Rice’s Tessellating Pentagons: Type 11

Rice’s Tessellating Pentagons: Type 11

Taylor Fountain

Math 401

09/16/2019

Assignment 1

Printed Type 11 Pentagon

Throughout the exploration of tessellations, pentagons have remained a point of interest as the lowest-order polygon whose regular variant cannot tile the Euclidean plane. This is due to the internal angles not being a divisor of 360°, preventing a full rotation from being formed without gaps or overlaps between them. With regular pentagons out of the picture (or the plane), the search for irregular pentagons that tiled the Euclidean plane commenced, resulting in the discovery of 15 distinct types convex pentagons that monohedrally fulfill the task.

Between 1976 and 1977, amateur mathematician Marjorie Rice discovered four new types of pentagons that tessellated the Euclidean plane. All 4 were 2-isohedral and demonstrate pgg symmetry (1 of the 17 types of wallpaper patterns), which is characterized by two distinct centers of 180° rotation and glide reflectional symmetry. Type 11 adhered to the constraint equations seen below (see left diagram for labeling of sides and angles), and consisted of an 8-tile primitive unit (see right diagram).

The code for the Type 11 Pentagon presented with only one modifier, which was of angle AA (angle E in our diagram). Setting the angle at a value greater than 130° or less than 110° resulted in quadrilateral and triangle that met at a point, so the range for possible values was fairly small. In the end, I set the angle to a value of 120°.

Code in OpenScad

I printed my extruded pentagon on the MakerBot printer. One error I encountered was the placing of the 8 tiles significantly increasing the time to print: by spacing out the tiles on the plate, the extruder had to lift several times when forming the base, whereas a more condensed arrangement of the tiles may have decreased the required print time. In all, the print took 2 hours and 31 minutes, so, while there was no print slot directly after mine, I should consider placing separate objects closer together in the future in order to decrease print time. In short: think more like a computer.

8-tile primitive group, exploded for clarity

An idea that occurred to me while viewing the primitive unit was that, when the adjacent pentagons that were reflected over side b were view as one piece, the unit could be comprised of four identical heptagons, each with one concave angle. Using this shift, some conclusions could be drawn about tiling the Euclidean plane with irregular heptagons containing concave angles.