n-sphere - Wikipedia
In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...
The HyperSphere - Massachusetts Institute of Technology
The mathematical objects that live on the sphere in four dimensional space -- the hypersphere -- are both beautiful and interesting. The four dimensional sphere is a unique object, with properties both similar to and surprisingly different from those of our ordinary sphere.
Four-dimensional space - Wikipedia
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world.
Calculating the Hypervolume of a 4D Hypersphere - Medium
2024年4月26日 · Before we derive the 4D hypersphere formula, we will first look at the method in action in only 3 dimensions, then scale this up to give us the 4-dimensional wonder. This all first starts with a cross-section of a sphere, a great circle:
Deriving the formula for the hypervolume of a 4D sphere.
2018年5月9日 · One can then apply this formula to the hypersphere, whose (hypersurface) volume you have determined to be $2\pi^2R^3$. By analogy with determining the area of a circle as the limit of a sequence of triangles of ever more finely divided base, the hypervolume of the hypersphere should be one fourth the common altitude of the hyperpyramids, $R ...
Hyperspheres Part I: The 4 Dimensional Hypersphere - GlobalSpec
2007年1月30日 · We can find the surface area of a 4 dimensional Hypersphere by integrating over the angular parts of the volume integral, just like we did for the three dimensional sphere. So we can find the volume and surface area of a n-Hypersphere of 1, 2, 3, and 4 dimensions.
Shapes of Space: The Hypersphere - Brown University
A hypersphere is the four-dimensional analog of a sphere. Although a sphere exists in 3-space, its surface is two-dimensional. Similarly, a hypersphere has a three-dimensional surface which curves into 4-space. Our universe could be the hypersurface of a hypersphere.
A 4-Dimensional Sphere, Animated - Foundalis
4D-spheres are bounded in 4D-space by definition: there is a 4D-sphere that includes them, namely, their own self. It is their projections on any subspace that includes time as one of its dimensions that have the hyperbolic character, hence, appear unbounded in that subspace.
Volume Hypersphere: 4D Sphere Equation - Physics Forums
2016年4月9日 · The volume of a 4D sphere can be calculated using the equation V = (π^2 * r^4)/2, where r is the radius of the sphere. This equation is derived from the general formula for the volume of an n-dimensional sphere.
hypersphere - David Darling
x 2 + y 2 + z 2 + w 2 = r 2 is the equation of a hypersphere, where w is measured along a fourth dimension at right angles to the x-, y -, and z -axes. The hypersphere has a hypervolume (analogous to the volume of a sphere) of π 2r 4 /2, and a surface volume (analogous to the sphere's surface area) of 2π 2r 3.