The Geometry and Art of Tessellation
Introduction
A plane tessellation is a pattern made up of one or more shapes completely covering a surface without
any gaps or overlaps. As an art 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. Archimedes and other ancient mathematicians investigated properties of regular
polygons and combinations of regular polygons that tessellated the plane.
As briefly mentioned in Chapter Three, Moorish artists performed some of the most extensive works with
mosaic designs. The Alhambra built by Moorish kings in the 12th and 13th centuries, is the most famous
building in the western world for its tessellation 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 have 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 the 20th century.
Today, the use of computers and sophisticated software has brought this art to the reach of more
audiences, with a lesser 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 is M. C. Escher who popularized this art in the
western hemisphere. Students in all grade levels, by applying some mathematical transformations in
software utilities, such as Tessellation Exploration and TesselMania, may create Escher-like designs that
would take several days for an artist to create using pencil drawings.
In this chapter we are interested in only tessellations whose vertex points are not located on the edge of
any polygon in the tessellation. A usual brick pattern has vertex points that are located on the edges of
rectangular bricks so such a tessellation is not in our interest.
5.1. The Monohedral Tessellations of Polygons
Figure 5.1.1 presents a base tile and the tessellation that was created using this tile. We say that this tile
tessellates the plane. By that we mean if we have congruent copies of the tile we can cover a surface
without any gaps or overlaps. We say this tessellation is monohedral because we have used only one type
of tile on it.
Figure 5.1.1
Polygons underlie most tessellations. In this section we will explore the following question:
Which polygons (convex or nonconvex) tessellate a plane?
The simplest polygon is the triangle. Draw a triangle (any triangle that you can imagine). Will this
triangle tessellate the plane? The answer is yes. (Try it by making some copies of your triangle and
placing them next to each other to cover a surface without gaps or overlaps.)
Any triangle tessellates the plane because the sum of its interior angles is 180º. Figure 5.1.2.a shows an
arbitrary triangle,  ABC, with three angles 1, 2, and 3. Figure 5.1.2.b shows that if we arrange
these three angles to be next to each other, then we have a half turn. Therefore, three copies of this
triangle can be arranged next to each other in such a way as to make a half turn (Figure 5.1.2c).
Consequently, if we have six copies of this triangle we can arrange them about a point in such a way that
the space around the point will become filled without any gaps or overlaps (Figure 5.1.2.d). If we
continue using more copies of  ABC, we are able to cover the plane without any gaps or overlaps (Figure
5.1.2.e). The following statement, then, is the first one that we establish about tessellating polygons:
Any triangle tessellates the plane.
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3
C
2
3
2
3
B
1
2
3
2
1
3
1
1
A
(a)
(b)
(c)
(d)
(e)
Figure 5.1.2
The next polygon under investigation is the quadrilateral. Can we say that any quadrilateral tessellates
the plane? The same as the previous case the answer is positive.
According to what was presented for polygons in Chapter Four, the sum of the interior angles of any
quadrilateral equals 360º. This means if we have four copies of the quadrilateral ABCD in Figure 5.1.3.a,
we can place them around a point to form 360º, covering the plane around that point without any gaps or
overlaps (Figure 5.1.3.b). Figure 5.1.3.c presents the tessellation created based on the quadrilateral ABCD.
D
4
C
3
2
1
B
A
(a)
4
3
1
2
(b)
(c)
Figure 5.1.3
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The same argument is valid for any nonconvex quadrilateral. The following is an example of such a
tessellation.
Figure 5.1.4
Therefore, the following statement is established:
Any quadrilateral tessellates the plane.
Now let us investigate whether or not any pentagon will tessellate. Figure 5.1.5 presents a tessellation
that is generated by pentagonal tiles. But our question is whether all pentagons will tessellate the plane.
Figure 5.1.5
In general, we cannot say that all pentagons tessellate the plane. For example, consider the regular
pentagon. Figure 5.1.6 shows regular pentagons in two different sets that are positioned in the plane so
that each set of pentagons share a common vertex. We notice that for three pentagons there is a gap and
for four pentagons they overlap. The problem is each interior angle of a regular pentagon, as we found in
Chapter Four, is 108º. 360º is not divisible by 108º. This leads us to the conclusion that it is not true to
say, “Any pentagon will tessellate the plane.”
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Figure 5.1.6
Figure 5.1.7 presents a hexagon, a heptagon, and an octagon. Do any of these figures tessellate the plane?
Use copies of these figures to justify the answer to this question. You may copy and cut them out in order
to experiment with the extra copies that you made from each.
Figure 5.1.7
We can continue our investigation and find that in general only triangles and quadrilaterals always
tessellate the plane. However, for each polygon we may be able to construct a specific case that
tessellates the plane. See Figure 5.1.8 for examples using heptagons and nonagons.
Figure 5.1.8
Exercise Set 5.1
1. What is a plane tessellation?
2. Where does the word tessellation originate?
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3. Who were the Moors? How did they contribute to the art of tessellation?
4. In general which polygons can tessellate the plane? Why?
5. Construct the triangle  ABC using Sketchpad. Construct point D outside of  ABC. We want to make
a copy of this triangle to be located in a new place where point D replaces point A. Mathematically
speaking, we want to translate ABC from A to D. For this, we select
points A and D but select A
first. Then we open the Transform menu and choose “Mark Vector”. Now, if we select  ABC and
choose “Translate” in the Transform menu and click on “OK” then the desired triangle will be
constructed.
Based on the above instructions, do the following (Hint: Use “Translate” function diagonally):
(a) Construct the scalene triangle  ABC. Use this triangle to tessellate the plane. For the first
step translate  ABC from point A to point B.
(b) Construct the parallelogram ABCD. Tessellate the plane using this parallelogram.
(c) Construct the right trapezoid ABCD (A trapezoid with a right angle). Tessellate the plane
using this trapezoid. We notice in this case that we should first translate some of its sides and not
the trapezoid itself.
(d) Construct the convex quadrilateral ABCD. Tessellate the plane using this quadrilateral.
(e) Construct the nonconvex quadrilateral ABCD. Tessellate the plane using this quadrilateral.
6. Construct a square and on one side of it construct an equilateral triangle and then hide this side. The
generated pentagon has all sides congruent but not all its angles are congruent. This pentagon tessellates
the plane. Construct this tessellation.
7. There is a special class of nonregular hexagons that tessellate the plane. Such a hexagon has the
property that its sides are pairwise parallel (and therefore opposite sides are congruent).
(a) Construct such a hexagon. For this, construct first a convex quadrilateral such as
quadrilateral ABCD. Then translate the three sides of BC, CD, and AD properly as is shown
in Figure 5.1.9 to construct the desired hexagon (Hide side AB).
(b) Tessellate the plane using this hexagon.
D
C
B
A
Figure 5.1.9
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