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Hopf fibration

h : S³ → S²

Stereographic projection into R³

Fibers of h after projection from N = (0, 0, 0, 1)

t = 0.000

r = √(1 − t²) = 1.000

Show full fiber projection (not on one slice) Show supporting torus Show projection point

Base space `S`²

Click a point to add its fiber.

Tools

t 0.000

Slice height t t = 0.000

Speed Loop

Load configuration

Selected fibers

No fibers selected yet. Click the base sphere or load a preset.

Spaces

S³ = { (z₁, z₂) ∈ C² : |z₁|² + |z₂|² = 1 }

Writing z₁ = x + iy and z₂ = z + it,

S³ = { (x, y, z, t) ∈ R⁴ : x² + y² + z² + t² = 1 }

The base is the unit sphere S² in R³.

Hopf map

h : S³ → S², h(z₁, z₂) = (2 Re(z₁ z₂), 2 Im(z₁ z₂), |z₁|² − |z₂|²)

Equivalently in real coordinates,

h(x, y, z, t) = (2(xz + yt), 2(yz − xt), x² + y² − z² − t²)

Fibers

The circle group S¹ acts on S³ by

e^iθ · (z₁, z₂) = (e^iθz₁, e^iθz₂)

This action preserves h, so each fiber h⁻¹(b) is an S¹-orbit. With θ = 2πs and p₀ = (x₀, y₀, z₀, t₀),

(x, y, z, t)(s) = (x₀ cos θ − y₀ sin θ, x₀ sin θ + y₀ cos θ, z₀ cos θ − t₀ sin θ, z₀ sin θ + t₀ cos θ)

A point b on the lower-left sphere selects the fiber h⁻¹(b). Its color agrees with the base point.

Stereographic projection

From the pole N = (0, 0, 0, 1), where t = 1,

σ(x, y, z, t) = (x/(1 − t), y/(1 − t), z/(1 − t)) ∈ R³

Fibers project to circles in R³, except for the fiber through N, which projects to a line. Distinct fibers remain linked.

Fixed-t slice

The slider chooses a height t ∈ [−1, 1]. The grey sphere is the level set

x² + y² + z² = 1 − t², r = √(1 − t²)

At t = 0 this is the unit 2-sphere. At t = ±1 the slice collapses to a pole of S³.

Markers are the intersections of the selected fibers with this level set. A generic fiber meets it in 0, 1, or 2 points.