Holey Trimmetry

Symmetry arising from symmetry isn’t surprising. But what about symmetry arising from asymmetry? You can find both among the rep-tiles, which are geometrical shapes that can be completely replaced by smaller copies of themselves. A square is a symmetrical rep-tile. It can be replaced by nine smaller copies of itself:

Rep-9 Square

If you trim the copies so that only five are left, you have a symmetrical seed for a symmetrical fractal:

Fractal cross stage #1

Fractal cross #2

Fractal cross #3

Fractal cross #4

Fractal cross #5

Fractal cross #6

Fractal cross (animated)

Fractal cross (static)

If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal:

Fractal Hex-Ring #1

Fractal Hex-Ring #2

Fractal Hex-Ring #3

Fractal Hex-Ring #4

Fractal Hex-Ring #5

Fractal Hex-Ring #6

Fractal Hex-Ring (animated)

Fractal Hex-Ring (static)

Now here’s an asymmetrical rep-tile, a nonomino or shape created from nine squares joined edge-to-edge:


It can be divided into twelve smaller copies of itself, like this:

Rep-12 Nonomino (discovered by Erich Friedman)

If you trim the copies so that only five are left, you have an asymmetrical seed for a familiar symmetrical fractal:

Fractal cross stage #1

Fractal cross #2

Fractal cross #3

Fractal cross #4

Fractal cross #5

Fractal cross #6

Fractal cross (animated)

Fractal cross (static)

If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal:

Fractal Hex-Ring #1

Fractal Hex-Ring #2

Fractal Hex-Ring #3

Fractal Hex-Ring #4

Fractal Hex-Ring #5

Fractal Hex-Ring (animated)

Fractal Hex-Ring (static)

Elsewhere other-available:

Square Routes Re-Re-Visited

Rep-tilian Rites

A pentomino is one of the shapes created by laying five squares edge-to-edge. There are twelve of them (not counting reflections) and this is the P-pentomino:


But it’s not just a pentomino, it’s also a rep-tile, or a shape that can divided into smaller copies of itself. There are two ways of doing this (I’ve rotated the pentomino 90° to make the images look better):



Once you’ve divided the shape into four copies, you can divide the copies, then the copies of the copies, and the copies of the copies of the copies, and so on for ever:



And if you’ve got a reptile, you can turn it into a fractal. Simply divide the shape, discard one or more copies, and continue:


Pentomino-based fractal stage 1


Pentomino-based fractal stage 2


Pentomino-based fractal stage 3


Stage 4


Stage 5


Stage 6


Stage 7


Stage 8


Stage 9


Stage 10

Here are some more fractals created using the same divide-and-discard process:



Animated version



Animated version













You can also use variants on a standard rep-tile dissection, like rotating the copies or trying different patterns of dissection at different levels to see what new shapes appear:































Get Your Prox Off #2

Serendipity is the art of making happy discoveries by accident. I made a mistake writing a program to create fractals and made the happy discovery of an attractive new fractal. And also of a new version of an attractive fractal I had seen before.

As I described in Get Your Prox Off, you can create a fractal by 1) moving a point towards a randomly chosen vertex of a polygon, but 2) forbidding a move towards the nearest vertex or the second-nearest vertex or third-nearest, and so on. If the polygon is a square, the four possible basic fractals look like this (note that the first fractal is also produced by banning a move towards a vertex that was chosen in the previous move):


v = 4, ban = prox(1)
(ban move towards nearest vertex)


v = 4, ban = prox(2)
(ban move towards second-nearest vertex)


v = 4, ban = prox(3)


v = 4, ban = prox(4)

This program has to calculate what might be called the order of proximity: that is, it creates an array of distances to each vertex, then sorts the array by increasing distance. I was using a bubble-sort, but made a mistake so that the program ran through the array only once and didn’t complete the sort. If this happens, the fractals look like this (note that vertex 1 is on the right, with vertex 2, 3 and 4 clockwise from it):

v = 4, ban = prox(1), sweep = 1


v = 4, ban = prox(2), sweep = 1


v = 4, ban = prox(3), sweep = 1


(Animated version of v4, ban(prox(3)), sw=1)


v = 4, ban = prox(4), sweep = 1

Note that in the last case, where ban = prox(4), a bubble-sort needs only one sweep to identify the most distant vertex, so the fractal looks the same as it does with a complete bubble-sort.

These new fractals looked interesting, so I had the idea of adjusting the number of sweeps in the incomplete bubble-sort: one sweep or two or three and so on (with enough sweeps, the bubble-sort becomes complete, but more sweeps are needed to complete a sort as the number of vertices increases). If there are two sweeps, then ban(prox(1)) and ban(prox(2)) look like this:


v = 4, ban = prox(1), sweep = 2


v = 4, ban = prox(2), sweep = 2

But the fractals produced by sweep = 2 for ban(prox(3)) and ban(prox(4)) are identical to the fractals produced by a complete bubble sort. Now, suppose you add a central point to the polygon and treat that as an additional vertex. If the bubble-sort is incomplete, a ban(prox(1)) fractal with a central point looks like this:


v = 4+c, ban = prox(1), sw = 1


v = 4+c, ban = prox(1), sw = 2

When sweep = 3, an attractive new fractal appears:


v = 4+c, ban = prox(1), sw = 3


v = 4+c, ban = prox(1), sw = 3 (animated)

If you ban two vertices, the nearest and second-nearest, i.e. ban(prox(1), prox(2)), a complete bubble-sort produces a familiar fractal:


v = 4+c, ban = prox(1), prox(2)

And here is ban(prox(2), prox(4)), with a complete bubble-sort:


v = 4, ban = prox(2), prox(4)

If the bubble-sort is incomplete, sweep = 1 and sweep = 2 produce these fractals for ban(prox(1), prox(2)):


v = 4, ban = prox(1), prox(2), sw = 1


v = 4, ban = prox(1), prox(2), sw = 2*

*The second of those fractals is identical to v = 4, ban(prox(2), prox(3)) with a complete bubble-sort.

Here is ban(prox(1), prox(5)) with a complete bubble-sort:


v = 4, ban = prox(1), prox(5)

Now try ban(prox(1), prox(5)) with an incomplete bubble-sort:


v = 4, ban = prox(1), prox(5), sw = 1


v = 4, ban = prox(1), prox(5), sw = 2

When sweep = 3, the fractal I had seen before appears:


v = 4, ban = prox(1), prox(5), sw = 3


v = 4, ban = prox(1), prox(5), sw = 3 (animated)

Where had I seen it before? While investigating this rep-tile (a shape that can be tiled with smaller versions of itself):


L-triomino rep-tile


L-triomino rep-tile (animated)

The rep-tile is technically called an L-triomino, because it looks like a capital L and is one of the two distinct shapes you can create by joining three squares at the edges. You can create fractals from an L-triomino by dividing it into four copies, discarding one of the copies, then repeating the divide-and-discard at smaller and smaller scales:


L-triomino fractal stage #1


L-triomino fractal stage #2


L-triomino fractal stage #3


L-triomino fractal stage #4


L-triomino fractal stage #5


L-triomino fractal (animated)


L-triomino fractal (close-up)

And here’s part of the ban(prox(1), prox(5)) fractal for comparison:



So you can get to the same fractal (or versions of it), by two apparently different routes: random movement of a point inside a square or repeatedly dividing-and-discarding the sub-copies of an L-triomino. That’s serendipity!

Previously pre-posted:

Get Your Prox Off

Radical Sheet

If you take a sheet of standard-sized paper and fold it in half from top to bottom, the folded sheet has the same proportions as the original, namely √2 : 1. In other words, if x = √2 / 2, then 1 / x = √2:

√2 = 1.414213562373…, √2 / 2 = 0.707106781186…, 1 / 0.707106781186… = 1.414213562373…

So you could say that paper has radical sheet (the square or other root of a number is also called its radix and √ is known as the radical sign). When a rectangle has the proportions √2 : 1, it can be tiled with an infinite number of copies of itself, the first copy having ½ the area of the original, the second ¼, the third ⅛, and so on. The radical sheet below is tiled with ten diminishing copies of itself, the final two having the same area:



You can also tile a radical sheet with six copies of itself, two copies having ¼ the area of the original and four having ⅛:



This tiling is when you might say the radical turns crucial, because you can create a fractal cross from it by repeatedly dividing and discarding. Suppose you divide a radical sheet into six copies as above, then discard two of the ⅛-sized rectangles, like this:


Stage 1

Then repeat with the smaller rectangles:


Stage 2


Stage 3


Stage 4


Stage 5


Animated version


Fractile cross

The cross is slanted, but it’s easy to rotate the original rectangle and produce an upright cross:



The Art Grows Onda

Anyone interested in recreational mathematics should seek out three compendiums by Ian Stewart: Professor Stewart’s Cabinet of Mathematical Curiosities (2008), Professor Stewart’s Hoard of Mathematical Treasures (2009) and Professor Stewart’s Casebook of Mathematical Mysteries (2014). They’re full of ideas and puzzles and are excellent introductions to the scope and subtlety of maths. I first came across Alexander’s Horned Sphere in one of them. I also came across this simpler shape that packs infinity into a finite area:


I call it a horned triangle or unicorn triangle and it reminds me of a wave curling over, like Katsushika Hokusai’s The Great Wave off Kanagawa (c. 1830) (“wave” is unda in Latin and onda in Spanish).

The Great Wave off Kanagawa by Katsushika Hokusai (1760–1849)

The Great Wave off Kanagawa by Katsushika Hokusai (1760–1849)

To construct the unicorn triangle, you take an equilateral triangle with sides of length 1 and erect a triangle with sides of length 0.5 on one of its corners. Then on the corresponding corner of the new triangle you erect a triangle with sides of length 0.25. And so on, for ever.



When you double the sides of a polygon, you quadruple the area: a 1×1 square has an area of 1, a 2×2 square has an area of 4. Accordingly, when you halve the sides of a polygon, you quarter the area: a 1×1 square has an area of 1, a 0.5 x 0.5 square has an area of 0.25 or 1/4. So if the original triangle of the unicorn triangle above has an area of 1 rather than sides of 1, the first triangle added has an area of 0.25 = 1/4, the next an area of 0.0625 = 1/16, and so on. The infinite sum is this:

1/4 + 1/16 + 1/256 + 1/1024 + 1/4096 + 1/16384…

Which equals 1/3. This becomes important when you see the use made of the shape in Stewart’s book. The unicorn triangle is a rep-tile, or a shape that can be divided into smaller copies of the same shape:



An equilateral triangle can be divided into four copies of itself, each 1/4 of the original area. If an equilateral triangle with an area of 4 is divided into three unicorn triangles, each unicorn has an area of 1 + 1/3 and 3 * (1 + 1/3) = 4.

Because it’s a rep-tile, a unicorn triangle is also a fractal, a shape that is self-similar at smaller and smaller scales. When one of the sub-unicorns is dropped, the fractals become more obvious:




Elsewhere other-posted:

Rep-Tiles Revisited

Hex Appeal

A polyiamond is a shape consisting of equilateral triangles joined edge-to-edge. There is one moniamond, consisting of one equilateral triangle, and one diamond, consisting of two. After that, there are one triamond, three tetriamonds, four pentiamonds and twelve hexiamonds. The most famous hexiamond is known as the sphinx, because it’s reminiscent of the Great Sphinx of Giza:


It’s famous because it is the only known pentagonal rep-tile, or shape that can be divided completely into smaller copies of itself. You can divide a sphinx into either four copies of itself or nine copies, like this (please open images in a new window if they fail to animate):



So far, no other pentagonal rep-tile has been discovered. Unless you count this double-triangle as a pentagon:


It has five sides, five vertices and is divisible into sixteen copies of itself. But one of the vertices sits on one of the sides, so it’s not a normal pentagon. Some might argue that this vertex divides the side into two, making the shape a hexagon. I would appeal to these ancient definitions: a point is “that which has no part” and a line is “a length without breadth” (see Neuclid on the Block). The vertex is a partless point on the breadthless line of the side, which isn’t altered by it.

But, unlike the sphinx, the double-triangle has two internal areas, not one. It can be completely drawn with five continuous lines uniting five unique points, but it definitely isn’t a normal pentagon. Even less normal are two more rep-tiles that can be drawn with five continuous lines uniting five unique points: the fish that can be created from three equilateral triangles and the fish that can be created from four isosceles right triangles:



Rep It Up

When I started to look at rep-tiles, or shapes that can be divided completely into smaller copies of themselves, I wanted to find some of my own. It turns out that it’s easy to automate a search for the simpler kinds, like those based on equilateral triangles and right triangles.

right triangle rep-tiles




(Please open the following images in a new window if they fail to animate)


triangle mosaic

Previously pre-posted (please peruse):

Rep-Tile Reflections

Know Your Limaçons

Front cover of The Penguin Dictionary of Curious and Interesting Geometry by David WellsThe Penguin Dictionary of Curious and Interesting Geometry, David Wells (1991)

Mathematics is an ocean in which a child can paddle and an elephant can swim. Or a whale, indeed. This book, a sequel to Wells’ excellent Penguin Dictionary of Curious and Interesting Mathematics, is suitable for both paddlers and plungers. Plumbers, even, because you can dive into some very deep mathematics here.

Far too deep for me, I have to admit, but I can wade a little way into the shallows and enjoy looking further out at what I don’t understand, because the advantage of geometry over number theory is that it can appeal to the eye even when it baffles the brain. If this book is more expensive than its prequel, that’s because it needs to be. It’s a paperback, but a large one, to accommodate the illustrations.

Fortunately, plenty of them appeal to the eye without baffling the brain, like the absurdly simple yet mindstretching Koch snowflake. Take a triangle and divide each side into thirds. Erect another triangle on each middle third. Take each new line of the shape and do the same: divide into thirds, erect another triangle on the middle third. Then repeat. And repeat. For ever.

A Koch snowflake (from Wikipedia)

A Koch snowflake (from Wikipedia)

The result is a shape with a finite area enclosed by an infinite perimeter, and it is in fact a very early example of a fractal. Early in this case means it was invented in 1907, but many of the other beautiful shapes and theorems in this book stretch back much further: through Étienne Pascal and his oddly organic limaçon (which looks like a kidney) to the ancient Greeks and beyond. Some, on the other hand, are very modern, and this book was out-of-date on the day it was printed. Despite the thousands of years devoted by mathematicians to shapes and the relationship between them, new discoveries are being made all the time. Knots have probably been tied by human beings for as long as human beings have existed, but we’ve only now started to classify them properly and even find new uses for them in biology and physics.

Which is not to say knots are not included here, because they are. But even the older geometry Wells looks at would be enough to keep amateur and recreational mathematicians happy for years, proving, re-creating, and generalizing as they work their way through variations on all manner of trigonomic, topological, and tessellatory themes.

Previously pre-posted (please peruse):

Poulet’s Propeller — discussion of Wells’ Penguin Dictionary of Curious and Interesting Numbers (1986)

Rep-Tile Reflections

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

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.

The rep-tile known as the fish

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.