# Tessellations, a first look Exploration

From EscherMath

**Objective:**

- Introduction to some basic ideas about tessellations.
- Manipulatives are used to help develop our intuition.
- Explore how many tessellations we can make using just one type of geometric shape.

## Materials

- Pattern blocks

- Printed copy of the Tessellations, a first look Exploration.

## Tessellations

When polygons are fitted together to fill a plane with no gaps or overlaps, the pattern is called a tessellation. You have seen them in floor tilings, quilts, art designs, etc. Tessellation patterns can be made from one shape or from more than one shape; here our investigation will use one shape at a time.

We will start with a hands-on approach, and use pattern blocks to explore tessellations.

- Use the green tiles to create a tessellation by equilateral triangles. Sketch one tessellation. How many possibilities are there? How many possibilities are there if vertices are only allowed to meet other vertices?
- Use the blue tiles to create a tessellation by rhombuses. Sketch one tessellation. How many possibilities are there? How many possibilities are there if vertices are only allowed to meet other vertices?
- Use the red tiles to create a tessellation by isosceles trapezoids. Sketch one tessellation. How many possibilities are there? How many possibilities are there if vertices are only allowed to meet other vertices?
- Use the yellow tiles to create a tessellation by hexagons. Sketch one tessellation. How many possibilities are there? How many possibilities are there if vertices are only allowed to meet other vertices?
- Look at Escher's Regular Division of the Plane Drawings numbers 1-21 (Visions of Symmetry pg 116-132). Make a list of the geometric shapes that are used to tessellate the plane. For example: in sketch 1, Escher bases his drawing on a tessellation by parallelograms.
- Find a pentagon that will tessellate. Sketch the tessellation.
- Find a pentagon that will not tessellate. Explain why not (i.e. try to explain what goes wrong when one tries to tessellate the plane using this pentagon).
- Find a hexagon that is not regular, but which will tessellate. Do you think any hexagon will tessellate? Explain your answer.

**Handin:**
A sheet with answers to all questions.