Τα Στοιχεία του Ευκλείδου, ια΄
κεʹ. Κύβος ἐστὶ σχῆμα στερεὸν ὑπὸ ἓξ τετραγώνων ἴσων περιεχόμενον.
κϛʹ. ᾿Οκτάεδρόν ἐστὶ σχῆμα στερεὸν ὑπὸ ὀκτὼ τριγώνων ἴσων καὶ ἰσοπλεύρων περιεχόμενον.
κζʹ. Εἰκοσάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ εἴκοσι τριγώνων ἴσων καὶ ἰσοπλεύρων περιεχόμενον.
κηʹ. Δωδεκάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ δώδεκα πενταγώνων ἴσων καὶ ἰσοπλεύρων καὶ ἰσογωνίων περιεχόμενον.
Euclid’s Elements, Book 11
25. A cube is a solid figure contained by six equal squares.
26. An octahedron is a solid figure contained by eight equal and equilateral triangles.
27. An icosahedron is a solid figure contained by twenty equal and equilateral triangles.
28. A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.
Q. Each face of a convex polyhedron can serve as a base when the solid is placed on a horizontal plane. The center of gravity of a regular polyhedron is at the center, therefore it is stable on any face. Irregular polyhedrons are easily constructed that are unstable on certain faces; that is, when placed on a table with an unstable face as the base, they topple over. Is it possible to make a model of an irregular convex polyhedron that is unstable on every face?
A. No. If a convex polyhedron were unstable on every face, a perpetual motion machine could be built. Each time the solid toppled over onto a new base it would be unstable and would topple over again.
— From “Ridiculous Questions” in Martin Gardner’s Mathematical Magical Show (1965), chapter 10.
Papyrocentric Performativity Presents:
Brought to Book – A Book of English Essays, selected by W.E. Williams (Pelican 1942)
Glamourdämmerung – Treasures of Nirvana, Gillian G. Gaar (Carlton 2011)
Highway to Hell – The Road, Cormac McCarthy (2006)
Solids and Shadows – An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes, Koji Miyazaki (Wiley-Interscience 1987) (posted @ Overlord of the Über-Feral)
Magna Mater Marina – The Illustrated World Encyclopedia of Marine Fish and Sea Creatures, Amy-Jane Beer and Derek Hall (Lorenz Books 2007) (@ O.o.t.Ü.-F.)
Or Read a Review at Random: RaRaR
Two, three, four – or rather, two, three, ∞. Polygons are closed shapes in two dimensions (e.g., the square), polyhedra closed shapes in three dimensions (the cube), and polytopes closed shapes in four or more (the hypercube). You could spend a lifetime exploring any one of these geometries, but unless you take psychedelic drugs or brain-modification becomes much more advanced, you’ll be able to see only two of them: the geometries of polygons and polyhedra. Polytopes are beyond imagining but you can glimpse their shadows here – literally, because we can represent polytopes by the shadows they cast in 3-space or by the shadows of their shadows in 2-space.
Elsewhere Miyazaki doesn’t have to convey wonder and beauty by shadows: not only is this book full of beautiful shapes, it’s beautifully designed too and the way it alternates black-and-white pages with colour actually increases the power of both. It isn’t restricted to pure mathematics either: Miyazaki also looks at the modern and ancient art and architecture inspired by geometry, and at geometry in nature: the dodecahedral pollen of Gypsophilum elegans (Showy Baby’s-Breath), for example, and the tetrahedral seeds of the Water Chestnut (Trapa spp.), which the Japanese spies and assassins called the ninja used as natural caltrops. A regular tetrahedron always lies on a flat surface with a vertex facing directly upward, and when a pursued ninja scattered the sharply pointed tetrahedral seeds of the Water Chestnut, they were regular enough to injure “the soles of feet of his pursuers”.
The slightly odd English there is another example of what I like about this book, because it proves the parochialism of language and the universality of mathematics. Miyazaki’s mathematics, as far as I can tell, is flawless, like that of many other Japanese mathematicians, but his self-translated English occasionally isn’t. Japanese mathematics was highly developed before Japan fell under strong Western influence. It would continue to develop if the West disappeared tomorrow. Language is something we have to absorb intuitively from the particular culture we’re born into, but mathematics is learnt and isn’t tied to any particular culture. That’s why it’s accessible in the same way to minds everywhere in the world. Miyazaki’s pictures and prose are an extended proof of all that, and the book is actually more valuable because it was written by a Japanese speaker. I think it’s probably more attractively designed for the same reason: the skill with which the pictures have been selected and laid out reflects something characteristically Japanese. Elegance and simplicity perhaps sum it up, and elegance and simplicity are central to mathematics and some of the greatest art.
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.
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.
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. 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.”
The dodecahedron itself, consisting of twelve regular dodecahedrons, is replete with the golden ratio, long regarded as of special significance in aesthetics. 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. 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.