Connected Mathematics 3, abstract geometry, 7th grade, p. 84, problem 40c -- please help!

Question

Hi all:

 

Our daughter had this interesting problem in her last math batch which we can't solve. Could you please help?

 

Can you find side lengths for a pentagon that will tile a surface? Explain why or why not.

 

We tried to first consider the case of regular pentagons. To calculate individual angles, we first calculate the angle sum for the whole pentagon by splitting it into three triangles (3x180=540). Then each of the angles is 540/5, so 108 degrees. Since there must not be any gaps between the tiles, let's consider an intersection between two regular pentagonal tiles that share a side. The sum of the two angles at the nexus will be 216 degrees, so the side gap angle will be 360 (full circle) minus 216, which gives us 143, so it cannot be filled with another regular pentagon.

 

Where we got completely stumped is with irregular pentagons (all sides are not equal). We are not even sure how to approach the problem. We tried to use reflectional symmetry to join irregular pentagons to each other, but can't really prove anything logically, just have a feeling it will not work to tile a floor with them without gaps.

 

Could you please help? This looks like such a great problem!

 

Thanks in advance! Best, Elena ;-)

 

 

Answer 1

In advance, the only regular polygons that tessellate with themselves are the triangle, square, and hexagon.

Tessellation with other regular polygons can be done with combinations of regular polygons (got a soccer ball handy?) 

You "solved" the issue of the regular pentagon by noting the none of the angle arrangements add to 360°

 

As to creating a tile (the mathematical term is "tessellation") with irregular polygons, Google the works of Marius "M.C." Escher.

 

A YouTube with a fun project:

https://www.youtube.com/watch?v=Lc4LGZwlcvs