Lesz is More

• Matematyka jest najpotężniejszym intelektualnym wehikułem, jaki kiedykolwiek został skonstruowany, za pomocą którego uciekamy przed czasem, lecz nie ma powodu przypuszczać, że mogłaby kiedyś umożliwić tego rodzaju ucieczkę, jaką ucieleśnia pogoń za Absolutem. — Leszek Kołakowski

• Mathematics is the most powerful intellectual vehicle that has ever been constructed, by means of which we flee ahead of time, but there is no reason to suppose that it could someday enable the kind of escape embodied by the pursuit of the Absolute. — Leszek Kołakowski

Koch Rock

The Koch snowflake, named after the Swedish mathematician Helge von Koch, is a famous fractal that encloses a finite area within an infinitely long boundary. To make a ’flake, you start with an equilateral triangle:

Koch snowflake stage #1 (with room for manœuvre)

Next, you divide each side in three and erect a smaller equilateral triangle on the middle third, like this:

Koch snowflake #2

Each original straight side of the triangle is now 1/3 longer, so the full perimeter has also increased by 1/3. In other words, perimeter = perimeter * 1⅓. If the perimeter of the equilateral triangle was 3, the perimeter of the nascent Koch snowflake is 4 = 3 * 1⅓. The area of the original triangle also increases by 1/3, because each new equalitarian triangle is 1/9 the size of the original and there are three of them: 1/9 * 3 = 1/3.

Now here’s stage 3 of the snowflake:

Koch snowflake #3, perimeter = 4 * 1⅓ = 5⅓

Again, each straight line on the perimeter has been divided in three and capped with a smaller equilateral triangle. This increases the length of each line by 1/3 and so increases the full perimeter by a third. 4 * 1⅓ = 5⅓. However, the area does not increase by 1/3. There are twelve straight lines in the new perimeter, so twelve new equilateral triangles are erected. However, because their sides are 1/9 as long as the original side of the triangle, they have 1/(9^2) = 1/81 the area of the original triangle. 1/81 * 12 = 4/27 = 0.148…

Koch snowflake #4, perimeter = 7.11

Koch snowflake #5, p = 9.48

Koch snowflake #6, p = 12.64

Koch snowflake #7, p = 16.85

Koch snowflake (animated)

The perimeter of the triangle increases by 1⅓ each time, while the area reaches a fixed limit. And that’s how the Koch snowflake contains a finite area within an infinite boundary. But the Koch snowflake isn’t confined to itself, as it were. In “Dissecting the Diamond”, I described how dissecting and discarding parts of a certain kind of diamond could generate one side of a Koch snowflake. But now I realize that Koch snowflakes are everywhere in the diamond — it’s a Koch rock. To see how, let’s start with the full diamond. It can be divided, or dissected, into five smaller versions of itself:

Dissectable diamond

When the diamond is dissected and three of the sub-diamonds are discarded, two sub-diamonds remain. Let’s call them sub-diamonds 1 and 2. When this dissection-and-discarding is repeated again and again, a familiar shape begins to appear:

Koch rock stage 1

Koch rock #2

Koch rock #3

Koch rock #4

Koch rock #5

Koch rock #6

Koch rock #7

Koch rock #8

Koch rock #9

Koch rock #10

Koch rock #11

Koch rock #12

Koch rock #13

Koch rock (animated)

Dissecting and discarding the diamond creates one side of a Koch triangle. Now see what happens when discarding is delayed and sub-diamonds 1 and 2 are allowed to appear in other parts of the diamond. Here again is the dissectable diamond:

Dia-flake stage 1

If no sub-diamonds are discarded after dissection, the full diamond looks like this when each sub-diamond is dissected in its turn:

Dia-flake #2

Now let’s start discarding sub-diamonds:

Dia-flake #3

And now discard everything but sub-diamonds 1 and 2:

Dia-flake #4

Dia-flake #5

Dia-flake #6

Dia-flake #7

Dia-flake #8

Dia-flake #9

Dia-flake #10

Now full Koch snowflakes have appeared inside the diamond — count ’em! I see seven full ’flakes:

Dia-flake #11

Dia-flake (animated)

But that isn’t the limit. In fact, an infinite number of full ’flakes appear inside the diamond — it truly is a Koch rock. Here are examples of how to find more full ’flakes:

Dia-flake 2 (static)

Dia-flake 2 (animated)

Dia-flake 3 (static)

Dia-flake 3 (animated)

Previously pre-posted:

Dissecting the Diamond — other fractals in the dissectable diamond

Prose Shows

I don’t know about you, but this is exactly what I like to see in the opening paragraph of an essay engaging issues around William S. Burroughs and the cult of rock’n’roll dot dot dot…

Naked Lunch is inseparable from its author William S. Burroughs, which tends to happen with certain major works. The book may be the only Burroughs title many literature buffs can name. In terms of name recognition, Naked Lunch is a bit like Miles Davis’ Kind of Blue, which also arrived in 1959. Radical for its time, Kind of Blue now sounds quaint, though it is undeniably a masterwork. — William S. Burroughs and the Cult of Rock ’n’ Roll, Casey Rae

Did you spot it? Didja?

Previously pre-posted:

The Hum of Heresy
The Conqueror Term
Bill Self

Dissecting the Diamond

Pre-previously on O.o.t.Ü.-F., I dilated the delta. Now I want to dissect the diamond. In geometry, a shape is dissected when it is completely divided into smaller shapes of some kind. If the smaller shapes are identical (except for size) to the original, the original shape is called a rep-tile (because it can be tiled with repeating versions of itself). If the smaller identical shapes are equal in size to each other, the rep-tile is regular; if the smaller shapes are not equal, the rep-tile is irregular. This diamond is an irregular rep-tile or irrep-tile:

Dissectable diamond

Dissected diamond

As you can see, the diamond can be dissected into five smaller versions of itself, two larger ones and three smaller ones. This makes it a rep-5 irrep-tile. And the smaller versions, or sub-diamonds, can themselves be dissected ad infinitum, like this:

Dissected diamond stage #1

Dissected diamond #2

Dissected diamond #3

Dissected diamond #4

Dissected diamond #5

Dissected diamond #6

Dissected diamond #7

Dissected diamond #8

Dissected diamond #9

Dissected diamond (animated)

The full dissected diamond is a fractal, or shape that is similar to itself at varying scales. However, the fractality of the diamond becomes most obvious when you dissect-and-discard. That is, first you dissect the diamond, then you discard one (or more) of the sub-diamonds, like this:

Diamond fractal (retaining sub-diamonds 1,2,3,4) stage #1

1234-Diamond #2

1234-Diamond #3

1234-Diamond #4

1234-Diamond #5

1234-Diamond #6

1234-Diamond #7

1234-Diamond #8

1234-Diamond #9

1234-Diamond (animated)

Here are some more fractals created by dissecting and discarding one sub-diamond:

Diamond fractal (retaining sub-diamonds 1,2,4,5)

1245-Diamond (anim)


2345-Diamond (anim)

The 2345-diamond fractal has variants created by mirroring one or more sub-diamonds, so that the orientation of the sub-dissections changes. Here is one of the variants:

2345-Diamond (variation)

2345-Diamond (variant) (anim)

And here is a fractal created by dissecting and discarding two sub-diamonds:

Diamond fractal (retaining sub-diamonds 1,2,3)

123-Diamond (anim)

Again, the fractal has variants created by mirroring one or more of the sub-diamonds:

123-Diamond (variant #1)

123-Diamond (variant #2)

123-Diamond (variant #3)

123-Diamond (variant #4)

Some more fractals created by dissecting and discarding two sub-diamonds:


125-Diamond (anim)


134-Diamond (anim)


235-Diamond (anim)


135-Diamond (anim)

A variant of the 135-Diamond fractal looks like one side of a Koch snowflake:

135-Diamond (variant #1) — like Koch snowflake

135-Diamond (variant #2)

Finally, here are some colour variants of the full dissected diamond:

Full diamond colour variants (anim)

Elsewhere other-engageable:

Dilating the Delta

Total Score

The number 23 is always (and trivially) equal to some running total of the digits of its roots in base 2. In other bases, that’s not always true (n.b. numbers inside square brackets represent single digits in that base):

√23 = 23^(1/2) = 100.1100101110111011100111010101110111000001000... in base 2
23 = digsum(100.110010111011101110011101010111011)
23^(1/2) = 11.21011101110011111122022101121121... in base 3
23 = digsum(11.2101110111001111112202)
23^(1/2) = 4.8832850[10]89028... in base 11
23 = digsum(4.883)
23^(1/2) = 4.[14]5[15]53[14]0[12]0[14]5[13]... in base 18
23 = digsum(4.[14]5)
23^(1/2) = 4.[19]29[13][19]4[11][23][19][11][20]... in base 24
23 = digsum(4.[19])
23^(1/2) = 4.[19][22]9[21][17]5[12][10]456... in base 25
23 = digsum(4.[19])

23^(1/3) = 10.11011000000001111010101010011000101000110000001100000010010000101011... in base 2
23 = digsum(10.1101100000000111101010101001100010100011000000110000001001)
23^(1/3) = 2.21121001121111121022212100220... in base 3
23 = digsum(2.2112100112111112102)
23^(1/3) = 2.312000132222212022030003... in base 4
23 = digsum(2.31200013222221)
23^(1/3) = 2.6600365246121403... in base 8
23 = digsum(2.660036)
23^(1/3) = 2.753154453877080... in base 9
23 = digsum(2.75315)
23^(1/3) = 2.93120691571[10]001[10]... in base 11
23 = digsum(2.931206)
23^(1/3) = 2.[12]9[13]0[11]74[11]61[14]2... in base 15
23 = digsum(2.[12]9)
23^(1/3) = 2.[13]807[10][10]98[10]303... in base 16
23 = digsum(2.[13]8)
23^(1/3) = 2.[21]2[10][10][13][11][21][23][15][24][21]... in base 25
23 = digsum(2.[21])
23^(1/3) = 2.[21][24][11][20][24][22][23][25]0[11][11]... in base 26
23 = digsum(2.[21])

23^(1/4) = 10.0011000010011111110100101010011000001001011110001110101... in base 2
23 = digsum(10.001100001001111111010010101001100000100101111)
23^(1/4) = 2.1411772251404570... in base 8
23 = digsum(2.141177)
23^(1/4) = 2.1634161832077814... in base 9
23 = digsum(2.163416)
23^(1/4) = 2.33[15]2[14][13]967[10]6[12]5... in base 17
23 = digsum(2.33[15])
23^(1/4) = 2.6[15][19][11][31][17][10][18][21]30[27]... in base 34
23 = digsum(2.6[15])
23^(1/4) = 2.[12]9[63][18][41][32][37][56][58][60]1[17]... in base 64
23 = digsum(2.[12]9)
23^(1/4) = 2.[21]9[26]6[54][21][20]3[64][86][110]... in base 111
23 = digsum(2.[21])
23^(1/4) = 2.[21][30][66][22][73][19]3[15][51][24]8... in base 112
23 = digsum(2.[21])
23^(1/4) = 2.[21][52][36][111][32][104][66][40][95][33]5... in base 113
23 = digsum(2.[21])
23^(1/4) = 2.[21][74][50][62][27]19[100][70][48][89]... in base 114
23 = digsum(2.[21])
23^(1/4) = 2.[21][96][108]2[101][62][43][18][71][113][37]... in base 115
23 = digsum(2.[21])

23^(1/5) = 1.110111110100011010011101000111111011111011000... in base 2
23 = digsum(1.11011111010001101001110100011111101)
23^(1/5) = 1.313310122131013323323010... in base 4
23 = digsum(1.31331012213101)
23^(1/5) = 1.[10]5714140[10][11][11]61... in base 12
23 = digsum(1.[10]57)
23^(1/5) = 1.[11]45210[12]3974[12]0[11]... in base 13
23 = digsum(1.[11]452)
23^(1/5) = 1.[22][17][15]788[12][20][10][16]5... in base 26
23 = digsum(1.[22])

And in base 10:

23^(1/7) = 1.565065607960239...
23 = digsum(1.56506)

23^(1/11) = 1.32982177397055...
23 = digsum(1.3298)

23^(1/25) = 1.133624213096260543...
23 = digsum(1.13362421)

23^(1/43) = 1.075642836327515...
23 = digsum(1.07564)

23^(1/51) = 1.0634095245502272...
23 = digsum(1.063409)

23^(1/59) = 1.054581462032154...
23 = digsum(1.05458)

23^(1/74) = 1.043282031364111825...
23 = digsum(1.04328203)

23^(1/78) = 1.041017545329593513...
23 = digsum(1.04101754)

23^(1/81) = 1.039468791371841...
23 = digsum(1.03946)

23^(1/85) = 1.037576979258809...
23 = digsum(1.03757)

23^(1/86) = 1.0371320245405187874...
23 = digsum(1.037132024)

23^(1/101) = 1.031531403111493041428...
23 = digsum(1.03153140311)

Dilating the Delta

A circle with a radius of one unit has an area of exactly π units = 3.141592… units. An equilateral triangle inscribed in the unit circle has an area of 1.2990381… units, or 41.34% of the area of the unit circle.

In other words, triangles are cramped! And so it’s often difficult to see what’s going on in a triangle. Here’s one example, a fractal that starts by finding the centre of the equilateral triangle:

Triangular fractal stage #1

Next, use that central point to create three more triangles:

Triangular fractal stage #2

And then use the centres of each new triangle to create three more triangles (for a total of nine triangles):

Triangular fractal stage #3

And so on, trebling the number of triangles at each stage:

Triangular fractal stage #4

Triangular fractal stage #5

As you can see, the triangles quickly become very crowded. So do the central points when you stop drawing the triangles:

Triangular fractal stage #6

Triangular fractal stage #7

Triangular fractal stage #8

Triangular fractal stage #9

Triangular fractal stage #10

Triangular fractal stage #11

Triangular fractal stage #12

Triangular fractal stage #13

Triangular fractal (animated)

The cramping inside a triangle is why I decided to dilate the delta like this:

Triangular fractal

Circular fractal from triangular fractal

Triangular fractal to circular fractal (animated)

Formation of the circular fractal (animated)

And how do you dilate the delta, or convert an equilateral triangle into a circle? You use elementary trigonometry to expand the perimeter of the triangle so that it lies on the perimeter of the unit circle. The vertices of the triangle don’t move, because they already lie on the perimeter of the circle, but every other point, p, on the perimeter of the triangles moves outward by a fixed amount, m, depending on the angle it makes with the center of the triangle.

Once you have m, you can move outward every point, p(1..i), that lies between p on the perimeter and the centre of the triangle. At least, that’s the theory between the dilation of the delta. In practice, all you need is a point, (x,y), inside the triangle. From that, you can find the angle, θ, and distance, d, from the centre, calculate m, and move (x,y) to d * m from the centre.

You can apply this technique to any fractal created in an equilateral triangle. For example, here’s the famous Sierpiński triangle in its standard form as a delta, then as a dilated delta or circle:

Sierpiński triangle

Sierpiński triangle to circular Sierpiński fractal

Sierpiński triangle to circle (animated)

But why stop at triangles? You can use the same elementary trigonometry to convert any regular polygon into a circle. A square inscribed in a unit circle has an area of 2 units, or 63.66% of the area of the unit circle, so it too is cramped by comparison with the circle. Here’s a square fractal that I’ve often posted before:

Square fractal, jump = 1/2, ban on jumping towards any vertex twice in a row

It’s created by banning a randomly jumping point from moving twice in a row 1/2 of the distance towards the same vertex of the square. When you dilate the fractal, it looks like this:

Circular fractal from square fractal, j = 1/2, ban on jumping towards vertex v(i) twice in a row

Circular fractal from square (animated)

And here’s a related fractal where the randomly jumping point can’t jump towards the vertex directly clockwise from the vertex it’s previously jumped towards (so it can jump towards the same vertex twice or more):

Square fractal, j = 1/2, ban on vertex v(i+1)

When the fractal is dilated, it looks like this:

Circular fractal from square, i = 1

Circular fractal from square (animated)

In this square fractal, the randomly jumping point can’t jump towards the vertex directly opposite the vertex it’s previously jumped towards:

Square fractal, ban on vertex v(i+2)

And here is the dilated version:

Circular fractal from square, i = 2

Circular fractal from square (animated)

And there are a lot more fractals where those came from. Infinitely many, in fact.

Purple Poesy


It is with pleasure that we announce our ability to offer to the public the papers of the Re-Echo Club. This club, somewhat after the order of the Echo Club, late of Boston, takes pleasure in trying to better what is done. On the occasion of the meeting of which the following gems of poesy are the result, the several members of the club engaged to write up the well-known tradition of the Purple Cow in more elaborate form than the quatrain made famous by Mr. Gelett Burgess:

“I NEVER saw a Purple Cow,
I never hope to see one;
But I can tell you, anyhow,
I’d rather see than be one.”



Oh, Cow of rare rapturous vision,
Oh, purple, impalpable Cow,
Do you browse in a Dream Field Elysian,
Are you purpling pleasantly now?
By the side of wan waves do you languish?
Or in the lithe lush of the grove?
While vainly I search in my anguish,
Bovine of mauve!

Despair in my bosom is sighing,
Hope’s star has sunk sadly to rest;
Though cows of rare sorts I am buying,
Not one breathes a balm to my breast.
Oh, rapturous rose-crowned occasion
When I such a glory might see!
But a cow of a purple persuasion
I never would be.

Elsewhere other-engageable:

The Purple Cow Parodies
Diversions of the Re-Echo Club
Such Nonsense! An Anthology (c. 1918) — with this and other parodies

Square Routes Re-Re-Re-Re-Revisited

Pre-previously in my post-passionate portrayal of polygonic performativity, I’ve usually looked at what happens when a moving point is banned from jumping twice-in-a-row (and so on) towards the same vertex of a square or other polygon. But what happens when the point isn’t banned but compelled to do something different? For example, if the point usually jumps 1/2 of the distance towards the vertex for the second (third, fourth…) time, you could make it jump 2/3 of the way, like this:

usual jump = 1/2, forced jump = 2/3

And here are the fractals created when the vertex currently chosen is one or two places clockwise from the vertex chosen before:

usual jump = 1/2, forced jump = 2/3, vertex-inc = +1

j1 = 1/2, j2 = 2/3, vi = +2

Or you can make the point jump towards a different vertex to the one chosen, without recording the different vertex in the history of jumps:

v1 = +0, v2 = +1, j = 1/2

v1 = +0, v2 = +1, vi = +2

v1 = 0, v2 = +2

v1 = 0, v2 = +2, vi = +1

Or you can make the point jump towards the center of the square:

v1 = 0, v2 = center, j = 1/2

v1 = 0, v2 = center, vertex-inc = +1

v1 = 0, v2 = center, vertex-inc = +2

And so on:

v1 = +1, v2 = +1, vi = +1

v1 = +1, v2 = +1, vi = +2

v1 = +0, v2 = +1, reverse test

v1 = +0, v2 = +1, vi = +1, reverse test

v1 = +0, v2 = +1, vi = +2, reverse test

v1 = +0, v2 = +2, reverse test

v1 = +0, v2 = +2, vi = +1, reverse test

v1 = +2, v2 = +2, vi = +1, reverse test

j1 = 1/2, j2 = 2/3, vi = +0,+0 (record previous two jumps in history)

j1 = 1/2, j2 = 2/3, vi = +0,+2

j1 = 1/2, j2 = 2/3, vi = +2,+2

j1 = 1/2, j2 = 2/3, vi = +0,+0,+0 (previous three jumps)

Previously pre-posted (please peruse):

Square Routes
Square Routes Revisited
Square Routes Re-Revisited
Square Routes Re-Re-Revisited
Square Routes Re-Re-Re-Revisited

Lost Lustre

Adonis, M. Cytheris, and M. Menelaus, is indescribable; the eyes are pained as they gaze upon it; yet there is said to be an unnamed species from the emerald mountains of Bogota, of which a single specimen is in a private cabinet in London, which is far more lustrous than these.” — The Romance of Natural History (1861), Philip Henry Gosse