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Four-Dimensional Geometry


Four-dimensional geometry is Euclidean geometry extended into one additional dimension. The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., hypercube, hyperplane, hypersphere. n-dimensional polyhedra are called polytopes. The four-dimensional cases of general n-dimensional objects are often given special names, such as those summarized in the following table.

The surface area of a hypersphere in n dimensions is given by

 S_n=(2pi^(n/2)R^(n-1))/(Gamma(1/2n)),
(1)

where Gamma(n) is the gamma function, giving the first few values as

S_1=2
(2)
S_2=2piR
(3)
S_3=4piR^2
(4)
S_4=2pi^2R^3,
(5)

with coefficients 2, 2, 4, 2, 8/3, 1, 16/15, ... (OEIS A072478 and A072479).

The volume is given by

 V_n=(pi^(n/2)R^n)/(Gamma(1+1/2n)),
(6)

giving the first few values as

V_1=2R
(7)
V_2=piR^2
(8)
V_3=4/3piR^3
(9)
V_4=1/2pi^2R^4,
(10)

with coefficients 2, 1, 4/3, 1/2, 8/15, 1/6, 16/105, ... (OEIS A072345 and A072346).


See also

Dimension, High-Dimensional Solid, Hypercube, Hypersphere

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References

Hinton, C. H. The Fourth Dimension. Pomeroy, WA: Health Research, 1993.Manning, H. The Fourth Dimension Simply Explained. Magnolia, MA: Peter Smith, 1990.Manning, H. Geometry of Four Dimensions. New York: Dover, 1956.Neville, E. H. The Fourth Dimension. Cambridge, England: Cambridge University Press, 1921.Rucker, R. von Bitter. The Fourth Dimension: A Guided Tour of the Higher Universes. Boston, MA: Houghton Mifflin, 1984.Sloane, N. J. A. Sequences A072345, A072346, A072478, and A072479 in "The On-Line Encyclopedia of Integer Sequences."Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, 1958.

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Four-Dimensional Geometry

Cite this as:

Weisstein, Eric W. "Four-Dimensional Geometry." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Four-DimensionalGeometry.html

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