Map the 3-sphere into 3D space and see how fibers become circles and lines
The 3-sphere lives in four dimensions, out of sight. Stereographic projection brings it into the 3D space we can see, and turns every Hopf fiber into a circle.
Toggle the Stereographic projection box to switch between a shadow of S³ (three of its four coordinates) and the true projection. Move the latitude slider and watch fibers near the projection point stretch toward infinity.
We cannot directly see the 3-sphere — it lives in 4-dimensional space. To visualize it, we use stereographic projection, which maps all of S³ (minus one point) faithfully into ordinary 3D space R³.
The projection is conformal (angle-preserving), which means circles on S³ map to circles in R³. This is why Hopf fibers appear as beautiful circles after projection, not distorted curves.
Toggle between viewing the fibers on S³ (using only three of the four coordinates) and the full stereographic projection. Notice how the projection spreads fibers out from a compact ball into the full space, with fibers near the projection point stretching toward infinity.
Key insight: Stereographic projection maps circles to circles (or to straight lines, when the circle passes through the projection point). The fiber of the south pole passes through this point, which is why it appears as a straight line.
Pick base points from the menu and watch how the fiber's size and position change. The formulas that draw each circle are in the drawer below.
Given a point on S² with spherical coordinates (θ, φ), the corresponding Hopf fiber on S³ is parameterized by t ∈ [0, 2π):
(x₁, y₁, x₂, y₂) = (cos(θ/2)cos(t), cos(θ/2)sin(t), sin(θ/2)cos(t+φ), sin(θ/2)sin(t+φ))
The stereographic projection from S³ ⊂ R⁴ to R³ removes one dimension:
(x, y, z, w) → (x, y, z) / (1 - w)
Together these formulas produce the circles you see in the 3D viewer. Explore how the fiber shape changes as you vary the base point.
Spot a mistake or have an idea?