Understanding Higher Dimensions: Lessons from a S³ Projection

S² Stereographic Projection — A Guided Tour

A globe on a sheet of glass

Picture a glass plane tangent to the south pole of a unit globe. From the north pole \(N\), draw a ray through a point\(P=(x,y,z)\in S^2\). Where that ray hits the plane is \(\sigma_N(P)\in\mathbb{R}^2\). This gives the conformal map\[ \sigma_N(x,y,z)=\Big(\tfrac{x}{1-z},\,\tfrac{y}{1-z}\Big),\qquad z\neq 1. \]The north pole itself flies to infinity, so the plane is the sphere minus one point.

Inverting the map

Given a planar point \(U=(u,v)\in\mathbb{R}^2\) with \(\rho^2=u^2+v^2\), the inverse projection is\[ (u,v)\mapsto\Big( \tfrac{2u}{1+\rho^2},\; \tfrac{2v}{1+\rho^2},\; \tfrac{\rho^2-1}{\rho^2+1}\Big). \]Near the origin we sit close to the south pole; large radii push us near the north pole on \(S^2\).

Angle-preserving, not size-preserving

Stereography is conformal: it preserves angles but distorts lengths and areas. The planar metric pulls back to\[ \sigma_N^{*}(du^2+dv^2)=\Big(\tfrac{2}{1+\rho^2}\Big)^{\!2}\,ds_{S^2}^2. \]so an infinitesimal ruler in the plane scales by \(\lambda(\rho)=\tfrac{2}{1+\rho^2}\). Areas scale by \(\lambda(\rho)^2\). That's why regions far from the origin look exaggerated: we are peering toward the missing north pole.

Great circles become circles (or lines)

Intersect any plane through the sphere's center with \(S^2\): you get a great circle. Because stereography preserves circles, the image is either a Euclidean circle or a straight line (a circle passing through infinity). In the preview, the principal great circles (ZX/ZY/XY) appear as perfect circles or a line, depending on the chosen pole.

Meridians & parallels in the plane

Use spherical angles \(\theta\in[0,\pi]\) (polar from \(+z\)) and \(\phi\in(-\pi,\pi]\) (azimuth):\[ (x,y,z)=(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta). \]Under \(\sigma_N\),\[ (u,v)=\Big( \tfrac{\sin\theta\cos\phi}{1-\cos\theta},\ \tfrac{\sin\theta\sin\phi}{1-\cos\theta}\Big)=\Big( \cot\tfrac{\theta}{2}\cos\phi,\ \cot\tfrac{\theta}{2}\sin\phi \Big). \]So a parallel (fixed \(\theta\)) projects to a circle of radius \(\cot(\tfrac{\theta}{2})\) centered at the origin; as\(\theta\to 0\) the radius blows up (the parallel near the north pole escapes to infinity). A meridian (fixed \(\phi\)) becomes a straight line from the origin.

The complex plane and Möbius magic

Identify the target plane with \(\mathbb{C}\) via \(\zeta=u+iv\). Then\[ \zeta=\frac{x+iy}{1-z}=\cot\Big(\tfrac{\theta}{2}\Big) e^{i\phi}. \]Rotations of the sphere around the \(z\)-axis act as complex phase\(\zeta\mapsto e^{i\alpha}\zeta\). More generally, any 3D rotation becomes a Möbius (fractional linear) transform of the Riemann sphere \(\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\). This is why stereography is beloved in complex analysis: the sphere and the extended complex plane are the same picture.

A conformal compass for the UI

The preview’s options “meridians” and “parallels” draw these two orthogonal families. Near the origin (south pole contact), the grid looks nearly square—angles visibly preserved. As you move outward, circles stretch but intersections stay right-angled: a perfect visual certificate of conformality.

Geodesics and why straight is rare

On \(S^2\), geodesics are great circles. In the plane, only those great circles passing through the north pole map to straight lines; other great circles become Euclidean circles. So a “straight” shortest path on the sphere seldom looks straight on the map.

Inversion & the circle at infinity

Stereography intertwines with circle inversion. If \(C\) is a circle through \(N\), its image is a line in the plane, and conversely lines lift to circles through \(N\). The equator \(z=0\) becomes the unit circle when we scale the plane so that the south pole projects to the unit disk—another common normalization you can mentally compare with the preview.

Area distortion, quantified

A patch of area \(dA_{\text{plane}}\) shows on the sphere as\[ dA_{S^2}=\Big(\tfrac{1+\rho^2}{2}\Big)^{\!2} dA_{\text{plane}}, \qquad \rho=\sqrt{u^2+v^2}. \]So “equal squares” on the map don’t correspond to equal spherical areas—the familiar Mercator-like trade-off but for a perfect, circle-preserving map.

From S² to S³ intuition

Everything you're seeing on S² is the exact 1-dimension-down analogue of our S³ picture: the conformal factor generalizes to \(\big(\tfrac{2}{1+r^2}\big)^2\) in 3D, great circles map to circles/lines, and the "missing pole" becomes a line at infinity in the S³ stereographic view.

Visualizing S³ with Shadow and Stereographic Projections — Narrative

The 3-Sphere, Quaternions, and Rotations

The 3-sphere \(S^3\) is the unit sphere in \(\mathbb{R}^4\): \( w^2+x^2+y^2+z^2=1 \). A unit quaternion \(q=w+x\,\mathbf{i}+y\,\mathbf{j}+z\,\mathbf{k}\) encodes a 3D rotation. Indeed, \( S^3\cong\mathrm{SU}(2) \) double-covers \( \mathrm{SO}(3) \). The demo exposes \( (w,x,y,z) \) directly. For example, \( q(t)=(\cos t,\sin t,0,0) \)is a rotation about the \(x\)-axis; such points trace a red great circle on \(S^3\). Green/blue loops correspond to the \(y/z\) axes. These geodesics are mutually orthogonal.

Dual Projections: Shadow vs. Stereographic View

The left pane shows a linear 4D→3D shadow (drop \(w\) after 4D rotations). The right pane shows the stereographic image from \(N=(1,0,0,0)\), which preserves circles:\[ \sigma_N(w,x,y,z)=\frac{(x,y,z)}{1-w}. \]The two panes are synchronized to represent the same configuration of \(S^3\).

In the shadow view, the R/G/B great circles intersect at the ball’s center. In the stereographic view, they appear as linked round circles (with the exceptional pole fiber a straight line).

What Each Projection Shows

Shadow (left): linear compression into a solid ball; good for 4D rotation intuition but less clear for topology.
Stereographic (right): circle-preserving map to \(\mathbb{R}^3\); it exposes linkage and the torus structure of fibers.

The Hopf Fibration and Fiber Circles

The Hopf map \( h:S^3\to S^2 \) sends \( q\mapsto q\,\mathbf{i}\,q^{-1} \). The preimage \( h^{-1}(p) \) of a point \( p\in S^2 \) is a circle (a Hopf fiber). In stereography, every fiber is a round circle (or a line for the pole fiber). Any two distinct fibers link exactly once (Hopf link).

Nested Tori and Hopf Fiber Structure

A circle of base points on \(S^2\) lifts to a torus in \(S^3\) (a Clifford torus). Vary latitude to fill \(S^3\) by nested tori; stereography sends them to nested ring tori in \(\mathbb{R}^3\), foliated by Villarceau-like fiber circles. In the UI, constant-\(\mu\) selects these tori.

Hopf Coordinates \((\mu,\eta,\xi)\) and Curve Families

We use\((w,x,y,z)=(\cos\mu\cos\xi,\,\cos\mu\sin\xi,\,\sin\mu\cos\eta,\,\sin\mu\sin\eta)\). Constant \(\mu\) fixes a torus; constant \(\eta\) or constant \(\xi\) sweep orthogonal circle families on it; fibers cut the torus diagonally with \(\eta-\xi=\mathrm{const}\).

Bringing it Together

Use quaternion and Hopf controls to see where a point lies in both projections, which fiber it belongs to, and how constant-\(\mu\), \(\eta\), \(\xi\) families weave the torus grid while fibers cut across it.

Mathematical Foundation

1. Quaternions and S³

Unit quaternions \(q=(w,x,y,z)\) with \(\|q\|^2=w^2+x^2+y^2+z^2=1\) form the Lie group\(\,\text{SU}(2)\cong S^3\), a double cover of the spatial rotation group \(\text{SO}(3)\). A 3D vector \(\mathbf{v}\in\mathbb{R}^3\cong\text{Im}\,\mathbb{H}\) is rotated by:

In the viewer, the left panel shows an orthographic "shadow" of \(S^3\): we rotate in the 4D planes \((w,x),(w,y),(w,z)\) and then drop the \(w\) coordinate. The right panel shows the stereographic image in \(\mathbb{R}^3\).

1.1 Explicit \(\mathrm{SU}(2)\to\mathrm{SO}(3)\) action

For a unit quaternion \(q=w+x\,\mathbf{i}+y\,\mathbf{j}+z\,\mathbf{k}\), the adjoint action on pure imaginaries is a rotation with matrix:

Writing \(w=\cos(\theta/2),\; \|\mathbf{u}\|=\sin(\theta/2)\) yields the Rodrigues form with axis\(\hat{\mathbf{n}}=\mathbf{u}/\|\mathbf{u}\|\).

2. Stereographic Projection

With north pole N=(1,0,0,0), the stereographic map \(\sigma_N:S^3\setminus\{N\}\to\mathbb{R}^3\) is:

From the south pole S=(-1,0,0,0) replace \(1-w\) by \(1+w\). The inverse of \(\sigma_N\) is:

Geodesics on \(S^3\) are great circles; their stereographic images are round circles or lines in \(\mathbb{R}^3\).

3. Hopf Fibration and Clifford Tori

Identify \(S^3\subset\mathbb{C}^2\) by \(|z_1|^2+|z_2|^2=1\). Hopf coordinates:

For fixed \((\mu,\eta)\) varying \(\xi\) gives a Hopf fiber (a great circle). Under stereography, fibers become linked round circles on Clifford tori; varying \(\mu\) sweeps nested tori.

3.2 Stereographic torus radii \(R(\mu),\,r(\mu)\)

With \(q=(\cos\mu\cos\xi,\cos\mu\sin\xi,\sin\mu\cos\eta,\sin\mu\sin\eta)\):

Eliminating \(\xi\) yields a ring torus about the \(X\)-axis:

So \( R(\mu)=\tfrac12(\tan\mu+\cot\mu)\) and \( r(\mu)=\tfrac12|\tan\mu-\cot\mu|\).

3.3 Any two distinct Hopf fibers link once

With \(h(q)=q\,\mathbf{i}\,q^{-1}\), regular fibers have linking number equal to the Hopf invariant:

4. Kepler Problem and Runge–Lenz Vector

For \(H = \tfrac{\mathbf{p}^2}{2m} - \tfrac{k}{r}\) with\(\mathbf{L} = \mathbf{r}\times\mathbf{p}\), the (quantum) RL vector is

For \(H<0\), define

This explains hydrogen's degeneracy \(E_n=-\tfrac{mk^2}{2\hbar^2 n^2}\) with multiplicity \(n^2\).

5. S³ Realizations of Kepler Symmetry

Fock's S³: Compactify momentum space at fixed \(E<0\) to a 3-sphere; \(\mathbf{L}\) and a scaled \(\mathbf{A}\)generate SO(4) on S³.

Kustaanheimo–Stiefel (KS): For \(u\in\mathbb{H}\) with \(r=\|u\|^2\),

Kepler motion becomes uniform great-circle motion on S³; projecting back yields conics.

6. What the Viewer Shows

• Great circles (RGB): orthogonal SU(2) factors inside SO(4).
• Hopf fibers on tori: a cyclic phase at fixed energy.
• Quaternion point \(q\): a point on \(S^3\cong\mathrm{SU}(2)\) tracing the symmetry manifold.

7. From S³ Geodesics to Conic Sections

KS maps uniform great-circle motion to Kepler conics:

Additional Notes

Conformal factor (S³): For \(X\in\mathbb{R}^3\) with \(r=\|X\|\),

Great circles → circles/lines: a 2-plane through \(0\in\mathbb{R}^4\) intersects \(S^3\) in a circle; its stereographic image is a circle (or line if the plane hits the pole).