### The 3-sphere: Extrinsic and Intrinsic Forms

Figure 1. An extract from Einstein’s 1917 paper on cosmology.

The circle in two dimensions and the sphere in three are just two members of an infinite family of hyper-surfaces. By analogy with the circle ${\mathbb{S}^1}$ in the plane ${\mathbb{R}^2}$ and the sphere ${\mathbb{S}^2}$ in three-space ${\mathbb{R}^3}$, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in ${\mathbb{R}^4}$ but can also be envisaged as a non-Euclidean manifold in ${\mathbb{R}^3}$.

Spherical Surfaces in Several Dimensions

A unit sphere is the set of points in a space that are unit distance from the origin. We can define spheres in several dimensions:

$\displaystyle \begin{array}{rcl} \mbox{0-dimensional sphere,}\ \mathbb{S}^0 && \{ x\in\mathbb{R}^1 : x^2 = 1 \} \\ \mbox{1-dimensional sphere,}\ \mathbb{S}^1 && \{ (x,y)\in\mathbb{R}^2 : x^2+y^2 = 1 \} \\ \mbox{2-dimensional sphere,}\ \mathbb{S}^2 && \{ (x,y,z)\in\mathbb{R}^3 : x^2+y^2+z^2 = 1 \} \\ \mbox{3-dimensional sphere,}\ \mathbb{S}^3 && \{ (x,y,z,w)\in\mathbb{R}^4 : x^2+y^2+z^2+w^2 = 1 \} \end{array}$

We can also define the unit balls obtained by “filling in” the spheres. The ${(n+1)}$-ball is the set of points on or within the ${n}$-sphere. Thus,

$\displaystyle \begin{array}{rcl} \mbox{1-dimensional ball,}\ \mathbb{B}^1 && \{ x\in\mathbb{R}^1 : x^2 \le 1 \} \\ \mbox{2-dimensional ball,}\ \mathbb{B}^2 && \{ (x,y)\in\mathbb{R}^2 : x^2+y^2 \le 1 \} \\ \mbox{3-dimensional ball,}\ \mathbb{B}^3 && \{ (x,y,z)\in\mathbb{R}^3 : x^2+y^2+z^2 \le 1 \} \\ \mbox{4-dimensional ball,}\ \mathbb{B}^4 && \{ (x,y,z,w)\in\mathbb{R}^4 : x^2+y^2+z^2+w^2 \le 1 \} \end{array}$

The 0-sphere comprises just two points ${\{-1,+1\}}$ on the real line. The 1-ball is the closed interval ${[-1,+1]}$. The 1-sphere is the unit circle in the Euclidean plane ${\mathbb{R}^2}$. The 2-ball is the solid disk in the plane. The 2-sphere ${\mathbb{S}^2}$ is the spherical surface in 3-space. The 3-ball is the solid sphere in 3-space.

Figure 2. Two 2-balls (disks) distorted into hemispheres can be merged to form a 2-sphere.

We can describe the 3-sphere extrinsically as a hyper-surface ${\mathbb{S}^3}$ embedded in 4-space ${\mathbb{R}^4}$. If we fix the value of the coordinate ${w}$, we get the set ${H(w_0) := \{ (x,y,z,w_0)\}}$. This is equivalent to a 2-sphere in the 3-dimensional cross-section of ${\mathbb{R}^4}$. However, we can also treat ${\mathbb{S}^3}$ and other spheres intrinsically.

Constructing Spheres from Balls

We can construct a sphere ${\mathbb{S}^{n+1}}$ by taking two copies of the ${n}$-ball ${\mathbb{B}^n}$ and glueing their boundaries. For example, we take two copies of ${\mathbb{B}^1}$, each of which is a line segment. We distort them to get two semicircular arcs, each having two boundary points. We can equate the boundaries to form a circle ${\mathbb{S}^1}$.

To form the 2-sphere ${\mathbb{S}^2}$, we distort two copies of the unit disk ${\mathbb{B}^2}$ into hemispherical caps, and join them along their boundary circles, which we take as the equator (Fig. 2).

Moving up One Dimension

Stepping up one dimension, we take two copies of ${\mathbb{B}^3}$, each of which is a solid globe (Fig. 3). If we map each point on the surface of the left globe to the corresponding point on the surface of the right one, we obtain a manifold which has no boundary. It is homeomorphic to the unit 3-sphere ${\mathbb{S}^3}$. It is simply connected, with a path from any point in one ball to any point in the other. The centres of the two balls may be regarded as the “poles” of this manifold.

Figure 3. Two 3-balls (filled-in 2-spheres) can be merged, by ‘glueing’ their spherical boundaries together, to form a 3-sphere.

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.

Millennium Problem

One of the Millennium Problems, the Poincaré Conjecture, states that the 3-sphere is, up to homeomorphism, the only compact, simply connected, 3-dimensional manifold without boundary. This conjecture was proved in 2003 by Grigori Perelman. The 3-sphere is also homeomorphic to the one-point compactification of 3-space, ${\mathbb{R}^3\cup\{\infty\}}$.

Einstein’s Cosmology

To obtain a steady-state solution to his relativistic equations, Einstein introduced an term involving a quantity called the cosmological constant. He was then able to find a solution for a spatially finite universe. The topological form of space was a 3-sphere. Indeed Einstein explicitly wrote the equation

$\displaystyle \xi_1^2 + \xi_2^2 + \xi_3^2 + \xi_4^2 = R^2$

which describes a 3-sphere ${\mathbb{S}^3}$ embedded in 4-space. An extract from Einstein’s 1917 paper is shown in Fig. 1 above. Observations later showed that the universe is not in a steady state, but is expanding. By introducing the constant, Einstein missed the chance to predict the expanding universe using general relativity.

It was later said by George Gamow that Einstein described the introduction of the cosmological constant as the biggest blunder of his life. However, his discovery that his field equations could accommodate a term representing gravitational repulsion was profound. The cosmological constant is essential in explaining the acceleration of the expansion.

Trailer

Long before Einstein, or Gauss, or Newton, the poet Dante described a structure of the universe, which has the form of a hyper-sphere. But that is another story (which we will tell soon in another post).

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A UCD course on recreational mathematics, AweSums: The Wonder, Utility and Fun of Mathematics, will be presented this autumn by Prof Peter Lynch — there are still some places, and registration is open at www.ucd.ie/lifelonglearning

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2022 Berkeley Lecture, Maynooth University
“Levels of Infinity”
Speaker: Peter Lynch
Thursday 29 September 2022, 6:30pm
John Hume Lecture Theatre 4, North Campus

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