Hyperbolic Geometry And The Poincare Disk

Perhaps the most famous example of mathematical art is the work of Escher. “The six largest (three white angles and three black devils) are arranged about the centre and radiate from it. The disc is divided into six sections in which, turn and turn about, the angles on a black background and then the devils on a white one, gain the upper hand. In this way, heaven and hell change places six times.” M. C. Escher Speaking about the first work in his Circle Limit Series Escher says: ”The limit is no longer a point, but a line which borders the whole complex and gives it a logical boundary. In this way one creates, as it where, a universe, a geometrical enclosure. If the progressive reduction in size radiates in all directions at an equal rate, then the limit becomes circle.” M. C. Escher This is a kind tessellation of hyperbolic space which we are going to try to understand. Hyperbolic space is a kind of Non-Euclidian geometry. So lets start with Euclid. Euclid founded his geometry on five postulates. The first four are quite simple, although they sound little odd form a modern perspective. The idea is that you […]
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There is a visual proof of Pythagoras theorem. I’ve made an interactive version. Recall the famous theorem about the sides of a right angle triangle: where c is the side of the hypotenuse, a and b are the other two sides. The two squares at the bottom have the same area because they both have sides a+b. The one on the left is the c square plus four triangles. The one on the right is the a square plus the b square plus four triangles. Both squares have four equal triangles and the same area, so whats left over must also be equal. Drag the anchor point (the little circle) to change the triangle. Years ago I made a painting with this proof:
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Wallpaper Patterns As Slices Through Space

Many interesting patterns in 2 dimensions can be understood as a slice through a lattice in a higher dimension. The simplest example is a slice through a three dimensional checker board. If you slice the corner off a cube at just the right angle you get an equilateral triangle. Slice at the same angle, but deeper into the cube, and you get a hexagon. So the pattern of hexagons and triangles, so common in Islamic art, is just a slice through a 3d checkerboard. This animation illustrates the idea:
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