A tessellation (or tiling) is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane.

A tessellation (or tiling) is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane.

A tessellation (or tiling) is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane.

A tessellation (or tiling) is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane.

A tessellation (or tiling) is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane.

Question: Can the quadrilateral below, which has no congruent angles or sides, be used to tessellate the plane?

A tessellation (or tiling) is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane.

Question: Can the quadrilateral below, which has no congruent angles or sides, be used to tessellate the plane?

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Tessellations

Tessellations

A tessellation (or tiling) is a special type of pattern that consists of geometric figures that fit without gaps or overlaps to cover the plane.

Why can’t regular pentagons alone tessellate the plane?

Of particular interest to us are tessellations composed of polygons.

regular hexagons equilateral triangles

non-regular hexagons non-regular pentagons

regular hexagons, squares, and equilateral trianglesparallelograms

Combinations of regular polygons that can meet at a vertex The interior angles of the polygons meeting at a vertex must add to 360 degrees. There are seventeen combinations of regular polygons whose interior angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

4.6.12

3.3.4.12

3.10.15

3.4.3.12

3.7.42

3.12.12

3.9.18

3.8.24

4.5.20

4.8.8

5.5.10

6.6.6

3.3.4.12 3.4.3.12

3.3.6.63.6.3.6

4.4.4.4

3.4.4.6 3.4.6.4 3.3.3.3.6

3.3.3.4.4 3.3.4.3.4 3.3.3.3.3.3

3.3.4.12 3.4.3.12

3.3.6.63.6.3.6

4.4.4.4

3.4.4.6 3.4.6.4 3.3.3.3.6

3.3.3.4.4 3.3.4.3.4 3.3.3.3.3.3

3.3.4.12 3.4.3.12

3.3.6.63.6.3.6

4.4.4.4

3.4.4.6 3.4.6.4 3.3.3.3.6

3.3.3.4.4 3.3.4.3.4 3.3.3.3.3.3

3.3.4.12 3.4.3.12

3.3.6.63.6.3.6

4.4.4.4

3.4.4.6 3.4.6.4 3.3.3.3.6

3.3.3.4.4 3.3.4.3.4 3.3.3.3.3.3

Combinations of regular polygons that can meet at a vertex The interior angles of the polygons meeting at a vertex must add to 360 degrees. There are seventeen combinations of regular polygons whose interior angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

3.10.15

Combinations of regular polygons that can meet at a vertex The interior angles of the polygons meeting at a vertex must add to 360 degrees. There are seventeen combinations of regular polygons whose interior angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

However, only eleven of these can be used to tessellate the plane. In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size.

3.12.12

Combinations of regular polygons that can meet at a vertex The interior angles of the polygons meeting at a vertex must add to 360 degrees. There are seventeen combinations of regular polygons whose interior angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

However, only eleven of these can be used to tessellate the plane. In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size.

There are only four with 3 polygons at a vertex:

3.122 - semi-regular, truncated hexagonal tiling 4.6.12 - semi-regular, truncated trihexagonal tiling 4.82 - semi-regular, truncated square tiling 63 - regular, hexagonal tiling

4.6.123.12.12

4.8.8 6.6.6

Combinations of regular polygons that can meet at a vertex The interior angles of the polygons meeting at a vertex must add to 360 degrees. There are seventeen combinations of regular polygons whose interior angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

However, only eleven of these can be used to tessellate the plane. In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size.

M.C. Escher is a Dutch artist famous for his artwork that involves tessellations.

How to create an Escher-like tessellation with Sketchpad

Tessellating a shape with translation

Start with a parallelogram Create a design on one side and hide the side

Mark the vector

((((

and decorate.and translate the design to the opposite side of the parallelogram

Start with a parallelogram. Create a design on one side and hide the side.

Mark the vector

Mark the vector and translate the design to the opposite side of the parallelogram

((((

and decorate.

(((( ((((

and translate the whole figure repeatedly.

((((

(((( (((( ((((

(((( (((( ((((

(((( (((( ((((

((((

((((

(((( (((( ((((

(((( (((( ((((

(((( (((( ((((

((((

((((

and translate repeatedlyMark the vector

(((( (((( ((((

(((( (((( ((((

(((( (((( ((((

((((

((((

(((( (((( ((((

(((( (((( ((((

(((( (((( ((((

((((

((((

Tessellating a shape with rotations

Construct an

equilateral triangle

Cut a shape out

of one side

Rotate the cut out 60 (or -60) and hide unwanted segments

Shade in the resulting polygon

Rotate and translate, change some colors, then hide all points

Construct an

equilateral triangle

Cut a shape out

of one side

Construct the midpoint of one side, cut out a shape from one endpoint to the midpoint,

and rotate it 180° around the midpoint

Rotate the entire shape 60° around a vertex and hide unwanted segments

Repeat the rotation and hide unwanted points and segments

Creating a Pinwheel (Kaleidoscope)

Homework #15

Use Geometer’s Sketchpad to construct each of the following. Email your constructions to ckoppelm@kennesaw.edu by noon Wednesday.

1.Construct a quadrilateral with no congruent sides or angles. Tessellate the plane with this quadrilateral. (Your construction must have at least four rows with at least 4 repetitions of the quadrilateral).

2.Create a novel tessellation. Be creative!!!!

3. Create a pinwheel. Be artistic!!!!

The last two are optional. The best of each (judged by the class) will win a prize.