SERIES H // HOPF FIBRATION
Hopf fibration posters
Shapes from the fourth dimension. Hundreds of circles, each linked with every other like links of a chain, brought down into our space as nested tori of light.
Read the guide: The Hopf fibration explained →
MATHEMATICS
The maths behind the picture
- 01
A circle in four dimensions
The three-sphere S³ lives in four-dimensional space. In 1931 Heinz Hopf found that it can be divided entirely into circles, so that every pair of circles is linked like two links of a chain.
S¹ → S³ → S² - 02
Every point, one circle
Each point on an ordinary sphere corresponds to one circle in S³. The program picks points along parallels or a spiral and draws the circle for each.
(z₁, z₂) ↦ (2z₁z̄₂, |z₁|² − |z₂|²) - 03
Down into our space
Stereographic projection brings the fourth dimension down into ours, the way a map brings the Earth onto paper. Circles stay circles, and each one stays linked with every other without ever touching it.
p ↦ (x, y, z) / (1 − w)
HISTORY
Where this idea comes from
- 1931
Heinz Hopf discovers the map from the three-sphere to the sphere and proves that all its circles are linked.
- 1975
Physicists recognise the Hopf fibration in the magnetic monopole and in the quantum mechanics of spin.
- 2008
Jos Leys, Étienne Ghys and Aurélien Alvarez make the films "Dimensions", showing the Hopf fibration to the world.
- today
Here every piece draws thousands of real circles from the fourth dimension, computed point by point.