Strange Shapes

Geometry Level 4

For how many values of $n$ does there exist a convex $n$ -sided polygon that can be tiled using only squares and equilateral triangles--at least one of each--of side length 1?

6 7 8 9 10 11 12

1 solution

Julian Poon
Dec 20, 2016

Consider what internal angles this n-gon must have. Its internal angles must be "shared" by the triangles and squares that makes it up. Hence, the angles this n-gon can have is limited to 60 (traiangle), 120 (2 triangles), 90 (square), 150 (triangle and square). This n-gon cannot have internal angles more than 180 or if not the n-gon is no longer convex.

The restriction that the polygon must contain at least one square and one triangle gives the minimum of n to be n=5, since the maximum number of edges two polygons can share is 1. Now it surfaces that it would be useful to find the maximum n possible. To maximise "n", we must maximise the sum of its internal angles. Since the maximum internal angle the n-gon can have is 150, to maximise the sum of internal angles, all of its angles must equal to 150. This results in 12 being the maximum value of n.

Now, since we have the lowest n (5) and the largest n (12) that is possible, we now prove that every n between 5 and 12 inclusive is possible by spewing explicit examples of such polygons:

Great image.

For the part of maximizing $n$ , I think it would be easier to talk about the exterior angle. IE The exterior angle is at least $30 ^ \circ$ , and sums up to $360 ^ \circ$ , hence there are at most 12 angles/sides. Yes, ultimately it's the same argument, just presenting the algebra differently.

Calvin Lin Staff - 4 years, 5 months ago

Yeah, great imaging! Another way to equation the problem is, if the n-gon is convex and its external angles is composed of a squares and b triangles, than a and b must satisfy simultaneously the follow diofantine inequalities:

360-(90a+60b)<180 90a+60b<360 with the restriction that a and b (both) greater or equal than 1.

Cláudio Dias - 4 years, 5 months ago

I guess an extension problem could be to determine how many of each $n$ -sided polygons are possible to construct from squares and triangles.

Sharky Kesa - 4 years, 5 months ago

Strange Shapes

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