Tessellations, a first look Exploration

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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
Printer.svg Printed version of the Tessellations, a first look Exploration, dimmed: File:Tessellations-PatternBlocks.pdf
  • 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.

  1. 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?
  2. 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?
  3. 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?
  4. 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?
  5. 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.
  6. Find a pentagon that will tessellate. Sketch the tessellation.
  7. 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).
  8. 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.