Not about the Math, Part 1

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If I told you that there’s math in this article, would that scare you away? Well, look at the pictures first if it makes you less apprehensive, and maybe they’ll lure you in for a closer look. You won’t be sorry. The math will be minimal, anyway: not a formula to be found, and only one symbol used once. Hey, I struggled with the subject in school too, right through grade 12, and only discovered some of the interesting bits some years after that.

It turns out that all of space, whether it be flat (two dimensions) or solid (three dimensions), can be filled with shapes which neither overlap nor have holes between them (higher dimensions too, but those are a little hard for us 3d-dwellers to visualize). These shapes are called tiles, just like those in your bathroom, which are usually square or rectangular. How many different tiles are there? Only ∞; in other words, an infinite number. I FOUND THEM.

There was something about living in Georgia in the early 2000s, single, with enough free time outside of work to explore Tbilisi and marvel at its details, especially in media like chased metal or weaving. That was then, that was me. I loved being here; still do. Now, married and running a guest house/farm in the mountains, I certainly don’t have the time I did back then.

Patterns are a big thing for me, and have been since my early childhood. Fractals, a kind of geometry with unending detail found both in math and in nature, had also been a fascination of mine for some years before I reached Georgia. This country was the catalyst for my fractal tiling discoveries, in a way I can only attempt to describe. I have now lived here longer than anywhere else in the world, and it’s time to write about my finds here in the weekly format available to me since early 2011.

Squares and equilateral triangles (all angles the same size, all sides the same length) themselves can tile the plane, that is, 2d space; cubes, 3d space. All three of these shapes can also be broken down into smaller copies of themselves: the square and triangle, four copies each; the cube, eight. Because of this property, in math they are called the regular reptiles (from “replicating tiles”). All of my tiles are tiles of these three shapes, too, arising from this very self-replicating property of theirs.

The exciting part was realizing that the find has infinite variations for each of the three parent shapes. Now, there are different “kinds” of infinity, too. One is called countable: you simply start at zero and count up from there towards infinity, never reaching it. Another is uncountable: the infinity of numbers between zero and one, for example. Between ANY two such decimals as well, no matter how close together they are, exist an infinite number more. My tiles are a countable infinity, “smaller” than the uncountables yet still an infinity.

Also… if there are infinite beautiful finds in the tile sets, there are also infinite ugly ones, and infinite mediocre ones too. At the moment, it takes human esthetic sense to sort these kinds out. But give an AI some training on the many beautiful ones which we have found so far, and it may well “learn” from our choices. They should not be too “dusty”, containing pieces too small, form lost in the process. An infinity of them, in fact, will consist of only one piece.

So what, you may ask? Well, so tile! Anywhere you need a repeating pattern to cover a wall or table, drape a window or a person, you may use a tiling. If none you like in the commercial world please you, make or find one from the infinity of MY tiles. Choose any colors you like for it, print it on the material you need, wallpaper, cloth, ceramic sheets, whatever. Your own customized décor.

Next week, some more examples and types.

Tony Hanmer has lived in Georgia since 1999, in Svaneti since 2007, and been a weekly writer for GT since early 2011. He runs the “Svaneti Renaissance” Facebook group, now with over 1700 members, at www.facebook.com/groups/SvanetiRenaissance/

He and his wife also run their own guest house in Etseri:

www.facebook.com/hanmer.house.svaneti

Tony Hanmer