The 3-Sphere

So far we have seen geometries in 2D and 3D, which are the dimensions we are familiar with.  But mathematician have found interesting geometries in higher dimensions and it would be great if we could visualize them.  In particular a huge collection of interesting surfaces come from the 3-sphere ${\mathbb{S}}^3$ in ${\mathbb{R}}^4$.  Though the developers of Houdini probably did not aim for creating an environment beyond 3D Euclidean space, it turns out as a surprise that Houdini is a natural tool to explore the 3-sphere.

Definition

The 3-sphere ${\mathbb{S}}^3$ is defined as a subset of unit vectors in ${\mathbb{R}}^4$
$${\mathbb{S}}^3=\{(x,y,z,w)\in {\mathbb{R}}^4|x^2+y^2+z^2+w^2=1\}.$$
Just like the unit spheres in other dimensions, the tangent vectors $v=(\dot{x},\dot{y},\dot{z},\dot{w})\in {\mathbb{R}}^4$ at a point $p_0=(x_0,y_0,z_0,w_0)\in {\mathbb{S}}^3$ satisfies $⟨v,p_0{⟩}_{{\mathbb{R}}^4}=0$, that is, the tangent plane at $p_0$ is a 3-dimensional hyperplane with normal vector being $p_0$.  The inner product of tangent vectors (also known as the metric) on the 3-sphere inherits from the ${\mathbb{R}}^4$ inner product, which gives us the notion of measures such as length, angle, area, and volume, and therefore defines geodesics (shortest paths), polygons (with edges being geodesics) and Riemannian curvatures (deviation of sum of exterior angles from $2\pi$ per unit area of polygon oriented in a particular direction).

Stereographic Projection

By the stereographic projection one has an identification between ${\mathbb{S}}^3$ and ${\mathbb{R}}^3\cup \{\mathrm{\infty }\}$:

$$\mathtt{\text{S3toR3}}((x,y,z,w))=(\frac{x}{1-w},\frac{y}{1-w},\frac{z}{1-w})$$
and its iverse
$$\mathtt{\text{R3toS3}}(P=(x,y,z))=(\frac{2x}{1+|P{|}^2},\frac{2y}{1+|P{|}^2},\frac{2z}{1+|P{|}^2},\frac{-1+|P{|}^2}{1+|P{|}^2}).$$

Each point in ${\mathbb{R}}^3\cup \{\mathrm{\infty }\}$ represents a unique point in ${\mathbb{S}}^3$ and vice versa, hence we visualize geometries in ${\mathbb{S}}^3$ by mapping them in ${\mathbb{R}}^3\cup \{\mathrm{\infty }\}$ through the stereographic projection. The stereographic projection is particularly nice because it is conformal, hence the angles we see after projection are the same as they would look in 4D!

Another remarkable fact about $\mathtt{\text{S3toR3}}$ is that it maps minimal surfaces (soap films extremizing area) in ${\mathbb{S}}^3$ to Willmore surfaces in ${\mathbb{R}}^3$ (shapes of elastic surfaces extremizing bending energy). [J.L. Weiner 1978]

Nevertheless, the length is not preserved in stereographic projections; the objects in ${\mathbb{S}}^3$ closer to $(0,0,0,1)$ will look much larger after projection.  Put it differently, the seemingly infinitely large space ${\mathbb{R}}^3\cup \{\mathrm{\infty }\}$ is actually not that large after $\mathtt{\text{R3toS3}}$.  In fact ${\mathbb{S}}^3$ is compact.  In topology $\mathtt{\text{R3toS3}}:{\mathbb{R}}^3\to {\mathbb{S}}^3\setminus \{(0,0,0,1)\}$ is called Alexandroff’s one-point compactification.

One-point compactification is just the abstract way of convincing oneself ${\mathbb{R}}^3\cup \{\mathrm{\infty }\}$ can be viewed as a closed and bounded set, with stereographic projection the concrete way of doing so.  This brings a nice picture that ${\mathbb{S}}^3$ is in fact the union of 2 solid tori with their boundary torus surfaces glued together. (The complement of a solid torus in ${\mathbb{R}}^3$ is another solid torus after one-point compactification.)

${\mathbb{S}}^3$ is the set of unit quaternion

Elements in ${\mathbb{S}}^3$ are naturally viewed as quaternions with unit length.  This makes ${\mathbb{S}}^3$ a (non-abelian) group with quaternionic multiplication (multiplications of unit quaternions are unit quaternions).  This group is in fact a double cover of 3D rotation group $SO(3)$ because each unit quaternion $q$ represents a 3D rotation $v\mapsto qv\bar{q}$ and that $q$ and $-q$ represent the same 3D rotation.

Rotations in 4D

To explore a 2-spherical globe in 3D, you apply 3D rotation to the sphere.  To explore around a 3-sphere, you apply rotations in 4D.

4D rotations ($SO(4)$) have 6 degrees of freedom.  They can be represented by a pair of unit quaternions: given $(q_1,q_2)\in {\mathbb{S}}^3\times {\mathbb{S}}^3$, the map $\psi \mapsto q_1\psi {\bar{q}}_2$ from $\mathbb{H}\to \mathbb{H}$ is a 4D rotation.  The representation ${\mathbb{S}}^3\times {\mathbb{S}}^3\to SO(4)$ is a double cover that $(q_1,q_2)$ rotation is the same as $(-q_1,-q_2)$ rotation.  Composition of 4D rotation is implemented in ${\mathbb{S}}^3\times {\mathbb{S}}^3$ as $(q_1,q_2)\circ (q_3,q_4)=(q_1,q_2)\cdot (q_3,q_4)=(q_1{q}_3,q_2{q}_4)$.

Let’s look at some special subgroups of 4D rotations.  One example is $\{(q_1,q_2)\in {\mathbb{S}}^3|q_1=q_2\}$.  It rotates 4-vectors as $\psi \mapsto q\psi \bar{q}$.  When $\psi \in {\mathbb{S}}^3$ and visualized via $\mathtt{\text{S3toR3}}$, it becomes just the 3D rotations.

Another example is $\{(q_1,q_2)\in {\mathbb{S}}^3|q_1=\overline{q_2}\}$.  In $\mathtt{\text{S3toR3}}$ visualization view it behaves as a sphere inversion in the direction of $\pm \mathrm{I}\mathrm{m}(q_1)$.

Another interesting subgroup is $\{(e^{i\theta },1)|\theta \in [0,2\pi ]\}$.  The trajectory of $e^{i\theta }\psi$ of a given $\psi$ becomes a circle that wind both “along” and “around” a torus.  The trajectory of $\psi e^{i\theta }$ forming from another subgroup $\{(1,e^{i\theta })\}$ is similar but with another orientation.  Note that given any random ${\psi }_1$, ${\psi }_2\in {\mathbb{S}}^3$, the circles $\{e^{i\theta }{\psi }_1|\theta \in [0,2\pi ]\}$ and $\{e^{i\theta }{\psi }_2\}|\theta \in [0,2\pi ]\}$ are always interlinked (if they are not the same circles).

Hopf fibration

Hopf fibration says that ${\mathbb{S}}^3$ is in fact the disjoint union of circles, and the set of these circles is a 2-sphere.  That is, there is a smooth map $\pi :{\mathbb{S}}^3\to {\mathbb{S}}^2$, called the Hopf map, and the preimage of each point in ${\mathbb{S}}^2$ is a circle in ${\mathbb{S}}^3$.

A concrete example of a Hopf map is $\psi \stackrel{\pi }{\mapsto }\bar{\psi }i\psi$ for $\psi \in {\mathbb{S}}^3$.  The result is just a rotation of the unit vector $i$ in 3D by $\bar{\psi }\in {\mathbb{S}}^3$ so the result lies in ${\mathbb{S}}^2$.  One can check that $\pi$ is onto (covers the whole 2-sphere) and for each $s=\bar{\psi }i\psi \in {\mathbb{S}}^2$, the preimage ${\pi }^{-1}(s)=\{e^{i\theta }\psi |\theta \in [0,2\pi ]\}$.  These circles are also called the Hopf fibers.