A rep-tile, or repeat-tile, is a two-dimensional shape that can be divided completely into copies of itself. A square, for example, can be divided into smaller squares: four or nine or sixteen, and so on. Rectangles are the same. Triangles can be divided into two copies or three or more, depending on their precise shape. Here are some rep-tiles, including various rep-triangles:
Various rep-tiles — click for larger image
Some are simple, some are complex. Some have special names: the sphinx and the fish are easy to spot. I like both of those, particularly the fish. It would make a good symbol for a religion: richly evocative of life, eternally sub-divisible of self: 1, 9, 81, 729, 6561, 59049, 531441… I also like the double-square, the double-triangle and the T-tile in the top row. But perhaps the most potent, to my mind, is the half-square in the bottom left-hand corner. A single stroke sub-divides it, yet its hypotenuse, or longer side, represents the mysterious and mind-expanding √2, a number that exists nowhere in the physical universe. But the half-square itself is mind-expanding. All rep-tiles are. If intelligent life exists elsewhere in the universe, perhaps other minds are contemplating the fish or the sphinx or the half-square and musing thus: “If intelligent life exists elsewhere in the universe, perhaps…”
Mathematics unites human minds across barriers of language, culture and politics. But perhaps it unites minds across barriers of biology too. Imagine a form of life based on silicon or gas, on unguessable combinations of matter and energy in unreachable, unobservable parts of the universe. If it’s intelligent life and has discovered mathematics, it may also have discovered rep-tiles. And it may be contemplating the possibility of other minds doing the same. And why confine these speculations to this universe and this reality? In parallel universes, in alternative realities, minds may be contemplating rep-tiles and speculating in the same way. If our universe ends in a Big Crunch and then explodes again in a Big Bang, intelligent life may rise again and discover rep-tiles again and speculate again on their implications. The wildest speculation of all would be to hypothesize a psycho-math-space, a mental realm beyond time and matter where, in mathemystic communion, suitably attuned and aware minds can sense each other’s presence and even communicate.
Credo in Piscem…
So meditate on the fish or the sphinx or the half-square. Do you feel the tendrils of an alien mind brush your own? Are you in communion with a stone-being from the far past, a fire-being from the far future, a hive-being from a parallel universe? Well, probably not. And even if you do feel those mental tendrils, how would you know they’re really there? No, I doubt that the psycho-math-space exists. But it might and science might prove its existence one day. Another possibility is that there is no other intelligent life, never has been, and never will be. We may be the only ones who will ever muse on rep-tiles and other aspects of mathematics. Somehow, though, rep-tiles themselves seem to say that this isn’t so. Particularly the fish. It mimics life and can spawn itself eternally. As I said, it would make a good symbol for a religion: a mathemysticism of trans-biological communion. Credo in Piscem, Unum et Infinitum et Æternum. “I believe in the Fish, One, Unending, Everlasting.” That might be the motto of the religion. If you want to join it, simply wish upon the fish and muse on other minds, around other stars, who may be doing the same.
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 
You should read these:
Parallel Universes – Max Tegmark
The Mathematical Universe
Math unifies all things because the universe is inherently mathematical. Equations work not because they are useful approximations of reality, but because they are reality.
Before one recognize this, one may naively make statements like this:
“A single stroke sub-divides it, yet its hypotenuse, or longer side, represents the mysterious and mind-expanding √2, a number that exists nowhere in the physical universe.”
√2, like i (the “imaginary” unit) is just as real as any other number, and the above instances demonstrate, can repeatably be found in nature.
Math unifies all things because the universe is inherently mathematical. Equations work not because they are useful approximations of reality, but because they are reality.
Obviously, this depends what you mean by “reality”. I would make a distinction between maths-1, maths-as-the-ineluctable-essence-of-all-that-is-and-ever-could-be, and maths-2, maths-as-fallible-human-activity (or alien-activity). Physical reality partakes of, or is a sub-set of, maths-1, but maths-1 is infinitely bigger and maths-2 can discover more in maths-1 than it can in physical reality.
√2, like i (the “imaginary” unit) is just as real as any other number, and the above instances demonstrate, can repeatably be found in nature.
Not in nature as physical reality. You cannot discover indefinite digits of √2 there any more than you can discover indefinite digits of π using Buffon’s needle. √2 and π may exist fully in the mind of God or a Borgesian infini-text, but we can’t find them in physical reality and maths-2 can express them only to the physical limits of our technology.
“Obviously, this depends what you mean by “reality”. I would make a distinction between maths-1, maths-as-the-ineluctable-essence-of-all-that-is-and-ever-could-be, and maths-2, maths-as-fallible-human-activity (or alien-activity). Physical reality partakes of, or is a sub-set of, maths-1, but maths-1 is infinitely bigger and maths-2 can discover more in maths-1 than it can in physical reality.”
Indeed, I would make that distinction, though probably not with those names. The second is to the first like science is to the natural world.
“‘√2, like i (the “imaginary” unit) is just as real as any other number, and the above instances demonstrate, can repeatably be found in nature.’
Not in nature as physical reality.”
When you say that, what do you mean? What do you think “in nature as physical reality” entails?
“You cannot discover indefinite digits of √2 there any more than you can discover indefinite digits of π using Buffon’s needle.”
Calculation via computer is just as “natural” as calculation with Buffon’s needle. The former just allows a greater degree of precision than the latter.
This is diving into deep waters like consciousness, meaning, purpose, intentionality, determinism, etc.
“‘√2, like i (the “imaginary” unit) is just as real as any other number, and the above instances demonstrate, can repeatably be found in nature.’
Not in nature as physical reality.”
When you say that, what do you mean? What do you think “in nature as physical reality” entails?
My short definition: Physical reality is what’s left of the universe when mind and consciousness are subtracted from it. If they can be. (Universe = our corner of the hypothetical multiverse.) Certain mathematical structures have effects independent of consciousness and purpose (e.g., see the 13- and 17-year cicadas), but I don’t think √2 is one of them.
“You cannot discover indefinite digits of √2 there any more than you can discover indefinite digits of π using Buffon’s needle.”
Calculation via computer is just as “natural” as calculation with Buffon’s needle. The former just allows a greater degree of precision than the latter.
I think there’s a much bigger epistemological etc difference between the two than that. Buffon’s needle is not a mathematical structure or data-generator in the way that a computer program is. It’s analogue, it’s improvised, it’s not designed and purposive, it’s not “debuggable”, etc. And the quantitative difference in precision and speed is so great that it effectively becomes qualitative.
Thanks for the links, btw. Some v. interesting stuff, reminiscent of Rucker, Gödel, Escher, Bach, Hindu cosmology, etc. The idea of an infinite multiverse that realizes everything is frightening, tho’, because it would contain infinite evil and horror: infinite Hillary Clintons making infinite speeches, infinite Paulo Coelhos writing infinite books, infinite copies of The Best of Guns n’ Roses, etc. A comment on Tegmark’s idea that maths-1 (“all mathematical structures”) is all realized somewhere physically: In that case, I find it hard to understand how √2 or π is being realized as such, rather than as a random structure that happens to correspond to what human beings recognize as a meaningful, non-random structure. Which comes back to consciousness, intentionality, the mind of God, and so on.
I take it back about the links and Mark Tegmark’s ideas. They are interesting, but there’s a lot of waffle. Like a lot of Ashkenazi stuff, the closer you get, the smaller it looks.