Tessellations on the Web

A set of lessons that integrate
technology, hands-on activities, and mathematical depth

http://www.math.csusb.edu/faculty/susan/tess/

California Math Council Southern Section meeting
Nov. 8, 1997

Susan Addington
Math Dept.	e-mail: `susan@math.csusb.edu`
California State University	phone: (909) 880-5362
San Bernardino, CA 92407	fax: (909) 880-7119

Contents

Local considerations: What do the pieces of a tessellation look like close up?
- A. Angles at a vertex; impossibilities
- B. Angles in an n-gon
- C. Tessellations with regular polygons
- D. Tessellations with other special polygons
Global considerations: What does the overall pattern look like?
- A. Repeating and non-repeating tessellations
- B. Tessellations with symmetry

1. Definitions

A tessellation (of the plane) is a set of shapes that cover the whole plane (to infinity) without gaps or overlaps. Tiling is another word for tessellation, but tessellation sounds fancier. The shalation, but tessellation sounds fancier. The shapes making up the tessellation are called tiles .
A tessellation does not have to be a repeating pattern.
The shapes in a tessellation do not have to be polygons.

A polygon is a plane figure with straight sides.
The sides of a polygon are called edges and the corners are called vertices (singular: vertex ).
An n-gon is a polygon with n edges (for example, a 4-gon is also called a quadrilateral).
An equilateral polygon is one in which all edges have the same length.
An equiangular polygon is one in which all angles have the same measure.
A regular polygon is one that is both equilateral and equiangular. (They are the most symmetric polygons.)

A tessellation whose tiles are polygons is edge-to-edge if each vertex of each tile meets other tiles only at vertices. A checkerboard is an edge-to-edge tiling. The usual way of stacking bricks in a wall is not edge-to-edge.

1. Local considerations: What do the pieces of a tessellation look like close up?

A. Angles at a vertex; impossibilities

Without using a protractor or a ruler, figure out all the lengths and angles of the pattern block pieces. Consider the side of the square to be 1 unit long. Remember that the an be 1 unit long. Remember that the angle of a complete rotation is 360 degrees.

Extensions: describing the shapes of polygons by sides and angles, designing new tiles in the computer program ptile. (Free Mac software; some instructions, but parts apply only to the Math Dept. computers at CSUSB.

Make a tessellation using pattern blocks. Look at several of the vertices, and make a general statement about the sum of angles at a vertex of a tessellation. What can you say if a vertex meets another tile in the middle of an edge?
Which of the following tessellation descriptions are possible or impossible? There are three choices of answer:
- Impossible
- Can do it, and here's an example
- Can't tell without further work
Here are facts about the angles in the polygons (or figure them out yourself with part B.)

equilateral triangle 60^o
square 90^o
regular pentagon 108^o pentagon 108^o
regular hexagon 120^o
regular 12-gon 150^o

a. Is it possible to have a tessellation using regular pentagons?
b. Is it possible to have a tessellation using house-shaped pentagons? (that is, a square topped by a 45-45-90 triangle)

c. Is it possible to have a tessellation in which 4 triangles and a square meet at every vertex?
d. Is it possible to have a tessellation in which 3 triangles and 2 squares meet at every vertex?
e. Is it possible to have a tessellation in which 2 triangles and 2 regular hexagons meet at every vertex?

f. Is it possible to have a tessellation in which 2 triangles, a square, and a regular 12-gon meet at every vertex?

Note: some of these tessellations are impossible even though the angles add up correctly. So the moral of this story is that just looking at a small part (a vertex) of a tessellation isn't enough to understand the whole thing.

B. Angles in an n-gon

The sum of angles in any plane triangle is 180^o. What is the sum of angles in a quadrilateral? Start with the pattern blocks that have 4 sides, since you already know the angles. In general, divide the quadrilateral into triangles and make sure all the angles are covered and no extra angles sneak in.
Use the sames sneak in.
Use the same idea to find the sum of angles in polygons with more sides.
Generalize to polygons with n sides.

Extension: Do the reasoning and result still work with nonconvex polygons?

In a regular (or equiangular) polygon, all the angles are the same.
What is any angle in a regular 4-gon? A regular 5-gon? A regular 6-gon? Generalize to a regular n-gon.

Extension: Plot the relationship between number of sides and the angle in a regular n-gon on a graph. Solve "inverse" problems:
Is there a regular polygon with angle 165^o?
Is there a regular polygon with angle 170^o?
Is there a regular polygon with angle > 175^o?
Is there a regular polygon with angle > 185^o?
Is there a regular polygon with angle < 60^o?

C. Tessellations with regular polygons
In math as in life, the more rules there are, the fewer options you have. So if you want to answer a question such as "What are all the possible tessellations?", you have more hope of succeeding if you require your tessellations to follow some rules. (Otherwise there will be too many to keep track of.) Here is one popular set of rules.
Rules for semiregular tessellations

All tiles are regular polygons
The polygons meet edge-to-edge
Every vertex of the tessBR>
Every vertex of the tessellation looks the same: it has the same number of the same types of polygons meeting in the same way.

Find all the semiregular tessellations that use only one type of regular polygon. (These are called regular tessellations.)
Find all the semiregular tessellations that use equilateral triangles and squares.
Find all the semiregular tessellations that use equilateral triangles and regular hexagons.
Find all the semiregular tessellations that use regular octagons and any other regular polygons.

Extensions: Classify all the semiregular tessellations. (There aren't that many.) The hard part is making an organized list of all the possible combinations of angles at a vertex. Then weed out the ones that don't work.

Notation: Vertex type. Suppose you had to describe all the tessellations you found over the phone. Here is an efficient way. Since all the vertices look the same, you could just describe what happens at one vertex. Imagine that it's a pole, and you walk over the tiles as you walk around the pole. A checkerboard would be described as "square, square, sd as "square, square, square, square." But this is a semiregular tessellation, so the only kind of quadrilateral allowed is a square, so you could just say "4,4,4,4". (Usually periods are used instead of commas: 4.4.4.4.)
Make the tessellations 3.3.3.3.3.3, 6.6.6, 3.3.3.4.4.

Describe the tessellations you found in 1-4 with this notation.

Extension: Try making 4.4.4 with snap-together polygons. (Hint: it pops up.) In general, you get polyhedra.

D. Extensions: Tessellations with other special polygons or other rules.

Other interesting tiles:

Triangles other than equilateral: isosceles, scalene
Quadrilaterals other than squares, including ones with no sides or angles equal.
Ominoes: An omino is a polygon made by attaching a number of squares edge-to-edge. A domino is made with 2 squares, a triomino with 3 squares, a quadromino with 4 squares, a pentomino with 5 squares, etc. Use only one shape or several.
Star polygons. Try symmetric stars with the angles at the points equal. What are the other angles? What other shapes would be needed to tessellate with a given star?
Penrose tiles. These are based on the golden ratio. Generally there are matching rules for what edges can be attached to others. You then get non-repeating tessellations.

Other interesting rules:

Use the semiregul
- Use the semiregular rules, but allow 2 or more vertex types.
- Edge-to-edge tessellations (not necessarily with regular polygons) where every vertex is divided into equal angles. For example, if there are 3 tiles meeting at a vertex, the angles must be 120.
3. Global considerations: What does the overall pattern look like?

A. Repeating and non-repeating tessellations

If a pattern doesn't repeat, how can you be sure it will go on forever? Try making some patterns with Penrose tiles. You will find that you sometimes reach dead ends, where no tile will fit. Some kind of rule for continuing to build the pattern will often ensure that you won't hit a dead end.
Here is an example of a random pattern of a tessellation by 45-45-90 triangles.
Start with a checkerboard tessellation. Pick a central square, and trace a spiral path that expands outwards from the central square. Each time you land on a new square, flip a coin. Cut the square in half diagonally with slope +1 if the coin lands heads, and with slope -1 if the coin lands tails.

B. Tessellations with symmetry

If a pattern has some kind of symmetry, then it repeats in some way. For example, in a pattern with one line of reflection symmetry, the pattern on one side of the line is a repeat of the pattern on the other side.
There n the other side.
There are 4 ways a pattern in the plane can be symmetric: reflection, rotation, translation, and glide reflection. You can also have combinations of these. For example, a pattern could have a rectangular grid of lines of reflection symmetry, or translations in both horizontal and vertical directions.
See the pages "The four types of symmetry in the plane.

Symmetric patterns in the plane can be classified by how many directions of translation symmetry.
- No translation symmetry. These are called rosette patterns. There are two types: those with only rotations, and those with rotations and reflections.
- 1 direction of translation symmetry. These are called frieze , or strip patterns. The various combinations of types of symmetry give 7 different symmetry types.
- 2 directions of translation symmetry. These are called wallpaper patterns. The various combinations of types of symmetry give 17 different symmetry types.
Having two directions of symmetry mo directions of symmetry means that the whole pattern can be generated from a small piece (actually a parallelogram), repeated over and over by sliding it backwards and forwards (one of the directions of translation symmetry, and up and down (the other direction) to cover the whole plane. That's what is done in the next activity. Included are 3 of the 17 types of wallpaper pattern; directions for the others can be found on the Web.

Make a wallpaper-type tessellation by altering the sides of a simple starting shape.

You can do this with paper and scissors and tape, by creating a template to trace, or cutting copies to paste on your picture.
Or you can do it with a computer paint program, following Suzanne Alejandre's tutorial.
Or you can avoid all work and use the program Tesselmania (but you might not learn anything.)
1. A pattern with two directions of translation symmetry, and no other symmetries (wallpaper type ). See Suzanne's tessellation tutorials.
2. A pattern with two directions of translation symmetry, and which can be generated by repeating rotations over and over (wallpaper type 442). See A Tessellation with Rotation ellation with Rotation Symmetry.
3. A pattern with two directions of translation symmetry, and which can be generated by repeating reflections over and over (wallpaper type *2222). See A Tessellation with Reflection Symmetry.

equilateral triangle	60^o
square	90^o
regular pentagon	108^o pentagon	108^o
regular hexagon	120^o
regular 12-gon	150^o