Back to Drac’

draconic, adj. /drəˈkɒnɪk/ pertaining to, or of the nature of, a dragon. [Latin draco, -ōnem, < Greek δράκων dragon] — The Oxford English Dictionary

In Curvous Energy, I looked at the strange, beautiful and complex fractal known as the dragon curve and showed how it can be created from a staid and sedentary square:

A dragon curve


Here are the stages whereby the dragon curve is created from a square. Note how each square at one stage generates a pair of further squares at the next stage:

Dragon curve from squares #1


Dragon curve from squares #2


Dragon curve from squares #3


Dragon curve from squares #4


Dragon curve from squares #5


Dragon curve from squares #6


Dragon curve from squares #7


Dragon curve from squares #8


Dragon curve from squares #9


Dragon curve from squares #10


Dragon curve from squares #11


Dragon curve from squares #12


Dragon curve from squares #13


Dragon curve from squares #14


Dragon curve from squares (animated)


The construction is very easy and there’s no tricky trigonometry, because you can use the vertices and sides of each old square to generate the vertices of the two new squares. But what happens if you use lines rather than squares to generate the dragon curve? You’ll discover that less is more:

Dragon curve from lines #1


Dragon curve from lines #2


Dragon curve from lines #3


Dragon curve from lines #4


Dragon curve from lines #5


Each line at one stage generates a pair of further lines at the next stage, but there’s no simple way to use the original line to generate the new ones. You have to use trigonometry and set the new lines at 45° to the old one. You also have to shrink the new lines by a fixed amount, 1/√2 = 0·70710678118654752… Here are further stages:

Dragon curve from lines #6


Dragon curve from lines #7


Dragon curve from lines #8


Dragon curve from lines #9


Dragon curve from lines #10


Dragon curve from lines #11


Dragon curve from lines #12


Dragon curve from lines #13


Dragon curve from lines #14


Dragon curve from lines (animated)


But once you have a program that can adjust the new lines, you can experiment with new angles. Here’s a dragon curve in which one new line is at an angle of 10°, while the other remains at 45° (after which the full shape is rotated by 180° because it looks better that way):

Dragon curve 10° and 45°


Dragon curve 10° and 45° (animated)


Dragon curve 10° and 45° (coloured)


Here are more examples of dragon curves generated with one line at 45° and the other line at a different angle:

Dragon curve 65°


Dragon curve 65° (anim)


Dragon curve 65° (col)


Dragon curve 80°


Dragon curve 80° (anim)


Dragon curve 80° (col)


Dragon curve 135°


Dragon curve 135° (anim)


Dragon curve 250°


Dragon curve 250° (anim)


Dragon curve 250° (col)


Dragon curve 260°


Dragon curve 260° (anim)


Dragon curve 260° (col)


Dragon curve 340°


Dragon curve 340° (anim)


Dragon curve 340° (col)


Dragon curve 240° and 20°


Dragon curve 240° and 20° (anim)


Dragon curve 240° and 20° (col)


Dragon curve various angles (anim)


Previously pre-posted:

Curvous Energy — a first look at dragon curves

Rigging in the Trigging

Here’s a simple pattern of three triangles:

Three-Triangle Pattern


Now replace each triangle in the pattern with the same pattern at a smaller scale:

Replacing triangles


If you keep on doing this, you create what I’ll call a trigonal fractal (trigon is Greek for “triangle”):

Trigonal Fractal stage #3 (click for larger)


Trigonal Fractal stage #4


Trigonal Fractal stage #5


Trigonal Fractal #6


Trigonal Fractal #7


Trigonal Fractal #8


Trigonal Fractal (animated) (click for larger)


You can use the same pattern to create different fractals by rotating the replacement patterns in different ways. I call this “rigging the trigging” and here are some of the results:




You can also use a different seed-pattern to create the fractals:

Trigonal fractal (animated)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)


Note: The title of this incendiary intervention is of course a paranomasia on the song “Frigging in the Rigging”, also known as “Good Ship Venus” and performed by the Sex Pistols on The Great Rock ’n’ Roll Swindle (1979).

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)


Pi and By

Here’s √2 in base 2:

√2 = 1.01101010000010011110... (base=2)

And in base 3:

√2 = 1.10201122122200121221... (base=3)

And in bases 4, 5, 6, 7, 8, 9 and 10:

√2 = 1.12220021321212133303... (b=4)
√2 = 1.20134202041300003420... (b=5)
√2 = 1.22524531420552332143... (b=6)
√2 = 1.26203454521123261061... (b=7)
√2 = 1.32404746317716746220... (b=8)
√2 = 1.36485805578615303608... (b=9)
√2 = 1.41421356237309504880... (b=10)

And here’s π in the same bases:

π = 11.00100100001111110110... (b=2)
π = 10.01021101222201021100... (b=3)
π = 03.02100333122220202011... (b=4)
π = 03.03232214303343241124... (b=5)
π = 03.05033005141512410523... (b=6)
π = 03.06636514320361341102... (b=7)
π = 03.11037552421026430215... (b=8)
π = 03.12418812407442788645... (b=9)
π = 03.14159265358979323846... (b=10)

Mathematicians know that in all standard bases, the digits of √2 and π go on for ever, without falling into any regular pattern. These numbers aren’t merely irrational but transcedental. But are they also normal? That is, in each base b, do the digits 0 to [b-1] occur with the same frequency 1/b? (In general, a sequence of length l will occur in a normal number with frequency 1/(b^l).) In base 2, are there as many 1s as 0s in the digits of √2 and π? In base 3, are there as many 2s as 1s and 0s? And so on.

It’s a simple question, but so far it’s proved impossible to answer. Another question starts very simple but quickly gets very difficult. Here are the answers so far at the Online Encyclopedia of Integer Sequences (OEIS):

2, 572, 8410815, 59609420837337474 – A049364

The sequence is defined as the “Smallest number that is digitally balanced in all bases 2, 3, … n”. In base 2, the number 2 is 10, which has one 1 and one 0. In bases 2 and 3, 572 = 1000111100 and 210012, respectively. 1000111100 has five 1s and five 0s; 210012 has two 2s, two 1s and two 0s. Here are the numbers of A049364 in the necessary bases:

10 (n=2)
1000111100, 210012 (n=572)
100000000101011010111111, 120211022110200, 200011122333 (n=8410815)
11010011110001100111001111010010010001101011100110000010, 101201112000102222102011202221201100, 3103301213033102101223212002, 1000001111222333324244344 (n=59609420837337474)

But what number, a(6), satisfies the definition for bases 2, 3, 4, 5 and 6? According to the notes at the OEIS, a(6) > 5^434. That means finding a(6) is way beyond the power of present-day computers. But I assume a quantum computer could crack it. And maybe someone will come up with a short-cut or even an algorithm that supplies a(b) for any base b. Either way, I think we’ll get there, π and by.

Phrock and Roll

What does a fractal phallus look like?

Millions of people have axed this corely key question.

The Overlord of the Über-Feral can answer it — keyly, corely and comprehensively dot dot dot

And here is the answer: Phrallic Frolics

Tright Treeing

Here is a very simple tree with two branches:

Two-branch tree


These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

Two-branch tree stage 1


2-Tree stage 2


2-Tree stage 3


2-Tree stage 4


2-Tree (animated)


If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1, draw a left branch to node 2 and colour the node green.
2. Return to node 1.
3. Draw a right branch to node 3 and colour the node green.
4. Finish.

Now try a slightly less simple tree with branches that fork twice:

Four-branch tree (static)


These are the steps that a simple computer program follows to draw the tree, with a red arrow indicating where the computer’s focus is at each stage:

4-Tree #1


4-Tree #2


4-Tree #3


4-Tree #4


4-Tree #5


4-Tree #6


4-Tree #7


4-Tree #8


4-Tree #9


4-Tree #10


4-Tree #11


4-Tree (animated)


If you had to give the computer an explicit instruction at each stage, the instructions might look something like this:

1. Start at node 1 and draw a left branch to node 2.
2. Draw a left branch to node 3 and colour it green.
3. Return to node 2.
4. Draw a right branch to node 4 and colour it green.
5. Return to node 2.
6. Return to node 1.
7. Draw a right branch to node 5.
8. Draw a left branch to node 6.
9. Draw a left branch to node 7 and colour it green.
10. Return to node 6.
11. Draw a left branch to node 8 and colour it green.
12. Finish.

It’s easy to see that the list of instructions would be much bigger for a tree with branches that fork three times, let alone four times or you. But you don’t need to give a full set of explicit instructions: you can use a program, or a list of instructions using variables. Suppose the tree has branches that fork f times. If f = 4, you will need an array variable level() with four values, level(1), level(2), level(3) and level(4). Now follow these instructions:

1. li = 1, level(1) = 0, level(2) = 0, ... level(f+1) = 0
2. level(li) = level(li) + 1
3. If level(li) = 1, draw a branch to the left and jump to step 7
4. If level(li) = 2, draw a branch to the right and jump to step 7
5. li = li - 1 (note that this line is reached if the tests fail in lines 3 and 4)
6. If li > 0, jump to step 2, otherwise jump to step 11
7. If li = f, draw a green node and jump to step 5
9. li = li + 1
10. Jump to step 2
11. Finish.

By changing the value of f, a computer can use those eleven basic instructions to draw any size of tree (I’ve left out details like changes in the length of branches and so on). When f = 4, the tree will look like this:

16-Tree (static)


16-Tree (animated)


With simple adjustments, the program can be used for other shapes whose underlying structure can be represented symbolically as a tree. The program is in fact a fractalizer, that is, it draws a fractal. So if you use a version of the program to draw fractals based on right-triangles, you can say you are “tright treeing” (tright = triangle-that-is-right).

Here is some tright treeing. Start with a simple isoceles right-triangle. It can be divided into smaller isoceles right-triangles by finding the midpoint of the hypotenuse, then repeating:

Right-triangle rep-2 stage 1


Right-triangle #2


Tright #3


Tright #4


Tright #5


Tright #6


Tright #7


Tright #7 (no internal lines)


You can distort the isoceles right-triangle in interesting ways by finding the midpoint of a side other than the hypotenuse, like this:

Right-triangle (distorted) #1


Distorted tright #2


Distorted tright #3


Distorted tright #4


Distorted tright #5


Distorted tright #6


Distorted tright #7


Distorted tright #8


Distorted tright #9


Distorted tright #10


Distorted tright #11


Distorted tright #12


Distorted tright #13


Distorted tright (animated)


Here’s a different right-triangle. When you divide it regularly, it looks like this:

Right-triangle rep-3 stage 1


Rep-3 Tright #2


3-Tright #3


3-Tright #4


3-Tright #5


3-Tright #6


3-Tright #7


3-Tright #8


3-Tright #9


3-Tright (one colour)


When you distort the divisions, you can create interesting fractals (click on images for larger versions):

Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright


Distorted 3-Tright (animated)


And when four of the distorted right-triangles (rep-2 or rep-3) are joined in a diamond, you can create shapes like these:

Creating a diamond #1


Creating a diamond #2


Creating a diamond #3


Creating a diamond #4


Creating a diamond (animated)


Rep-3 right-triangle diamond (divided)


Rep-3 right-triangle diamond (single colour)


Distorted rep-3 right-triangle diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond


Distorted 3-tright diamond (animated)


Distorted rep-2 right-triangle


Distorted 2-tright diamond


Distorted 2-tright diamond


Distorted 2-tright diamond


Distorted 2-tright diamond


Distorted 2-tright diamond (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

An N-Finity

10111 in base 2
212 in base 3
113 in base 4
43 in base 5
35 in base 6
32 in base 7
27 in base 8
25 in base 9
23 in base 10
21 in base 11
1B in base 12
1A in base 13
19 in base 14
18 in base 15
17 in base 16
16 in base 17
15 in base 18
14 in base 19
13 in base 20
12 in base 21
11 in base 22
10 in base 23
N in all bases >= 24

√23 = 4.79583152331…

Mice Thrice

Twice before on Overlord-in-terms-of-Core-Issues-around-Maximal-Engagement-with-Key-Notions-of-the-Über-Feral, I’ve interrogated issues around pursuit curves. Imagine four mice or four beetles each sitting on one corner of a square and looking towards the centre of the square. If each mouse or beetle begins to run towards the mouse or beetle to its left, it will follow a curving path that takes it to the centre of the square, like this:

vertices = 4, pursuit = +1


The paths followed by the mice or beetles are pursuit curves. If you arrange eight mice clockwise around a square, with a mouse on each corner and a mouse midway along each side, you get a different set of pursuit curves:

v = 4 + 1 on the side, p = +1


Here each mouse is pursuing the mouse two places to its left:

v = 4+s1, p = +2


And here each mouse is pursuing the mouse three places to its left:

v = 4+s1, p = +3


Now try a different arrangement of the mice. In the square below, the mice are arranged clockwise in rows from the bottom right-hand corner. That is, mouse #1 begins on the bottom left-hand corner, mouse #2 begins between that corner and the centre, mouse #3 begins on the bottom left-hand corner, and so on. When each mice runs towards the mouse three places away, these pursuit curves appear:

v = 4 + 1 internally, p = +3


Here are some more:

v = 4 + i1, p = +5


v = 4 + i2, p = +1


v = 4 + i2, p = +2


So far, all the mice have eventually run to the centre of the square, but that doesn’t happen here:

v = 4 + i2, p = 4


Here are more pursuit curves for the v4+i2 mice, using an animated gif:

v = 4 + i2, p = various (animated — open in new tab for clearer image)


And here are more pursuit curves that don’t end in the centre of the square:

v = 4 + i4, p = 4


v = 4 + i4, p = 8


v = 4 + i4, p = 12


v = 4 + i4, p = 16


But the v4+i4 pursuit curves more usually look like this:

v = 4 + i4, p = 7


Now try adapting the rules so that mice don’t run directly towards another mouse, but towards the point midway between two other mice. In this square, the odd- and even-numbered mice follow different rules. Mice #1, #3, #5 and #7 run towards the point midway between the mice one and two places away, while ice #2, #4, #6 and #8 run towards the point midway between the mice two and seven places away:

v = 4 + s1, p(1,3,5,7) = 1,2, p(2,4,6,8) = 2,7


I think the curves are very elegant. Here’s a slight variation:

v = 4 + s1, p1 = 1,3, p2 = 2,7


Now try solid curves:

v = 4 + s1, p1 = 1,3, p2 = 2,7 (red)


v = 4 + s1, p1 = 1,3, p2 = 2,7 (yellow-and-blue)


And some variants:

v = 4 + s1, p1 = 1,7, p2 = 1,2


v = 4 + s1, p1 = 2,3, p2 = 2,5


v = 4 + s1, p1 = 5,6, p2 = 1,3


v = 4 + s1, p1 = 5,6, p2 = 1,4


v = 4 + s1, p1 = 5,6, p2 = 1,6


Elsewhere other-posted:

Polymorphous Pursuit
Persecution Complex

Horn Again

Pre-previously on Overlord-in-terms-of-Core-Issues-around-Maximal-Engagement-with-Key-Notions-of-the-Über-Feral, I interrogated issues around this shape, the horned triangle:

unicorn_reptile_static

Horned Triangle (more details)


Now I want to look at the tricorn (from Latin tri-, “three”, + -corn, “horn”). It’s like a horned triangle, but has three horns instead of one:

Tricorn, or three-horned triangle


These are the stages that make up the tricorn:

Tricorn (stages)


Tricorn (animated)


And there’s no need to stop at triangles. Here is a four-horned square, or quadricorn:

Quadricorn


Quadricorn (animated)


Quadricorn (coloured)


And a five-horned pentagon, or quinticorn:

Quinticorn, or five-horned pentagon


Quinticorn (anim)


Quinticorn (col)


And below are some variants on the shapes above. First, the reversed tricorn:

Reversed Tricorn


Reversed Tricorn (anim)


Reversed Tricorn (col)


The nested tricorn:

Nested Tricorn (anim)


Nested Tricorn (col)


Nested Tricorn (red-green)


Nested Tricorn (variant col)


The nested quadricorn:

Nested Quadricorn (anim)


Nested Quadricorn


Nested Quadricorn (col #1)


Nested Quadricorn (col #2)


Finally (and ferally), the pentagonal octopus or pentapus:

Pentapus (anim)


Pentapus


Pentapus #2


Pentapus #3


Pentapus #4


Pentapus #5


Pentapus #6


Pentapus (col anim)


Elsewhere other-engageable:

The Art Grows Onda — the horned triangle and Katsushika Hokusai’s painting The Great Wave off Kanagawa (c. 1830)