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

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.

Radical Sheet

If you take a sheet of standard-sized paper and fold it in half from top to bottom, the folded sheet has the same proportions as the original, namely √2 : 1. In other words, if x = √2 / 2, then 1 / x = √2:

√2 = 1.414213562373…, √2 / 2 = 0.707106781186…, 1 / 0.707106781186… = 1.414213562373…

So you could say that paper has radical sheet (the square or other root of a number is also called its radix and √ is known as the radical sign). When a rectangle has the proportions √2 : 1, it can be tiled with an infinite number of copies of itself, the first copy having ½ the area of the original, the second ¼, the third ⅛, and so on. The radical sheet below is tiled with ten diminishing copies of itself, the final two having the same area:

papersizes

papersizes_static

You can also tile a radical sheet with six copies of itself, two copies having ¼ the area of the original and four having ⅛:

paper_6div_static

paper_6div

This tiling is when you might say the radical turns crucial, because you can create a fractal cross from it by repeatedly dividing and discarding. Suppose you divide a radical sheet into six copies as above, then discard two of the ⅛-sized rectangles, like this:

paper_cross_1

Stage 1


Then repeat with the smaller rectangles:

paper_cross_2

Stage 2


paper_cross_3

Stage 3


paper_cross_4

Stage 4


paper_cross_5

Stage 5


paper_cross

Animated version

paper_cross_static

Fractile cross

The cross is slanted, but it’s easy to rotate the original rectangle and produce an upright cross:

paper_cross_upright

paper_cross_upright_static

Pair on a D-String

What’s special about the binary number 10011 and the ternary number 1001120221? To answer the question, you have to see double. 10011 contains all possible pairs of numbers created from 0 and 1, just as 1001120221 contains all possible pairs created from 0, 1 and 2. And each pair appears exactly once. Now try the quaternary number 10011202130322331. That contains exactly one example of all possible pairs created from 0, 1, 2 and 3.

But there’s something more: in each case, the number is the smallest possible number with that property. As the bases get higher, that gets less obvious. In quinary, or base 5, the smallest number containing all possible pairs is 10011202130314042232433441. The digits look increasingly random. And what about base 10? There are 100 possible pairs of numbers created from the digits 0 to 9, starting with 00, 01, 02… and ending with …97, 98, 99. To accommodate 100 pairs, the all-pair number in base 10 has to be 101 digits long. It’s a string of digits, so let’s call it a d-string:

1, 0, 0, 1, 1, 2, 0, 2, 1, 3, 0, 3, 1, 4, 0, 4, 1, 5, 0, 5, 1, 6, 0, 6, 1, 7, 0, 7, 1, 8, 0, 8, 1, 9, 0, 9, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 5, 5, 6, 5, 7, 5, 8, 5, 9, 6, 6, 7, 6, 8, 6, 9, 7, 7, 8, 7, 9, 8, 8, 9, 9, 1

Again, the digits look increasingly random. They aren’t: they’re strictly determined. The d-string is in harmony. As the digits are generated from the left, they impose restrictions on the digits that appear later. It might appear that you could shift larger digits to the right and make the number smaller, but if you do that you no longer meet the conditions and the d-string collapses into dischord.

Now examine d-strings containing all possible triplets created from the digits of bases 2, 3 and 4:

1, 0, 0, 0, 1, 0, 1, 1, 1, 0 in base 2 = 558 in base 10

1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 2, 1, 1, 1, 2, 0, 1, 2, 1, 2, 2, 0, 2, 2, 2, 1, 0 in base 3 = 23203495920756 in base 10

1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 2, 1, 0, 3, 0, 0, 3, 1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 0, 1, 3, 1, 2, 2, 0, 2, 2, 1, 2, 3, 0, 2, 3, 1, 3, 2, 0, 3, 2, 1, 3, 3, 0, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 1, 0 in base 4 = 1366872334420014346556556812432766057460 in base 10

Note that there are 8 possible triplets in base 2, so the all-triplet number has to be 10 digits long. In base 10, there are 1000 possible triplets, so the all-triplet number has to be 1002 digits long. Here it is:

1, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 2, 1, 0, 3, 0, 0, 3, 1, 0, 4, 0, 0, 4, 1, 0, 5, 0, 0, 5, 1, 0, 6, 0, 0, 6, 1, 0, 7, 0, 0, 7, 1, 0, 8, 0, 0, 8, 1, 0, 9, 0, 0, 9, 1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 0, 1, 3, 1, 1, 4, 0, 1, 4, 1, 1, 5, 0, 1, 5, 1, 1, 6, 0, 1, 6, 1, 1, 7, 0, 1, 7, 1, 1, 8, 0, 1, 8, 1, 1, 9, 0, 1, 9, 1, 2, 2, 0, 2, 2, 1, 2, 3, 0, 2, 3, 1, 2, 4, 0, 2, 4, 1, 2, 5, 0, 2, 5, 1, 2, 6, 0, 2, 6, 1, 2, 7, 0, 2, 7, 1, 2, 8, 0, 2, 8, 1, 2, 9, 0, 2, 9, 1, 3, 2, 0, 3, 2, 1, 3, 3, 0, 3, 3, 1, 3, 4, 0, 3, 4, 1, 3, 5, 0, 3, 5, 1, 3, 6, 0, 3, 6, 1, 3, 7, 0, 3, 7, 1, 3, 8, 0, 3, 8, 1, 3, 9, 0, 3, 9, 1, 4, 2, 0, 4, 2, 1, 4, 3, 0, 4, 3, 1, 4, 4, 0, 4, 4, 1, 4, 5, 0, 4, 5, 1, 4, 6, 0, 4, 6, 1, 4, 7, 0, 4, 7, 1, 4, 8, 0, 4, 8, 1, 4, 9, 0, 4, 9, 1, 5, 2, 0, 5, 2, 1, 5, 3, 0, 5, 3, 1, 5, 4, 0, 5, 4, 1, 5, 5, 0, 5, 5, 1, 5, 6, 0, 5, 6, 1, 5, 7, 0, 5, 7, 1, 5, 8, 0, 5, 8, 1, 5, 9, 0, 5, 9, 1, 6, 2, 0, 6, 2, 1, 6, 3, 0, 6, 3, 1, 6, 4, 0, 6, 4, 1, 6, 5, 0, 6, 5, 1, 6, 6, 0, 6, 6, 1, 6, 7, 0, 6, 7, 1, 6, 8, 0, 6, 8, 1, 6, 9, 0, 6, 9, 1, 7, 2, 0, 7, 2, 1, 7, 3, 0, 7, 3, 1, 7, 4, 0, 7, 4, 1, 7, 5, 0, 7, 5, 1, 7, 6, 0, 7, 6, 1, 7, 7, 0, 7, 7, 1, 7, 8, 0, 7, 8, 1, 7, 9, 0, 7, 9, 1, 8, 2, 0, 8, 2, 1, 8, 3, 0, 8, 3, 1, 8, 4, 0, 8, 4, 1, 8, 5, 0, 8, 5, 1, 8, 6, 0, 8, 6, 1, 8, 7, 0, 8, 7, 1, 8, 8, 0, 8, 8, 1, 8, 9, 0, 8, 9, 1, 9, 2, 0, 9, 2, 1, 9, 3, 0, 9, 3, 1, 9, 4, 0, 9, 4, 1, 9, 5, 0, 9, 5, 1, 9, 6, 0, 9, 6, 1, 9, 7, 0, 9, 7, 1, 9, 8, 0, 9, 8, 1, 9, 9, 0, 9, 9, 2, 2, 2, 3, 2, 2, 4, 2, 2, 5, 2, 2, 6, 2, 2, 7, 2, 2, 8, 2, 2, 9, 2, 3, 3, 2, 3, 4, 2, 3, 5, 2, 3, 6, 2, 3, 7, 2, 3, 8, 2, 3, 9, 2, 4, 3, 2, 4, 4, 2, 4, 5, 2, 4, 6, 2, 4, 7, 2, 4, 8, 2, 4, 9, 2, 5, 3, 2, 5, 4, 2, 5, 5, 2, 5, 6, 2, 5, 7, 2, 5, 8, 2, 5, 9, 2, 6, 3, 2, 6, 4, 2, 6, 5, 2, 6, 6, 2, 6, 7, 2, 6, 8, 2, 6, 9, 2, 7, 3, 2, 7, 4, 2, 7, 5, 2, 7, 6, 2, 7, 7, 2, 7, 8, 2, 7, 9, 2, 8, 3, 2, 8, 4, 2, 8, 5, 2, 8, 6, 2, 8, 7, 2, 8, 8, 2, 8, 9, 2, 9, 3, 2, 9, 4, 2, 9, 5, 2, 9, 6, 2, 9, 7, 2, 9, 8, 2, 9, 9, 3, 3, 3, 4, 3, 3, 5, 3, 3, 6, 3, 3, 7, 3, 3, 8, 3, 3, 9, 3, 4, 4, 3, 4, 5, 3, 4, 6, 3, 4, 7, 3, 4, 8, 3, 4, 9, 3, 5, 4, 3, 5, 5, 3, 5, 6, 3, 5, 7, 3, 5, 8, 3, 5, 9, 3, 6, 4, 3, 6, 5, 3, 6, 6, 3, 6, 7, 3, 6, 8, 3, 6, 9, 3, 7, 4, 3, 7, 5, 3, 7, 6, 3, 7, 7, 3, 7, 8, 3, 7, 9, 3, 8, 4, 3, 8, 5, 3, 8, 6, 3, 8, 7, 3, 8, 8, 3, 8, 9, 3, 9, 4, 3, 9, 5, 3, 9, 6, 3, 9, 7, 3, 9, 8, 3, 9, 9, 4, 4, 4, 5, 4, 4, 6, 4, 4, 7, 4, 4, 8, 4, 4, 9, 4, 5, 5, 4, 5, 6, 4, 5, 7, 4, 5, 8, 4, 5, 9, 4, 6, 5, 4, 6, 6, 4, 6, 7, 4, 6, 8, 4, 6, 9, 4, 7, 5, 4, 7, 6, 4, 7, 7, 4, 7, 8, 4, 7, 9, 4, 8, 5, 4, 8, 6, 4, 8, 7, 4, 8, 8, 4, 8, 9, 4, 9, 5, 4, 9, 6, 4, 9, 7, 4, 9, 8, 4, 9, 9, 5, 5, 5, 6, 5, 5, 7, 5, 5, 8, 5, 5, 9, 5, 6, 6, 5, 6, 7, 5, 6, 8, 5, 6, 9, 5, 7, 6, 5, 7, 7, 5, 7, 8, 5, 7, 9, 5, 8, 6, 5, 8, 7, 5, 8, 8, 5, 8, 9, 5, 9, 6, 5, 9, 7, 5, 9, 8, 5, 9, 9, 6, 6, 6, 7, 6, 6, 8, 6, 6, 9, 6, 7, 7, 6, 7, 8, 6, 7, 9, 6, 8, 7, 6, 8, 8, 6, 8, 9, 6, 9, 7, 6, 9, 8, 6, 9, 9, 7, 7, 7, 8, 7, 7, 9, 7, 8, 8, 7, 8, 9, 7, 9, 8, 7, 9, 9, 8, 8, 8, 9, 8, 9, 9, 9, 1, 0

Consider the quadruplet number in base 10. There are 10000 possible quadruplets, so the all-quadruplet number is 10003 digits long. And so on. In general, the “all n-tuplet” number in base b contains b^n n-tuplets and is (b^n + n-1) digits long. If b = 10 and n = 4, the d-string starts like this:

1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 1, 0, 0, 3, 0, 0, 0, 3, 1, 0, 0, 4, 0, 0, 0, 4, 1, 0, 0, 5, 0, 0, 0, 5, 1, 0, 0, 6, 0, 0, 0, 6, 1, 0, 0, 7, 0, 0, 0, 7, 1, 0, 0, 8, 0, 0, 0, 8, 1, 0, 0, 9, 0, 0, 0, 9, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 1, 2, 1, 0, 1, 3, 0, 0, 1, 3, 1, 0, 1, 4, 0, 0, 1, 4, 1, 0, 1, 5, 0, 0, 1, 5, 1, 0, 1, 6, 0, 0, 1, 6, 1, 0, 1, 7, 0, 0, 1, 7, 1, 0, 1, 8, 0, 0, 1, 8, 1, 0, 1, 9, 0, 0, 1, 9, 1, 0, 2, 0, 1, 0, 2, 1, 1, 0, 2, 2, 0, 0, 2, 2, 1, 0, 2, 3, 0, 0, 2, 3, 1, 0, 2, 4, 0, 0, 2, 4, 1, 0, 2, 5, 0, 0, 2, 5, 1, 0, 2, 6…

What about when n = 100? Now the d-string is ungraspably huge – too big to fit in the known universe. But it starts with 1 followed by a hundred 0s and every digit after that is entirely determined. Perhaps there’s a simple way to calculate any given digit, given its position in the d-string. Either way, what is the ontological status of the d-string for n=100? Does it exist in some Platonic realm of number, independent of physical reality?

Some would say that it does, just like √2 or π or e. I disagree. I don’t believe in a Platonic realm. If the universe or multiverse ceased to exist, numbers and mathematics in general would also cease to exist. But this isn’t to say that mathematics depends on physical reality. It doesn’t. Nor does physical reality depend on mathematics. Rather, physical reality necessarily embodies mathematics, which might be defined as “entity in interrelation”. Humans have invented small-m mathematics, a symbolic way of expressing the physical embodiment of big-m mathematics.

But small-m mathematics is actually more powerful and far-ranging, because it increases the number, range and power of entities and their interaction. Where are √2 and π in physical reality? Nowhere. You could say that early mathematicians saw their shadows, cast from a Platonic realm, and deduced their existence in that realm, but that’s a metaphor. Do all events, like avalanches or thunderstorms, exist in some Platonic realm before they are realized? No, they arise as physical entities interact according to laws of physics. In a more abstract way, √2 and π arise as entities of another kind interact according to laws of logic: the concepts of a square and its diagonal, of a circle and its diameter.

The d-strings discussed above arise from the interaction of simpler concepts: the finite set of digits in a base and the ways in which they can be combined. Platonism is unnecessary: the arc and spray of a fountain are explained by the pressure of the water, the design of the pipes, the arrangement of the nozzles, not by reference to an eternal archetype of water and spray. In small-m mathematics, there are an infinite number of fountains, because small-m mathematics opens a door to a big-U universe, infinitely larger and richer than the small-u universe of physical reality.

Rep-Tile Reflections

A rep-tile, or repeat-tile, is a two-dimensional shape that can be divided completely into copies of itself. A square, for example, can be divided into smaller squares: four or nine or sixteen, and so on. Rectangles are the same. Triangles can be divided into two copies or three or more, depending on their precise shape. Here are some rep-tiles, including various rep-triangles:

Various rep-tiles

Various rep-tiles — click for larger image

Some are simple, some are complex. Some have special names: the sphinx and the fish are easy to spot. I like both of those, particularly the fish. It would make a good symbol for a religion: richly evocative of life, eternally sub-divisible of self: 1, 9, 81, 729, 6561, 59049, 531441… I also like the double-square, the double-triangle and the T-tile in the top row. But perhaps the most potent, to my mind, is the half-square in the bottom left-hand corner. A single stroke sub-divides it, yet its hypotenuse, or longer side, represents the mysterious and mind-expanding √2, a number that exists nowhere in the physical universe. But the half-square itself is mind-expanding. All rep-tiles are. If intelligent life exists elsewhere in the universe, perhaps other minds are contemplating the fish or the sphinx or the half-square and musing thus: “If intelligent life exists elsewhere in the universe, perhaps…”

Mathematics unites human minds across barriers of language, culture and politics. But perhaps it unites minds across barriers of biology too. Imagine a form of life based on silicon or gas, on unguessable combinations of matter and energy in unreachable, unobservable parts of the universe. If it’s intelligent life and has discovered mathematics, it may also have discovered rep-tiles. And it may be contemplating the possibility of other minds doing the same. And why confine these speculations to this universe and this reality? In parallel universes, in alternative realities, minds may be contemplating rep-tiles and speculating in the same way. If our universe ends in a Big Crunch and then explodes again in a Big Bang, intelligent life may rise again and discover rep-tiles again and speculate again on their implications. The wildest speculation of all would be to hypothesize a psycho-math-space, a mental realm beyond time and matter where, in mathemystic communion, suitably attuned and aware minds can sense each other’s presence and even communicate.

The rep-tile known as the fish

Credo in Piscem…

So meditate on the fish or the sphinx or the half-square. Do you feel the tendrils of an alien mind brush your own? Are you in communion with a stone-being from the far past, a fire-being from the far future, a hive-being from a parallel universe? Well, probably not. And even if you do feel those mental tendrils, how would you know they’re really there? No, I doubt that the psycho-math-space exists. But it might and science might prove its existence one day. Another possibility is that there is no other intelligent life, never has been, and never will be. We may be the only ones who will ever muse on rep-tiles and other aspects of mathematics. Somehow, though, rep-tiles themselves seem to say that this isn’t so. Particularly the fish. It mimics life and can spawn itself eternally. As I said, it would make a good symbol for a religion: a mathemysticism of trans-biological communion. Credo in Piscem, Unum et Infinitum et Æternum. “I believe in the Fish, One, Unending, Everlasting.” That might be the motto of the religion. If you want to join it, simply wish upon the fish and muse on other minds, around other stars, who may be doing the same.