Fract-Hills

The Farey sequence is a fascinating sequence of fractions that divides the interval between 0/1 and 1/1 into smaller and smaller parts. To find the Farey fraction a[i] / b[i], you simply find the mediant of the Farey fractions on either side:

• a[i] / b[i] = (a[i-1] + a[i+1]) / (b[i-1] + b[i+1])

Then, if necessary, you reduce the numerator and denominator to their simplest possible terms. So the sequence starts like this:

• 0/1, 1/1

To create the next stage, find the mediant of the two fractions above: (0+1) / (1+1) = 1/2

• 0/1, 1/2, 1/1

For the next stage, there are two mediants to find: (0+1) / (1+2) = 1/3, (1+1) / (2+3) = 2/3

• 0/1, 1/3, 1/2, 2/3, 1/1

Note that 1/2 is the mediant of 1/3 and 2/3, that is, 1/2 = (1+2) / (3+3) = 3/6 = 1/2. The next stage is this:

• 0/1, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1/1

Now 1/2 is the mediant of 2/5 and 3/5, that is, 1/2 = (2+3) / (5+5) = 5/10 = 1/2. Further stages go like this:

• 0/1, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 1/1

• 0/1, 1/6, 1/5, 2/9, 1/4, 3/11, 2/7, 3/10, 1/3, 4/11, 3/8, 5/13, 2/5, 5/12, 3/7, 4/9, 1/2, 5/9, 4/7, 7/12, 3/5, 8/13, 5/8, 7/11, 2/3, 7/10, 5/7, 8/11, 3/4, 7/9, 4/5, 5/6, 1/1

• 0/1, 1/7, 1/6, 2/11, 1/5, 3/14, 2/9, 3/13, 1/4, 4/15, 3/11, 5/18, 2/7, 5/17, 3/10, 4/13, 1/3, 5/14, 4/11, 7/19, 3/8, 8/21, 5/13, 7/18, 2/5, 7/17, 5/12, 8/19, 3/7, 7/16, 4/9, 5/11, 1/2, 6/11, 5/9, 9/16, 4/7, 11/19, 7/12, 10/17, 3/5, 11/18, 8/13, 13/21, 5/8, 12/19, 7/11, 9/14, 2/3, 9/13, 7/10, 12/17, 5/7, 13/18, 8/11, 11/15, 3/4, 10/13, 7/9, 11/14, 4/5, 9/11, 5/6, 6/7, 1/1

The Farey sequence is actually a fractal, as you can see more easily when it’s represented as an image:

Farey fractal stage #1, representing 0/1, 1/2, 1/1

Farey fractal stage #2, representing 0/1, 1/3, 1/2, 2/3, 1/1

Farey fractal stage #3, representing 0/1, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1/1

Farey fractal stage #4, representing 0/1, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 1/1

Farey fractal stage #5

Farey fractal stage #6

Farey fractal stage #7

Farey fractal stage #8

Farey fractal stage #9

Farey fractal stage #10

Farey fractal (animated)

That looks like the slope of a hill to me, so you could call it a Farey fract-hill. But Farey fract-hills or Farey fractals aren’t confined to the unit interval, 0/1 to 1/1. Here are Farey fractals for the intervals 0/1 to n/1, n = 1..10:

Farey fractal for interval 0/1 to 1/1

Farey fractal for interval 0/1 to 2/1, beginning 0/1, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1/1, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 2/1

Farey fractal for interval 0/1 to 3/1, beginning 0/1, 1/3, 1/2, 2/3, 1/1, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 2/1, 7/3, 5/2, 8/3, 3/1

Farey fractal for interval 0/1 to 4/1, beginning
0/1, 1/3, 1/2, 2/3, 1/1, 4/3, 3/2, 5/3, 2/1, 7/3, 5/2, 8/3, 3/1, 10/3, 7/2, 11/3, 4/1

Farey fractal for interval 0/1 to 5/1, beginning 0/1, 1/1, 5/4, 10/7, 5/3, 7/4, 2/1, 7/3, 5/2, 8/3, 3/1, 13/4, 10/3, 25/7, 15/4, 4/1, 5/1

Farey fractal for interval 0/1 to 6/1, beginning 0/1, 1/2, 1/1, 4/3, 3/2, 5/3, 2/1, 5/2, 3/1, 7/2, 4/1, 13/3, 9/2, 14/3, 5/1, 11/2, 6/1

Farey fractal for interval 0/1 to 7/1, beginning 0/1, 7/5, 7/4, 2/1, 7/3, 21/8, 14/5, 3/1, 7/2, 4/1, 21/5, 35/8, 14/3, 5/1, 21/4, 28/5, 7/1

Farey fractal for interval 0/1 to 8/1, beginning 0/1, 1/2, 1/1, 3/2, 2/1, 5/2, 3/1, 7/2, 4/1, 9/2, 5/1, 11/2, 6/1, 13/2, 7/1, 15/2, 8/1

Farey fractal for interval 0/1 to 9/1, beginning 0/1, 1/1, 3/2, 2/1, 3/1, 7/2, 4/1, 13/3, 9/2, 14/3, 5/1, 11/2, 6/1, 7/1, 15/2, 8/1, 9/1

Farey fractal for interval 0/1 to 10/1, beginning 0/1, 5/4, 5/3, 2/1, 5/2, 3/1, 10/3, 15/4, 5/1, 25/4, 20/3, 7/1, 15/2, 8/1, 25/3, 35/4, 10/1

The shape of the slope is determined by the factorization of n:

n = 12 = 2^2 * 3

n = 16 = 2^4

n = 18 = 2 * 3^2

n = 20 = 2^2 * 5

n = 25 = 5^2

n = 27 = 3^3

n = 32 = 2^5

n = 33 = 3 * 11

n = 42 = 2 * 3 * 7

n = 64 = 2^6

n = 65 = 5 * 13

n = 70 = 2 * 5 * 7

n = 77 = 7 * 11

n = 81 = 3^4

n = 96 = 2^5 * 3

n = 99 = 3^2 * 11

n = 100 = 2^2 * 5^2

Farey fractal-hills, n = various

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Jumping Jehosophracts!

As I’ve shown pre-previously on Overlord-in-terms-of-issues-around-the-Über-Feral, you can create interesting fractals by placing restrictions on a point jumping inside a fractal towards a randomly chosen vertex. For example, the point can be banned from jumping towards the same vertex twice in a row, and so on.

But you can use other restrictions. For example, suppose that the point can jump only once or twice towards any vertex, that is, (j = 1,2). It can then jump towards the same vertex again, but not the same number of times as it previously jumped. So if it jumps once, it has to jump twice next time; and vice versa. If you use this rule on a pentagon, this fractal appears:

v = 5, j = 1,2 (black-and-white)


v = 5, j = 1,2 (colour)


If the point can also jump towards the centre of the pentagon, this fractal appears:

v = 5, j = 1,2 (with centre)


And if the point can also jump towards the midpoints of the sides:

v = 5, j = 1,2 (with midpoints)


v = 5, j = 1,2 (with midpoints and centre)


And here the point can jump 1, 2 or 3 times, but not once in a row, twice in a row or thrice in a row:

v = 5, j = 1,2,3


v = 5, j = 1,2,3 (with centre)


Here the point remembers its previous two moves, rather than just its previous move:

v = 5, j = 1,2,3, hist = 2 (black-and-white)


v = 5, j = 1,2,3, hist = 2


v = 5, j = 1,2,3, hist = 2 (with center)


v = 5, j = 1,2,3, hist = 2 (with midpoints)


v = 5, j = 1,2,3, hist = 2 (with midpoints and centre)


And here are hexagons using the same rules:

v = 6, j = 1,2 (black-and-white)


v = 6, j = 1,2


v = 6, j = 1,2 (with centre)


And octagons:

v = 8, j = 1,2


v = 8, j = 1,2 (with centre)


v = 8, j = 1,2,3, hist = 2


v = 8, j = 1,2,3, hist = 2


v = 8, j = 1,2,3,4 hist = 3


v = 8, j = 1,2,3,4 hist = 3 (with center)


The Hex Fractor

Pre-previously on Overlord-in-terms-of-issues-around-the-Über-Feral, I looked at the fractals created when various restrictions are placed on a point jumping at random half-way towards the vertices of a square. For example, the point can be banned from jumping towards the same vertex twice in a row or towards the vertex to the left of the vertex it has just jumped towards, and so on.

Today I want to look at what happens to a similar point moving inside pentagons and hexagons. If the point can’t jump twice towards the same vertex of a pentagon, this is the fractal that appears:

Ban second jump towards same vertex (v + 0)


Ban second jump towards same vertex (color)


If the point can’t jump towards the vertex immediately to the left of the one it’s just jumped towards, this is the fractal that appears:

Ban jump towards v + 1


Ban jump towards v + 1 (color)


And this is the fractal when the ban is on the vertex two places to the left:

Ban jump towards v + 2


Ban jump towards v + 2 (color)


You can also ban more than one vertex:

Ban jump towards v + 0,1


Ban jump towards v + 1,2


Ban jump towards v + 1,4


Ban jump towards v + 1,4 (color)


Ban jump towards v + 2,3


And here are fractals created in similar ways inside hexagons:

Ban jump towards v + 0,1


Ban jump towards v + 0,3


Ban jump towards v + 0,1,2


Ban jump towards v + 0,1,2 (color)


Ban jump towards v + 0,1,4


Ban jump towards v + 0,1,5


Ban jump towards v + 0,2,4


Ban jump towards v + 0,2,4 (color)


Ban jump towards v + 1,2,3


Ban jump towards v + 1,2,3 (color)


Ban jump towards v + 1,2,4


Ban jump towards v + 1,2,4, (color)


Ban jump towards v + 1,3,5


Ban jump towards v + 1,3,5 (color)


Ban jump towards v + 1,2


Ban jump towards v + 1,2


Ban jump towards v + 1,3


Ban jump towards v + 1,3 (color)


Ban jump towards v + 1,5


Ban jump towards v + 1,5 (color)


Ban jump towards v + 2,3


Ban jump towards v + 2,3 (color)


Ban jump towards v + 2,4


Ban jump towards v + 2,4 (color)


Elsewhere other-accessible:

Square Routes Re-Verticed

Bat out of L

Pre-previously on Overlord-in-terms-of-the-Über-Feral, I’ve looked at intensively interrogated issues around the L-triomino, a shape created from three squares that can be divided into four copies of itself:

An L-triomino divided into four copies of itself


I’ve also interrogated issues around a shape that yields a bat-like fractal:

A fractal full of bats


Bat-fractal (animated)


Now, to end the year in spectacular fashion, I want to combine the two concepts pre-previously interrogated on Overlord-in-terms-of-the-Über-Feral (i.e., L-triominoes and bats). The L-triomino can also be divided into nine copies of itself:

An L-triomino divided into nine copies of itself


If three of these copies are discarded and each of the remaining six sub-copies is sub-sub-divided again and again, this is what happens:

Fractal stage 1


Fractal stage 2


Fractal #3


Fractal #4


Fractal #5


Fractal #6


Et voilà, another bat-like fractal:

L-triomino bat-fractal (static)


L-triomino bat-fractal (animated)


Elsewhere other-posted:

Tri-Way to L
Bats and Butterflies
Square Routes
Square Routes Revisited
Square Routes Re-Revisited
Square Routes Re-Re-Revisited

Tridentine Math

The Tridentine Mass is the Roman Rite Mass that appears in typical editions of the Roman Missal published from 1570 to 1962. — Tridentine Mass, Wikipedia

A 30°-60°-90° right triangle, with sides 1 : √3 : 2, can be divided into three identical copies of itself:

30°-60°-90° Right Triangle — a rep-3 rep-tile…


And if it can be divided into three, it can be divided into nine:

…that is also a rep-9 rep-tile


Five of the sub-copies serve as the seed for an interesting fractal:

Fractal stage #1


Fractal stage #2


Fractal stage #3


Fractal #4


Fractal #5


Fractal #6


Fractal #6


Tridentine cross (animated)


Tridentine cross (static)


This is a different kind of rep-tile:

Noniamond trapezoid


But it yields the same fractal cross:

Fractal #1


Fractal #2


Fractal #3


Fractal #4


Fractal #5


Fractal #6


Tridentine cross (animated)


Tridentine cross (static)


Elsewhere other-available:

Holey Trimmetry — another fractal cross

Square Routes Re-Re-Revisited

This is an L-triomino, or shape created from three squares laid edge-to-edge:

When you divide each square like this…

You can create a fractal like this…

Stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Stage #9


Stage #10


Animated fractal


Here are more fractals created from the triomino:

Animated


Static


Animated


Static


Animated


Static


And here is a different shape created from three squares:

And some fractals created from it:

Animated


Static


Animated


Static


Animated


Static


And a third shape created from three squares:

And some fractals created from it:

Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Previously pre-posted (please peruse):

Tri-Way to L
Square Routes
Square Routes Revisited
Square Routes Re-Revisited

Bats and Butterflies

I’ve used butterfly-images to create fractals. Now I’ve found a butterfly-image in a fractal. The exciting story begins with a triabolo, or shape created from three isoceles right triangles:


The triabolo is a rep-tile, or shape that can be divided into smaller copies of itself:


In this case, it’s a rep-9 rep-tile, divisible into nine smaller copies of itself. And each copy can be divided in turn:


But what happens when you sub-divide, then discard copies? A fractal happens:

Fractal crosses (animated)


Fractal crosses (static)


That’s a simple example; here is a more complex one:

Fractal butterflies #1


Fractal butterflies #2


Fractal butterflies #3


Fractal butterflies #4


Fractal butterflies #5


Fractal butterflies (animated)


Some of the gaps in the fractal look like butterflies (or maybe large moths). And each butterfly is escorted by four smaller butterflies. Another fractal has gaps that look like bats escorted by smaller bats:

Fractal bats (animated)

Fractal bats (static)


Elsewhere other-posted:

Gif Me Lepidoptera — fractals using butterflies
Holey Trimmetry — more fractal crosses

Holey Trimmetry

Symmetry arising from symmetry isn’t surprising. But what about symmetry arising from asymmetry? You can find both among the rep-tiles, which are geometrical shapes that can be completely replaced by smaller copies of themselves. A square is a symmetrical rep-tile. It can be replaced by nine smaller copies of itself:

Rep-9 Square

If you trim the copies so that only five are left, you have a symmetrical seed for a symmetrical fractal:

Fractal cross stage #1


Fractal cross #2


Fractal cross #3


Fractal cross #4


Fractal cross #5


Fractal cross #6


Fractal cross (animated)


Fractal cross (static)


If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal:

Fractal Hex-Ring #1


Fractal Hex-Ring #2


Fractal Hex-Ring #3


Fractal Hex-Ring #4


Fractal Hex-Ring #5


Fractal Hex-Ring #6


Fractal Hex-Ring (animated)


Fractal Hex-Ring (static)


Now here’s an asymmetrical rep-tile, a nonomino or shape created from nine squares joined edge-to-edge:

Nonomino


It can be divided into twelve smaller copies of itself, like this:

Rep-12 Nonomino (discovered by Erich Friedman)


If you trim the copies so that only five are left, you have an asymmetrical seed for a familiar symmetrical fractal:

Fractal cross stage #1


Fractal cross #2


Fractal cross #3


Fractal cross #4


Fractal cross #5


Fractal cross #6


Fractal cross (animated)


Fractal cross (static)


If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal:

Fractal Hex-Ring #1


Fractal Hex-Ring #2


Fractal Hex-Ring #3


Fractal Hex-Ring #4


Fractal Hex-Ring #5


Fractal Hex-Ring (animated)


Fractal Hex-Ring (static)


Elsewhere other-available:

Square Routes Re-Re-Visited

Square Routes Re-Revisited

This is a very simple fractal:


It has four orientations:


Any orientation can be turned into any other by a rotation of 90°, 180° or 270°, either clockwise or anticlockwise. If you mix orientations and rotations, you can create much more complex fractals. Here’s a selection of them:

Animated fractal

Static fractal


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Animated fractal

Static


Elsewhere other-posted:

Square Routes
Square Routes Revisited

He Say, He Sigh, He Sow #44

H. Rider Haggard describes fractals:

Out of the vast main aisle there opened here and there smaller caves, exactly, Sir Henry said, as chapels open out of great cathedrals. Some were large, but one or two — and this is a wonderful instance of how nature carries out her handiwork by the same unvarying laws, utterly irrespective of size — were tiny. One little nook, for instance, was no larger than an unusually big doll’s house, and yet it might have been a model for the whole place, for the water dropped, tiny icicles hung, and spar columns were forming in just the same way. — King Solomon’s Mines, 1885, ch. XVI, “The Place of Death”