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Octagonal Star Geometric Progression

Repeating patterns of kites and squares, arranged around a central octagonal star. I really like this particular arrangement of shapes, and hope to some day exploit it further. There are many options as to folding and pleat assignment here, and I hope you’ll allow me the indulgence of sticking to the one that seems most appropriate to my own tastes.

I continue to be inspired by Islamic Art, and the wealth of geometric patterns that exist within the body of that work. However, I’m still in touch with my geometric roots, and my love of fractals- thusly patterns that are able to grow endlessly serve two purposes for me.

1.) They are wonderful patterns that often match designs from ancient buildings and artwork

2.) they are geometric tessellations of a non-euclidean space- something with a fractal dimension that I’m unable to calculate.

This factors in somewhat with the concepts of tessellations of hyperbolic space, but it’s not quite the same. I don’t have the math knowledge to explain it, but it would be something along the lines of a warped dimensional topology which, if met, would allow this piece to be tiled with equal sized pieces, or something to that effect.

I like the shapes, really, more than anything else. I hope you enjoy it too.

This little piece of paper is most certainly destined for some additional decoration, but I haven’t made any good decisions on what it should be. suggestions are quite welcome!

3 Comments

  1. Be careful. Non-Euclidean spaces really have nothing to do with fractals. When we look at, say, a tessellation of hyperbolic space, it often looks fractal because we’re taking some weird-ass space and mashing it down into the Euclidean plane so that we can make a normal picture of it. This is why those lovely prints of Escher where he has snakes or bats tessellated in the hyperbolic plane look so strange. He’s using the Poincare disk model of the hyperbolic plane, and this makes it look to us as if the bats or snakes are getting smaller and smaller as they get infinitely closer to the boundary circle. This is really just a projection of a space–that boundary circle is actually the “circle at infinity.” Being on the circle means being at infinity, so you can never get there.

    Now, fractals also exhibit (usually) things getting smaller and smaller “to infinity” in some sense, but it’s really a totally different ballgame. You’re right in stating that with fractals it’s all about the “dimension” of the object, but this is an easy concept to be confused by. There are many different ways to measure “dimension.” The one we’re all used to is called topoligical dimension and it is just the number of degrees of freedom you have if you were a point living in your object. (So the top. dim. of a line is one and of a plane is two.) As a result, topological dimension is always a whole number. But when people consider fractals they usually use Hausdorff dimension which requires graduate level math to really understand. For normal objects, like lines and circles, Hausdorf dim = topological dim. But for weird objects, like the Koch snowflake and the Sierpinski triangle, we have Haus. dim > top. dim. What gives these objects unusual Haus dim are things like the fact that the Koch curve is infinitely long, even though it has as a beginning and an ending point. And the Sierpinski triangle has zero area, even though the iterative process that creates it doesn’t seem to remove all the area.

    Hyperbolic plane tessellations do not, generally, have these kind of properties, so their Haus dim and top dim would be equal. That doesn’t mean you couldn’t study fractals in the hyperbolic plane, but they’d look pretty crazy!

    I hope that made some sense.

  2. Administrator says

    Ah, thank you so much for clearing that up for me! I had a conversation about tessellations in hyperbolic space with someone before, with regards to fractional dimensions and all that, but I never really made a connection between the quasi-fractal nature.

    If I understand what you’re saying correctly, it’s not that they are decreasing in size at all- there’s no iterative nature or anything like that; but I’m looking at a “standard” tessellation that is warped, as if I’m looking through a fisheye lens. (well, except it extends to infinity, which I suppose would be a good definition of hyperbolic space?)

    This means then that there is precious little which is different, other than the way it is percieved to me by forcing it into a euclidean space, rather than viewing it from a hyperbolic perspective. So if I were to paint it on the inside of a circular hyperbolic-ish container, then it would appear correctly if I looked into the container head-on.

    I’ve always been fascinated by fractal dimensions, after reading about Peano curves when I was a kid.

    I think I’m clear on my misconceptions here, hopefully. It’s always good to have a mathematician chime in and straighten things out for you!

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