Pentagon Tessellation Exploration

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Time-30.svg

Objective: Learn about tessellations by convex pentagons.

Materials needed: Protractor, one regular pentagon.

Tessellating with pentagons

  1. Can you use your regular pentagon to tessellate?
  2. Find a pentagon which tessellates and draw the tessellation.

Corner angles

  1. Measure the corner angles of your regular pentagon. What are they?
  2. Explain why the regular pentagon cannot tessellate.
  3. What is the angle sum of a pentagon? (Hint: cut it into three triangles)
  4. Divide the angle sum by 5 to get the corner angle of a regular pentagon. Does this match your measured value?

Convex pentagon tessellations

Which convex pentagons can tessellate the plane?

There are 15 known types. To see them, run Jaap's Tiling Viewer Applet and then open the menus for "Pentagon Tiliings", "Convex", and "15 convex pentagon tile types". Pick one of the types and use the controls on the right to change the parameters.

The first five, with names beginning PS, have been known for a long time. In 1968, Kershner found three more types and "proved" (incorrectly!) that he had found them all. There is a bit more discussion of the history on Jaap's page.

  1. Which type is the tessellation you made in question 2?
  2. When was the most recent convex pentagon tile type discovered?

Convex hexagon tessellations

Read David King's page on hexagon tessellations, and try this interactive applet to explore the convex hexagon tessellations.

  1. How many types of convex hexagon tessellation are there?

Handin: A sheet with answers to all questions.