral triangles surrounding a
point and end with a combination of an equilateral triangle or regular heptagon, and a regular 42-gon
around a point.
Figure 5. The 21 arrangements of regular polygons around a point
These 21 arrangements are the result of 17 possible combinations of regular polygons around a vertex,
and can be found using the following formula.
The second step is to determine which of these arrangements can create a tessellation. We are only
interested in those tessellations in which all the vertex points of each of them have identical sets of
polygons in the same order. It can be shown that besides the three regular tessellations, there are only
eight tessellations that are formed by regular polygons and the arrangement of polygons at every vertex
point is identical (Figure 6).
These eight tessellations are called semiregular tessellations. The three regular and the eight semiregular
tessellations are referred to as the Archimedean or uniform tilings. It is worth mentioning that there are an
infinite number of patterns that are made by regular polygons but do not have the same arrangement of
angles at every vertex point.
Figure 6. The semiregular tessellations
4. Isometry and Symmetry
An isometry or rigid transformation is a function on the plane which preserves distances. This means it
moves figures without changing their sizes or shapes. There are four types of isometrics; translation,
rotation, reflection and glide reflection.
The following examples are about translations.
Figure 7. Two examples of translation with two different grid sizes Using Tessellation Exploration
The following figure presents two examples, a half-turn and a 90 rotation, which are performed in
Tessellation Exploration.
Figure 8. Two examples of rotations Using Tessellation Exploration
The following figure demonstrates two examples of reflections, a vertical line reflection and a horizontal
line reflection that are performed in Tessellation Exploration.
Figure 9. Two examples of reflections using Tessellation Exploration
The following figure shows examples for both a horizontal and a vertical glide reflection, which are
constructed using Tessellation Exploration.
Figure 10. Two examples of glide reflections using Tessellation Exploration
We say a figure has symmetry if there exists an isometry such that the image of the figure under that
isometry coincides with the figure in its original position. For instance, consider an equilateral triangle.
Then each altitude is a line of reflection. Therefore, an equilateral triangle has reflective symmetry with
three lines of reflection. Moreover, it has rotational symmetry through angles /3 and 2/3 about its
centriod. We say the centriod is a three-fold center of symmetry. These five symmetries, including the
identity isometry, that maps every point onto itself, makes a symmetry group of order six. It is called the
dihedral group of order 6. The order refers to the number of isometries in a group.
The idea of a symmetry group can be extended to tessellations. Let us consider the regular triangle
tessellation. Then the set of six symmetries for any triangle is a set of symmetries for the tessellations
(the identity symmetry is the only symmetry common in all symmetries of all triangles). Therefore, we
have infinite number of symmetries created for reflections about the altitudes of each triangle and
rotations about the centroid of each triangle and rotations about the centroid of each triangle. We have
more symmetries than this. Now, each side of each triangle is a reflection line. Each vertex is a six-fold
center of symmetry. The midpoint of each side is also a two-fold center of symmetry.
It is not difficult to see that this tessellation has also transitional and glide reflection symmetries. For the
transitional symmetry we note that the transition cannot be achieved by moving the triangles to the
adjacent ones. This is not the case for the square and regular hexagon tessellations. The reason for this is
for the latter tessellations, the boundary of each tile consists of parallel opposite sides, which is not the
case for the triangle tessellation.
5. Creating Interesting Prototiles for Monohedral Tessellations
The information presented in the previous sections can be applied in order to create artistic tessellations.
For instance, consider a square. We are interested in modifying this square and creating a pleasing motif.
We may modify one side, reflect, and then translate the modified side to the opposite side (glide
reflection). The same idea enables us to modify another side and glide reflect it to its opposite.
The following figures show the process of making a motif which has been created based on these two
glide reflections and the resulting tessellation.
Figure 11. A monohedral tessellation using modified square based on two gilide reflections.
The following tessellation is based on a glide reflection of one side to another, and a rotation about the
midpoint of the third side of an equilateral triangle.
Figure 12. A monohedral tessellation using a modified triangle based on a glide reflection a rotation.
6. Tessellation Exploration, TesselMania, and Tile Composer
The software Tessellation Exploration, TesselMania (and its new version TesselMania Deluxe), and Tile
Composer are educational tools that introduce isometries and their applications to tessellations. In
TesselMania, the transformations and their corresponding symbols are as follows.
T = translation
G = glide reflection
C = 180 rotation
C3 = 120 rotation
C4 = 90 rotation
C6 = 60 rotation
There are choices of squares, rectangles, triangles, regular hexagons, and a specific quadrilateral for the
underlying prototile of the tessellation. The software includes tools for drawing and coloring the
prototile.
The software Tessellation Exploration is in fact the complete version of TesselMania from mathematics
point of view. However, for educational purposes students in elementary level are more comfortable to
work with former version. On the main menu of this software click on “Create a New Tessellation”.
Then “Choose a Base Shape” screen appears along with a triangle, quadrilateral, pentagon, and hexagon.
Select one of them, for example, the quadrilateral, and click “Quadrilateral” and then “Next” to continue.
Then the “Choose the Moves” screen appears. Now we select the transformations that we would like to
use in our tessellation. The translations that we choose determine the following:
1. How the sides of the polygon are transformed to create the base tile.
2. How the base tile will be transformed to create the tessellation.
On “Choose the Move” you have the choice of selecting a single transformation or a certain combination
of transformations. We choose, for instance, “Slides” tab and click “Create My Own”. A tile appears on
the “Tessellation Creator”, ready for us to shape it and add our artwork. This is exactly what Escher had
in his mind when he started creating his Pegasus tiling!
To shape the tile you need to use the following tools:
“Resize the Tile” tool. Click
,
, and
. The first tool,
and drag a corner handle, which appears around the tile. As the tile is
resized, the tessellation on the left side of the screen changes. The next tool,
tool. There are two ways to shape the tile using
points to the tile’s outline. The last tool,
added click on
previous form.
, is the
, is the “Shape the Tile”
: You can move a vertex point, and you can add
, is the “Delete Points” tool. To delete points that you have
and then click on an unwanted point. The point is removed and the side returns to its
To add artwork you need to use the paint tools that are available on the screen. Since we are mainly
concerned about the geometric constructions of tessellations and not the art involved in them we will not
explain these tools in this book and leave it to the reader to explore them. Using a square and two
translations we created the following tessellation, which is not as pleasing as the Pegasus, but assumes the
same geometric properties as it does.
Figure 13
For more examples, consider another square tessellation but this time we would like to use two rotations.
The following figures show the process of making a base tile that is created based on two rotations and
the resulting tessellation.
Figure 14
In paper “Modularity in Medieval Persian Mosaics: Textual, Empirical, Analytical, and Theoretical
Considerations” by Reza Sarhangi, Slavik Jablan, and Radmila Sazdanovic we read about the Persian
mosaics that “The widespread approach for constructing and arranging pattern designs in ceramic mosaic
during the 10th and 11th centuries was the use of squares. The economy of energy and space for molding,
casting, painting, and baking tiles forced artisans to use single-colored square tiles. The four colors
available were light blue, navy blue, brown and black. The color scheme improved rapidly by increasing
the number of colors made through different combinations of metals. It is natural to assume that a
practical way of achieving new patterns from these squares for some artisans would be to cut them in
different formats and assemble them such a way that different colors replace one other in new
arrangements. In this way the artisans could rely on the color contrast of cut-tiles to emphasize designs,
rather than use a compass and straightedge. An elementary example would be to consider congruent
squares in two colors of black and white. If we cut an isosceles right triangle with sides equal to one half
of the side of these squares in one color and exchange it with the other triangle with opposite color, we
have two two-color “modules” where one is the negative of the other. Now if we also include the two
original single-color square designs, then we have four modules to work with to create patterns”.
The following figure presents four modules that are created using tiles in two colors.
Figure 15. Four modules created based on two different colors of congruent squares.
The Tile Composer is a
software utility that is
created by Chris K.
Palmer based on the idea
of modularity.
This
software is able to
compose “tiles” based
on
the
idea
of
modularity and saves
them in its libraries. The
tiles from these libraries
are then used in the
second environment to
tessellate a plane.
Figure 16