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- A 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
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 …
How Do You Derive the Line Element on a 3-Sphere? - Physics …
Jan 20, 2008 · The line element on a 3-sphere is defined using the Pythagorean theorem in four dimensions. It takes into account the three spatial dimensions and the time dimension, and is …
Quaternions and metric of the 3-sphere - Physics Forums
Nov 16, 2010 · What is the 3-sphere? The 3-sphere, also known as the 3-dimensional hypersphere, is a mathematical object that extends the concept of a sphere into a higher …
Exploring the Boundary of the 3-Sphere: A Topological Analysis
Nov 29, 2020 · In summary, the conversation discusses the topic of the 3-sphere, a manifold without boundary. The participants raise the question of whether there is a simple analytical …
How Many Parameters Are Needed to Parametrize the 3-Sphere?
Jun 25, 2018 · The 3-sphere can be parametrized using four coordinates, known as hyperspherical coordinates. These coordinates are similar to spherical coordinates, but with an …
Parameterizing the 3-Sphere: 3 Angles? - Physics …
Jun 2, 2017 · What are the three angles used to parameterize the 3-sphere? The three angles used to parameterize the 3-sphere are longitude, latitude, and azimuth. These angles represent the position of a point on the 3-sphere in …
Riemann tensor, Ricci tensor of a 3 sphere - Physics Forums
Jan 9, 2012 · What you have at the top is not the metric of a 3-sphere, but simply the metric of R^3 in spherical polar coordinates. A 3-sphere metric will have r fixed and 3 angles to describe …
What Are the Fascinating Rotations of 3-Spheres in 4D Space?
Oct 28, 2015 · A 3-sphere is a sphere in 4D space. A 3-sphere -- or any other object in 4space with non-zero volume -- can have two planes of rotation. These planes are always at right …
Derivation of Metric for 3-Sphere Embedded in 4-Space - Physics …
Apr 13, 2010 · In summary, the metric for a 3-sphere embedded in 4-space is ds^2=dr^2+R^2 sin^2(\frac{r}{R})(d\theta^2+sin^2\theta d\phi^2) where r is the distance from some "pole" and …
Rotations of 3-Spheres: Exploring 4D Space & Beyond - Physics …
Oct 28, 2015 · A 3-sphere, or any other object with non-zero volume in 4D space, can have two planes of rotation that are always at right angles to each other. To make it easier to …
3-sphere with Ricci flow - Physics Forums
Sep 15, 2024 · TL;DR Summary: does the center of a 3 sphere remain stable with Ricci flow? 1. Consider you have a 3-sphere, Ricci flow says it contracts to a point in finite time.
Projecting 3-Sphere onto a Plane | Stereographic Projection
Jul 15, 2011 · Projecting a 3-sphere onto a plane can be useful in visualizing and understanding the properties of the 3-sphere. It allows for a more intuitive understanding of the object and …
Metric in Manifold Homework: 3-sphere in 4D Euclidean Space
Sep 22, 2014 · The metric of a 3-sphere in 4D Euclidean space is calculated using the Pythagorean theorem, just like a regular sphere in three-dimensional space. However, in 4D …
How Does the 3-Sphere Map to the Bloch Sphere in Quantum …
May 23, 2011 · The Bloch Sphere is a subset of the 3-sphere, representing the quantum states of a qubit. The north and south poles of the Bloch Sphere correspond to the equator of the 3 …
Lie Algebra Structure of 3-Sphere: Left Invariant Vector Fields
Sep 24, 2009 · The Lie algebra structure of the 3-sphere refers to the set of vector fields that are left invariant under the action of a Lie group that is isomorphic to the 3-sphere. This allows us …
Visualizing higher dimensional spheres - Physics Forums
Jun 26, 2014 · Here's a better way to visualize rotation of a 3-sphere. The 3-sphere can be parameterized with polar coordinates as sin(x)e^it, cos(x)e^iu, with 0<=x<pi/2, -pi<=t,u<pi. …
Possible illogicalness of a 3-sphere shape of the Universe's space?
May 6, 2020 · In summary, the conversation discusses the use of metaphors to depict the universe as a 2-sphere or a 3-sphere. It also considers mapping the three dimensions of …
What's the difference between a two sphere and a three sphere?
Mar 17, 2005 · -The main difference between a 2-sphere and a 3-sphere is their number of diemsnions (this is obvious I suppose). -The idea of a 3-sphere is hard to grasp, the best way …
What is Andrew Cheong's Model of the Universe? - Physics Forums
Jul 28, 2005 · The model described is known as the "3-sphere model" or "hyperspherical model" of the universe, but it has been ruled out as the accepted model due to its inability to explain …
Is the Universe a 3-sphere or a 4-sphere? - Physics Forums
May 14, 2019 · No. Even for the case where a universe that is spatially closed (i.e., spatially a 3-sphere) recollapses to a big crunch, time is not "circular"--the end of the universe does not …
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