A regular
polygon is a shape with all congruent sides and all congruent angles. For example, an equilateral triangle is a
regular polygon because it has 3 congruent sides and 3 congruent angles, and a
square is a regular polygon because it has 4 congruent sides and 4 congruent angles.
Regular Tessellations
Some regular
polygons can be used to make a tile pattern (or a tessellation), and others
cannot. In order for a regular polygon
to be tileable, its interior angle must divide evenly into 360°, and the
interior angle of a regular polygon with n sides is θ = 1/n(n
– 2)180°. A regular triangle, which has n
= 3 sides, has an interior angle θ = 1/3(3 – 2)180° =
60°, which divides evenly into 360°, which means it can be used to make a tile
pattern. A square, which has n = 4
sides, has an interior angle θ = 1/4(4 – 2)180° =
90°, which divides evenly into 360°, which means it also can be used to make a
tile pattern. However, a regular
pentagon, which has n = 5 sides, has an interior angle θ = 1/5(5
– 2)180° = 108°, which does not divide evenly into 360°, which means
it cannot be used to make a tile pattern.
A regular hexagon, which has n = 6 sides, has an interior angle θ = 1/6(6
– 2)180° = 120°, which divides evenly into 360°, which means it can
be used to make a tile pattern.
Interior Angles of
Regular Polygons
Since the
next biggest factor of 360° after 120° is 180°, which is too big to be an
interior angle of a regular polygon, there are no other regular polygons that
can be used to make a tile pattern.
Therefore, the equilateral triangle, the square, and the hexagon are the
only regular polygons that can be used to make a tile pattern.
Regular Polygons That
Can Be Tiled
Semi-Regular Tessellations
If the
requirement that all the regular polygons in the tile pattern must have the
same number of sides is removed, then there are a few more tile patterns that
can be obtained called semi-regular tessellations. Semi-regular tessellations (or Archimedean
tessellations) are tile patterns that contain two or more regular polygons with
the same order around each vertex. For
example, at my house growing up we had a brick driveway consisting of octagons
and squares. This was a semi-regular
tessellation because it was a tile pattern containing regular polygons in which
each vertex was surrounded by 2 regular octagons and 1 square.
Brick Driveway Pattern
Consisting of
Octagons and Squares
One way to
find all the possible semi-regular tessellations is to examine which
combinations of regular polygons fit snugly around a single vertex. In order for this to happen, each interior
angle, which for a regular polygon with n sides is θ = 1/n(n
– 2)180°, must add up to 360°. If
there are 3 regular polygons around one vertex with sides n1, n2,
and n3; then 1/n1(n1 –
2)180° + 1/n2(n2 – 2)180° + 1/n3(n3
– 2)180° = 360°. After distributing, the
equation becomes 180° – 360°/n1 + 180° – 360°/n2
+ 180° – 360°/n3 = 360°; after combining like terms, the
equation becomes -360°/n1 – 360°/n2
– 360°/n3 = -180°; and after dividing by -360°, the
equation simplifies to
1/n1 + 1/n2 + 1/n3
= 1/2
This holds
true for the known three hexagons example (1/6 + 1/6
+ 1/6 = 1/2 or (6, 6, 6)), and
for the brick driveway example consisting of 1 square and 2 octagons (1/4
+ 1/8 + 1/8 = 1/2
or (4, 8, 8)). Using trial and error and
the fact that n is an integer and n ≥ 3, the only possible solutions are (3, 7,
42), (3, 8, 24), (3, 9, 18), (3, 10, 15), (3, 12, 12), (4, 5, 20), (4, 6, 12),
(4, 8, 8), (5, 5, 10), and (6, 6, 6).
All Combinations of 3
Polygons Fitting
Snugly around a Single
Vertex
However, even
though all of these solutions represent all the possible ways for 3 regular
polygons to fit snugly around a single vertex, not all of them can be used to
make a tessellation. One final
requirement for a tessellation is that the number of different sides must
divide evenly into the third side. For
example, the solution (5, 5, 10), which represents a pentagon, pentagon, and
decagon, fits snugly around a single vertex but cannot be tessellated. The first pentagon, with 5 sides, would need
to alternately share sides with the other pentagon and decagon, 2 different
shapes, but 2 does not divide evenly into 5, and so these 3 shapes cannot be
tessellated. (See here for more details.)
Attempt at a (5, 5,
10) Tessellation Results in Some Gaps
This
requirement eliminates all the above solutions except for (3, 12, 12), (4, 6,
12), (4, 8, 8), and (6, 6, 6).
(3, 12, 12)
(4, 6, 12)
(4, 8, 8)
The same
logic can be applied to find semi-regular tessellations with 4 or more regular
polygons. In general, if there are k
regular polygons around one vertex with sides with sides n1, n2,
… nk; then ∑ k p=1 1/np(np
– 2)180° = 360°. After distributing, the
equation becomes ∑ k p=1180° – ∑ k
p=1360°/np = 360°, which is 180°k – ∑ k
p=1360°/np = 360°; after some rearranging, the
equation becomes -∑ k p=1360°/np
= -180k
+ 360°;
and after dividing by -360°, the equation simplifies to ∑ k
p=11/np = 1/2(k
– 2) or
1/n1 + 1/n2 + …
+ 1/nk= 1/2(k
– 2)
For k = 3, 1/n1
+ 1/n2 + 1/n3 = 1/2
(as proved above); for k = 4,
1/n1 + 1/n2 + 1/n3
+ 1/n4 = 1
for k = 5,
1/n1 + 1/n2 + 1/n3
+ 1/n4 + 1/n5 = 3/2
and for k =
6,
1/n1 + 1/n2 + 1/n3
+ 1/n4 + 1/n5 + 1/n6
= 2
Since n ≥ 3,
there are no integer solutions when k ≥ 7 (geometrically, 7 or more equilateral
triangles cannot fit around a single vertex).
Once again
using trial and error and the fact that n is an integer and n ≥ 3, the only
possible solutions for k ≥ 4 are (3, 3, 4, 12), (3, 3, 6, 6), (3, 4, 4, 6), (4,
4, 4, 4), (3, 3, 3, 3, 6), (3, 3, 3, 4, 4), and (3, 3, 3, 3, 3, 3). For some of these solutions, a new ordering
produces a new unique pattern, and so (3, 4, 3, 12), (3, 6, 3, 6), (3, 4, 6, 4),
and (3, 3, 4, 3, 4) can be added to the list.
All Combinations of More
Than 3 Polygons
Fitting Snugly around
a Single Vertex
However, the
requirement for semi-regular tessellations that there is the same order around
each vertex eliminates (3, 3, 4, 12), (3, 3, 6, 6), (3, 4, 4, 6), and (3, 4, 3,
12); leaving only the solutions (3, 4, 6, 4), (3, 6, 3, 6), (4, 4, 4, 4), (3,
3, 3, 3, 6), (3, 3, 3, 4, 4), (3, 3, 4, 3, 4), and (3, 3, 3, 3, 3, 3).
(3, 4, 6, 4)
(3, 6, 3, 6)
(3, 3, 3, 3, 6)
(3, 3, 3, 4, 4)
(3, 3, 4, 3, 4)
Since (6, 6,
6), (4, 4, 4, 4), and (3, 3, 3, 3, 3, 3) are regular tessellations, that leaves
8 possible semi-regular tessellation patterns: 1 triangle and 2 dodecagons (3,
12, 12); 1 square, 1 hexagon, and 1 dodecagon (4, 6, 12); 1 square and 2
octagons (4, 8, 8); 1 triangle, 2 squares, and 1 hexagon (3, 4, 6, 4); 2
triangles and 2 hexagons (3, 6, 3, 6); 4 triangles and 1 hexagon (3, 3, 3, 3,
6); and 3 triangles and 2 squares (3, 3, 3, 4, 4) and (3, 3, 4, 3, 4). (See here for more details.)
The Eight Semi-Regular
Tessellations
Quasi-Regular Tessellations
If the
semi-regular tessellation requirement that there is the same order around each
vertex is removed, then there are countless other variations of tile patterns
that can be obtained called quasi-regular tessellations. Many quasi-regular tessellations can be
formed by modifying an existing semi-regular tessellation. For example, since 1 regular hexagon can be
formed from 6 equilateral triangles, the semi-regular tessellation of (6, 6, 6)
can be transformed to a quasi-regular tessellation in which some of the
vertices are (3, 3, 6, 6) and others are (3, 3, 3, 3, 3, 3); and the
semi-regular tessellation of (4, 6, 12) can be transformed to a quasi-regular
tessellation in which some of the vertices are (3, 3, 4, 12) and others are (3,
3, 3, 3, 3, 3).
A regular hexagon
formed by
6 equilateral
triangles
A regular dodecagon
formed by
1 regular hexagon, 6
squares,
and 6 equilateral
triangles
In addition,
1 regular dodecagon can be formed from 1 regular hexagon, 6 squares, and 6
equilateral triangles, so the semi-regular tessellation of (3, 12, 12) can be
transformed to a quasi-regular tessellation in which some of the vertices are
(3, 3, 3, 4, 4) and some are (3, 4, 6, 4).
There are countless other variations of quasi-regular tessellations, but
all of them use the same vertex combinations found in regular tessellations and
semi-regular tessellations.
Some (of Many) Quasi-Regular
Tessellations
Conclusion
There are many
ways to tile an area with shapes all consisting of segments of the same
length. Three of those ways are by
regular tessellations, in which all the shapes are the same. These include 3 hexagons (6, 6, 6), 4 squares
(4, 4, 4, 4), and 6 triangles (3, 3, 3, 3, 3, 3). Eight other ways are by semi-regular
tessellations, in which some of the shapes are different, but all of the shapes
follow the same order around each vertex used in the tessellation. These include 1 triangle and 2 dodecagons (3,
12, 12); 1 square, 1 hexagon, and 1 dodecagon (4, 6, 12); 1 square and 2
octagons (4, 8, 8); 1 triangle, 2 squares, and 1 hexagon (3, 4, 6, 4); 2
triangles and 2 hexagons (3, 6, 3, 6); 4 triangles and 1 hexagon (3, 3, 3, 3,
6); and 3 triangles and 2 squares (3, 3, 3, 4, 4) and (3, 3, 4, 3, 4). Finally, there are several more ways to tile
an area with shapes all consisting of segments of the same length using
quasi-regular tessellations, in which some of the shapes are different and in
which some of the vertices have different orders of shapes around them.
