Solids and Shadows

Thursday, 24th Oct, 2013 by Krilling for Company in Aesthetics, Book Reviews, Geometry, Mathematics, Polyhedra and tagged 2-space, 3-space, An Adventure in Multidimensional Space: The Art and Geometry of Polygons, and Polytopes, art and architecture, beautiful shapes, closed shapes, dodecahedral pollen, elegance and simplicity, English language, four dimensions, geometry in nature, Gypsophilum elegans, inspired by geometry, Japanese mathematician, Japanese mathematics, Japanese spies and assassins, Koji Miyazaki, math, mathematics, maths, natural caltrops, ninja, Polygons, polyhedra, polytope shadows, polytopes, simplicity and elegance, tetrahedral seeds of the Water Chestnut, the universality of mathematics, three dimensions, three-space, two dimensions, two-space | 2 Comments

An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes, Koji Miyazaki (Wiley-Interscience 1987)

Two, three, four – or rather, two, three, ∞. Polygons are closed shapes in two dimensions (e.g., the square), polyhedra closed shapes in three dimensions (the cube), and polytopes closed shapes in four or more (the hypercube). You could spend a lifetime exploring any one of these geometries, but unless you take psychedelic drugs or brain-modification becomes much more advanced, you’ll be able to see only two of them: the geometries of polygons and polyhedra. Polytopes are beyond imagining but you can glimpse their shadows here – literally, because we can represent polytopes by the shadows they cast in 3-space or by the shadows of their shadows in 2-space.

A four-dimensional shape in two dimensions (see Tesseract)

Elsewhere Miyazaki doesn’t have to convey wonder and beauty by shadows: not only is this book full of beautiful shapes, it’s beautifully designed too and the way it alternates black-and-white pages with colour actually increases the power of both. It isn’t restricted to pure mathematics either: Miyazaki also looks at the modern and ancient art and architecture inspired by geometry, and at geometry in nature: the dodecahedral pollen of Gypsophilum elegans (Showy Baby’s-Breath), for example, and the tetrahedral seeds of the Water Chestnut (Trapa spp.), which the Japanese spies and assassins called the ninja used as natural caltrops. A regular tetrahedron always lies on a flat surface with a vertex facing directly upward, and when a pursued ninja scattered the sharply pointed tetrahedral seeds of the Water Chestnut, they were regular enough to injure “the soles of feet of his pursuers”.

Polyhedral plankton by Ernst Haeckel

The slightly odd English there is another example of what I like about this book, because it proves the parochialism of language and the universality of mathematics. Miyazaki’s mathematics, as far as I can tell, is flawless, like that of many other Japanese mathematicians, but his self-translated English occasionally isn’t. Japanese mathematics was highly developed before Japan fell under strong Western influence. It would continue to develop if the West disappeared tomorrow. Language is something we have to absorb intuitively from the particular culture we’re born into, but mathematics is learnt and isn’t tied to any particular culture. That’s why it’s accessible in the same way to minds everywhere in the world. Miyazaki’s pictures and prose are an extended proof of all that, and the book is actually more valuable because it was written by a Japanese speaker. I think it’s probably more attractively designed for the same reason: the skill with which the pictures have been selected and laid out reflects something characteristically Japanese. Elegance and simplicity perhaps sum it up, and elegance and simplicity are central to mathematics and some of the greatest art.

Another four-dimensional shape in two dimensions (see 120-cell)

Go with the Floe

Thursday, 17th Oct, 2013 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated fractal, animated fractals, animated gif, animated gifs, floe, floes, fractal, fractals, frost, frost-flowers, frozen water, geometrical, geometry, ice, ice-floe, ice-floes, math, mathematics, maths, recreational math, recreational mathematics, recreational maths, simple fractal, simple fractals | Leave a comment

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:

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

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

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:

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:

Ice floes (see Owen Kanzler)

Frost on window (see Kenneth G. Libbrecht)

Know Your Limaçons

Friday, 20th Sep, 2013 by Krilling for Company in Book Reviews, Fractals, Geometry, Mathematics and tagged Étienne Pascal, David Wells, fractal, fractals, geometer, geometrical, geometry, knots, Koch snowflake, limaçon of Pascal, math, mathematics, maths, recreational math, recreational mathematics, recreational maths, tessellation, The Penguin Dictionary of Curious and Interesting Geometry, topology, trigonometry | Leave a comment

The 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)

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

Thursday, 12th Sep, 2013 by Krilling for Company in Geometry, Mathematics, Pursuit Curves and tagged animated gif, animated gifs, beetle problem, geometrical, geometry, logarithmic spiral, logarithmic spirals, math, math animation, math animations, mathematics, maths, maths animation, maths animations, mice problem, mouse problem, pursuit curve, Pursuit Curves, recreational math, recreational mathematics, recreational maths, spiral, spirals | Leave a comment

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

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

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:

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

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

Neuclid on the Block

Friday, 17th May, 2013 by Krilling for Company in Civilization, Genetics, Geometry, History, Mathematics, Philosophy, Philosophy of mathematics, Prime numbers and tagged abstraction, accounting, agriculture, Alexandrian mathematician, Alexandrian mathematics, animal counting, Archimedes cattle problem, architecture, Arithmetic, Babylonian math, Babylonian mathematics, Babylonian maths, Beethoven, black mathematician Jonathan Farley, brain, brain as mathematical engine, brains as mathematical engines, counting, Euclid's Elements, exercitium arithmeticae occultum nescientis se numerare animi, farmers, farming, genes for mathematical ability, genes for mathematics, genetics, Greek mathematician, Greek mathematics, Greek text of Euclid, hammer-blow against reality, hammer-blows against reality, HBD, human bio-diversity, human brain, infinity, innumeracy, internal logic, invention of writing, π, π as 3, π in the Bible, language and mathematics, Leibniz, line, lines, logic, logical, math, mathematical proof, mathematical reality, mathematician John Derbyshire, mathematics, mathematics and language, mathematics and writing, maths, Mesopotamia, Mesopotamian mathematics, mightier than a million suns, Monkey math, music, myth and math, myths and maths, Neuclid on the Block, non-human primates, number, numbers, numerals, Penguin Dictionary of Curious and Interesting Numbers by David Wells, philosophy, philosophy of mathematics, physical reality, pi, pi as 3, pi in the Bible, Pirahã language, point, points, prime numbers, primes, proof, reality, surface, surfaces, symbol, symbols, writing and mathematics | Leave a comment

How many blows does it take to demolish a wall with a hammer? It depends on the wall and the hammer, of course. If the wall is reality and the hammer is mathematics, you can do it in three blows, like this:

α’. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν.
β’. Γραμμὴ δὲ μῆκος ἀπλατές.
γ’. Γραμμῆς δὲ πέρατα σημεῖα.

1. A point is that of which there is no part.
2. A line is a length without breadth.
3. The extremities of a line are points.

That is the astonishing, world-shattering opening in one of the strangest – and sanest – books ever written. It’s twenty-three centuries old, was written by an Alexandrian mathematician called Euclid (fl. 300 B.C.), and has been pored over by everyone from Abraham Lincoln to Bertrand Russell by way of Edna St. Vincent Millay. Its title is highly appropriate: Στοιχεῖα, or Elements. Physical reality is composed of chemical elements; mathematical reality is composed of logical elements. The second reality is much bigger – infinitely bigger, in fact. In his Elements, Euclid slipped the bonds of time, space and matter by demolishing the walls of reality with a mathematical hammer and escaping into a world of pure abstraction.

• Continue reading Neuclid on the Block

O Apollo

Friday, 17th May, 2013 by Krilling for Company in Fractals, Geometry, Mathematics, Music, Poetry, Swinburne and tagged "The Last Oracle", A.C. Swinburne, Algernon Charles Swinburne, animated fractal, animated fractals, animated gif, animated gifs, Apollo, Apollonian gasket, Apollonian net, Beethoven, fractal, fractals, Greek mathematician Apollonius of Perga, intra-Apollonian gasket, intra-Apollonian net, math, mathematics, music, musician, poem, poems, Poems and Ballads Second Series (1878), poet, Poetry, Swinburne, the god Apollo | Leave a comment

One of Swinburne’s most powerful, but least-known, poems is “The Last Oracle”, from Poems and Ballads, Second Series (1878). A song in honour of the god Apollo, it begins in lamentation:

Years have risen and fallen in darkness or in twilight,
   Ages waxed and waned that knew not thee nor thine,
While the world sought light by night and sought not thy light,
   Since the sad last pilgrim left thy dark mid shrine.
Dark the shrine and dumb the fount of song thence welling,
   Save for words more sad than tears of blood, that said:
Tell the king, on earth has fallen the glorious dwelling,
   And the watersprings that spake are quenched and dead.
Not a cell is left the God, no roof, no cover
   In his hand the prophet laurel flowers no more.

And ends in exultation:

         For thy kingdom is past not away,
            Nor thy power from the place thereof hurled;
         Out of heaven they shall cast not the day,
            They shall cast not out song from the world.
         By the song and the light they give
         We know thy works that they live;
         With the gift thou hast given us of speech
         We praise, we adore, we beseech,
         We arise at thy bidding and follow,
            We cry to thee, answer, appear,
   O father of all of us, Paian, Apollo,
            Destroyer and healer, hear! (“The Last Oracle”)

The power, grandeur and beauty of this poem remind me of the music of Beethoven. Swinburne is also, on a smaller scale and in a different medium, one of the geniuses of European art. He and Beethoven were both touched by Apollo, but Apollo was more than the god of music and poetry: he also presided mathematics. But then mathematics is much more visible, or audible, in music and poetry than it is in other arts. Rhythm, harmony, scansion, melody and rhyme are mathematical concepts. Music is built of notes, poetry of stresses and rhymes, and the rules governing them are easier to formalize than those governing, say, sculpture or prose. Nor do poetry and music have to make sense or convey explicit meaning like other arts. That’s why I think a shape like this is closer to poetry or music than it is to painting:

(Image from Wikipedia.)

This shape has formal structure and beauty, but it has no explicit meaning. Its name has a divine echo: the Apollonian gasket or net, named after the Greek mathematician Apollonius of Perga (c.262 BC–c.190 BC), who was named after Apollo, god of music, poetry and mathematics. The Apollonian gasket is a fractal, but the version above is not as fractal as it could be. I wondered what it would look like if, like fleas preying on fleas, circles appeared inside circles, gaskets within gaskets. I haven’t managed to program the shape properly yet, but here is my first effort at an Intra-Apollonian gasket:

(If the image does not animate or looks distorted, please try opening it in a new window)

When the circles are solid, they remind me of ice-floes inside ice-floes:

Simpler gaskets can be interesting too:

Tri Again

Friday, 3rd May, 2013 by Krilling for Company in Fractals, Geometry, Mathematics, Our Lady of the Gutter and tagged animated fractal, animated fractals, animated gif, animated gifs, arrowhead fractal, arrowhead fractals, equilateral triangle, equilateral triangles, fractal, fractal method, fractal methods, fractals, geometric, geometry, math, mathematics, maths, orientation, orientations, Polish mathematician Wacław Sierpiński, random jump, Sierpiński triangle, triangle, Triangles, triangular fractal, triangular fractals | Leave a comment

All roads lead to Rome, so the old saying goes. But you may get your feet wet, so try the Sierpiński triangle instead. This fractal is named after the Polish mathematician Wacław Sierpiński (1882-1969) and quite a few roads lead there too. You can create it by deleting, jumping or bending, inter alia. Here is method #1:

Divide an equilateral triangle into four, remove the central triangle, do the same to the new triangles.

Here is method #2:

Pick a corner at random, jump half-way towards it, mark the spot, repeat.

And here is method #3:

Bend a straight line into a hump consisting of three straight lines, then repeat with each new line.

Each method can be varied to create new fractals. Method #3, which is also known as the arrowhead fractal, depends on the orientation of the additional humps, as you can see from the animated gif above. There are eight, or 2 x 2 x 2, ways of varying these three orientations, so eight fractals can be produced if the same combination of orientations is kept at each stage, like this (some are mirror images — if an animated gif doesn’t work, please open it in a new window):

If different combinations are allowed at different stages, the number of different fractals gets much bigger:

• Continuing viewing Tri Again.

V for Vertex

Thursday, 25th Apr, 2013 by Krilling for Company in Fractals, Geometry, Mathematics, Randomness and tagged equilateral triangle, equilateral triangles, fractal, fractal hexagon, fractal pentagon, fractal pentagons, fractal polygon, fractal polygons, fractal program, fractal square, fractal squares, fractal triangle, fractals, geometrical, geometry, math, mathematics, maths, pentagon, Pentagons, program, Programming, random, Sierpiński gasket, Sierpiński sieve, Sierpiński triangle, simple fractal, simple fractals, square, Squares, triangle, Triangles | Leave a comment

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):

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:

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.

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

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:

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:

Live and Let Dice

Thursday, 28th Mar, 2013 by Krilling for Company in Geometry, Mathematics, Polyhedra, Probability, Statistics and tagged Big Bang, caltrop, caltrops, center of gravity, centre of gravity, cube, cubic dice, cubic die, death-dice, death-die, death-face, death-penalty, dice odds, dice-roll, dice-rolling, dice-rolls, die-rolling, dodecahedral dice, dodecahedral die, dodecahedron, edge, edges, execution, face, faces, faces and vertices, fair dice, geometric, geometry, hexahedral dice, hexahedral die, hexahedron, icosahedral dice, icosahedral die, icosahedron, life-imprisonment, math, mathematics, maths, octahedral dice, octahedral die, octahedron, odds, Plato, Plato's cosmology, Platonic polyhedra, Platonic polyhedron, polyhedra, polyhedral, polyhedron, probability, probability theory, roll a die, rolling dice, solid shape, solid shapes, spiked weapon, tetrahedral dice, tetrahedral die, tetrahedron, three-dimensional shape, three-dimensional shapes, Tycholatric, Tycholatry, vertex, vertices | Leave a comment

How many ways are there to die? The answer is actually five, if by “die” you mean “roll a die” and by “rolled die” you mean “Platonic polyhedron”. The Platonic polyhedra are the solid shapes in which each polygonal face and each vertex (meeting-point of the edges) are the same. There are surprisingly few. Search as long and as far as you like: you’ll find only five of them in this or any other universe. The standard cubic die is the most familiar: each of its six faces is square and each of its eight vertices is the meeting-point of three edges. The other four Platonic polyhedra are the tetrahedron, with four triangular faces and four vertices; the octahedron, with eight triangular faces and six vertices; the dodecahedron, with twelve pentagonal faces and twenty vertices; and the icosahedron, with twenty triangular faces and twelve vertices. Note the symmetries of face- and vertex-number: the dodecahedron can be created inside the icosahedron, and vice versa. Similarly, the cube, or hexahedron, can be created inside the octahedron, and vice versa. The tetrahedron is self-spawning and pairs itself. Plato wrote about these shapes in his Timaeus (c. 360 B.C.) and based a mathemystical cosmology on them, which is why they are called the Platonic polyhedra.

Tetrahedron

Hexahedron

Octahedron

Dodecahedron

Icosahedron

They make good dice because they have no preferred way to fall: each face has the same relationship with the other faces and the centre of gravity, so no face is likelier to land uppermost. Or downmost, in the case of the tetrahedron, which is why it is the basis of the caltrop. This is a spiked weapon, used for many centuries, that always lands with a sharp point pointing upwards, ready to wound the feet of men and horses or damage tyres and tracks. The other four Platonic polyhedra don’t have a particular role in warfare, as far as I know, but all five might have a role in jurisprudence and might raise an interesting question about probability. Suppose, in some strange Tycholatric, or fortune-worshipping, nation, that one face of each Platonic die represents death. A criminal convicted of a serious offence has to choose one of the five dice. The die is then rolled f times, or as many times as it has faces. If the death-face is rolled, the criminal is executed; if not, he is imprisoned for life.

The question is: Which die should he choose to minimize, or maximize, his chance of getting the death-face? Or doesn’t it matter? After all, for each die, the odds of rolling the death-face are 1/f and the die is rolled f times. Each face of the tetrahedron has a 1/4 chance of being chosen, but the tetrahedron is rolled only four times. For the icosahedron, it’s a much smaller 1/20 chance, but the die is rolled twenty times. Well, it does matter which die is chosen. To see which offers the best odds, you have to raise the odds of not getting the death-face to the power of f, like this:

3/4 x 3/4 x 3/4 x 3/4 = 3/4 ^4 = 27/256 = 0·316…

5/6 ^6 = 15,625 / 46,656 = 0·335…

7/8 ^8 = 5,764,801 / 16,777,216 = 0·344…

11/12 ^12 = 3,138,428,376,721 / 8,916,100,448,256 = 0·352…

19/20 ^20 = 37,589,973,457,545,958,193,355,601 / 104,857,600,000,000,000,000,000,000 = 0·358…

Those represent the odds of avoiding the death-face. Criminals who want to avoid execution should choose the icosahedron. For the odds of rolling the death-face, simply subtract the avoidance-odds from 1, like this:

1 – 3/4 ^4 = 0·684…

1 – 5/6 ^6 = 0·665…

1 – 7/8 ^8 = 0·656…

1 – 11/12 ^12 = 0·648…

1 – 19/20 ^20 = 0·642…

So criminals who prefer execution to life-imprisonment should choose the tetrahedron. If the Tycholatric nation offers freedom to every criminal who rolls the same face of the die f times, then the tetrahedron is also clearly best. The odds of rolling a single specified face f times are 1/f ^f:

1/4 x 1/4 x 1/4 x 1/4 = 1/4^4 = 1 / 256

1/6^6 = 1 / 46,656

1/8^8 = 1 / 16,777,216

1/12^12 = 1 / 8,916,100,448,256

1/20^20 = 1 / 104,857,600,000,000,000,000,000,000

But there are f faces on each polyhedron, so the odds of rolling any face f times are 1/f ^(f-1). On average, of every sixty-four (256/4) criminals who choose to roll the tetrahedron, one will roll the same face four times and be reprieved. If a hundred criminals face the death-penalty each year and all choose to roll the tetrahedron, one criminal will be reprieved roughly every eight months. But if all criminals choose to roll the icosahedron and they have been rolling since the Big Bang, just under fourteen billion years ago, it is very, very, very unlikely that any have yet been reprieved.

Rep-Tile Reflections

Saturday, 23rd Mar, 2013 by Krilling for Company in Astronomy, Geometry, Mathematics, Philosophy, Philosophy of mathematics, Rep-Tiles, Science and tagged alternative realities, alternative reality, √2, Big Bang, Big Crunch, Credo in Piscem Unum et Infinitum et Æternum, double-square, double-triangle, extraterrestrial intelligence, gas-based life, geometric, geometrical, geometry, half-square, hypotenuse, intelligence, intelligent life, math, math-based religion, math-religion, mathematical religion, mathemystic, mathemystic communion, mathemystical, mathemysticism, maths, maths-based religion, mental realm, metaphysical, metaphysics, mind, minds, mystic communism, parallel universe, parallel universes, powers of 9, powers of nine, psycho-math-space, rectangle, rectangles, Religion, rep-tile, Rep-Tiles, rep-triangle, rep-triangles, repeat-tile, repeat-tiles, repeating tile, repeating tiles, repetition-tile, repetition-tiles, right triangle, silicon-based life, speculation, speculatory, square, square root of 2, square root of two, Squares, tessellate, tessellation, tessellations, the fish, the sphinx, trans-biological communion, triangle, Triangles, universe | 5 Comments

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 — 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.

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.

Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

მეფე (2) • მსოფლიოს (3) • მათემატიკა (5) •

Category Archives: Geometry