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

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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

Living Culler

When you replace a square with four smaller squares, each a quarter the size of the original, the smaller squares occupy the same area, because 4 * ¼ = 1. If you discard one sub-square, then divide each of the three remaining sub-squares into four sub-sub-square, discard one sub-sub-quare and repeat, you create fractals like those I looked at in Squaring and Paring. The fractals stay within a fixed boundary.

Square replaced with four smaller squares, each ¼th the size of the original


Animated fractal


Static fractal


This time I want to look at a slightly different process. Replace a square with nine smaller squares each a quarter the size of the original. Now the sub-squares occupy a larger area than the original, because 9 * ¼ = 2¼. If you discard — or cull — sub-squares and repeat, the resultant fractal grows beyond the original boundary. Indeed, sub-squares start to overlap, so you can use colours to represent how often a particular pixel has been covered with a square. Here is an example of this process in action:

Square replaced with nine smaller squares, each ¼th the size of the original


Animated fractal


Static fractal #1


Static fractal #2


Here are the individual stages of a more complex fractal that uses the second process:

Stage 1


Stage 2


Stage 3


Stage 4


Stage 5


Stage 6


Stage 7


Stage 8


Stage 9 (compare Fingering the Frigit and Performativizing the Polygonic)


Stage 10


Animated version


Static version #1


Static version #2


And here are some more of the fractals you can create in a similar way:


Static version #1

Static version #2


Static version #2

Static version #2

Static version #3





Various fractals in an animated gif


Squaring and Paring

Squares are often thought to be the most boring of all shapes. Yet every square holds a stunning secret – something that in legend prompted a mathematical cult to murder a traitor. If each side of a square is one unit long, how long is the square’s diagonal, that is, the line from one corner to the opposite corner?

By Pythagoras’ theorem, the answer is this:

• x^2 = 1^2 + 1^2
• x^2 = 2
• x = √2

But what is √2? Pythagoras and his followers thought that all numbers could be represented as either whole numbers or ratios of whole numbers. To their dismay, so it’s said, they discovered that they were wrong. √2 is an irrational number – it can’t be represented as a ratio. In modern notation, it’s an infinitely decimal that never repeats:

• √2 = 1·414213562373095048801688724209698…

A modern story, unattested in ancient records, says that the irrationality of √2 was a closely guarded secret in the Pythagorean cult. When Hippasus of Metapontum betrayed the secret, he was drowned at sea by enraged fellow cultists. Apocryphal or not, the story shows that squares aren’t so boring after all.

Nor are they boring when they’re caught in the fract. Divide one square into nine smaller copies of itself:


Discard three of the copies like this:

Stage 1
Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)


Then do the same to each of the sub-squares:

Stage 1


And repeat:

Stage 3


Stage 4


Stage 5


Stage 6


The result is a fractal of endlessly subdividing contingent hexagons:

Animated vesion


Retain squares 1, 2, 4, 6, 8, 9 (reading left-to-right, bottom-to-top)


Here are a few more of the fractals you can create by squaring and paring:

Retain squares 1, 3, 5, 7, 9 (reading left-to-right, bottom-to-top)


Retain squares 2, 4, 5, 6, 8


Retain squares 1, 2, 4, 5, 6, 8, 9


Retain squares 1, 4, 6, 7, 10, 11, 13, 16


Retain squares 1, 3, 6, 7, 8, 9, 10, 11, 14, 16


Retain squares 2, 3, 5, 6, 8, 9, 11, 12, 14, 15


Retain squares 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25


Retain squares 1, 3, 7, 8, 11, 12, 14, 15, 18, 19, 23, 25


Retain squares 1, 5, 7, 8, 9, 12, 14, 17, 18, 19, 21, 25


Retain squares 2, 3, 4, 6, 7, 9, 10, 11, 15, 16, 17, 19, 20, 22, 23, 24


Retain squares 1, 2, 5, 6, 7, 9, 13, 17, 19, 20, 21, 24, 25


Previously pre-posted (please peruse):

M.i.P. Trip

Square Routes Revisited

Take a square, divide it into four smaller squares, and discard the smaller square on the top right. Do the same to each of the subsquares, dividing it into four sub-subsquares, then discarding the one on the top right. And repeat with the sub-subsquares. And the sub-sub-squares. And the sub-sub-sub-squares. And so on. The result is a fractal like this:

sq2x2_123_1

Stage 1


sq2x2_123_2

Stage 2


sq2x2_123_3

Stage 3


sq2x2_123_4

Stage 4


sq2x2_123

Animated fractal


sq2x2_123_static

Final fractal (static)


It looks as though this procedure isn’t very fertile. But you can enrich it by rotating each of the subsquares in a different way, so that the discarded sub-subsquare is different. Here’s an example:

Stage 1


Stage 2


Stage 3


Stage 4


Stage 5


Stage 6


Stage 7


Animated fractal


Final fractal (static)


Here are more examples of how rotating the subsquares in different ways produces different fractals:

Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Animated fractal

Static fractal


Previously pre-posted:

Square Routes — first look at this kind of fractal

Corralling Chaos

All the best people brood incessantly on the fact that a point inside a square jumping half-way towards a randomly chosen vertex will not create a fractal. Inside a triangle, yes: a fractal appears. Inside a pentagon too. But not inside a square:

Point jumping half-way towards a randomly chosen vertex


Instead, the interior of the square fills with random points: it crawls with chaos, you might say. However, fractals appear inside a square if the point is restricted in some way: banned from jumping towards a vertex twice in a row; banned from jumping towards the second-nearest vertex; and so on. Those restrictions are what might be called soft, because they take place in software (or in the brain of someone following the rule as a game or piece of performance art). Here’s what might be called a hard restriction that creates a fractal: the point cannot jump towards a randomly vertex if its jump passes over any part of the red upright cross:

Point cannot pass over red lines


I call this a barrier fractal. It’s obvious that the point cannot jump from one corner of the square towards the opposite corner, which creates bare space stretching from each vertex towards the tips of the upright cross. Less obvious is the way in which this bare space “cascades” into other parts of the square, creating a repeatedly branching and shrinking pattern.



When the barrier is a circle, a similar fractal appears:


If the point can also jump towards the center of the circle, this is what happens:

“Down through the aether I saw the accursed earth turning, ever turning, with angry and tempestuous seas gnawing at wild desolate shores and dashing foam against the tottering towers of deserted cities.” — “The Crawling Chaos” (1921), Winifred Jackson and H. P. Lovecraft.


Now here’s an upright cross with a gap in the middle:


Here’s an upright cross when the point can also jump towards the center of the cross:


A slanted cross with a central attractor:


And a single horizontal stroke:


A slanted stroke — note pentagons:


Even if the barrier is small and set on an edge of the square, it affects the rest of the square:


A more attractive example of edge-affects-whole:


Circles away from the edges


Detail of previous image






Here the point can also jump towards the center of the square’s edges:


A more subtle barrier fractal uses the previous jumps of the point to restrict its next jump. For example, if the point cannot jump across the line created by its previous-but-one jump, it moves like this:

Jump can’t cross track of last-but-one jump (animated gif)


The fractal itself looks like this:


Rule: on jump #3, cannot jump across the line created by jump #1; on jump #4, cannot cross the line created by jump #2; and so on.



And this is the fractal if the point cannot jump across the line created by its previous-but-two jump:

Rule: on jump #4, cannot jump across the line created by jump #2; on jump #5, cannot cross the line created by jump #3; and so on



Jumper to Jumper

Previously I’ve looked at fractals created by a point moving half-way towards the random chosen vertex of a polygon. But you can also choose an initial vertex, then choose a new vertex by adding a random number to that initial vertex. Then repeat. For example, if the polygon is a square and the initial vertex is v = 1, then choose v + 3 = 4 or v – 1 = 3, and so on.

You can then ban or un-ban the choice of vertex-jump as you can ban or un-ban direct choices of vertex. These two methods of random choice are effectively the same, but one can be simpler to program than the other. That’s why I’ve come across some new fractals by using vertex-jumps. Here they are:

vertices = 4, vertex-jump = (1,2,3,4), ban on same choice twice in a row


vertices = 4, vertex-jump = (1,2,3,4), ban on 2 in row (black-and-white version)


v = 4, vj = (1,2,3,4), ban on choice c + 2 from previous choice c


v = 4, vj = (1,2,3,4), ban c + 2 (animated gif)


vj = (1,2,3,4), ban c + 2 (black-and-white)


vj = (1,2,3,4), ban c + 0 at time t+1 unless c + 0 at time t-1


vj = (1,2,3,4), ban c + 0 at t+1, unless c + 0 at t-1 (black-and-white)


vj = (1,2,3,4,5), ban c + 0


vj = (0,1,2,3,4), ban c + 0


vj = (0,1,2,3,4), ban c + 0 (black-and-white)


vj = (1,2,3,4), ban c + 2 at t+1 unless c + 2 at t-1 (animated gif)


vj = (1,2,3,4), ban c + various at t+1 unless c + various at t-1 (animated gif)


vj = (1,2,3,4,5), ban c + 0 at t+1 unless c + 0 at t-1


vj = (-2,-1,0,1,2), ban c + 0


vj = (-2,-1,0,1,2), ban c + 0 (black-and-white)


vj = (0,1,2,3,4), ban c + va unless c + va


v = 5, vj = (1,2,3,4), ban c + 0


v = 5, vj = (1,2,3,4), ban c + 2


v = 5, vj = (0,1,2,3), ban c + 3


v = 6, vj = (0,1,2,3), ban c + 2


v = 6, vj = va, ban c + va (animated gif)


Appointment with Distality

distal, adj. Anat. Situated away from the centre of the body, or from the point of origin (said of the extremity or distant part of a limb or organ); terminal. Opp. to proximal. [← stem of dist- (in distant adj.) + -al, after dorsal, ventral, etc.] — Oxford English Dictionary

When a point jumps inside a triangle, moving halfway towards a randomly chosen vertex each time, a fractal known as the Sierpiński triangle appears:
chaos_triangle

Point jumping halfway towards random vertex of a triangle


chaos_triangle_bw

Point jumping inside triangle (black-and-white version)


But when a point moves at random in the same way inside a square, no fractal appears. Instead, the interior of the square gradually fills with a haze of pixels:
random_fill

Point jumping halfway towards random vertex of a square


Now trying imposing restrictions on the point jumping inside a square. If it can’t jump towards a vertex twice in a row, this fractal appears:
select_1_0

Ban consecutive jumps towards same vertex


select_1_0_bw

Ban consecutive jumps towards same vertex (black-and-white version)


Suppose the vertices are numbered from 1 to 4 and the point can’t jump towards the vertex one lower than the previously chosen vertex. That is, if it jumps towards vertex 3, it can’t jump next towards vertex 2, but it can jump towards vertices 1, 3, or 4 (if the vertex is 1, it’s banned from moving towards vertex 4, i.e. 1-1 = 0 = 4). Now this fractal appears:
select_1_1

Ban jump towards vertex v-1


select_1_1_bw


This is the fractal when the point can’t jump towards the vertex two places lower than the one it has just jumped towards:
select_1_2

Ban jump towards vertex v-2


select_1_2_bw


But if you can ban, you can also un-ban. Suppose the point jumps towards vertex v at time t and is then banned from jumping towards vertex v-2 at time t+1 unless it had jumped towards vertex v-1 at time t-1. This interesting fractal appears:
select_2_1_1_2

Ban jump v-2 at t+1 unless jump v-1 at t-1


Here are some more fractals using the ban / un-ban technique:
select_2_1_various

Ban / un-ban various


select_2_1_0_1

Ban jump v+0 at t+1 unless jump v+1 at t-1


select_2_1_1_3

Ban jump v+1 at t+1 unless jump v+3 at t-1


select_2_1_2_0

Ban jump v+0 at t+1 unless jump v+2 at t-1


select_2_1_2_2

Ban jump v+2 at t+1 unless jump v+2 at t-1


select_1_2_various

Ban / un-ban various


You can also impose or lift bans based not on the vertex the point jumps towards, but on the distance the point jumps. For example, take the radius r of the circle circumscribing the square and divide it into four segments, 0 to ¼r, ¼r to ½r, ½r to ¾r, and ¾r to r. When the point is going to jump towards vertex v, test whether its jump will land in the same segment, measured from the center of the circle, as it currently occupies. If it does, ban the jump and choose another vertex. Or unban the vertex if the point occupied segment s + x at time t-1. Here are some of the fractals produced using this technique:
dist_2_1_various

Ban / un-ban based on distance jumped


dist_center_1_0

Ban jump into segment s+0 of 4


dist_center_1_1

Ban jump into segment s+1 from center


dist_center_1_2

Ban jump into segment s+2


dist_center_-2_1_2_2

Ban jump into s+2 at t+1 unless jump into s+2 at at t-1


dist_xy_1_0

Ban jump into s+0 from present point


dist_xy_1_2

Ban jump into s+2 from present point


dist_xy_1_3

Ban jump into s+3 from present point


dist_xy_2_1_1_0

Ban jump into s+0 at t+1 unless jump into s+1 at at t-1


It’s easy to think of variants on all these themes, but I’ll leave them as an exercise for the interested reader.

The Swing’s the Thing

Order emerges from chaos with a triangle or pentagon, but not with a square. That is, if you take a triangle or a pentagon, chose a point inside it, then move the point repeatedly halfway towards a vertex chosen at random, a fractal will appear:

triangle

Sierpiński triangle from point jumping halfway to randomly chosen vertex


pentagon

Sierpiński pentagon from point jumping halfway to randomly chosen vertex


But it doesn’t work with a square. Instead, the interior of the square slowly fills with random points:

square

Square filling with point jumping halfway to randomly chosen vertex


As I showed in Polymorphous Perverticity, you can create fractals from squares and randomly moving points if you ban the point from choosing the same vertex twice in a row, and so on. But there are other ways. You can take the point, move it towards a vertex at random, then swing it around the center of the square through some angle before you mark its position, like this:

square_sw90

Point moves at random, then swings by 90° around center


square_sw180

Point moves at random, then swings by 180° around center


You can also adjust the distance of the point from the center of the square using a formula like dist = r * rmdist, where dist is the distance, r is the radius of the circle in which the circle is drawn, and rm takes values like 0.1, 0.25, 0.5, 0.75 and so on:

square_dist_rm0_05

Point moves at random, dist = r * 0.05 – dist


square_dist_rm0_1

Point moves at random, dist = r * 0.1 – dist


square_dist_rm0_2

Point moves at random, dist = r * 0.2 – dist


But you can swing the point while applying a vertex-ban, like banning the previously chosen vertex, or the vertex 90° or 180° away. In fact, swinging the points converts one kind of vertex ban into the others.

square_ban0

Point moves at random towards vertex not chosen previously


square_ban0_sw405

Point moves at random, then swings by 45°


square_ban0_sw360

Point moves at random, then swings by 360°


square_ban0_sw697

Point moves at random, then swings by 697.5°


square_ban0_sw720

Point moves at random, then swings by 720°


square_ban0_sw652

Point moves at random, then swings by 652.5°


square_ban0_swing_va_animated

Animated angle swing


You can also reverse the swing at every second move, swing the point around a vertex instead of the center or around a point on the circle that encloses the square. Here are some of the fractals you get applying these techniques.
square_ban0_sw45_rock

Point moves at random, then swings alternately by 45°, -45°


square_ban0_sw90_rock

Point moves at random, then swings alternately by 90°, -90°


square_ban0_sw135_rock

Point moves at random, then swings alternately by 135°, -135°


square_ban0_sw180_rock

Point moves at random, then swings alternately by 180°, -180°


square_ban0_sw225

Point moves at random, then swings alternately by 225°, -225°


square_ban0_sw315

Point moves at random, then swings alternately by 315°, -315°


square_ban0_sw360_rock

Point moves at random, then swings alternately by 360°, -360°


square_swing_vx0_va_animated

Animated alternate swing


square_circle_sw45

Point moves at random, then swings around point on circle by 45°


square_circle_sw67

Point moves at random, then swings around point on circle by 67.5°


square_circle_sw90

Point moves at random, then swings around point on circle by 90°


square_circle_sw112

Point moves at random, then swings around point on circle by 112.5°


square_circle_sw135

Point moves at random, then swings around point on circle by 135°


square_circle_sw180

Point moves at random, then swings around point on circle by 180°


square_circle_sw_animated

Animated circle swing


Square Routes

One of the pleasures of exploring an ancient city like York or Chester is that of learning new routes to the same destination. There are byways and alleys, short-cuts and diversions. You set off intending to go to one place and end up in another.

Maths is like that, even at its simplest. There are many routes to the same destination. I first found the fractal below by playing with the L-triomino, or the shape created by putting three squares in the shape of an L. You can divide it into four copies of the same shape and discard one copy, then do the same to each of the sub-copies, then repeat. I’ve decided to call it the hourglass fractal:

l-triomino_124

Hourglass fractal (animated)


l-triomino_124_upright_static1

Hourglass fractal (static)


Then I unexpectedly came across the fractal again when playing with what I call a proximity fractal:
v4_ban15_sw3_anim

Hourglass animated (proximity fractal)


v4_ban15_sw3_col

(Static image)


Now I’ve unexpectedly come across it for a third time, playing with a very simple fractal based on a 2×2 square. At first glance, the 2×2 square yields only one interesting fractal. If you divide the square into four smaller squares and discard one square, then do the same to each of the three sub-copies, then repeat, you get a form of the Sierpiński triangle, like this:

sq2x2_123_1

Sierpiński triangle stage 1


sq2x2_123_2

Sierpiński triangle #2


sq2x2_123_3

Sierpiński triangle #3


sq2x2_123_4

Sierpiński triangle #4


sq2x2_123

Sierpiński triangle animated


sq2x2_123_static

(Static image)


The 2×2 square seems too simple for anything more, but there’s a simple way to enrich it: label the corners of the sub-squares so that you can, as it were, individually rotate them 0°, 90°, 180°, or 270°. One set of rotations produces the hourglass fractal, like this:

sq2x2_123_013_1

Hourglass stage 1


sq2x2_123_013_2

Hourglass #2


sq2x2_123_013_3

Fractal #3


sq2x2_123_013_4

Hourglass #4


sq2x2_123_013_5

Hourglass #5


sq2x2_123_013_6

Hourglass #6


sq2x2_123_013

Hourglass animated


sq2x2_123_013_static

(Static image)


Here are some more fractals from the 2×2 square created using this technique (I’ve found some of them previously by other routes):

sq2x2_123_022


sq2x2_123_022_static

(Static image)


sq2x2_123_031


sq2x2_123_031_static

(Static image)


sq2x2_123_102


sq2x2_123_102_static

(Static image)


sq2x2_123_2011


sq2x2_123_201_static

(Static image)


sq2x2_123_211


sq2x2_123_211_static

(Static image)


sq2x2_123_213


sq2x2_123_213_static

(Static image)


sq2x2_123_033_-111


sq2x2_123_033_-111_static

(Static image)


sq2x2_123_201_1-11_static

(Static image)


sq2x2_200_1-11_static

(Static image)


sq2x2_123_132

(Static image)