Map the 3-sphere into 3D space and see how fibers become circles and lines
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 north pole passes through this point, which is why it appears as a straight line.
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.