MTE3103 – Geometry
What are tessellations?
A tessellation is a way to tile a floor (that goes on forever) with shapes so
that there is no overlapping and no gaps.
Patterns covering the plane by fitting together replicas of the same basic
shape.
In geometrical terminology a tessellation is the pattern resulting from the
arrangement of regular polygons to cover a plane without any interstices
(gaps) or overlapping. The patterns are usually repeating.
Activity 1 :
Try to use one shape (any type of regular polygon) and create a
tessellation. How many types of regular polygon create a tessellation?
There are three types of tessellation.
1) Regular Tessellations
Regular tessellations are made up entirely of congruent regular polygons all meeting
vertex to vertex. There are only three regular tessellations which use a network of
equilateral triangles, squares and hexagons.
RULE #1: The tessellation must tile a floor (that goes on forever)
with no overlapping or gaps.
RULE #2: The tiles must be regular polygons - and all the same.
RULE #3: Each vertex must look the same.
What's a vertex?
where all the "corners" meet!
What can we tessellate using these rules?
Triangles?
Notice what happens at each vertex!
The interior angle of each equilateral triangle is
60 degrees.....
60 + 60 + 60 + 60 + 60 + 60 = 360 degrees
Squares?
What happens at each vertex?
90 + 90 + 90 + 90 = 360 degrees again!
So, we need to use regular polygons that add up to 360 degrees.
Will pentagons work?
The interior angle of a pentagon is 108 degrees. . .
108 + 108 + 108 = 324 degrees . . . Nope!
Hexagons?
120 + 120 + 120 = 360 degrees Yep!
Heptagons?
No way!! Now we are getting overlaps!
Octagons?
In fact, all polygons with more than six sides will overlap! So, the only
regular polygons that tessellate are triangles, squares and hexagons!
Activity 2 : Use any two types of shapes to create a tessellation.
2) Semi-regular Tessellations
Semi-regular tessellations are made up with two or more types of regular polygon
which are fitted together in such a way that the same polygons in the same cyclic
order surround every vertex. There are eight semi-regular tessellations which
comprise different combinations of equilateral triangles, squares, hexagons,
octagons and dodecagons.
(These tessellations are made by using two or more different regular
polygons. The rules are still the same. Every vertex must have the exact
same configuration)
Those using triangles and hexagons-
Examples:
3, 3, 3, 3, 6
3, 6, 3, 6
These tessellations are both made up of hexagons and triangles, but their
vertex configuration is different.
To name a tessellation, simply work your way around one vertex counting
the number of sides of the polygons that form that vertex.
The trick is to go around the vertex in order so that the smallest numbers
possible appear first.
That's why we wouldn't call our 3, 3, 3, 3, 6 tessellation a 3, 3, 6, 3, 3!
Here's another tessellation made up of hexagons and triangles.
Can you see why this isn't an official semi-regular tessellation?
It breaks the vertex rule! Do you see where?
Here are some tessellations using squares and triangles:
3, 3, 3, 4, 4 3, 3, 4, 3, 4
Can you see why this one won't be a semi-regular tessellation?
MORE SEMI-REGULAR TESSELLATIONS
What others semi-regular tessellations can you think of?
3) Non-regular Tessellations
Non-regular tessellations are those in which there is no restriction on the order of
the polygons around vertices. There is an infinite number of such tessellations.
Taking account of the above mathematical definitions it will be readily
appreciated that most patterns made up with one or more polyiamonds are not
strictly tessellations because the component polyiamonds are not regular
polygons. The patterns might more accurately be called mosaics or tiling
patterns. Regular tessellations in the mathematical sense are possible, however,
with the moniamond, the triangular tetriamond and the hexagonal hexiamond.
Semi-regular tesselations are possible with combinations of the moniamond and
the hexagonal hexiamond.
Tessellations can be created by performing one or more of three basic
operations, translation, rotation and reflection, on a polyiamond (see Figure).
Translation - sliding the polyiamond along the plane. The translation operation
can be applied to all polyiamonds.
Rotation - rotating the polyiamond in the plane. The rotation operation can be
applied to all polyiamonds which do not possess circular symmetry, for example
the hexagonal hexiamond, which remains unchanged following rotation through
60o or multiples thereof.
Reflection - reflecting the polyiamond in the plane, as if being viewed in a mirror.
The reflection operation is limited to polyiamonds which are enantiomorphic. An
enantiomorphic polyiamond is one which cannot be superimposed on its
reflection, its mirror image.
Simple tessellations are those in which only the translation operation is used.
Complex tessellations are those in which one or both of the rotation and
reflection operations is used with the translation operation.
A single or multiple of a polyiamond may be combined to form a figure which is
capable of tessellating the plane using only the translation operation. This figure
will be called the unit cell.
A particular unit cell may be filled by multiples of different polyiamonds. Gardner
described how five pairs of heptiamonds could be used to fill the same unit cell
tessellation pattern. You will be able to find many other examples in the
illustrations later.
Tessellations may be further classified according to how the unit cells
containing one or more polyiamonds are arranged. If the unit cells are
arranged such that a regular repeating pattern is produced the tessellation
is termed periodic.
If the arrangement produces an irregular or random pattern the
tessellation is termed aperiodic.
Another arrangement which produces a tessellation with a centre of
circular symmetry is termed radial - such tessellations, with the exception
of special cases, are complex and will comprise two three or six unit cells
each containing an infinite number of poyiamonds.
All tesselations which are regular belong to a set of seventeen different
symmetry groups which exhaust all the ways in which patterns can be
repeated endlessly in two dimensions.
Polyiamonds of odd order cannot provide simple tessellations. Every
polyiamond of odd order is by definition unbalanced. The rotation and
reflection operations must be used in order to provide balanced unit cells
for tessellation.
