Tiling the Plane with Pentagons (?), Part 1
- David Chandler
Can all pentagons tile the plane? (Series of two demonstrations)
This pair of demonstrations shows that there are pentagons that will tile the plane, but that not all pentagons will do so. This pentagon was constructed with two sides parallel. (Move each of the vertices to see that this is true.) The sum of the angles of a pentagon is always 540° (= 360° + 180°), so it is impossible for one each of all the angles to come together at a single point. Note in the demonstration that there are two kinds of vertices in the pattern: one where two angles add up to a straight angle and one where the remaining three angles come together. "Consecutive interior angles" for parallel lines are supplementary (add to 180°), so as long as we require that two sides of the pentagon be parallel, there will always be a pair of angles that can come together to form 180°. Two sets of these, or one set plus an edge, add up to 360°, so we are able to wrap around a point. The remaining three angles add to 360°, so they also will wrap around a point. The following demonstration takes off the restriction that two sides are parallel, so as soon as the parallelism is disrupted, the tiling no longer works. Not even regular pentagons will not tile the plane. Later in this chapter we see that the angles of a regular pentagon are always 108°. This is not divisible into 360°, so tiling cannot work.