![]() In other words, there’s a chain of dominations. But that’s not the end of it: (4,4) is dominated by (4,5) but dominates (3,4), which in its turn dominates (2,4). Whatever individual number is in (4,5) must be greater than whatever individual number is in (4,4).Ĭonversely, you can say that (4,4) is dominated by (4,5). If we label squares by row and column, you can say that square (4,5), just above the lower righthand corner, dominates square (4,4), because (4,5) is on the dominant side of the inequality sign between the two squares ( futōshiki, 不等式, means “inequality” in Japanese). There are no numbers at all in the futoshiki, so where do you start? Well, first let’s establish some vocabulary for discussing futoshiki. You work out what those missing numbers are by using the inequality signs scattered over the futoshiki. ![]() At first, most or all of the numbers are missing. It’s a number-puzzle where you use logic to re-create a 5×5 square in which every row and column contains the numbers 1 to 5. I’m calling it a fylfy fractal:ĭivide-and-discard fractals in the four triangles of a divided square stage #1įinally, you can adjust the fylfy fractals so that each point in the square becomes the equivalent point in a circle:įutoshiki is fun. The resulting four-fractal shape is variously called a swastika, a gammadion, a cross cramponnée, a Hakenkreuz and a fylfot. Now try dividing a square into four right triangles, then turning each of the four triangles into a divide-and-discard fractal. You can create more attractive and interesting fractals by rotating the sub-triangles clockwise or anticlockwise. Alas, it’s not a very attractive or interesting fractal: If you divide and discard one of the sub-copies, then carry on dividing-and-discarding with the sub-copies and sub-sub-copies and sub-sub-sub-copies, you get the fractal seen below. Here it is as a rep-4 rep-tile, tiled with four smaller copies of itself: Hexagon fractal, triangular to circular (animated)Īn equilateral triangle is a rep-tile, because it can be tiled completely with smaller copies of itself. Grid fractal, triangular to circular (animated) Here’s what I call a rep9-tri grid fractal: You can create other fractals by dividing-and-discarding sub-triangles from a rep-9 equilaterial triangle. When you’ve reduced the diamonds to dust, you’ve got a fractal, a shape that repeats itself at smaller and smaller scales:Īfter that, you can convert the fractal-within-a-triangle into a fractal-within-a-circle:ĭiamond fractal, triangular to circular (animated) ![]() Now do the same to the remaining six: divide each into nine sub-triangles and discard three of the sub-triangles. Six sub-triangles left after three are discardedīut why stop there? Once you’ve discarded three triangles, six triangles are left. Then you discard three of the sub-triangles, like this:Įquilateral triangle divided into nine sub-triangles First, you divide an equilateral triangle into nine smaller equilateral triangles. But if you look at the Mitsubishi diamonds with a mathematical eye, you can see how to create them in two simple steps. Those are the three diamonds of Mitsubishi, whose name itself means “three diamonds” or “three rhombi” in Japanese (see 三菱). It’s one of the most famous and easily recognizable logos in the world: Triangular fractal to circular fractal (animated) But they get more obvious like this:Īnd when you have a fractal created using an equilateral triangle, it’s easy to expand the fractal into a circle, like this: Here are variant colorings of the stage-3 tiled triangle:īut where are the fractals? In one way, you’ve already seen them. ![]() (please open in new window if image is distorted)Įquilateral triangle tiled with 1-√3-2 triangles (stage 2)Įquilateral triangle tiled with 1-√3-2 triangles (stage 3) And it’s easy to find that space when you realize that a standard equilateral triangle can be divided into six 1-√3-2 triangles:Įquilateral triangle divided into six 1-√3-2 trianglesĮquilateral triangle tiled with 1-√3-2 triangles (stage 1) Once you’ve got a rep-tile, you can create fractals. And if it’s rep3, that is, can be divided into three identical copies of itself, then it’s also rep9, rep27, and so on: The horizontal side, b, has a length of √3 = 1.73205080757… So the right triangle is 1-√3-2. If the vertical side, a, is 1, then the hypotenuse, c, is 2, because the length that fits once into a fits twice into c. Now you can prove the exact values of a, b, and c. You might be able to guess by eye, but could you prove your guess? Now try the same right triangle tiled with three identical copies of itself: But what are the exact values of a, b, and c? Here’s a right triangle, where a^2 + b^2 = c^2. ![]()
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