3-sphere

A higher-dimensional analogue of a sphere in four-dimensional space

The 3-sphere, also known as a hypersphere, is a higher-dimensional analogue of a sphere. It is a three-dimensional surface that exists in four-dimensional Euclidean space. The 3-sphere is an important object in the field of topology and has applications in various areas of mathematics and physics.

Definition

The 3-sphere, denoted as \( S^3 \), is defined as the set of points in four-dimensional space \( \mathbb{R}^4 \) that are equidistant from a fixed central point. Mathematically, it can be expressed as:

\[ S^3 = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1^2 + x_2^2 + x_3^2 + x_4^2 = r^2 \}. \]

Here, \( r \) is the radius of the 3-sphere, and the center is typically taken to be the origin \( (0, 0, 0, 0) \).

Properties

The 3-sphere is a compact, smooth manifold without boundary. It is a three-dimensional manifold, meaning that locally, it resembles three-dimensional Euclidean space. However, globally, it has a different topology.

Topology

The 3-sphere is a simply connected space, meaning that any loop on the 3-sphere can be continuously contracted to a point. This property is significant in the study of homotopy and homology in algebraic topology.

Geometry

The geometry of the 3-sphere can be understood through its curvature. The 3-sphere has constant positive curvature, which distinguishes it from the flat geometry of Euclidean space and the negative curvature of hyperbolic space.

Applications

The 3-sphere appears in various areas of mathematics and physics. In mathematics, it is used in the study of Lie groups, particularly the group \( \text{SU}(2) \), which is homeomorphic to the 3-sphere. In physics, the 3-sphere is relevant in the context of general relativity and cosmology, where it can model closed universes.

Visualizations

Visualizing the 3-sphere is challenging due to its existence in four-dimensional space. However, certain projections and mappings can help illustrate its structure. The Hopf fibration is a notable example, where the 3-sphere is decomposed into circles.

Related pages

• Sphere
• Topology
• Manifold
• Hopf fibration

Gallery

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  Coordinate representation of a hypersphere.

• Visualization of a hypersphere.

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  Illustration of the Hopf fibration.

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  Toroidal coordinates related to the 3-sphere.

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