Dime Time

Everyone knows the shapes for one and two dimensions, far fewer know the shapes for three and four dimensions, let alone five, six and seven. And what shapes are those? The shapes that answer this question:

• How many equidistant points are possible in 1d, 2d, 3d, 4d…?

In one dimension it’s obvious that the answer is 2. In other words, you can get only two equidistant points, (a,b), on a straight line. Point a must be as far from point b as point b is from point a. You can’t add a third point, c, such that (a,b,c) are equidistant. Not on a straight line in 1d. But suppose you bend the line into a circle, so that you’re working in two dimensions. It’s easy to place three equidistant points, (a,b,c), on a circle.

equidistant points on a circle

Three equidistant points around a circle forming the vertices of an equilateral triangle

And it’s also easy to see that the three points will form the vertices of an equilateral triangle. Now try adding a fourth point, d. If you place it in the center of the triangle, it will be equidistant from (a,b,c), but it will be nearer to (a,b,c) than they are to each other. So you can have only 2 equidistant points in 1d and 3 equidistant points in 2d.

But what are the co-ordinates of the equidistant points in 1d and 2d? Suppose (a,b) in 1d are given the co-ordinates (0) and (1), so that a is 1 unit distant from b. When you move to 2d and add point c, the co-ordinates for (a,b) become (0,0) and (1,0). They’re still 1 unit distant from each other. But what are the co-ordinates for c? Start by placing c exactly midway between a and b, so that it has the co-ordinates (0.5,0) and is 0.5 units distant from both a and b. Now, if you move c in the first dimension, it will become nearer either to a or b: (0.49,0) or (0.51,0) or (0.48,0) or (0.52,0)…

But if you move c in the second dimension, it will always be equidistant from a and b, because (a,b) stay in the first dimension, as it were, and c moves equally away from both into the second dimension. So where in 2d will c be 1 unit distant from both a and b just as a and b are 1 unit distant from each other in 1d? You can see the answer here:

equilateral_triangle heightHeight of an equilateral triangle

The co-ordinates for c are (0.5,√3/2) or (0.5,0.8660254…), because the second co-ordinate satisfies the Pythagorean equation 1^2 = 0.5^2 + (√3/2)^2 = 0.25 + 0.75. That is, to find the second co-ordinate of c for 2d, you find the answer to √(1 – 0.5^2) = √(1-0.25) = √0.75 = √(3/4) = √3/2 = 0.8660254….

But you can’t add a fourth point, d, in 2d such that (a,b,c,d) are equidistant. So let’s move to 3d for the points (a,b,c,d). Begin with point d in the center of the triangle formed by (a,b,c), where it will have the co-ordinates (0.5,√3/6,0) = (0.5,0.28867513…,0) and will be equidistant from (a,b,c). But d will be nearer to (a,b,c) than they are to each other. However, if you move d in the third dimension, it will be moving equally away from (a,b,c). So where in 3d will d be 1 unit from (a,b,c)? By analogy with 2d, the third co-ordinate for d will satisfy the generalized Pythagorean equation √(1 – 0.5^2 – (√3/6)^2). And √6/3 = √(1 – 0.5^2 – (√3/6)^2) = 0.81649658… So point d will have the co-ordinates (0.5,√3/6,√6/3) = (0.5, 0.288675135…, 0.816496581…).

And the four points (a,b,c,d) will be the vertices of a three-dimensional shape called the tetrahedron:

Rotating tetrahedron

Rotating tetrahedron

But you can’t add a fifth point, e, in 3d such that (a,b,c,d,e) are equidistant. So let’s move to 4d, the fourth dimension, for the points (a,b,c,d,e). Begin with point e in the center of the tetrahedron formed by (a,b,c,d), where it will have the co-ordinates (0.5,√3/6,√6/12,0) = (0.5,0.28867513…, 0.2041241…, 0) and will be equidistant from (a,b,c,d). But e will be nearer to (a,b,c,d) than they are to each other. However, if you move e in the fourth dimension, it will be moving equally away from (a,b,c,e). So where in 4d will e be 1 unit from (a,b,c,d)? By analogy with 2d and 3d, the co-ordinate for 4d will satisfy the equation √(1 – 0.5^2 – (√3/6)^2 – (√6/12)^2). And √10/4 = √(1 – 0.5^2 – (√3/6)^2 – (√6/12)^2) = 0.79056941… So point e will have the co-ordinates (0.5,√3/6,√6/3,√10/4) = (0.5, 0.288675135…, 0.816496581…, 0.79056941…).

And the five points (a,b,c,d,e) will be the vertices of a four-dimensional shape called variously the hyperpyramid, the 5-cell, the pentachoron, the 4-simplex, the pentatope, the pentahedroid and the tetrahedral pyramid. It’s impossible for 3d creatures like human beings (at present) to visualize the hyperpyramid, but we can see its 3d shadow, as it were. And here is the 3d shadow of a rotating hyperpyramid:

Rotating hyperpyramid or 5-cell

Rotating hyperpyramid

N.B. Wikipedia reveals the mathematically beautiful fact that the “simplest set of coordinates [for a hyperpyramid] is: (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2), (φ,φ,φ,φ), with edge length 2√2, where φ is the golden ratio.”

So that’s the hyperpyramid, with 5 points in 4d. But you can’t add a sixth point, f, in 4d such that (a,b,c,d,e,f) are equidistant. You have to move to 5d. And it should be clear by now that in any dimension nd, the maximum possible number of equidistant points, p, in that dimension will be p = n+1. And here are the co-ordinates for p in dimensions 1 to 10 (the co-ordinates are given in full for 1d to 4d, then for 5d to 10d only the co-ordinates of the additional point are given):

d1: (0), (1)
d2: (0,0), (1,0), (0.5,0.866025404)
d3: (0,0,0), (1,0,0), (0.5,0.866025404,0), (0.5,0.288675135,0.816496581)
d4: 0.5, 0.288675135, 0.204124145, 0.790569415
d5: 0.5, 0.288675135, 0.204124145, 0.158113883, 0.774596669
d6: 0.5, 0.288675135, 0.204124145, 0.158113883, 0.129099445, 0.763762616
d7: 0.5, 0.288675135, 0.204124145, 0.158113883, 0.129099445, 0.109108945, 0.755928946
d8: 0.5, 0.288675135, 0.204124145, 0.158113883, 0.129099445, 0.109108945, 0.0944911183, 0.75
d9: 0.5, 0.288675135, 0.204124145, 0.158113883, 0.129099445, 0.109108945, 0.0944911183, 0.0833333333, 0.745355992
d10: 0.5, 0.288675135, 0.204124145, 0.158113883, 0.129099445, 0.109108945, 0.0944911183, 0.0833333333, 0.0745355992, 0.741619849

In each dimension d, the final co-ordinate, cd+1, of the additional point satisfies the generalized Pythagorean equation cd+1 = √(1 – c1^2 – c2^2 – … cd^2).

Readers’ advisory: I am not a mathematician and the discussion above cannot be trusted to be free of errors, whether major or minor.

Lorn This Way

There was a young man of Cape Horn,
Who wished that he’d never been born;
     And he wouldn’t have been,
     If his father had seen
That the end of the rubber was torn.

(Possibly by Swinburne)

Proxi-Performative Post-Scriptum

The toxic title of this para-poetic post (incorporating key archaic adjective “lorn”, meaning “desolate, forsaken”) is a radical reference to core Lady Gaga single “Born This Way” (which, to the best of my recollection, I haven’t heard but have heard of…). I originally intended to call the post “Torn This Way”, but decided that this adversely and anticlimactically anticipated the punchline of the limerick.

Think Inc

This is a T-square fractal:

T-square fractal

Or you could say it’s a T-square fractal with the scaffolding taken away, because there’s nothing to show how it was made. And how is a T-square fractal made? There are many ways. One of the simplest is to set a point jumping 1/2 of the way towards one or another of the four vertices of a square. If the point is banned from jumping towards the vertex two places clockwise (or counter-clockwise) of the vertex, v[i=1..4], it’s just jumped towards, you get a T-square fractal by recording each spot where the point lands.

You also get a T-square if the point is banned from jumping towards the vertex most distant from the vertex, v[i], it’s just jumped towards. The most distant vertex will always be the diagonally opposite vertex, or the vertex, v[i+2], two places clockwise of v[i]. So those two bans are functionally equivalent.

But what if you don’t talk about bans at all? You can also create a T-square fractal by giving the point three choices of increment, [0,1,3], after it jumps towards v[i]. That is, it can jump towards v[i+0], v[i+1] or v[i+3] (where 3+2 = 5 → 5-4 = 1; 3+3 = 6 → 2; 4+1 = 5 → 1; 4+2 = 6 → 2; 4+3 = 7 → 3). Vertex v[i+0] is the same vertex, v[i+1] is the vertex one place clockwise of v[i], and v[i+3] is the vertex two places clockwise of v[i].

So this method is functionally equivalent to the other two bans. But it’s easier to calculate, because you can take the current vertex, v[i], and immediately calculate-and-use the next vertex, without having to check whether the next vertex is forbidden. In other words, if you want speed, you just have to Think Inc!

Speed becomes important when you add a new jumping-target to each side of the square. Now the point has 8 possible targets to jump towards. If you impose several bans on the next jump, e.g the point can’t jump towards v[i+2], v[i+3], v[i+5], v[i+6] and v[i+7], you will have to check for five forbidden targets. But using the increment-set [0,1,4] you don’t have to check for anything. You just inc-and-go:

inc = 0, 1, 4

Here are more fractals created with the speedy inc-and-go method:

inc = 0, 2, 3

inc = 0, 2, 5

inc = 0, 3, 4

inc = 0, 3, 5

inc = 1, 4, 7

inc = 2, 4, 7

inc = 0, 1, 4, 7

inc = 0, 3, 4, 5

inc = 0, 3, 4, 7

inc = 0, 4, 5, 7

inc = 1, 2, 6, 7

With more incs, there are more possible paths for the jumping point and the fractals become more “solid”:

inc = 0, 1, 2, 4, 5

inc = 0, 1, 2, 6, 7

inc = 0, 1, 3, 5, 7

Now try applying inc-and-go to a pentagon:

inc = 0, 1, 2

(open in new window if blurred)

inc = 0, 2, 3

And add a jumping-target to each side of the pentagon:

inc = 0, 2, 5

inc = 0, 3, 6

inc = 0, 3, 7

inc = 1, 5, 9

inc = 2, 5, 8

inc = 5, 6, 9

And add two jumping-targets to each side of the pentagon:

inc = 0, 1, 7

inc = 0, 2, 12

inc = 0, 3, 11

inc = 0, 3, 12

inc = 0, 4, 11

inc = 0, 5, 9

inc = 0, 5, 10

inc = 2, 7, 13

inc = 2, 11, 13

inc = 3, 11, 13

After the pentagon comes the hexagon:

inc = 0, 1, 2

inc = 0, 1, 5

inc = 0, 3, 4

inc = 0, 3, 5

inc = 1, 3, 5

inc = 2, 3, 4

Add a jumping-target to each side of the hexagon:

inc = 0, 2, 5

inc = 0, 2, 9

inc = 0, 6, 11

inc = 0, 3, 6

inc = 0, 3, 8

inc = 0, 3, 9

inc = 0, 4, 7

inc = 0, 4, 8

inc = 0, 5, 6

inc = 0, 5, 8

inc = 1, 5, 9

inc = 1, 6, 10

inc = 1, 6, 11

inc = 2, 6, 8

inc = 2, 6, 10

inc = 3, 5, 7

inc = 3, 6, 9

inc = 6, 7, 11

A Ladd Inane

“I once received a letter from an eminent logician, Mrs. Christine Ladd-Franklin, saying that she was a solipsist, and was surprised that there were no others.” — Bertrand Russell, Human Knowledge: Its Scope and Limits (1948)

Peri-Performative Post-Scriptum

The title of this post is, of course, a radical reference to core Jethro Tull album Aladdin Sane (1973).

Boole(b)an #3

In the posts “Boole(b)an #1″ and “Boole(b)an #2” I looked at fractals created by certain kinds of ban on a point jumping (quasi-)randomly towards the four vertices, v=1..4, of a square. For example, suppose the program has a vertex-history of 2, that is, it remembers two jumps into the past, the previous jump and the pre-previous jump. There are sixteen possible combinations of pre-previous and previous jumps: [1,1], [1,2], 1,3] … [4,2], [4,3], [4,4].

Let’s suppose the program bans 4 of those 16 combinations by reference to the current possible jump. For example, it might ban [0,0]; [0,1]; [0,3]; [2,0]. To see what that means, let’s suppose the program has to decide at some point whether or not to jump towards v=3. It will check whether the combination of pre-previous and previous jumps was [3+0,3+0] = [3,3] or [3+0,3+1] = [3,4] or [3+0,3+3] = [3,2] or [3+2,3+0] = [1,3] (when the sum > 4, v = sum-4). If the previous combination is one of those pairs, it bans the jump towards v=3 and chooses another vertex; otherwise, it jumps towards v=3 and updates the vertex-history. This is the fractal that appears when those bans are used:

ban = [0,0]; [0,1]; [0,3]; [2,0]

And here are more fractals using a vertex-history of 2 and a ban on 4 of 16 possible combinations of pre-previous and previous jump:

ban = [0,0]; [0,1]; [0,3]; [2,2]

ban = [0,0]; [0,2]; [1,0]; [3,0]

ban = [0,0]; [0,2]; [1,1]; [3,3]

ban = [0,0]; [0,2]; [1,3]; [3,1]

ban = [0,0]; [1,0]; [2,2]; [3,0]

ban = [0,0]; [1,1]; [1,2]; [3,2]

Continue reading “Boole(b)an #3”…

Elsewhere other-engageable

Boole(b)an #1
Boole(b)an #2

Swan Klong

Ghìrkýthi mhlóSphálákhtthi mhlóDwèlrthi / The Swans of Queen Sphalaghd1

(Translated and edited by Simon Whitechapel)

In the black mirroring surface of the canal, these things: a sky very high and clear, the blue infinite interior of the skull of a god2 brooding beauty and pain into the world; the walls that line the canal, white marble, sharp-edged; the vivid mosaics of precious stone with which the walls are set: oval belladonnic eyes of emerald and obsidian in faces of sculpted mammoth ivory; the mouths of jade3-lipped bayadères leaking ruby threads of wine; the topaz fingers and wrists of rhodolite4-crowned lutanists; quartz-glistered sinews in the arms of capon5-plump eunuchs fanning the dances of turquoise6-skinned odalisques who prance and beck in frozen heated showers of opal-drop sweat.

But all, in the mirroring water, is grey.7

In the black mirroring surface of the canal, these things: a broad slab of basalt to which is bound a naked man8, white against the stone’s darkness, black-haired and bound with soft unbreakable bonds of purple silk; on his belly have been painted the red strokes and hooks and curls of the ideogram for death, like the roaring fist-talon and shank of a stooping hawk9 or the opening hungry maw of a leopard10, a splash of red tissues ringed and spiked and shaped by white barbs of teeth.

But all, in the mirroring water, is grey.

In the black mirroring surface of the canal, these things: five swans; their bright, forward-sweeping eyes are set in white, oval-skulled heads that are enwedged with yellow, black-rooted beaks; their white, smooth-feathered, twice-curved necks, slender as stems, are shivered on the ripples of the passage of bodies that are white seed-pods curling to smooth, hooked tails. Five swans, silent white swans. Their beaks are small and regular as the hardened gold heads of the ritual axes of the Temple of the Thanatocrator11, which sound tchlunk tchlunk tchlunk in the skulls of the sacrifices, opening slotted, red-welling ways for the prosempyreal passage of the soul.

But all, in the mirroring water, is grey.

In the black, mirroring surface of the canal, this: the white swans clustered on the white body of the sacrifice. Their necks dart and sway, sowing moist red blooms into the fertile milk of his skin. He strains against his bonds, but the necks fall, the heads hammer with steady, unconscious grace, opening the blooms to full flower. In the mirroring water they are beautiful, like strewn blossoms12 for the feet of the Thanatocrator, who dances his hatred into the waking dreams of the world. The sacrifice is dead and the swans are streaked and smirched and spotted with gore, like heavy white flowers in a garden of torture.

Their necks bend and sway, and their beaks open and close, but in the mirroring water their voices are silent, and how may we tell what they say?


1. Trained swans were the favored form of execution under the insane and semi-legendary nymphomane Queen Sphalaghd (1-143 Anno Dominæ; 1137-1279 Anno Secundi Imperii), whose extravagances came nigh to ruining the kingdom before, after many hesitations on and retreats from the threshold, she converted fully to the austere and life-denying doctrines of the Thorn-God in the final lustrum of her nigromantically prolonged life.13 She was canonized by the Temple thirteen years after her death.

2. An obvious reference to the Thorn-God, and in another context the Yihhian (mhló)Kiùlthi might be translated “(of) the god”, but the use of the non-hieratic noun marker gives the flavor of the indefinite article in English and contributes to the sense of brooding anonymity in the story.

3. This is believed to be a satirical reference to Yokh-Tsiolphë’s own religion (see note 8 below), that of the Moon-Deity, whose abstemious priestesses wore strongly colored make-up while performing their ritual dances under the full moon.

4. From its use in other texts retrieved oneirically from the Temple, the hieroglyph appears to refer to some rose-colored semi-precious mineral, and I have chosen to translate the word as “rhodolite”: coronemus nos rosis antequam marcescant (“let us crown us with roses before they be withered”, Sapientia Solomonis 2:8) was a sentiment accepted in its widest possible sense at the week-long feasts held during the long years of Queen Sphalagdh’s dissipation.

5. Possibly a castrated form not of the domestic hen (Gallus domesticus) but of the peacock (Pavo cristatus).

6. Again a possible satirical reference to the priestesses of the Moon-Deity.

7. Mirrors in the Temple were only of dark minerals, principally basalt, haematite, and black coral (Gorgonia spp), for the priests taught that color was one of the snares of sensuality by which the world entrapped men’s souls. Accordingly, possession of a fully reflecting mirror was an excommunicable offence for members of the Thorn-God’s congregation.

8. The story is believed to refer to the execution of a nobleman called Yokh-Tsiolphë (Yugg-Siurphë in some texts), who had offended the Queen either by refusing to sacrifice his eldest son and daughter to the Thorn-God or (as most scholars now believe) by falling under suspicion of having composed an anonymous pasquinade against the Thorn-God which was briefly circulated at the royal court in 38 A.D./1174 A.S.I. The execution would have been one of the earliest signs of the Queen’s growing regard for the Thorn-God.

9. The Yihhian here is a little unclear and the reference is perhaps to the Osprey (Pandion haliætus), which was second only to the Great Grey Shrike (Lanius excubitor) in the ornithomancy of the Temple.

10. The Yihhian kiuthi literally means “spotted one” and can refer to several species of animal; “leopard” seems the most appropriate translation in this context.

11. Niédýthithlà (mhló)Nhriúlr, literally “deathly lord(’s)”, was a title of the Thorn-God, but the foreign derivation of the words in Yihhian means it is perhaps best translated into English as “thanatocrator”.

12. Blossoms of gorse (Ulex spp) and other spinose plants were thrown beneath the feet of dancing priests during rituals at the Temple, and many of the Temple’s hymns refer to osmomancy, or divination by the scents released from the crushed petals.

13. She is said to have been planning another round of the puerile and puellar sacrifices with which she purchased her unnatural youth at the time of her death, occasioned when she slipped on trampled petals in the Temple of the Thorn-God whilst approaching the altar for blessing and was impaled on the silver thorns topping a newly erected altar-rail. Some contemporary commentators hinted at numerological significance in her death, saying that the priests of the Thorn-God had persuaded her that by laying down her life at that age she would regain it at the beginning of the next cosmic cycle. It is possible, therefore, that the encephalotomy and cardiotomy of her ritual mummification were feigned.

Witch Switch

Below is one of the best album-covers I’ve ever seen. It’s a triumph of subtlety and simplicity:

Burning Witch, Crippled Lucifer (1998)

The American blackened doom sludge-sters Burning Witch used Sorgen / Sorrow (1894-5), a painting by the Norwegian painter Theodor Kittelsen (1857-1914), to conjure an atmosphere of despair and darkness. Here is the original painting, skilfully combining snow, darkness and despair:

Theodor Kittelsen, Sorgen (1894-95)

But while the painting and album are good examples of less-is-more, the album is also an example of less-and-more. Part of its power comes from the contrast between the simplicity of the wandering figure and the complexity of the scripts used for the band’s name and album title:

Crippled Lucifer (detail)

Usually images are more detailed than writing. Here it’s the reverse. And while you can easily read the writing, despite its complexity, you can’t “read” the figure, despite its simplicity. Kittelsen’s skilful simplicity raised questions that can’t be answered. Is the figure male or female? Why is it sorrowful? Where is it going?

Well, you can say where it’s going in one sense: it’s walking from left-to-right. And that made me wonder whether the album could have become even starker in its contrasts. If you’re literate in Norwegian or English, you naturally read images from left-to-right, because that’s the direction of the Roman alphabet. On the album, you read the figure and the writing in the same direction. They contrast starkly in other ways, but they don’t contrast there. So let’s try making them contrast there too. Compare these two versions of the cover:

Crippled Lucifer (original cover)

Crippled Lucifer (figure-and-snowscape mirrored)

I think there’s something emptier and more despairing in the mirrored figure, walking from right-to-left. On the original cover, the figure is in some sense walking into the future, despite the weight of sorrow it carries. As we read from left to right along a piece of writing, what’s to the left of our eye is the past, and what’s to the right is the future. The figure carries the same implication. And because the figure moving towards the highly-complex-but-perfectly-intelligible band-name-and-title, there’s almost an implication that its story will be told, even if it’s moving towards death or suicide.

When the image is mirrored, all that disappears. Moving from right-to-left, the figure seems to be walking into the past, not the future. It’s no longer near or moving towards the complexity-and-intelligibility of the band-name-and-title. It’s abandoning the world more strongly: there’s no hope, no future, no implication that its story will be told.

I think the same happens, though less strongly, when the original painting is contrasted with a mirrored version:

Sorrow (original)

Sorrow (mirrored)

The contrast is less stark because, unlike the album-cover, there’s no complex patch of writing in the painting and the figure is moving away from what writing there is: the artist’s signature in the bottom left. In the original, the figure is abandoning identity and intelligibility by moving away from the signature. That’s why I’ve removed the signature in the mirrored version of the painting. It would be anomalous on the right, whether or not it was mirror-reversed, and it would be anomalous if it stayed on the left.

Finally, here’s a photo of two musicians in Sunn O))), the band into which Burning Witch eventually evolved:

Sunn O))) in black robes

In the original, Stephen O’Malley and Greg Anderson are walking from right-to-left. Here’s a mirrored version for comparison:

Sunn O))) photo (mirrored)

I think the original photo has more power, because the robed figures are walking against the grain, as it were — against the direction in which our Roman-alphabet-conditioned eyes read a photo.