The Overlord of the Über-Feral says: Welcome to my bijou bloguette. You can scroll down to sample more or simply:
• Read a Writerization at Random: RaWaR
• ¿And What Doth It Mean To Be Flesh?
Gweel & Other Alterities – Incunabula’s new edition
Tales of Silence & Sortilege – Incunabula’s new edition
If you’d like to donate to O.o.t.Ü.-F., please click here.
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:
Divide an equilateral triangle into four, remove the central triangle, do the same to the new triangles.
Here is method #2:
Pick a corner at random, jump half-way towards it, mark the spot, repeat.
And here is method #3:
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.
The two most powerful drugs in the world are DHMO and mathematics.
To create a simple fractal, take an equilateral triangle and divide it into four more equilateral triangles. Remove the middle triangle. Repeat the process with each new triangle and go on repeating it. You’ll end up with a shape like this, which is known as the Sierpiński triangle, after the Polish mathematician Wacław Sierpiński (1882-1969):
But you can also create the Sierpiński triangle one pixel at a time. Choose any point inside an equilateral triangle. Pick a corner of the triangle at random and move half-way towards it. Mark this spot. Then pick a corner at random again and move half-way towards the corner. And repeat. The result looks like this:
A simple program to create the fractal looks like this:
initial()
repeat
fractal()
altervariables()
until falsefunction initial()
v = 3 [v for vertex]
r = 500
lm = 0.5
endfuncfunction fractal()
th = 2 * pi / v
[the following loop creates the corners of the triangle]
for l = 1 to v
x[l]=xcenter + sin(l*th) * r
y[l]=ycenter + cos(l*th) * r
next l
fx = xcenter
fy = ycenter
repeat
rv = random(v)
fx = fx + (x[rv]-fx) * lm
fy = fy + (y[rv]-fy) * lm
plot(fx,fy)
until keypressed
endfuncfunction altervariables()
[change v, lm, r etc]
endfunc
In this case, more is less. When v = 4 and the shape is a square, there is no fractal and plot(fx,fy) covers the entire square.
When v = 5 and the shape is a pentagon, this fractal appears:
But v = 4 produces a fractal if a simple change is made in the program. This time, a corner cannot be chosen twice in a row:
function initial()
v = 4
r = 500
lm = 0.5
ci = 1 [i.e, number of iterations since corner previously chosen]
endfuncfunction fractal()
th = 2 * pi / v
for l = 1 to v
x[l]=xcenter + sin(l*th) * r
y[l]=ycenter + cos(l*th) * r
chosen[l]=0
next l
fx = xcenter
fy = ycenter
repeat
repeat
rv = random(v)
until chosen[rv]=0
for l = 1 to v
if chosen[l]>0 then chosen[l] = chosen[l]-1
next l
chosen[rv] = ci
fx = fx + (x[rv]-fx) * lm
fy = fy + (y[rv]-fy) * lm
plot(fx,fy)
until keypressed
endfunc
One can also disallow a corner if the corner next to it has been chosen previously, adjust the size of the movement towards the chosen corner, add a central point to the polygon, and so on. Here are more fractals created with such variations:
(This is a guest-review by Norman Foreman, B.A.)
Yr Wylan Ddu, Simon Whitechapel (Papyrocentric Press, ?)
If, like me, you froth at the mouth and roll on the floor biting the carpet when you hear the phrase “Pre-order now”, then relief is at hand. You might have thought that “pre-ordering now” was as logical as “ordering pre-now”. You were wrong. Here is a book that really can be pre-ordered now, because it doesn’t exist yet. If it ever does exist, it will cease to be pre-orderable now. In the meantime, you’re pre-ordering it whether you know it or not. In fact, the less you know, the more you’re pre-ordering it. All life-forms in the Universe, actual and otherwise, are pre-ordering it at this very moment, from the humblest virus to the mightiest hive-mind.
Yr Wylan Ddu (2003) by Slow Exploding Gulls
There’s no escape, in other words. And no more review, you might think, given that the book doesn’t exist yet. True, but I can review the title. It’s Welsh, it means “The Black Gull”, and it’s pronounced something like “Ear Will An Thee”. It was also originally the title of an album in 2003 by the Exeter electronistas Slow Exploding Gulls. Whether S.E.G. will object to the appropriation remains to be seen. If they do, it can be pointed out that Dirgelwch Yr Wylan Ddu, or Secret of the Black Gull, was the title of a children’s book by Idwal Jones (1890-1964) published in 1978.
Idwal Jones’ Secret of the Black Gull (1978)
There is nothing corresponding to “of” in the original title of that book, but then Welsh grammar doesn’t work like that. Yr Wylan Ddu contains some good examples of how it does work. It’s an active, almost clockwork or organic, phrase compared to its static English equivalent. In isolation, the Welsh words for “the”, “black” and “gull” would be y, du, and gwylan, pronounced something like “ee”, “dee” and “goo-ill-an” in southern Welsh. But put them together and they mutate in more ways than one: Yr Wylan Ddu (adjectives generally follow the noun in Welsh). The similarity between gwylan and “gull” isn’t a coincidence: the English word is borrowed from Celtic.
However, it is unlikely that Yr Wylan Ddu will actually be written in Welsh or any other Celtic language. First, Whitechapel doubtless feels that this would reduce his already small audience. Second, he doesn’t speak Welsh. Or write it. So the book will probably follow past trends and be written in English. It’s also safe to predict that it will refer to at least one black gull. So: pre-order now. And please carry on doing so until further notice.
“One of mighty union-smashing Maggie’s few big mistakes – along with increasing comprehensive education, letting third-world immigration and enforced multiculturalism rip, leaving the NHS and BBC ‘safe in our hands’, smashing the fisheries, selling out the Northern Irish Protestants, increasing welfarism, ending academic freedom and trying to push through the Poll Tax – was to be unfriendly to German reunification.” — Chris Brand, gFactor.
“Homosexual men are nature’s Petri dishes.” — Greg Cochran, West Hunter.
Portrait of Marguerite Kelsey (1928) by Meredith Frampton (1894-1984).
Previously pre-posted (please peruse):
“The IAE [International Art English] of the French press release is almost too perfect: It is written, we can only imagine, by French interns imitating American interns imitating American academics imitating French academics.” — “International Art English”, Alix Rule and David Levine
If you like minimalism, you should like binary. There is unsurpassable simplicity and elegance in the idea that any number can be reduced to a series of 1’s and 0’s. It’s unsurpassable because you can’t get any simpler: unless you use finger-counting, two symbols are the minimum possible. But with those two – a stark 1 and 0, true and false, yin and yang, sun and moon, black and white – you can conquer any number you please. 2 = 10[2]. 5 = 101. 100 = 1100100. 666 = 1010011010. 2013 = 11111011101. 9^9 = 387420489 = 10111000101111001000101001001. You can also perform any mathematics you please, from counting sheep to modelling the evolution of the universe.
1 + 0 = ∞
But one disadvantage of binary, from the human point of view, is that numbers get long quickly: every doubling in size adds an extra digit. You can overcome that disadvantage using octal or hexadecimal, which compress blocks of binary into single digits, but those number systems need more symbols: eight and sixteen, as their names suggest. There’s an elegance there too, but binary goes masked, hiding its minimalist appeal beneath apparent complexity. It doesn’t need to wear a mask for computers, but human beings can appreciate bare binary too, even with our weak memories and easily tiring nervous systems. I especially like minimalist binary when it’s put to work on those most maximalist of numbers: the primes. You can compare integers, or whole numbers, to minerals. Some are like mica or shale, breaking readily into smaller parts, but primes are like granite or some other ultra-hard, resistant rock. In other words, some integers are easy to divide by other integers and some, like the primes, are not. Compare 256 with 257. 256 = 2^8, so it’s divisible by 128, 64, 32, 16, 8, 4, 2 and 1. 257 is a prime, so it’s divisible by nothing but itself and 1. Powers of two are easy to calculate and, in binary, very easy to represent:
2^0 = 1 = 1
2^1 = 2 = 10[2]
2^2 = 4 = 100
2^3 = 8 = 1000
2^4 = 16 = 10000
2^5 = 32 = 100000
2^6 = 64 = 1000000
2^7 = 128 = 10000000
2^8 = 256 = 100000000
Primes are the opposite: hard to calculate and usually hard to represent, whatever the base:
02 = 000010[2]
03 = 000011
05 = 000101
07 = 000111
11 = 001011
13 = 001101
17 = 010001
19 = 010011
23 = 010111
29 = 011101
31 = 011111
37 = 100101
41 = 101001
43 = 101011
Maximalist numbers, minimalist base: it’s a potent combination. But “brimes”, or binary primes, nearly all have one thing in common. Apart from 2, a special case, each brime must begin and end with 1. For the digits in-between, the God of Mathematics seems to be tossing a coin, putting 1 for heads, 0 for tails. But sometimes the coin will come up all heads or all tails: 127 = 1111111[2] and 257 = 100000001, for example. Brimes like that have a stark simplicity amid the jumble of 83 = 1010011[2], 113 = 1110001, 239 = 11101111, 251 = 11111011, 277 = 100010101, and so on. Brimes like 127 and 257 are also palindromes, or the same reading in both directions. But less simple brimes can be palindromes too:
73 = 1001001
107 = 1101011
313 = 100111001
443 = 110111011
1193 = 10010101001
1453 = 10110101101
1571 = 11000100011
1619 = 11001010011
1787 = 11011111011
1831 = 11100100111
1879 = 11101010111
But, whether they’re palindromes or not, all brimes except 2 begin and end with 1, so they can be represented as rings, like this:
Those twelve bits, or binary digits, actually represent the thirteen bits of 5227 = 1,010,001,101,011. Start at twelve o’clock (digit 1 of the prime) and count clockwise, adding 1’s and 0’s till you reach 12 o’clock again and add the final 1. Then you’ve clocked around the rock and created the granite of 5227, which can’t be divided by any integers but itself and 1. Another way to see the brime-ring is as an Ouroboros (pronounced “or-ROB-or-us”), a serpent or dragon biting its own tail, like this:
Alchemical Ouroboros (1478)
Another alchemical Ouroboros (1599)
But you don’t have to start clocking around the rock at midday or midnight. Take the Ouroboprime of 5227 and start at eleven o’clock (digit 12 of the prime), adding 1’s and 0’s as you move clockwise. When you’ve clocked around the rock, you’ll have created the granite of 6709, another prime:
Other Ouroboprimes produce brimes both clockwise and anti-clockwise, like 47 = 101,111.
Clockwise
101,111 = 47
111,011 = 59
111,101 = 61
Anti-Clockwise
111,101 = 61
111,011 = 59
101,111 = 47
If you demand the clock-rocked brime produce distinct primes, you sometimes get more in one direction than the other. Here is 151 = 10,010,111:
Clockwise
10,010,111 = 151
11,100,101 = 229
Anti-Clockwise
11,101,001 = 233
11,010,011 = 211
10,100,111 = 167
10,011,101 = 157
The most productive brime I’ve discovered so far is 2,326,439 = 1,000,110,111,111,110,100,111[2], which produces fifteen distinct primes:
Clockwise (7 brimes)
1,000,110,111,111,110,100,111 = 2326439
1,100,011,011,111,111,010,011 = 3260371
1,110,100,111,000,110,111,111 = 3830207
1,111,101,001,110,001,101,111 = 4103279
1,111,110,100,111,000,110,111 = 4148791
1,111,111,010,011,100,011,011 = 4171547
1,101,111,111,101,001,110,001 = 3668593
Anti-Clockwise (8 brimes)
1,110,010,111,111,110,110,001 = 3768241
1,100,101,111,111,101,100,011 = 3342179
1,111,111,011,000,111,001,011 = 4174283
1,111,110,110,001,110,010,111 = 4154263
1,111,101,100,011,100,101,111 = 4114223
1,111,011,000,111,001,011,111 = 4034143
1,110,110,001,110,010,111,111 = 3873983
1,000,111,001,011,111,111,011 = 2332667
Appendix: Deciminimalist Primes
Some primes in base ten use only the two most basic symbols too. That is, primes like 11[10], 101[10], 10111[10] and 1011001[10] are composed of only 1’s and 0’s. Furthermore, when these numbers are read as binary instead, they are still prime: 11[2] = 3, 101[2] = 5, 10111[2] = 23 and 1011001[2] = 89. Here is an incomplete list of these deciminimalist primes:
11[10] = 1,011[2]; 11[2] = 3[10] is also prime.
101[10] = 1,100,101[2]; 101[2] = 5[10] is also prime.
10,111[10] = 10,011,101,111,111[2]; 10,111[2] = 23[10] is also prime.
101,111[10] = 11,000,101,011,110,111[2]; 101,111[2] = 47[10] is also prime.
1,011,001[10] = 11,110,110,110,100,111,001[2]; 1,011,001[2] = 89[10] is also prime.
1,100,101[10] = 100,001,100,100,101,000,101[2]; 1,100,101[2] = 101[10] is also prime.
10,010,101[10] = 100,110,001,011,110,111,110,101[2]; 10,010,101[2] = 149[10] is also prime.
10,011,101[10] = 100,110,001,100,000,111,011,101[2]; 10,011,101[2] = 157[10] is also prime.
10,100,011[10] = 100,110,100,001,110,100,101,011[2]; 10,100,011[2] = 163[10] is also prime.
10,101,101[10] = 100,110,100,010,000,101,101,101[2]; 10,101,101[2] = 173[10] is also prime.
10,110,011[10] = 100,110,100,100,010,000,111,011[2]; 10,110,011[2] = 179[10] is also prime.
10,111,001[10] = 100,110,100,100,100,000,011,001[2].
11,000,111[10] = 101,001,111,101,100,100,101,111[2]; 11,000,111[2] = 199[10] is also prime.
11,100,101[10] = 101,010,010,101,111,111,000,101[2]; 11,100,101[2] = 229[10] is also prime.
11,110,111[10] = 101,010,011,000,011,011,011,111[2].
11,111,101[10] = 101,010,011,000,101,010,111,101[2].
100,011,001[10] = 101,111,101,100,000,101,111,111,001[2]; 100,011,001[2] = 281[10] is also prime.
100,100,111[10] = 101,111,101,110,110,100,000,001,111[2].
100,111,001[10] = 101,111,101,111,001,001,010,011,001[2]; 100,111,001[2] = 313[10] is also prime.
101,001,001[10] = 110,000,001,010,010,011,100,101,001[2].
101,001,011[10] = 110,000,001,010,010,011,100,110,011[2]; 101,001,011[2] = 331[10] is also prime.
101,001,101[10] = 110,000,001,010,010,011,110,001,101[2].
101,100,011[10] = 110,000,001,101,010,100,111,101,011[2].
101,101,001[10] = 110,000,001,101,010,110,111,001,001[2].
101,101,111[10] = 110,000,001,101,010,111,000,110,111[2]; 101,101,111[2] = 367[10] is also prime.
101,110,111[10] = 110,000,001,101,101,000,101,011,111[2].
101,111,011[10] = 110,000,001,101,101,010,011,100,011[2]; 101,111,011[2] = 379[10] is also prime.
101,111,111[10] = 110,000,001,101,101,010,101,000,111[2]; 101,111,111[2] = 383[10] is also prime.
110,010,101[10] = 110,100,011,101,001,111,011,110,101[2].
110,100,101[10] = 110,100,011,111,111,111,010,000,101[2]; 110,100,101[2] = 421[10] is also prime.
110,101,001[10] = 110,100,100,000,000,001,000,001,001[2].
110,110,001[10] = 110,100,100,000,010,010,100,110,001[2]; 110,110,001[2] = 433[10] is also prime.
110,111,011[10] = 110,100,100,000,010,100,100,100,011[2]; 110,111,011[2] = 443[10] is also prime.
Saying “I don’t like maths” is like saying “I don’t eat carbon.”
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 