SERIES H // Hopf

Fibration

€39

Nested tori of hundreds of circles, where every circle is linked with every other.

Colour

OriginalInk

Format

Unframed printFramed print

Size

30 × 40 cm45 × 60 cm

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About the piece

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. The Hopf fibration explained

Drawn in code from seed: 00003
Series: Hopf
Variant: Fibration
Palette: Ember

Sizing

30 × 40 cm (12 × 16 in) or 45 × 60 cm (18 × 24 in), with a white border around the picture. Framed prints come in a black or natural wood frame (12 mm wide) with plexiglass, ready to hang.

Print & materials

Fine art matte paper (200 gsm) at 300 DPI, made to order. Every size is rendered natively - lines stay razor-sharp even at the largest size.

Shipping & payment

Free shipping worldwide. Every print is made to order within a few business days and shipped worldwide, packed to arrive flawless. You see the delivery estimate for your country when you order.

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Fine art paper

200 gsm enhanced matte

Printed on thick uncoated art paper at 300 DPI. Rich colour, razor-sharp lines, no glare.

Real wood frames

Black or natural wood

Wooden frame, 22 mm deep, with plexiglass that is lighter and safer than glass. Arrives ready to hang.

Made to order

Free shipping worldwide

Each print is made for you when you order and packed to arrive flawless, wherever you are.

The story

In 1931 Heinz Hopf showed that the 3-sphere, a sphere in four dimensions, can be divided into circles that are each linked with every other. Stereographic projection brings them down into our space, where they lie on nested tori.

MATHEMATICS

The maths behind the picture

Every piece is computed, not drawn. These are the real equations and numbers from the program that made this exact piece.

Read the guide: The Hopf fibration explained →

Equations

S¹ → S³ → S²(z₁, z₂) ↦ (2z₁z̄₂, |z₁|² − |z₂|²)p ↦ (x, y, z) / (1 − w)

Parameters

Seed: 3

Measured while drawing

Circles: 100 - 320

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.