These are a couple of Sean's messy drawings of tessellating shapes.
He told me the other night that the only regular polygons that tessellate are the triangle, the square and the hexagon. It had something to do with the interior angles of these polygons. I didn't understand what he meant and called for Brian of course.
Brian explains that only regular polygons with interior angles that are factors of 360 tessellate.
The boys watched me as I tried to find out the interior angle of each polygon, to see if it was really true that only the interior angles of the triangle, square and hexagon are factors of 360.
The interior angle of an equilateral triangle is of course 60deg, so that goes into 360deg 6 times. Tick.
The interior angle of a square is 90deg, and that goes into 360deg 4 times. Tick.
No protractor in sight, I ask Brian, er, what's the interior angle of a hexagon?
He tells me this:
So to find the angle of any regular polygon, you've gotta subtract the number of sides by 2, multiply by 180deg, and divide by the number of sides.
After my usual "are you sure?", I got hold of a calculator and started doing what Brian said. First, I checked if the formula worked for a square, it does.
So I find that the interior angle of a pentagon is 108deg, and that doesn't go into 360deg. Which means it doesn't tessellate.
Moving on to hexagon, the angle is 120deg, which goes 3 times into 360deg. Tick.
We've got all 3 of the regular polygons that Sean tells me are the only ones that tessellate. But do I stop? Nooooo....I'm anal that way....I continue calculating for the rest of the whatever-gons.
I was about to calculate the interior angle of a 14-sided regular polygon when Brian asks me, "What are you doing Mummy?"
Me: The next number that will go into 360deg is 180deg, so I'm looking for the polygon with that interior angle.
Brian: You can't! 180deg is a straight line!
I then calculated the angle for a 1,000,000-sided polygon, and the calculator showed, "179.99964".
Ohhhhh, you're right Brian, I said, feeling really dense. Unlike me, Brian doesn't gloat.
I can't believe I was gonna go on calculating interior angles of goodness knows how many more polygons. 180deg! Of course it's a straight line, what a Duh! mummy moment. Unbelievable.
*Note: Sean's first drawing shows tessellation of an octagon, but it only works if combined with a square of equal sides to the octagon's sides.