I DON’T NORMALLY COMMENT on the artwork that I create; a piece should succeed or fail on its own, without the assistance (or distraction) of commentary. That said, I’ll make an exception in this case, because it’s germane to the nature of the piece.
I have developed an interest recently in the geometric art featured in islamic mosques and temples. Ceilings and walls are typically tiled in intricate designs that are predicated on four, five, or six-sided tessellations of simple geometric figures. This process has been used for thousands of years to create extraordinarily complex designs, and the only tools required for the basic design work are a ruler and compass. Pentagonal tessellations are the most complex, because the pentagon is only bilaterally (rather than fully) symmetrical, and other regular polygons must be included for the designs to scale properly.
Interestingly, there are no designs based on the heptagon. There are ostensibly two reasons for this. The first is that no one has ever figured out how to draw a perfect heptagon using only a ruler a compass. Methods have been developed recently that come close, but to date, close is as close as anyone has gotten. The second reason is that, theoretically at least, it is mathematically impossible to tesselate a heptagon in Euclidean space.
I decided not to believe this, and after rather a great deal of work, I was able to develop a technique for tessellating heptagons. This piece is the result of that effort. Like the pentagon, heptagonal tessellation also requires another polygon to work—in this case, the pentagon. The key is that these pentagons are slightly elongated and therefore irregular. Technically, that breaks the rules, but geez, work with me on this. I also added diamonds (two triangles back to back, each exactly one-seventh of the basic heptagon) in order to add some visual interest and a bit of complexity, although they’re not actually necessary for the design to work. It’s a little difficult to see at Facebook resolutions, but hopefully there is enough detail for you to get the idea.
Apparently heptagons tesselate nicely on spheres, but my copy of Facebook hasn’t yet been upgraded to accept three-dimensional objects. Sorry about that. Also, plenty of other folks have already figured out how to do this, but I hoped to produce something that also had some esthetic value. Had I Googled the problem first, I might not have bothered.
Anyway, here’s wishing everyone a great deal of seasonal joy in the coming days, and a full measure of peace and gratitude as we enter a new year.