Polymorphous Pursuit

Suppose four mice are standing on the corners of a large square. Each mouse begins running at the same speed towards the mouse one place away, reckoning clockwise. The mice will meet at the centre of the square and the path taken by each mouse will be what is known as a pursuit curve:

v4_mi1

vertices = 4, mouse-increment = 1


v4_mi1_animated

v = 4, mi = 1 (animated)


As I showed in “Persecution Complex”, it’s easy to find variants on the basic pursuit curve. If mi = 2, i.e. each mouse runs towards the mouse two places away, the mice will run in straight lines direct to the centre of the square:

v4_mi2

v = 4, mi = 2


v4_mi2_animated

v = 4, mi = 2 (animated)


That variant is trivial, but suppose there are eight mice, four starting on the corners of the square and four starting on the midpoints of the sides. Mice starting on the corners will run different pursuit curves to those starting on the midpoints, because the corners are further from the centre than the midpoints are:

v4_si1_mi1

v = 4, si = 1, mi = 1


v4_si1_mi1_extra


If mi = 3, the pursuit curves look like this:

v4_si1_mi3

v = 4, si = 1, mi = 3


v4_si1_mi3_animated

v = 4, si = 1, mi = 3 (animated)


Suppose there are twelve mice, four on each corner and two more on each side. If each mouse runs towards the mouse four places away, then the pursuit curves don’t all meet in the centre of the square. Instead, they meet in groups of three at four points equidistant from the centre, like this:

v4_si2_mi4

v4_si2_mi4_curves

v = 4, si = 2, mi = 4


v4_si2_mi4_animated

v = 4, si = 2, mi = 4 (animated)


v4_si4_mi4_animated

v = 4, si = 4, mi = 4 (animated)


v4_si4_mi4_large

v = 4, si = 4, mi = 4 (zoom)


Now suppose each mouse become sophisticated and runs toward the combined positions of two other mice, one two places away, the other three places away, like this:

v4_si1_mi2_3

v = 4, si = 1, mi = (2, 3)


v4_si1_mi2_3_animated

v = 4, si = 1, mi = (2, 3) (animated)


These polypursuits, as they could be called, can have complicated central regions:

v4_si2_mi1_4

v = 4, si = 2, mi = (1, 4)


v4_si2_mi1_4_animated

v = 4, si = 2, mi = (1, 4) (animated)


v4_si_va_mi_va

v = 4, si = various, mi = various


And what if you have two teams of mice, running towards one or more mice on the other team? For example, suppose two mice, one from each team, start on each corner of a square. Each mouse on team 1 runs towards the mouse on team 2 that is one place away, while each mouse on team 2 runs towards the mouse on team 1 that is two places away. If the pursuits curves of team 1 are represented in white and the pursuit curves of team 2 in green, the curves look like this:

2v4_mi1_mi2

v = 4 * 2, vmi = 1, vmi = 2


2v4_mi1_mi2_green

v = 4 * 2, vmi = 1, vmi = 2


2v4_mi1_mi2_animated

v = 4 * 2, vmi = 1, vmi = 2 (animated)


Now suppose the four mice of team 1 start on the corners while the mice of team 2 start at the centre of the square.

v4_c4_vmi1_cmi2_white

v = 4, centre = 4, vmi = 1, cmi = 2 (white team)


v4_c4_vmi1_cmi2_green

v = 4, centre = 4, vmi = 1, cmi = 2 (green team)


v4_c4_vmi1_cmi2_both

v = 4, centre = 4, vmi = 1, cmi = 2 (both teams)


v4_c4_vmi1_cmi2_animated

v = 4, centre = 4, vmi = 1, cmi = 2 (animated)


Here are more variants on pursuit curves formed by two teams of mice, one starting on the corners, one at the centre:

v4_c4_vmi0_1_cmi0

v = 4, centre = 4, vmi = (0, 1), cmi = 0


v4_c4_vmi0_2_cmi0

v = 4, centre = 4, vmi = (0, 2), cmi = 0


v4_c4_vmi0_3_cmi0

v = 4, centre = 4, vmi = (0, 3), cmi = 0


2v4_mi1_mi2_both

Go with the Floe

Fractals are shapes that contain copies of themselves on smaller and smaller scales. There are many of them in nature: ferns, trees, frost-flowers, ice-floes, clouds and lungs, for example. Fractals are also easy to create on a computer, because you all need do is take a single rule and repeat it at smaller and smaller scales. One of the simplest fractals follows this rule:

1. Take a line of length l and find the midpoint.
2. Erect a new line of length l x lm on the midpoint at right angles.
3. Repeat with each of the four new lines (i.e., the two halves of the original line and the two sides of the line erected at right angles).

When lm = 1/3, the fractal looks like this:

stick1

(Please open image in a new window if it fails to animate)

When lm = 1/2, the fractal is less interesting:

stick2

But you can adjust rule 2 like this:

2. Erect a new line of length l x lm x lm1 on the midpoint at right angles.

When lm1 = 1, 0.99, 0.98, 0.97…, this is what happens:

stick3

The fractals resemble frost-flowers on a windowpane or ice-floes on a bay or lake. You can randomize the adjustments and angles to make the resemblance even stronger:

frostfloe

Ice floes (see Owen Kanzler)

Ice floes (see Owen Kanzler)

Frost on window (see Kenneth G. Libbrecht, )

Frost on window (see Kenneth G. Libbrecht)

Know Your Limaçons

Front cover of The Penguin Dictionary of Curious and Interesting Geometry by David WellsThe Penguin Dictionary of Curious and Interesting Geometry, David Wells (1991)

Mathematics is an ocean in which a child can paddle and an elephant can swim. Or a whale, indeed. This book, a sequel to Wells’ excellent Penguin Dictionary of Curious and Interesting Mathematics, is suitable for both paddlers and plungers. Plumbers, even, because you can dive into some very deep mathematics here.

Far too deep for me, I have to admit, but I can wade a little way into the shallows and enjoy looking further out at what I don’t understand, because the advantage of geometry over number theory is that it can appeal to the eye even when it baffles the brain. If this book is more expensive than its prequel, that’s because it needs to be. It’s a paperback, but a large one, to accommodate the illustrations.

Fortunately, plenty of them appeal to the eye without baffling the brain, like the absurdly simple yet mindstretching Koch snowflake. Take a triangle and divide each side into thirds. Erect another triangle on each middle third. Take each new line of the shape and do the same: divide into thirds, erect another triangle on the middle third. Then repeat. And repeat. For ever.

A Koch snowflake (from Wikipedia)

A Koch snowflake (from Wikipedia)

The result is a shape with a finite area enclosed by an infinite perimeter, and it is in fact a very early example of a fractal. Early in this case means it was invented in 1907, but many of the other beautiful shapes and theorems in this book stretch back much further: through Étienne Pascal and his oddly organic limaçon (which looks like a kidney) to the ancient Greeks and beyond. Some, on the other hand, are very modern, and this book was out-of-date on the day it was printed. Despite the thousands of years devoted by mathematicians to shapes and the relationship between them, new discoveries are being made all the time. Knots have probably been tied by human beings for as long as human beings have existed, but we’ve only now started to classify them properly and even find new uses for them in biology and physics.

Which is not to say knots are not included here, because they are. But even the older geometry Wells looks at would be enough to keep amateur and recreational mathematicians happy for years, proving, re-creating, and generalizing as they work their way through variations on all manner of trigonomic, topological, and tessellatory themes.


Previously pre-posted (please peruse):

Poulet’s Propeller — discussion of Wells’ Penguin Dictionary of Curious and Interesting Numbers (1986)

Persecution Complex

Imagine four mice sitting on the corners of a square. Each mouse begins to run towards its clockwise neighbour. What happens? This:

Four mice chasing each other

Four mice chasing each other


The mice spiral to the centre and meet, creating what are called pursuit curves. Now imagine eight mice on a square, four sitting on the corners, four sitting on the midpoints of the sides. Each mouse begins to run towards its clockwise neighbour. Now what happens? This:

Eight mice chasing each other

Eight mice chasing each other


But what happens if each of the eight mice begins to run towards its neighbour-but-one? Or its neighbour-but-two? And so on. The curves begin to get more complex:

square+midpoint+2


(Please open the following image in a new window if it fails to animate.)

square+midpoint+3


You can also make the mice run at different speeds or towards neighbours displaced by different amounts. As these variables change, so do the patterns traced by the mice:

• Continue reading Persecution Complexified

V for Vertex

To create a simple fractal, take an equilateral triangle and divide it into four more equilateral triangles. Remove the middle triangle. Repeat the process with each new triangle and go on repeating it. You’ll end up with a shape like this, which is known as the Sierpiński triangle, after the Polish mathematician Wacław Sierpiński (1882-1969):

Sierpinski triangle

But you can also create the Sierpiński triangle one pixel at a time. Choose any point inside an equilateral triangle. Pick a corner of the triangle at random and move half-way towards it. Mark this spot. Then pick a corner at random again and move half-way towards the corner. And repeat. The result looks like this:

triangle

A simple program to create the fractal looks like this:

initial()
repeat
  fractal()
  altervariables()
until false

function initial()
  v = 3 [v for vertex]
  r = 500
  lm = 0.5
endfunc

function fractal()
  th = 2 * pi / v
[the following loop creates the corners of the triangle]
  for l = 1 to v
    x[l]=xcenter + sin(l*th) * r
    y[l]=ycenter + cos(l*th) * r
  next l
  fx = xcenter
  fy = ycenter
  repeat
    rv = random(v)
    fx = fx + (x[rv]-fx) * lm
    fy = fy + (y[rv]-fy) * lm
    plot(fx,fy)
  until keypressed
endfunc

function altervariables()
[change v, lm, r etc]
endfunc

In this case, more is less. When v = 4 and the shape is a square, there is no fractal and plot(fx,fy) covers the entire square.

square

When v = 5 and the shape is a pentagon, this fractal appears:

pentagon

But v = 4 produces a fractal if a simple change is made in the program. This time, a corner cannot be chosen twice in a row:

square_used1

function initial()
  v = 4
  r = 500
  lm = 0.5
  ci = 1 [i.e, number of iterations since corner previously chosen]
endfunc

function fractal()
  th = 2 * pi / v
  for l = 1 to v
    x[l]=xcenter + sin(l*th) * r
    y[l]=ycenter + cos(l*th) * r
    chosen[l]=0
  next l
  fx = xcenter
  fy = ycenter
  repeat
    repeat
      rv = random(v)
    until chosen[rv]=0
    for l = 1 to v
      if chosen[l]>0 then chosen[l] = chosen[l]-1
    next l
    chosen[rv] = ci
    fx = fx + (x[rv]-fx) * lm
    fy = fy + (y[rv]-fy) * lm
    plot(fx,fy)
  until keypressed
endfunc

One can also disallow a corner if the corner next to it has been chosen previously, adjust the size of the movement towards the chosen corner, add a central point to the polygon, and so on. Here are more fractals created with such variations:

square_used1_center

square_used1_vi1

square_used1_vi2

square_used2

pentagon_lm0.6

pentagon_used1_5_vi1

hexagon_used1_6_vi3

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.