![]() ![]() The citizens of the disk could not reach the bounding circle - known as the absolute - in a finite time. ![]() The radius of the “unit disk“ with this metric is given by integrating from the centre to a point on the boundary of the disk:īut this integral diverges as, confirming for the disk-dwellers that their universe is infinite in extent. (In relativity, the parameter is the proper time.) The distance is given byįrom this, we can derive quantities known as Christoffel symbols,Īnd write down equations for the geodesics In differential geometry, the metric tensor contains all that is needed to determine shortest paths, or geodesics. The geometry of Poincaré’s disk is encapsulated in the metric giving the distance increment. To move from one point to another, the disk-dweller could follow the shortest path only if he or she travelled on a circular arc rather than along a Euclidean straight line. Thus, for two paths that we - with our external Euclidean perspective - would regard as equal in length, a disk-dweller would require more steps to cover the path near the boundary than that closer to the centre. ![]() Poincaré argued that creatures, whom we will call disk-dwellers, living on the disk would be unable to detect that the disk was finite, and would regard it as infinitely large, with geometric properties quite different from the Euclidean plane. He supposed that the temperature varied linearly from a fixed value at the centre of the disk to absolute zero on the boundary, and that lengths varied in proportion to the temperature. He envisioned a circular disk in the Euclidean plane, where distances were distorted to give it geometric properties quite different from those of Euclid’s Elements. Henri Poincaré described a beautiful geometric model with some intriguing properties. ![]()
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