Curvous Energy

Here is a strange and beautiful fractal known as a dragon curve:

A dragon curve (note: this is a twin-dragon curve or Davis-Knuth dragon)


And here is the shape generally regarded as the dullest and most everyday of all:

A square


But squares are square, so let’s go back to dragon-curves. This particular kind of dragon-curve looks a lot like a Chinese dragon. You can see the same writhing energy and scaliness:

Chinese dragon


Dragon-curve for comparison


Dragon-curves also look like some species of soft coral:

Red soft-coral


In short, dragon-curves are organic and lively, quite unlike the rigid, lifeless solidity of a square. But there’s more to a dragon-curve than immediately meets the eye. Dragon-curves are rep-tiles, that is, you can tile one with smaller copies of itself:

Dragon-curve rep-tiled with two copies of itself


Dragon-curve rep-4


Dragon-curve rep-8


Dragon-curve rep-16


Dragon-curve rep-32


Dragon-curve self-tiling (animated)


From the rep-32 dragon-curve, you can see that a dragon-curve can be surrounded by six copies of itself. Here’s an animation of the process:

Dragon-curve surrounded (anim)


And because dragon-curves are rep-tiles, they will tile the plane:

Dragon-curve tiling #1


Dragon-curve tiling #2


But how do you make these strange and beautiful shapes, with their myriad curves and curlicules, their energy and liveliness? It’s actually very simple. You start with the shape generally regarded as the dullest and most everyday of all:

A square


Then you see how the shape can be replaced by five smaller copies of itself:

Square overlaid by five smaller squares


Square replaced by five smaller squares


Then you set about replacing it with two of those smaller copies:

Replacing squares Stage #0


Replacing squares Stage #1


Then you do it again to each of the copies:

Replacing squares Stage #2


And again:

Replacing squares #3


And again:

Replacing squares #4


And keep on doing it:

Replacing squares #5


Replacing squares #6


Replacing squares #7


Replacing squares #8


Replacing squares #9


Replacing squares #10


Replacing squares #11


Replacing squares #12


Replacing squares #13


Replacing squares #14


Replacing squares #15


And in the end you’ve got a dragon-curve:

Dragon-curve built from squares


Dragon-curve built from squares (animated)


Square Routes Re-Re-Re-Revisited

Discovering something that’s new to you in recreational maths is good. But so is re-discovering it by a different route. I’ve long been passionate about what happens when a point is allowed to jump repeatedly halfway towards the randomly chosen vertices of a square. If the point can choose any vertex any number of times, the interior of the square fills slowly and completely with points, like this:

Point jumping at random halfway towards vertices of a square


However, if the point is banned from jumping towards the same vertex twice or more in a row, an interesting fractal appears:

Fractal #1 — ban on jumping towards vertex vi twice or more


If the point can’t jump towards the vertex one place clockwise of the vertex it’s just jumped towards, this fractal appears:

Fractal #2 — ban on jumping towards vertex vi+1


If the point can’t jump towards the vertex two places clockwise of the vertex it’s just jumped towards, this fractal appears (two places clockwise is also two places anticlockwise, i.e. the banned vertex is diagonally opposite):

Fractal #3 — ban on jumping towards vertex vi+2


Now I’ve discovered a new way to create these fractals. You take a filled square, divide it into smaller squares, then remove some of them in a systematic way. Then you do the same to the smaller squares that remain. For fractal #1, you do this:

Fractal #1, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #1 (animated)


For fractal #2, you do this:

Fractal #2, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #2 (animated)


For fractal #3, you do this:

Fractal #3, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #3 (animated)


If the sub-squares are coloured, it’s easier to understand how, say, fractal #1 is created:

Fractal #1 (coloured), stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #1 (coloured and animated)


The fractal is actually being created in quarters, with one quarter rotated to form the second, third and fourth quarters:

Fractal #1, quarter









Here’s an animation of the same process for fractal #3:

Fractal #3 (coloured and animated)


So you can create these fractals either with a jumping point or by subdividing a square. But in fact I discovered the subdivided-square route by looking at a variant of the jumping-point route. I wondered what would happen if you took a point inside a square, allowed it to trace all possible routes towards the vertices without marking its position, then imposed the restriction for Fractal #1 on its final jump, namely, that it couldn’t jump towards the vertex it jumped towards on its previous jump. If the point is marked after its final jump, this is what appears (if the routes chosen had been truly random, the image would be similar but messier):

Fractal #1, restriction on final jump


Then I imposed the same restriction on the point’s final two jumps:

Fractal #1, restriction on final 2 jumps


And final three jumps:

Fractal #1, restriction on final 3 jumps


And so on:

Fractal #1, restriction on final 4 jumps


Fractal #1, restriction on final 5 jumps


Fractal #1, restriction on final 6 jumps


Fractal #1, restriction on final 7 jumps


Here are animations of the same process applied to fractals #2 and #3:

Fractal #2, restrictions on final 1, 2, 3… jumps


Fractal #3, restrictions on final 1, 2, 3… jumps


The longer the points are allowed to jump before the final restriction is imposed on their n final jumps, the more densely packed the marked points will be:

Fractal #1, packed points #1


Packed points #2


Packed points #3


Eventually, the individual points will form a solid mass, like this:

Fractal #1, solid mass of points


Fractal #1, packed points (animated)


Previously pre-posted (please peruse):

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

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

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

Bent for the Pent

A triangle can be tiled with triangles and a square with squares, but a pentagon can’t be tiled with pentagons. At least, not in the same way, using smaller copies of the same shape. The closest you can get is this:

Pentaflake #1


If you further subdivide the pentagon, you create what is known as a pentaflake:

Pentaflake #2


Pentaflake #3


Pentaflake #4


Pentaflake (animated)


Pentaflake (static)


But if you bend the rules and use irregular smaller pentagons, you can tile a pentagon like this, creating what I called a pentatile:

Pentatile stage 1


Further subdivisions create an interesting final pattern:

Pentatile #2


Pentatile #3


Pentatile #4


Pentatile #5


Pentatile #6


Pentatile (animated)


Pentatile (static)


By varying the size of the central pentagon, you can create other patterns:

Pentatile #1 (animated)


Pentatile #2 (animated)

Pentatile #2







Pentatile with no central pentagon


And here are various pentatiles in an animated gif:


And here are some variations on the pentaflake:







Elsewhere other-posted:

Bent for the Rent (1976) — the title of the incendiary intervention above is of course a reference to the “first and last glitter-rock album” by England’s loudest band, Spinal In Terms Of Tap
Phrallic Frolics — more on pentaflakes

Tri Again (Again)

I didn’t expect to find the hourglass fractal playing with squares. I even less expected it playing with triangles. Isosceles right triangles, to be precise. Then again, I found it first playing with the L-triomino, which is composed of three squares. And an isosceles triangle is half of a square. So it all fits. This is an isosceles right triangle:
isosceles_right_triangle

Isosceles right triangle


It’s mirror-symmetrical, so it looks the same in a mirror unless you label one of the acute-angled corners in some way, like this:

right_triangle_chiral_1

Right triangle with labelled corner


right_triangle_chiral_2

Right triangle reflected


Reflection is how you find the hourglass fractal. First, divide a right triangle into four smaller right triangles.

right_triangle_div4

Right triangle rep-tiled


Then discard one of the smaller triangles and repeat. If the acute corners of the smaller triangles have different orientations, one of the permutations creates the hourglass fractal, like this:

right_triangle_div4_1

Hourglass #1


right_triangle_div4_2

Hourglass #2


right_triangle_div4_3

Hourglass #3


right_triangle_div4_4

Hourglass #4


right_triangle_div4_5

Hourglass #5


right_triangle_div4_6

Hourglass #6


right_triangle_div4_7

Hourglass #7


right_triangle_div4_8

Hourglass #8


right_triangle_div4_9

Hourglass #9


right_triangle_div4_123_010

Hourglass animated


Another permutation of corners creates what I’ve decided to call the crane fractal, like this:
right_triangle_div4_123_001

Crane fractal animated


right_triangle_div4_123_001_static

Crane fractal (static)


The crane fractal is something else that I first found playing with the L-triomino:

l-triomino_234

Crane fractal from L-triomino


Previously pre-posted:

Square Routes
Tri Again

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)


Tri-Way to L

The name is more complicated than the shape: L-triomino. The shape is simply three squares forming an L. And it’s a rep-tile — it can be divided into four smaller copies of itself.

l-triomino

An L-triomino — three squares forming an L


l-triomino_anim

L-triomino as rep-tile


That means it can also be turned into a fractal, as I’ve shown in Rep-Tiles Revisited and Get Your Prox Off #2. First you divide an L-triomino into four sub-copies, then discard one sub-copy, then repeat. Here are the standard L-triomino fractals produced by this technique:

l-triomino_123_134

Fractal from L-triomino — divide and discard


l-triomino_234


l-triomino_124


l-triomino_124_upright


l-triomino_124_upright_static1

(Static image)


l-triomino_124_upright_static2

(Static image)


But those fractals don’t exhaust the possibilities of this very simple shape. The standard L-triomino doesn’t have true chirality. That is, it doesn’t come in left- and right-handed forms related by mirror-reflection. But if you number its corners for the purposes of sub-division, you can treat it as though it comes in two distinct orientations. And when the orientations are different in the different sub-copies, new fractals appear. You can also delay the stage at which you discard the first sub-copy. For example, you can divide the L-triomino into four sub-copies, then divide each sub-copy into four more sub-copies, and only then begin discarding.

Here are the new fractals that appear when you apply these techniques:

l-triomino_124_exp

Delay before discarding


l-triomino_124_exp_static

(Static image)


l-triomino_124_tst2_static1

(Static image)


l-triomino_124_tst2_static2

(Static image)


l-triomino_124_tst1


l-triomino_124_tst1_static1

(Static image)


l-triomino_124_tst1_static2

(Static image)


l-triomino_134_adj1

Adjust orientation


l-triomino_134_adj2


l-triomino_134_adj3


l-triomino_134_adj3_tst3

(Static image)


l-triomino_134_adj4


l-triomino_134_exp_static

(Static image)


l-triomino_234_exp