mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 6-sphere coordinates are a

coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

for three-dimensional space obtained by inverting the 3D

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

across the unit 2-sphere

x^2+y^2+z^2=1

. They are so named because the loci where one coordinate is constant form

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

s tangent to the origin from one of six sides (depending on which coordinate is held constant and whether its value is positive or negative). This coordinate system exists independently from and has no relation to the 6-sphere. The three coordinates are :

u = \frac,\quad v = \frac,\quad w = \frac.

Since inversion is an

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

, the equations for ''x'', ''y'', and ''z'' in terms of ''u'', ''v'', and ''w'' are similar: :

x = \frac,\quad y = \frac,\quad z = \frac.

This coordinate system is

R

-separable for the 3-variable Laplace equation.

References

* Moon, P. and Spencer, D. E. ''6-sphere Coordinates.'' Fig. 4.07 in ''Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed.'' New York: Springer-Verlag, pp. 122–123, 1988. *
Six-Sphere Coordinates
by Michael Schreiber, the

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an Open source, open-source collection of Interactive computing, interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown t ...

. Three-dimensional coordinate systems Inversive geometry Orthogonal coordinate systems {{geometry-stub

See also

References