SERIES A // Attractors

Halvorsen

€39

Three wings folded around each other, a symmetry breaking in every instant.

Colour

OriginalRoseInk

Format

Unframed printFramed print

Size

30 × 40 cm45 × 60 cm

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About the piece

Chaos has a shape. Each picture is the path of a single point through space, computed hundreds of thousands of times by the same equations. What is a strange attractor?

Drawn in code from seed: 00002
Series: Attractors
Variant: Halvorsen
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

Named after Arne Halvorsen, the system reached chaos textbooks through physicist Julien Sprott's catalogue. Each variable pushes the next in a circle, x on y, y on z, z back on x, so the shape has threefold symmetry, like a clover leaf.

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: What is a strange attractor? →

Equations

dx/dt = −ax − 4y − 4z − y²dy/dt = −ay − 4z − 4x − z²dz/dt = −az − 4x − 4y − x²

Parameters

System: Halvorsen
a: 1.4

Measured while drawing

Start point: -8.73, -3.56, 1.404
Time step: 0.004
Integration steps: 260,000
Method: RK4
Lyapunov exponent λ: 0.694
View angle: 85.3° / -4.1° / -14.3°
Coverage (%): 35.5

What these numbers mean

λ = 0.694 > 0 means chaos: two paths starting 0.00000001 apart separate by a factor of e^(0.694·t). Their distance doubles every 1 time units, so very soon they have nothing in common.

HISTORY

Where this idea comes from

1890
Henri Poincaré, working on the three-body problem, first glimpses chaos: a tiny change at the start changes the whole outcome.
1963
Edward Lorenz at MIT reduces a model of the atmosphere to three equations and finds that weather cannot be predicted far ahead.
1971
David Ruelle and Floris Takens coin the term "strange attractor".
1972
Lorenz's talk "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" gives the butterfly effect its name.