We Can Circ It Out

It’s a pretty little problem to convert this triangular fractal…

Sierpiński triangle (Wikipedia)


…into its circular equivalent:

Sierpiński triangle as circle


Sierpiński triangle to circle (animated)


But once you’ve circ’d it out, as it were, you can easily adapt the technique to fractals based on other polygons:

T-square fractal (Wikipedia)

T-square fractal as circle


T-square fractal to circle (animated)


Elsewhere other-accessible…

Dilating the Delta — more on converting polygonic fractals to circles…

The Belles of El

Title page of Sir Henry Billingsley’s first English version of Euclid’s Elements, 1570, with personifications of Geometria, Astronomia, Arithmetica and Musica as beautiful young women


The Elements of Geometrie of the Moſt Aucient Philoſopher Evclide of Megara.

Faithfully (now first) tranʃlated into the Engliʃhe toung, by H. Billingſley, Citizen of London.

Whereunto are annexed certaine Scolies, Annotations, and Inuentions, of the best Mathematiciens, both of times past, and in this our age.

With a very fruitfull Præface made by M.I. Dee, ʃpecifying the chiefe Mathematicall Sciences, what they are, and wherunto commodious: where, alʃo, are diʃcloʃed certaine new Secrets Mathematicall and Mechanicall, untill theʃe our daies, greatly miʃʃed.

Imprinted at London by Iohn Daye.


The title of this incendiary intervention is a paronomasia on “The Bells of Hell…”, a British airmen’s song in terms of core issues around World War I.

Primal Stream

• 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 — A000668, Mersenne primes (primes of the form 2^n – 1), at the Online Encyclopedia of Integer Sequences

• 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933 — A000043, Mersenne exponents: primes p such that 2^p – 1 is prime. Then 2^p – 1 is called a Mersenne prime. […] It is believed (but unproved) that this sequence is infinite. The data suggest that the number of terms up to exponent N is roughly K log N for some constant K.

• The largest known prime number (as of May 2022) is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. — Largest known prime number

Fylfy Fractals

An equilateral triangle is a rep-tile, because it can be tiled completely with smaller copies of itself. Here it is as a rep-4 rep-tile, tiled with four smaller copies of itself:

Equilateral triangle as rep-4 rep-tile


If you divide and discard one of the sub-copies, then carry on dividing-and-discarding with the sub-copies and sub-sub-copies and sub-sub-sub-copies, you get the fractal seen below. Alas, it’s not a very attractive or interesting fractal:

Divide-and-discard fractal stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Stage #9


Divide-and-discard fractal (animated)


You can create more attractive and interesting fractals by rotating the sub-triangles clockwise or anticlockwise. Here are some examples:









Now try dividing a square into four right triangles, then turning each of the four triangles into a divide-and-discard fractal. The resulting four-fractal shape is variously called a swastika, a gammadion, a cross cramponnée, a Hakenkreuz and a fylfot. I’m calling it a fylfy fractal:

Divide-and-discard fractals in the four triangles of a divided square stage #1


Fylfy fractal #2


Fylfy fractal #3


Fylfy fractal #4


Fylfy fractal #5


Fylfy fractal #6


Fylfy fractal #7


Fylfy fractal #8


Fylfy fractal (animated)


Finally, you can adjust the fylfy fractals so that each point in the square becomes the equivalent point in a circle:



















Absolutely Sabulous

The Hourglass Fractal (animated gif optimized at ezGIF)


Performativizing Paronomasticity

The title of this incendiary intervention is a paronomasia on the title of the dire Absolutely Fabulous. The adjective sabulous means “sandy; consisting of or abounding in sand; arenaceous” (OED).

Elsewhere Other-Accessible

Hour Re-Re-Re-Re-Powered — more on the hourglass fractal
Allus Pour, Horic — an earlier paronomasia for the fractal

Game of Throwns

In “Scaffscapes”, I looked at these three fractals and described how they were in a sense the same fractal, even though they looked very different:

Fractal #1


Fractal #2


Fractal #3


But even if they are all the same in some mathematical sense, their different appearances matter in an aesthetic sense. Fractal #1 is unattractive and seems uninteresting:

Fractal #1, unattractive, uninteresting and unnamed


Fractal #3 is attractive and interesting. That’s part of why mathematicians have given it a name, the T-square fractal:

Fractal #3 — the T-square fractal


But fractal #2, although it’s attractive and interesting, doesn’t have a name. It reminds me of a ninja throwing-star or shuriken, so I’ve decided to call it the throwing-star fractal or ninja-star fractal:

Fractal #2, the throwing-star fractal


A ninja throwing-star or shuriken


This is one way to construct a throwing-star fractal:

Throwing-star fractal, stage 1


Throwing-star fractal, #2


Throwing-star fractal, #3


Throwing-star fractal, #4


Throwing-star fractal, #5


Throwing-star fractal, #6


Throwing-star fractal, #7


Throwing-star fractal, #8


Throwing-star fractal, #9


Throwing-star fractal, #10


Throwing-star fractal, #11


Throwing-star fractal (animated)


But there’s another way to construct a throwing-star fractal. You use what’s called the chaos game. To understand the commonest form of the chaos game, imagine a ninja inside an equilateral triangle throwing a shuriken again and again halfway towards a randomly chosen vertex of the triangle. If you mark each point where the shuriken lands, you eventually get a fractal called the Sierpiński triangle:

Chaos game with triangle stage 1


Chaos triangle #2


Chaos triangle #3


Chaos triangle #4


Chaos triangle #5


Chaos triangle #6


Chaos triangle #7


Chaos triangle (animated)


When you try the chaos game with a square, with the ninja throwing the shuriken again and again halfway towards a randomly chosen vertex, you don’t get a fractal. The interior of the square just fills more or less evenly with points:

Chaos game with square, stage 1


Chaos square #2


Chaos square #3


Chaos square #4


Chaos square #5


Chaos square #6


Chaos square (anim)


But suppose you restrict the ninja’s throws in some way. If he can’t throw twice or more in a row towards the same vertex, you get a familiar fractal:

Chaos game with square, ban on throwing towards same vertex, stage 1


Chaos square, ban = v+0, #2


Chaos square, ban = v+0, #3


Chaos square, ban = v+0, #4


Chaos square, ban = v+0, #5


Chaos square, ban = v+0, #6


Chaos square, ban = v+0 (anim)


But what if the ninja can’t throw the shuriken towards the vertex one place anti-clockwise of the vertex he’s just thrown it towards? Then you get another familiar fractal — the throwing-star fractal:

Chaos square, ban = v+1, stage 1


Chaos square, ban = v+1, #2


Chaos square, ban = v+1, #3


Chaos square, ban = v+1, #4


Chaos square, ban = v+1, #5


Game of Throwns — throwing-star fractal from chaos game (static)


Game of Throwns — throwing-star fractal from chaos game (anim)


And what if the ninja can’t throw towards the vertex two places anti-clockwise (or two places clockwise) of the vertex he’s just thrown the shuriken towards? Then you get a third familiar fractal — the T-square fractal:

Chaos square, ban = v+2, stage 1


Chaos square, ban = v+2, #2


Chaos square, ban = v+2, #3


Chaos square, ban = v+2, #4


Chaos square, ban = v+2, #5


T-square fractal from chaos game (static)


T-square fractal from chaos game (anim)


Finally, what if the ninja can’t throw towards the vertex three places anti-clockwise, or one place clockwise, of the vertex he’s just thrown the shuriken towards? If you can guess what happens, your mathematical intuition is much better than mine.


Post-Performative Post-Scriptum

I am not now and never have been a fan of George R.R. Martin. He may be a good author but I’ve always suspected otherwise, so I’ve never read any of his books or seen any of the TV adaptations.

Scaffscapes

A fractal is a shape that contains copies of itself on smaller and smaller scales. You can find fractals everywhere in nature. Part of a fern looks like the fern as a whole:

Fern as fractal (source)


Part of a tree looks like the tree as a whole:

Tree as fractal (source)


Part of a landscape looks like the landscape as a whole:

Landscape as fractal (source)


You can also create fractals for yourself. Here are three that I’ve constructed:

Fractal #1


Fractal #2


Fractal #3 — the T-square fractal


The three fractals look very different and, in one sense, that’s exactly what they are. But in another sense, they’re the same fractal. Each can morph into the other two:

Fractal #1 → fractal #2 → fractal #3 (animated)


Here are two more fractals taken en route from fractal #2 to fractal #3, as it were:

Fractal #4


Fractal #5


To understand how the fractals belong together, you have to see what might be called the scaffolding. The construction of fractal #3 is the easiest to understand. First you put up the scaffolding, then you take it away and leave the final fractal:

Fractal #3, scaffolding stage 1


Fractal #3, stage 2


Fractal #3, stage 3


Fractal #3, stage 4


Fractal #3, stage 5


Fractal #3, stage 6


Fractal #3, stage 7


Fractal #3, stage 8


Fractal #3, stage 9


Fractal #3, stage 10


Fractal #3 (scaffolding removed)


Construction of fractal #3 (animated)


Now here’s the construction of fractal #1:

Fractal #1, stage 1


Fractal #1, stage 2


Fractal #1, stage 3

Construction of fractal #1 (animated)


Fractal #1 (static)


And the constructions of fractals #2, #4 and #5:

Fractal #2, stage 1


Fractal #2, stage 2


Fractal #2, stage 3

Fractal #2 (animated)


Fractal #2 (static)


Fractal #4, stage 1


Fractal #4, stage 2


Fractal #4, stage 3

Fractal #4 (animated)


Fractal #4 (static)


Fractal #5, stage 1


Fractal #5, stage 2


Fractal #5, stage 3

Fractal #5 (animated)


Fractal #5