Why do some pentagons tile a plane while others don't?

Answer 1

The interior angle of a regular pentagon is 108 degrees. Three such angles make 324 degrees while four make 432 degrees. It is not possible to cover 360 degrees without gaps or overlap. So regular pentagons cannot tessellate. However, irregular pentagons can. Their angles must be such that they can meet 3 or 4 at a time and, between them, cover 360 degrees. This constrains the values that the angles can have. Also, in most case, if the vertices of two pentagons meet at a point, the sides which end up alongside one another must be equal or else the longer side must be equal to the shorter side plus another side of a third pentagon. And in this case, the second and third pentagon must meet at angles which are supplementary. All these requirements place such constraints on the shapes that the 15th tessellating pentagon was discovered less than 2 years ago. See, for example,

//www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile