Geometry
Name______________________
Tessellations Activity
Nov. 2/3, 2011
The Art & Beauty of Tessellations
A tessellation covers the plane with the same shape OR with a repeating set of shapes.
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If the shapes are all the same the tessellation is monohedral.
If it uses regular polygons the tessellation is called regular.
If there are two or more shapes involved but the arrangement at each vertex is the same the
tessellation is called semi-regular.
Objective: Find all the possible regular tessellations, and then all of the semi-regular tessellations.
Before we embark on this journey to find all these beautiful tessellations, we need to make sure we
understand the basic features of regular polygons.
In preparation, let’s list all the regular polygons with n ≤12 and their interior angles:
# of sides
Name of Polygon
Sum of Int. Angles
3
4
5
6
7
8
9
10
11
12
For any tessellation what is the requirement for arrangements at each vertex?
Draw a diagram to illustrate your point.
Measure of a Single Angle
Geometry
Now, let’s find all of the possible regular-monohedral tessellations. Interestingly, there are only three
regular-monohedral tessellations. Determine which these are with your groups.
# of
sides
Name of Polygon
Measure of
an Angle
Arrangement of angles
at each vertex
Sketch
Prove/argue (paragraph form with appropriate illustrations) why there are ONLY three regular tessellations.
Geometry
While there are only three regular-monohedral tessellations, there are 8 semi-regular tessellations.
Let’s determine which these are, but please be systematic about this: Start with all those that use two different
polygons and within this scenario, start with a triangle and with another shape, then move to a square etc.
After this, move to three polygons, etc.
Names of Polygons
Measure of angle
(for each polygon)
Arrangement at each vertex
Geometry
Group Exercises:
1) Can you tessellate a plane with a quadrilateral with angles 123 degrees, 45 degrees, 145 degrees, and 47
degrees? Can you tessellate a plane with a triangle of angles 65 degrees, 58 degrees, and 57 degrees? Can
you tessellate a plane with a pentagon of angles 123 degrees, 45 degrees, 145 degrees, 87 degrees, and 140
degrees? Am I just making these angles up?
2) Play with the possibilities for 2 and 3-uniform tessellations. Draw/sketch/illustrate these below.
Note: If there are different arrangements around the vertex the tessellation is 2-uniform (two different
vertex patterns) or 3-uniform, etc.