Back to the M. C. Escher meme: Platonic Realms (Math Academy Online) is one of those “Holy Shit!” websites I stumble upon every now and then. It started as a Univ. of Colorado-Boulder graduate student project back in the 1990s.
Platonic Realms grew into a destination for math students at the high school and undergraduate levels, offering an encyclopedia, articles, and special downloads.
The encyclopedia is amazing! If I ever want to take a refresher course in statistics, I’ll know where to go. Meanwhile, I’m looking at its Escher article (Mini Text) and like the partial-paragraph from the section on tessellations that I’ve copy and pasted below. Tessellations, as I recall from The Alhambra, are a covering of an infinite geometric plane without gaps or overlaps by congruent plane figures of one type or a few types.
In 1957 he wrote an essay on tessellations, in which he remarked:
In mathematical quarters, the regular division of the plane has been considered theoretically . . . Does this mean that it is an exclusively mathematical question? In my opinion, it does not. [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature thay are more interested in the way in which the gate is opened than in the garden lying behind it.
Whether or not this is fair to the mathematicians, it is true that they had shown that of all the regular polygons, only the triangle, square, and hexagon can be used for a tessellation … Escher exploited these basic patterns in his tessellations, applying what geometers would call reflections, glide reflections, translations, and rotations to obtain a greater variety of patterns
NOTE: Copy and paste continued from a different partial-paragraph…
In Reptiles the tessellating creatures playfully escape from the prison of two dimensions and go snorting about the desktop, only to collapse back into the pattern again. Escher used this reptile pattern in many hexagonal tessellations.