• Black Poincaré Section physics hoodie, pixel-art Hénon–Heiles Poincaré section design, worn by a model, front close-up
  • Black Poincaré Section physics hoodie, pixel-art Hénon–Heiles Poincaré section design, worn by a model, back close-up
  • Black Poincaré Section physics hoodie, pixel-art Hénon–Heiles Poincaré section design, front view
  • Black Poincaré Section physics hoodie, pixel-art Hénon–Heiles Poincaré section design, back view
  • Black Poincaré Section physics hoodie, pixel-art Hénon–Heiles Poincaré section design, turning on a model
  • Black Poincaré Section physics hoodie, ΨP Pixelated Physics neck label
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Poincaré Section HoodieISLANDS IN A CHAOTIC SEA | Pixelated Physics

Poincaré Section Hoodie – ISLANDS IN A CHAOTIC SEA | Pixelated Physics

$54.99 USD
$54.99 USD
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Color
Black
Size
S–5XL
Made
To order · shipping times

Henri Poincaré’s trick for seeing motion in four dimensions: don’t watch the whole orbit. Mark a dot each time it passes through one plane. A Poincaré section of the Hénon–Heiles system at E = 1/8: bright islands of regular orbits inside a scattered sea of chaotic ones. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

Color
Size

Size guideS–5XL · inches

Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

ShippingFree US

Free US shipping. $10 flat shipping outside the US.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

Henri Poincaré’s trick for seeing motion in four dimensions: don’t watch the whole orbit. Mark a dot each time it passes through one plane.

Michel Hénon and Carl Heiles applied it in 1964 to a simple model of a star moving in a galaxy. Every orbit here has the same energy, E = 1/8. Each was integrated numerically and cut at x = 0 going in the same direction.

Regular orbits leave dots on closed curves, the bright nested islands. Chaotic orbits scatter their dots over a region, the dim sea. Both kinds exist side by side at one energy, and an orbit never crosses from one to the other.

Raise the energy and the sea floods most of the islands. At E = 1/8 the islands still hold.

Islands in a chaotic sea.

Equations

\[H = \tfrac12\left(p_x^2+p_y^2\right) + \tfrac12\left(x^2+y^2\right) + x^2y - \tfrac13y^3\]
\[E = \tfrac18,\qquad \text{section } x = 0,\; p_x > 0\]
SymbolMeaningUnit
\(H\) Hamiltonian, total energy (scaled) \(\mathrm{1}\)
\(x, y\) position coordinates \(\mathrm{1}\)
\(p_x, p_y\) momenta \(\mathrm{1}\)
\(E\) energy of every orbit shown \(\mathrm{1}\)

Sources

  1. Poincaré (1890) Sur le problème des trois corps et les équations de la dynamique, Acta Math. 13, 1
  2. Hénon & Heiles (1964) The applicability of the third integral of motion: Some numerical experiments, Astron. J. 69, 73
  3. Lichtenberg & Lieberman, Regular and Chaotic Dynamics, 2nd ed. (1992)
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