Io + Gatto / I + Cat (1932) by Wanda Wulz (1903-84)
Stanislaw Ulam (pronounced OO-lam) was an American mathematician who was doodling one day in 1963 and created what is now called the Ulam spiral. It’s a spiral of integers on a square grid with the prime squares filled in and the composite squares left empty. At the beginning it looks like this (the blue square is the integer 1, with 2 to the east, 3 to the north-east, 4 to the north, 5 to the north-west, 6 to the west, and so on):
Ulam spiral
And here’s an Ulam spiral with more integers:
Ulam spiral at higher resolution
Here’s an animated version:
Here’s the true spiral again with 1 marked as a blue square:
Ulam spiral centred on 1
What happens when you try other numbers at the centre? Here’s 2 at the centre as a purple square, because it’s prime:
Ulam spiral centred on 2
And 3 at the centre, also purple because it’s also prime:
Ulam spiral centred on 3
And 4 at the centre, blue again because 4 = 2^2:
Ulam spiral centred on 4
And 5 at the centre, prime and purple:
Ulam spiral centred on 5
Each time the central number changes, the spiral shifts fractionally. Here’s an animation of the central number shifting from 1 to 41. If you watch, you’ll see patterns remaining stable, then breaking up as the numbers shift towards the center and disappear (the central number is purple if prime, blue if composite):
Ulam whirlpool, or WhirlpUlam
I think the animation looks like a whirlpool or whirlpUlam (prounced whirlpool-am), as numbers spiral towards the centre and disappear. You can see the whirlpUlam more clearly here:
WhirlpUlam again
Note that something interesting happens when the central number is 41. The spiral is bisected by a long line of prime squares, like this:
Ulam spiral centred on 41
The line is actually a visual representation of something David Wells wrote about in The Penguin Dictionary of Curious and Interesting Numbers (1986):
Euler discovered the excellent and famous formula x2 + x + 41, which gives prime values for x = 0 to 39.
Here are the primes generated by the formula:
41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601
You’ll see other lines appear and disappear as the whirlpUlam whirls:
Ulam spiral centred on 17
Primes in line: 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 (n=0..15)
Ulam spiral centred on 59
Primes in line: 59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499, 587, 683, 787 (n=0..13)
Ulam spiral centred on 163
Primes in line: 163, 167, 179, 199, 227, 263, 307, 359, 419, 487, 563, 647, 739, 839, 947, 1063, 1187, 1319, 1459, 1607 (n=0..19)
Ulam spiral centred on 233
Primes in line: 233, 241, 257, 281, 313, 353, 401, 457, 521, 593, 673, 761, 857 ((n=0..12)
Ulam spiral centred on 653
Primes in line: 653, 661, 677, 701, 733, 773, 821, 877, 941, 1013, 1093, 1181, 1277, 1381, 1493, 1613, 1741, 1877 (n=0..17)
Ulam spiral centred on 409,333
Primes in line: 409,333, 409337, 409349, 409369, 409397, 409433, 409477, 409529, 409589, 409657, 409733, 409817, 409909, 410009, 410117, 410233 (n=0..15)
Some bisect the centre, some don’t, because you could say that the Ulam spiral has six diagonals, two that bisect the centre (top-left-to-bottom-right and bottom-left-to-top-right) and four that don’t. You could also call them spokes:
If you look at the integers in the spokes, you can see that they’re generated by polynomial functions in which c stands for the central number:
North-west spoke: 1, 5, 17, 37, 65, 101, 145, 197, 257, 325, 401, 485, 577, 677, 785, 901, 1025, 1157, 1297, 1445, 1601, 1765, 1937, 2117, 2305, 2501, 2705, 2917... = c + (2n)^2
South-east spoke: 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625... = c+(2n+1)^2-1
NW-SE diagonal: 1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121, 145, 169, 197, 225, 257, 289, 325, 361, 401, 441, 485, 529, 577, 625, 677, 729, 785, 841, 901, 961, 1025, 1089, 1157, 1225, 1297, 1369, 1445, 1521, 1601, 1681 = c + n^2 + 1 - (n mod 2)
North-east spoke: 1, 3, 13, 31, 57, 91, 133, 183, 241, 307, 381, 463, 553, 651, 757, 871, 993, 1123, 1261, 1407, 1561, 1723, 1893, 2071... = c + (n+1)^2 - n - 1
South-west spoke: 1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163... = c + (2n)^2 + 2n
SW-NE diagonal: 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641... = c + n^2 + n
Elsewhere other-engageable:
• All posts interrogating issues around the Ulam spiral
Papyrocentric Performativity Presents:
• God Guide – A Guide to Tolkien, David Day (Octopus 1993)
• The Catcher and the Rye – The Biology of Flowers, Eigil Holm, ill. by Thomas Bredsdorff and Peter Nielsen (Penguin Nature Guides 1979)
• Dayzed and Contused – The Greatest Footballer You Never Saw: The Robin Friday Story, Paul McGuigan and Paolo Hewitt (Mainstream 1997)
Or Read a Review at Random: RaRaR
Aztec or Jacobean lily, Sprekelia formosissima (L.) (Mexico, Guatemala and Honduras)
He’s been mixing with the wrong people:
“Our supporters and our country has had a long time suffering in terms of football. […] Our country has been through some difficult moments recently in terms of unity but sport has the power to unite — and football in particular has the power to do that.” — England manager Gareth Southgate, BBC Sport, 10vii2018.
Elsewhere other-engageable:
• Oh My Guardian #6 — the latest in the award-winning series
• All posts interrogating issues around the Guardian-reading community and its affiliates
• Ex-term-in-ate! — interrogating arguably the keyliest and coreliest Guardianista phrase
• All posts interrogating issues around “in terms of”
Twice before on Overlord-in-terms-of-Core-Issues-around-Maximal-Engagement-with-Key-Notions-of-the-Über-Feral, I’ve interrogated issues around pursuit curves. Imagine four mice or four beetles each sitting on one corner of a square and looking towards the centre of the square. If each mouse or beetle begins to run towards the mouse or beetle to its left, it will follow a curving path that takes it to the centre of the square, like this:
vertices = 4, pursuit = +1
The paths followed by the mice or beetles are pursuit curves. If you arrange eight mice clockwise around a square, with a mouse on each corner and a mouse midway along each side, you get a different set of pursuit curves:
v = 4 + 1 on the side, p = +1
Here each mouse is pursuing the mouse two places to its left:
v = 4+s1, p = +2
And here each mouse is pursuing the mouse three places to its left:
v = 4+s1, p = +3
Now try a different arrangement of the mice. In the square below, the mice are arranged clockwise in rows from the bottom right-hand corner. That is, mouse #1 begins on the bottom left-hand corner, mouse #2 begins between that corner and the centre, mouse #3 begins on the bottom left-hand corner, and so on. When each mice runs towards the mouse three places away, these pursuit curves appear:
v = 4 + 1 internally, p = +3
Here are some more:
v = 4 + i1, p = +5
v = 4 + i2, p = +1
v = 4 + i2, p = +2
So far, all the mice have eventually run to the centre of the square, but that doesn’t happen here:
v = 4 + i2, p = 4
Here are more pursuit curves for the v4+i2 mice, using an animated gif:
v = 4 + i2, p = various (animated — open in new tab for clearer image)
And here are more pursuit curves that don’t end in the centre of the square:
v = 4 + i4, p = 4
v = 4 + i4, p = 8
v = 4 + i4, p = 12
v = 4 + i4, p = 16
But the v4+i4 pursuit curves more usually look like this:
v = 4 + i4, p = 7
Now try adapting the rules so that mice don’t run directly towards another mouse, but towards the point midway between two other mice. In this square, the odd- and even-numbered mice follow different rules. Mice #1, #3, #5 and #7 run towards the point midway between the mice one and two places away, while ice #2, #4, #6 and #8 run towards the point midway between the mice two and seven places away:
v = 4 + s1, p(1,3,5,7) = 1,2, p(2,4,6,8) = 2,7
I think the curves are very elegant. Here’s a slight variation:
v = 4 + s1, p1 = 1,3, p2 = 2,7
Now try solid curves:
v = 4 + s1, p1 = 1,3, p2 = 2,7 (red)
v = 4 + s1, p1 = 1,3, p2 = 2,7 (yellow-and-blue)
And some variants:
v = 4 + s1, p1 = 1,7, p2 = 1,2
v = 4 + s1, p1 = 2,3, p2 = 2,5
v = 4 + s1, p1 = 5,6, p2 = 1,3
v = 4 + s1, p1 = 5,6, p2 = 1,4
v = 4 + s1, p1 = 5,6, p2 = 1,6
Elsewhere other-posted:
Currently listening…
• Pigiz Ligiz, Pigs and Grapes (2003)
• Jag Rote Kill, West by West (1972)
• Ziel Lovkopf, Wir Dulder (1980)
• Louve (+), Tb Rehearsal Tapes (1993)
• Hord Voe, Nord/Sud (1966)
• Ozark Swamphony, Sonic Remedies (1960)
• Blutfloh, Die Zauberflohte (2000)
• Zwoir, Oromig (1996)
• Flitwick Youth, Six Sieves (1989)
• Iuscaic, L2-B3/J7 (1995)
• Tiertochter, Elmsfeuer EP (2005)
• Eothorn, Duchess Esmeralda (1973)
• E.F. Dall’Abaco, 12 Concerti (1972)
• Jamie Hendrix XPRNS, Mosaïk (1996)
Previously pre-posted:
Toxic Turntable #1 • #2 • #3 • #4 • #5 • #6 • #7 • #8 • #9 • #10 • #11 • #12 • #13 •
“Alice and the Caterpillar” by John Tenniel (1820-1914), from Lewis Carroll’s Alice in Wonderland (1865)
Ratschläge einer Raupe is one possible German translation of “Advice from a Caterpillar”, which is the title of chapter five of Alice in Wonderland. But the drawing above doesn’t need a translation. John Tenniel and Lewis Carroll were a classic combination, like Quentin Blake and J.P. Martin or Thomas Henry and Richmal Crompton. Tenniel drew fantastic things in a matter-of-fact way, which was just right.
But that makes me wonder about Ratschläge einer Raupe. In German, Rat-schlag means “piece of advice” and Ratschläge is the plural. At first glance, the title is more fun in German: it alliterates and trips off the the tongue in a way the English doesn’t. And Schlag literally means “blow, stroke”, which captures the behaviour of the caterpillar well. Like many of the characters Alice encounters in Wonderland, he is a prickly and aggressive interlocutor. “Advice from a Caterpillar” is plain by comparison.
So perhaps that makes it better: it’s a matter-of-fact title for a surreal chapter. Tenniel’s art echoes that.
Pre-previously on Overlord-in-terms-of-Core-Issues-around-Maximal-Engagement-with-Key-Notions-of-the-Über-Feral, I interrogated issues around this shape, the horned triangle:
Horned Triangle (more details)
Now I want to look at the tricorn (from Latin tri-, “three”, + -corn, “horn”). It’s like a horned triangle, but has three horns instead of one:
Tricorn, or three-horned triangle
These are the stages that make up the tricorn:
Tricorn (stages)
Tricorn (animated)
And there’s no need to stop at triangles. Here is a four-horned square, or quadricorn:
Quadricorn
Quadricorn (animated)
Quadricorn (coloured)
And a five-horned pentagon, or quinticorn:
Quinticorn, or five-horned pentagon
Quinticorn (anim)
Quinticorn (col)
And below are some variants on the shapes above. First, the reversed tricorn:
Reversed Tricorn
Reversed Tricorn (anim)
Reversed Tricorn (col)
The nested tricorn:
Nested Tricorn (anim)
Nested Tricorn (col)
Nested Tricorn (red-green)
Nested Tricorn (variant col)
The nested quadricorn:
Nested Quadricorn (anim)
Nested Quadricorn
Nested Quadricorn (col #1)
Nested Quadricorn (col #2)
Finally (and ferally), the pentagonal octopus or pentapus:
Pentapus (anim)
Pentapus
Pentapus #2
Pentapus #3
Pentapus #4
Pentapus #5
Pentapus #6
Pentapus (col anim)
Elsewhere other-engageable:
• The Art Grows Onda — the horned triangle and Katsushika Hokusai’s painting The Great Wave off Kanagawa (c. 1830)
Papyrocentric Performativity Presents:
• Bullets and Butterflies – Mad Dog Killers: The Story of a Congo Mercenary, Ivan Smith (Helion / 30° South Publishers 2012)
• Jaundiced on George – George Orwell: English Rebel, Robert Colls (Oxford University Press 2013)
• Crabsody in View – RSPB Handbook of the Seashore, Maya Plass (Bloomsbury 2013)
Or Read a Review at Random: RaRaR
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 