A regular hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into three more equilateral triangles and a regular hexagon. If you discard the three triangles and repeat, you create a fractal, like this:
Adjusting the sides of the internal hexagon creates new fractals:
Discarding a hexagon after each subdivision creates new shapes:
And you can start with another regular polygon, divide it into triangles, then proceed with the hexagons:
Papyrocentric Performativity Presents:
• Sky Story – The Cloud Book: How to Understand the Skies, Richard Hamblyn (David & Charles 2008)
• Wine Words – The Oxford Companion to Wine, ed. Janice Robinson (Oxford University Press 2006)
• Nu Worlds – Numericon, Marianne Freiberger and Rachel Thomas (Quercus Editions 2014)
• Thalassobiblion – Ocean: The Definitive Visual Guide, introduction by Fabien Cousteau (Dorling Kindersley 2014) (posted @ Overlord of the Über-Feral)
Or Read a Review at Random: RaRaR
In maths, one thing leads to another. I wondered whether, in a spiral of integers, any number was equal to the digit-sum of the numbers on the route traced by moving to the origin first horizontally, then vertically. To illustrate the procedure, here is a 9×9 integer spiral containing 81 numbers:
| 65 | 64 | 63 | 62 | 61 | 60 | 59 | 58 | 57 | | 66 | 37 | 36 | 35 | 34 | 33 | 32 | 31 | 56 | | 67 | 38 | 17 | 16 | 15 | 14 | 13 | 30 | 55 | | 68 | 39 | 18 | 05 | 04 | 03 | 12 | 29 | 54 | | 69 | 40 | 19 | 06 | 01 | 02 | 11 | 28 | 53 | | 70 | 41 | 20 | 07 | 08 | 09 | 10 | 27 | 52 | | 71 | 42 | 21 | 22 | 23 | 24 | 25 | 26 | 51 | | 72 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |
Take the number 21, which is three places across and up from the bottom left corner of the spiral. The route to the origin contains the numbers 21, 22, 23, 8 and 1, because first you move right two places, then up two places. And 21 is what I call a route number, because 21 = 3 + 4 + 5 + 8 + 1 = digitsum(21) + digitsum(22) + digitsum(23) + digitsum(8) + digitsum(1). Beside the trivial case of 1, there are two more route numbers in the spiral:
58 = 13 + 14 + 6 + 7 + 7 + 6 + 4 + 1 = digitsum(58) + digitsum(59) + digitsum(60) + digitsum(61) + digitsum(34) + digitsum(15) + digitsum(4) + digitsum(1).
74 = 11 + 12 + 13 + 14 + 10 + 5 + 8 + 1 = digitsum(74) + digitsum(75) + digitsum(76) + digitsum(77) + digitsum(46) + digitsum(23) + digitsum(8) + digitsum(1).
Then I wondered about other possible routes to the origin. Think of the origin as one corner of a rectangle and the number being tested as the diagonal corner. Suppose that you always move away from the starting corner, that is, you always move up or right (or up and left, and so on, depending on where the corners lie). In a x by y rectangle, how many routes are there between the diagonal corners under those conditions?
It’s an interesting question, but first I’ve looked at the simpler case of an n by n square. You can encode each route as a binary number, with 0 representing a vertical move and 1 representing a horizontal move. The problem then becomes equivalent to finding the number of distinct ways you can arrange equal numbers of 1s and 0s. If you use this method, you’ll discover that there are two routes across the 2×2 square, corresponding to the binary numbers 01 and 10:
Across the 3×3 square, there are six routes, corresponding to the binary numbers 0011, 0101, 0110, 1001, 1010 and 1100:
Across the 4×4 square, there are twenty routes:
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Across the 5×5 square, there are 70 routes:
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Across the 6×6 and 7×7 squares, there are 252 and 924 routes:
After that, the routes quickly increase in number. This is the list for n = 1 to 14:
1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600… (see A000984 at the Online Encyclopedia of Integer Sequences)
After that you can vary the conditions. What if you can move not just vertically and horizontally, but diagonally, i.e. vertically and horizontally at the same time? Now you can encode the route with a ternary number, or number in base 3, with 0 representing a vertical move, 1 a horizontal move and 2 a diagonal move. As before, there is one route across a 1×1 square, but there are three across a 2×2, corresponding to the ternary numbers 01, 2 and 10:
There are 13 routes across a 3×3 square, corresponding to the ternary numbers 0011, 201, 021, 22, 0101, 210, 1001, 120, 012, 102, 0110, 1010, 1100:
And what about cubes, hypercubes and higher?
“In 1997, Fabrice Bellard announced that the trillionth digit of π, in binary notation, is 1.” — Ian Stewart, The Great Mathematical Problems (2013).
Eye Bogglers: A Mesmerizing Mass of Amazing Illusions, Gianni A. Sarcone and Marie-Jo Waeber (Carlton Books 2011; paperback 2013)
A simple book with some complex illusions. It’s aimed at children but scientists have spent decades understanding how certain arrangements of colour and line fool the eye so powerfully. I particularly like the black-and-white tiger set below a patch of blue on page 60. Stare at the blue “for 15 seconds”, then look quickly at a tiny cross set between the tiger’s eyes and the killer turns colour.
So what’s not there appears to be there, just as, elsewhere, what’s there appears not to be. Straight lines seem curved; large figures seem small; the same colour seems light on the right, dark on the left. There are also some impossible figures, as made famous by M.C. Escher and now studied seriously by geometricians, but the only true art here is a “Face of Fruits” by Arcimboldo. The rest is artful, not art, but it’s interesting to think what Escher might have made of some of the ideas here. Mind is mechanism; mechanism can be fooled. Optical illusions are the most compelling examples, because vision is the most powerful of our senses, but the lesson you learn here is applicable everywhere. This book fools you for fun; others try to fool you for profit. Caveat spectator.
Simple but complex: The café wall illusion
Echinops sphaerocephalus, or Great Globe Thistle
A polyiamond is a shape consisting of equilateral triangles joined edge-to-edge. There is one moniamond, consisting of one equilateral triangle, and one diamond, consisting of two. After that, there are one triamond, three tetriamonds, four pentiamonds and twelve hexiamonds. The most famous hexiamond is known as the sphinx, because it’s reminiscent of the Great Sphinx of Giza:
It’s famous because it is the only known pentagonal rep-tile, or shape that can be divided completely into smaller copies of itself. You can divide a sphinx into either four copies of itself or nine copies, like this (please open images in a new window if they fail to animate):
So far, no other pentagonal rep-tile has been discovered. Unless you count this double-triangle as a pentagon:
It has five sides, five vertices and is divisible into sixteen copies of itself. But one of the vertices sits on one of the sides, so it’s not a normal pentagon. Some might argue that this vertex divides the side into two, making the shape a hexagon. I would appeal to these ancient definitions: a point is “that which has no part” and a line is “a length without breadth” (see Neuclid on the Block). The vertex is a partless point on the breadthless line of the side, which isn’t altered by it.
But, unlike the sphinx, the double-triangle has two internal areas, not one. It can be completely drawn with five continuous lines uniting five unique points, but it definitely isn’t a normal pentagon. Even less normal are two more rep-tiles that can be drawn with five continuous lines uniting five unique points: the fish that can be created from three equilateral triangles and the fish that can be created from four isosceles right triangles:
When I started to look at rep-tiles, or shapes that can be divided completely into smaller copies of themselves, I wanted to find some of my own. It turns out that it’s easy to automate a search for the simpler kinds, like those based on equilateral triangles and right triangles.
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Previously pre-posted (please peruse):
A hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into a hexagon and three more equilateral triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.
Previously pre-posted (please peruse):
A square can be divided into four right triangles. A right triangle can be divided into a square and two more right triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 