mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

for three-dimensional space obtained by inverting the 3D

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

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 () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

s tangent to the

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from one of six sides (depending on which coordinate is held constant and whether its value is positive or negative). They have nothing whatsoever to do with the 6-sphere, which is an object of considerable interest in its own right. The three coordinates are :

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

Since inversion is its own

inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...

, 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

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

See also

References