Oct 11, 2024
The n-hypersphere (often simply called the n-sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (
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The n-hypersphere (often simply called the n-sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (
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An orthographic projection of NASA's Blue Marble data set (1 km resolution global satellite composite). "MODIS observations of polar sea ice were combined with observations of Antarctica made by the National Oceanic and Atmospheric Administration’s AVHRR sensor—the Advanced Very High Resolution Radiometer." Image was generated using a custom C program for handling the Blue Marble files, with orthographic projection formulas from MathWorld. Source: Wikipedia
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Costa's surface
A complete minimal embedded boundaryless surface of genus 1. It has D4 symmetry and is topologically a thrice-punctured torus. This means, roughly:
No edges, no self-intersections.
Naturally resides in three-dimensional Euclidean space.
Zero mean curvature. Everywhere on it it is locally a saddle with equal maximal bends in opposite directions.
One "hole". I.e. no more than one loop may be drawn on the surface without separating the surface into disconnected regions.
The group of symmetries of the surface - the ways to "morph" it back to itself, loosely - is the same as that for the square in two dimensions.
If you puncture three holes in a torus you may then deform and stretch it into Costa's surface. The "edges" created around the holes after puncture go off "to infinity" in the process.
The only complete minimal embedded surfaces known before Costa's were the plane, catenoid and helicoid. Many more have been found since then. Wikimedia has very brief ogv animation (link).
Hat tip: Mathworld