3-sphere

3-sphere

The 3-sphere or glome is a higher-dimensional analogue of a sphere. It is an object in mathematics that extends the concept of a circle (1-sphere) and a sphere (2-sphere) to the fourth dimension. The 3-sphere is defined as the set of points in four-dimensional space that are at a fixed distance from a central point. This distance is known as the radius of the 3-sphere, and the central point is called the center. The 3-sphere is denoted as S^3 and is a fundamental object in the field of topology, particularly in the study of compact manifolds and homotopy groups.

Definition

Mathematically, the 3-sphere can be defined as the set of all points in four-dimensional Euclidean space \( \mathbb{R}^4 \) that are at a fixed distance \( r \) from a central point \( c \), where \( r \) is the radius of the 3-sphere. The equation for a 3-sphere in Cartesian coordinates is given by:

\[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 + (w - w_0)^2 = r^2 \]

where \( (x_0, y_0, z_0, w_0) \) are the coordinates of the center \( c \) and \( r \) is the radius.

Properties

The 3-sphere has several interesting properties that distinguish it from spheres in lower dimensions. It is a closed, bounded, and simply-connected space, which means it is compact and without boundary, and every loop in the 3-sphere can be continuously contracted to a point. The surface area of a 3-sphere of radius \( r \) is \( 2\pi^2r^3 \), and its volume is \( \frac{1}{2}\pi^2r^4 \), which are generalizations of the formulas for the surface area and volume of a 2-sphere.

Embeddings and Projections

Visualizing a 3-sphere can be challenging due to the limitations of our three-dimensional perception. However, mathematicians use various techniques to study and represent 3-spheres, such as stereographic projection and embedding in higher dimensions. Stereographic projection is a way to project the points of a 3-sphere onto three-dimensional space, providing a useful tool for visualization and analysis.

Applications

The concept of a 3-sphere has applications in various fields of science and mathematics. In physics, it appears in the study of the universe's shape in cosmology, where the 3-sphere model is one of the possible topologies for a closed universe. In topology, the 3-sphere plays a crucial role in the study of four-dimensional manifolds and in the formulation of the Poincaré conjecture, a famous problem concerning the characterization of 3-spheres.

See Also

• Sphere
• 4-manifold
• Homotopy groups
• Compact manifolds
• Stereographic projection