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

Hextra Texture

A hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into a hexagon and three more equilateral triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.




Previously pre-posted (please peruse):

Fractal Fourmulas

Fractal Fourmulas

A square can be divided into four right triangles. A right triangle can be divided into a square and two more right triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.








Tri Again

All roads lead to Rome, so the old saying goes. But you may get your feet wet, so try the Sierpiński triangle instead. This fractal is named after the Polish mathematician Wacław Sierpiński (1882-1969) and quite a few roads lead there too. You can create it by deleting, jumping or bending, inter alia. Here is method #1:

Sierpinski middle delete

Divide an equilateral triangle into four, remove the central triangle, do the same to the new triangles.

Here is method #2:

Sierpinski random jump

Pick a corner at random, jump half-way towards it, mark the spot, repeat.

And here is method #3:

Sierpinski arrowhead

Bend a straight line into a hump consisting of three straight lines, then repeat with each new line.

Each method can be varied to create new fractals. Method #3, which is also known as the arrowhead fractal, depends on the orientation of the additional humps, as you can see from the animated gif above. There are eight, or 2 x 2 x 2, ways of varying these three orientations, so eight fractals can be produced if the same combination of orientations is kept at each stage, like this (some are mirror images — if an animated gif doesn’t work, please open it in a new window):






If different combinations are allowed at different stages, the number of different fractals gets much bigger:

• Continuing viewing Tri Again.

Live and Let Dice

How many ways are there to die? The answer is actually five, if by “die” you mean “roll a die” and by “rolled die” you mean “Platonic polyhedron”. The Platonic polyhedra are the solid shapes in which each polygonal face and each vertex (meeting-point of the edges) are the same. There are surprisingly few. Search as long and as far as you like: you’ll find only five of them in this or any other universe. The standard cubic die is the most familiar: each of its six faces is square and each of its eight vertices is the meeting-point of three edges. The other four Platonic polyhedra are the tetrahedron, with four triangular faces and four vertices; the octahedron, with eight triangular faces and six vertices; the dodecahedron, with twelve pentagonal faces and twenty vertices; and the icosahedron, with twenty triangular faces and twelve vertices. Note the symmetries of face- and vertex-number: the dodecahedron can be created inside the icosahedron, and vice versa. Similarly, the cube, or hexahedron, can be created inside the octahedron, and vice versa. The tetrahedron is self-spawning and pairs itself. Plato wrote about these shapes in his Timaeus (c. 360 B.C.) and based a mathemystical cosmology on them, which is why they are called the Platonic polyhedra.

An animated gif of a tetrahedron


An animated gif of a hexahedron


An animated gif of an octahedron


An animated gif of a dodecahedron


An animated gif of an icosahedron


They make good dice because they have no preferred way to fall: each face has the same relationship with the other faces and the centre of gravity, so no face is likelier to land uppermost. Or downmost, in the case of the tetrahedron, which is why it is the basis of the caltrop. This is a spiked weapon, used for many centuries, that always lands with a sharp point pointing upwards, ready to wound the feet of men and horses or damage tyres and tracks. The other four Platonic polyhedra don’t have a particular role in warfare, as far as I know, but all five might have a role in jurisprudence and might raise an interesting question about probability. Suppose, in some strange Tycholatric, or fortune-worshipping, nation, that one face of each Platonic die represents death. A criminal convicted of a serious offence has to choose one of the five dice. The die is then rolled f times, or as many times as it has faces. If the death-face is rolled, the criminal is executed; if not, he is imprisoned for life.

The question is: Which die should he choose to minimize, or maximize, his chance of getting the death-face? Or doesn’t it matter? After all, for each die, the odds of rolling the death-face are 1/f and the die is rolled f times. Each face of the tetrahedron has a 1/4 chance of being chosen, but the tetrahedron is rolled only four times. For the icosahedron, it’s a much smaller 1/20 chance, but the die is rolled twenty times. Well, it does matter which die is chosen. To see which offers the best odds, you have to raise the odds of not getting the death-face to the power of f, like this:

3/4 x 3/4 x 3/4 x 3/4 = 3/4 ^4 = 27/256 = 0·316…

5/6 ^6 = 15,625 / 46,656 = 0·335…

7/8 ^8 = 5,764,801 / 16,777,216 = 0·344…

11/12 ^12 = 3,138,428,376,721 / 8,916,100,448,256 = 0·352…

19/20 ^20 = 37,589,973,457,545,958,193,355,601 / 104,857,600,000,000,000,000,000,000 = 0·358…

Those represent the odds of avoiding the death-face. Criminals who want to avoid execution should choose the icosahedron. For the odds of rolling the death-face, simply subtract the avoidance-odds from 1, like this:

1 – 3/4 ^4 = 0·684…

1 – 5/6 ^6 = 0·665…

1 – 7/8 ^8 = 0·656…

1 – 11/12 ^12 = 0·648…

1 – 19/20 ^20 = 0·642…

So criminals who prefer execution to life-imprisonment should choose the tetrahedron. If the Tycholatric nation offers freedom to every criminal who rolls the same face of the die f times, then the tetrahedron is also clearly best. The odds of rolling a single specified face f times are 1/f ^f:

1/4 x 1/4 x 1/4 x 1/4 = 1/4^4 = 1 / 256

1/6^6 = 1 / 46,656

1/8^8 = 1 / 16,777,216

1/12^12 = 1 / 8,916,100,448,256

1/20^20 = 1 / 104,857,600,000,000,000,000,000,000

But there are f faces on each polyhedron, so the odds of rolling any face f times are 1/f ^(f-1). On average, of every sixty-four (256/4) criminals who choose to roll the tetrahedron, one will roll the same face four times and be reprieved. If a hundred criminals face the death-penalty each year and all choose to roll the tetrahedron, one criminal will be reprieved roughly every eight months. But if all criminals choose to roll the icosahedron and they have been rolling since the Big Bang, just under fourteen billion years ago, it is very, very, very unlikely that any have yet been reprieved.

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.

It’s Only Rot’n’Roll…

It’s Only Rot’n’Roll

A Porphyropolyhedric Tribute to Clark Ashton Smith

Banal, mundane, and dreary. Something needs to be done about the writing of Clark Ashton Smith — and I’ve tried to do it. The problem seems to me that the writing of CAS has been Roman in the gloamin’: that is, its twilight mystery, touched with Grecian glamor, plods across the page in the Roman alphabet, which is highly functional, but aesthetically unadventurous. Has any edition of CAS in English tried to match the beauty and complexity of the text with the beauty and complexity of a font? Not to my knowledge. Calligraphy, in the wider sense, is peripheral, at best, to English literature and and even the hyperlogomania of a book like Finnegans Wake takes place on a highly restricted graphological stage. Imagine what Joyce could have done with other alphabets, other ideographies, to stir into his mad meadish Sternen-stew of polyglossemanticity! And imagine CAS printed, or hand-written, in a script that reflects something of the beauty and complexity of his language. The beauty and fluidity of Georgian or Arabic would suit his tales of Zothique, for example; the complexity and density of Devanagari or Tamil would suit his tales of Hyperborea: but best of all would be a script invented specifically for CAS.

I haven’t supplied that, but I’ve tried to point the way with what I call a CAS-Whole, or porphyropolyhedric tribute to Clark Ashton Smith. It consists of a dodecahedron of paper and purple matches that uses four invented scripts to capture the opening lines of five of CAS’s best stories. In Plato’s cosmology, four of the regular (or Platonic) polyhedrons — the tetrahedron, the hexahedron, the octahedron, and the icosahedron — represent the four elements of which the universe is composed. The final regular polyhedron, the dodecahedron, represents the universe as a whole.[1] Hence, “CAS-Whole”. The purple matches — creating a porphyro-polyhedron — recall CAS’s words in The Black Book: “Strange pleasures are known to him who flaunts the immarcesible purple of poetry before the color-blind.”[2]

The dodecahedron itself, consisting of twelve regular dodecahedrons, is replete with the golden ratio, long regarded as of special significance in aesthetics.[3] One face is entirely black and might be called panglossic, representing all possible scripts in all possible languages; another, on the opposite side of the CAS-Whole, is entirely white and might be called an’glossic, representing silence and the blank page. Between the two, in a kind of “Goldilocks zone” between too much meaning and too little, are ten faces enscribed in four invented scripts with the opening words, in English, of five of CAS’s stories. Eight faces use a single, unadulterated script of the four, spiralling to the centre; two faces combine the four scripts. Given that the scripts are used for standard English, the stories can all be deciphered with a little effort and ingenuity. We are used, when reading in our mother tongues, to understanding with little effort and ingenuity, so the CAS-Whole might be regarded as a reminder of something we should not so carelessly take for granted. Furthermore, like all the Platonic solids, the dodecahedron can serve as a die, so the CAS-Whole reflects those central CASean themes of chance and fortune. Due to my ineptitude and impatience, not all of the faces are good regular pentagons, but that too can be woven into the symbolism of the CAS-Whole. The dodecahedron is not perfect, but I am not CAS and perfect dodecahedra do not occur in nature. Nor will the die roll true: fortune is biased.[4] Critics have pointed out that almost all CAS’s stories about death, so I hope that, imperfect as it is, one might say of the CAS-Whole: “It’s only rot’n’roll — but I like it.”


1. “There still remained a fifth construction, which God used for embroidering the constellations on the whole heaven.” Timaeus, c. 360 B.C. See http://www.ellopos.net/elpenor/physis/plato-timaeus/triangles.asp?pg=3

2. The Black Book of Clark Ashton Smith, Arkham House, 1979. See http://www.eldritchdark.com/writings/bibliography/writings/nonfiction/35/the-black-book-of-clark-ashton-smith

3. For more on the golden ratio, or golden section, please see http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html

4. A biased coin can be thrown “fair”, using a simple technique that can be adapted to a biased dodecahedron. Suppose a coin is much likelier to land heads than tails (or vice versa). Simply toss it twice. If it lands HH or TT, toss again. Otherwise, use the first of the two throws: simple probability will prove that even on a biased coin, HT is as likely as TH. Similarly, for a a biased dodecahedron, roll it twelve times. If any face repeats during the twelve rolls, roll twelve times again. When you have a sequence of twelve different faces, choose the first face. Based on my (far from reliable) caculations, there are 8,916,100,448,256 ways to roll a dodecahedral die twelve times, of which 479,001,600 contain no repeating number. One would therefore have to roll the die 18,614 times, on average, to produce a sequence in which no number repeats.