science tumbled

Learn geometry with M. C. Escher.
We previously covered some of the math behind Islamic wall tilings. The artist M. C. Escher was hugely inspired by the patterns he saw on the walls at Alhambra in Spain. He used them as inspiration for his tilings of the plane. The above is his “angels and devils”, where he doesn’t tile a traditional plane, but a representation of a hyperbolic plane.
Euclidian geometry is the regular old geometry we all learn in school. Hyperbolic geometry is created by throwing out Euclid’s postulate that given a line L and a point P not on L, there is one and only one line through P which is parallel to L. In hyperbolic geometry, there is more than one such line. Infinitely many, in fact. This gives rise to something that is like our familiar geometry, but also quite unlike it. For instance, all triangles have angles that add up to less than 180 degrees.
Since hyperbolic geometry is so difficult for us to picture, a clever man invented the Poincaré disk, which models a hyperbolic plane inside a traditional Euclidian circle. The points on the circle aren’t part of the model, they represent infinity. All the lines in the model are either diameters of the circle or arcs of circles that are orthogonal to it. (Thus many straight lines appear curved in the model.) The farther away from the center you get, the longer each Euclidian line segment represents in the hyperbolic world it models. This means that what Escher is doing above is tiling the entire plane with two figures, the angel and the devil. The smallest figures above (near the edge) are identical in size and shape to the largest (in the center).
The parallel postulate was Euclid’s fifth axiom of geometry. For two thousand years, numerous “proofs” of the axiom were published, only to be shown to err in one way or another. Not until around 1830 did the idea that one could get perfectly fine geometries out of negating the postulate gain serious traction. Hyperbolic geometry is what results when you say there are infinitely many lines that cross through a given point and are parallel to a line that doesn’t go through the point. Elliptic geometry (think of the surface of a sphere) is what happens if you say there are none.
There is still debate about whether the shape of the universe is flat (Euclidian), positively curved (elliptic) or negatively curved (hyperbolic). Most believe the universe is flat, or very nearly flat.

We previously covered some of the math behind Islamic wall tilings. The artist M. C. Escher was hugely inspired by the patterns he saw on the walls at Alhambra in Spain. He used them as inspiration for his tilings of the plane. The above is his “angels and devils”, where he doesn’t tile a traditional plane, but a representation of a hyperbolic plane.

Euclidian geometry is the regular old geometry we all learn in school. Hyperbolic geometry is created by throwing out Euclid’s postulate that given a line L and a point P not on L, there is one and only one line through P which is parallel to L. In hyperbolic geometry, there is more than one such line. Infinitely many, in fact. This gives rise to something that is like our familiar geometry, but also quite unlike it. For instance, all triangles have angles that add up to less than 180 degrees.

Since hyperbolic geometry is so difficult for us to picture, a clever man invented the Poincaré disk, which models a hyperbolic plane inside a traditional Euclidian circle. The points on the circle aren’t part of the model, they represent infinity. All the lines in the model are either diameters of the circle or arcs of circles that are orthogonal to it. (Thus many straight lines appear curved in the model.) The farther away from the center you get, the longer each Euclidian line segment represents in the hyperbolic world it models. This means that what Escher is doing above is tiling the entire plane with two figures, the angel and the devil. The smallest figures above (near the edge) are identical in size and shape to the largest (in the center).

The parallel postulate was Euclid’s fifth axiom of geometry. For two thousand years, numerous “proofs” of the axiom were published, only to be shown to err in one way or another. Not until around 1830 did the idea that one could get perfectly fine geometries out of negating the postulate gain serious traction. Hyperbolic geometry is what results when you say there are infinitely many lines that cross through a given point and are parallel to a line that doesn’t go through the point. Elliptic geometry (think of the surface of a sphere) is what happens if you say there are none.

There is still debate about whether the shape of the universe is flat (Euclidian), positively curved (elliptic) or negatively curved (hyperbolic). Most believe the universe is flat, or very nearly flat.

dailymeh posted this on September 5, 2011