Tessellating Around A Point With Regular Polygons

You have been given a supply of regular polygons with 3, 4, 5, 6, 8, 10 and 12 sides. I will refer to these as 3-, 4-, 5-, 6-, 8-, 10- and 12-gons.

1. Which combinations of these regular polygons will fit together, without gaps or overlaps, around a single point? Assume that the point around which they fit is a corner of each of the polygons. For example, three hexagons is one solution. The polygons may be all of the same type or you can mix and match Remember, the polygons are to fit just around a single point, not tessellate the entire plane.

2. Make a list of the successful combinations. There are twelve possible combinations, not counting rearrangements.

3. Although you cannot check this with actual polygons, convince your neighbour that one equilateral triangle, one regular 10-gon and one regular 15-gon will fit around a point.

4. If an equilateral triangle, a regular 9-gon and one other regular polygon fit around a point, how many sides does the third polygon have?

5. For each of the combinations that you listed in Question 2, perform the following rather strange procedure: Add the reciprocals of the number of sides of the polygons. So for example, if you found that one regular a-gon, three regular b-gons and two regular c-gons will fit around a point, compute 1/a + 3/b + 2/c. Hmmm. Can you generalize this result? Can you convince your neighbour why this should be so?

6. If one regular a-gon, one regular b-gon and one regular c-gon fit around a point you will have discovered that 1/a + 1/b + 1/c = 1/2. Check that a = 3, b = 7, c = 42 is a solution to this equation. Rewrite this numerical information as a statement about regular polygons.

7. Can you find all possible solutions to the equation in Question 6? There are solutions in Questions 3, 4 and 6. There are two more solutions As a hint, both of these solutions are triples of the form (a, b, c) with c = a*b with a < b.

Philip Mallinson

PCMI July 2001