“Geometry is not true, it is advantageous”*…
The tesseract is a four dimensional cube. It has 16 edge points v=(a,b,c,d), with a,b,c,d either equal to +1 or -1. Two points are connected, if their distance is 2. Given a projection P(x,y,z,w)=(x,y,z) from four dimensional space to three dimensional space, we can visualize the cube as an object in familar space. The effect of a linear transformation like a rotation
| 1 0 0 0 | R(t) = | 0 1 0 0 | | 0 0 cos(t) sin(t) | | 0 0 -sin(t) cos(t) |
in 4d space can be visualized in 3D by viewing the points v(t) = P R(t) v in R3.
* Henri Poincare
As we follow the bouncing ball, we might spare a thought for Felix Klein; he died on this date in 1925. A mathematician of broad gauge, he is best remembered for his work in non-Euclidean geometry, perhaps especially for his work on synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Program, which profoundly influenced mathematics. He created the Klein bottle, a one-sided closed surface–a non-orientable surface with no inside and no outside– that cannot be constructed in Euclidean space.