PropertiesThe 3-dimensional surface volume of a 3-sphere of radius r is
while the 4-dimensional hypervolume (the content of the 4-dimensional region, or ball, bounded by the 3-sphere) is
Every non-… See moreTopological constructionThere are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification.
A 3-sphere can be constructed topologically by … See more
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- Higher-dimensional analogue of a sphereA 3-sphere is a higher-dimensional analogue of a sphere1. It is the set of points equidistant from a fixed central point in 4-dimensional Euclidean space1. In mathematical terms, it is the set of all points a fixed distance from some center point in four-dimensional space2. The 3-sphere is also an example of a 3-manifold3.Learn more:✕This summary was generated using AI based on multiple online sources. To view the original source information, use the "Learn more" links.In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point.en.wikipedia.org/wiki/3-sphereJust as a two-dimensional sphere is the set of all points a fixed distance from some center point in ordinary three-dimensional space, a three-dimensional sphere (or “three-sphere”) is the set of all points a fixed distance from some center point in four-dimensional space.www.quantamagazine.org/what-is-the-geometry-of …This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space.en.wikipedia.org/wiki/Three-dimensional_space
n-sphere - Wikipedia
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3 -sphere into 3 -space. This image shows three coordinate …
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What does the 3-sphere look like? - Mathematics Stack Exchange
Oct 4, 2021 · A 3-sphere cannot fit into 3-dimensional euclidean space, just as a 2-sphere cannot be cut from a piece of paper. So while you can't just "look at it", here are two nice …
The 3-sphere: Extrinsic and Intrinsic Forms - ThatsMaths
Sep 22, 2022 · The 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected: any loop on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere.
Exploring the 3-Sphere: A Beginner's Guide to Higher Dimensions
Jun 4, 2014 · The 3-sphere has many theoretical applications in mathematics, physics, and cosmology. It is also used in computer graphics and animation to represent four-dimensional …
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3-Dimensional Space | The spherical space $S^3$
What is the 3-sphere? The space $S^3$ is the 3-dimensional analogue of the usual sphere in 2 dimensions. It is an isotropic space (all the directions play the same role).
3-sphere - Academic Kids
A 3-sphere is a compact, 3-dimensional manifold without boundary. It is also simply-connected. What this means, loosely speaking, is that any loop, or circular path, on the 3-sphere can be …
What is the 3-sphere? - jthatch.com
What is the 3-sphere? According to that Wikipedia page, the 3-sphere is the "higher-dimensional analogue of a sphere". A (regular) sphere is the boundary of a 3D ball, where a ball is a solid 3D shape. If the ball is a solid, like a marble, …
Henri Poincar¿, His Conjecture, Copacabana and Higher …
Jun 9, 2004 · For 100 years mathematicians have been trying to prove a conjecture that was first proposed by Henri Poincar¿ relating to an object known as the three-dimensional sphere, or 3 …
3-sphere | EPFL Graph Search
In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a …
3-manifold - Wikipedia
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3- manifold can be thought of as a possible shape of the universe. Just as …
Why is the shape of the universe assumed to be a 3-manifold?
A 3-sphere is three dimensional. In order to visualize it (if it can be called that) you need to embed it in 4-dimensional space, but the object itself is perfectly well defined without that larger …
Most people are familiar with the use of latitude and longitude in locating and naming points on the surface of the Earth, which, mathematically at least, is considered to be a perfectly round …
Glome - Hi.gher. Space
The glome, also known as the 3-sphere, is the 4-dimensional equivalent of a 3D sphere. It consists of a curved 3-manifold that forms circular intersections with planes, and spherical …
3-Sphere - SpringerLink
Jan 1, 2014 · The fourth kind of sphere is the 3-sphere S 3. Unlike S 0, S 1, and S 2, we cannot picture S 3 because human vision is 3-dimensional — it is impossible to picture S 3 within …
3-sphère — Wikipédia
En mathématiques, et plus précisément en géométrie, une 3-sphère est l'analogue d'une sphère en dimension quatre. C'est l'ensemble des points équidistants d'un point central fixé dans un …
3-sphere - Academic Dictionaries and Encyclopedias
Stereographic projection of the hypersphere s parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in …
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