Pentagonal tiling #tessellations, Part 3

Apprenticeship

Most of us learn the easy/best way. Look at the masters, follow their path and learn all that we can from them. Replicate their artwork. It is a long process, especially without any direction or assistance from a teacher. This is where I’m at right now — copying / learning from the pentagon symmetry system seekers: Reinhardt, Kershner, James, Rice, Stein, Mann, McLoud, and Von Derau. As I did for a while, copying M.C. Escher’s tessellations, decades ago, although I no longer need MCE inspiration to create a tessellation.

Notes about pentagonal tiling

A few notes I’ve kept from my observations, all in my own vocabulary, as I do not know the mathematician’s jargon:

  1. Very many of the pentagonal tilings can be recreated using the P2 symmetry group. A series of four two-way rotation points on a skewable plane. This gives the resulting tiles quite a range of possible outcomes.
  2. If using the P3 symmetry group, a hexagon is the primary grid behind many pentagonal tilings. This primary hexagon is subdivided in many ways; in 2, in 3, in 4, or in 9 equilateral parts.
  3. As if the whole tiling plane is constructed on a peg board, with pegs and bungee cords, all can be shifted around in a soup of pentagons. Like broken ice, floating, flowing, rearranging itself with its neighbours.
  4. Some tilings are created with a single pentagon, repeated using various rotation and translation scenarios. Some systems employ two, four or even eight different shapes. It is surprising that they have been able to classify these symmetries into only fifteen groups. I still can’t wrap my head around the rules or restrictions. I can draw these pentagonal tiles, but it doesn’t mean that I understand them. This is the identical quote I came up with 30 years ago, “drawing tessellations and understanding tessellations are two different beasts”.
  5. Those last two pentagonal symmetry groups are the most difficult to figure out. Still, my quest is not complete.

Below are my latest efforts at plowing through all the pentagonal symmetry systems. More to come next season.

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