• Black Lissajous physics hoodie, pixel-art Lissajous figure 3:2 phase π/2 design, worn by a model, front close-up
  • Black Lissajous physics hoodie, pixel-art Lissajous figure 3:2 phase π/2 design, worn by a model, back close-up
  • Black Lissajous physics hoodie, pixel-art Lissajous figure 3:2 phase π/2 design, front view
  • Black Lissajous physics hoodie, pixel-art Lissajous figure 3:2 phase π/2 design, back view
  • Black Lissajous physics hoodie, pixel-art Lissajous figure 3:2 phase π/2 design, turning on a model
  • Black Lissajous physics hoodie, ΨP Pixelated Physics neck label
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Lissajous HoodieINTEGER RATIOS CLOSE | Pixelated PhysicsHoodie

Lissajous Hoodie – INTEGER RATIOS CLOSE | Pixelated Physics

$54.99 USD
$54.99 USD
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Shipping calculated at checkout. Free US shipping · $10 flat shipping outside the US.
Color
Black
Size
S–5XL
Made
To order · shipping times

What does a 3:2 frequency ratio look like? Combine two simple harmonic motions at right angles and the moving point traces a Lissajous figure. When the frequencies form a whole-number ratio, here 3:2 with a quarter-cycle phase shift, the curve closes and repeats. An irrational ratio never closes. Jules Lissajous showed them with mirrors in 1857. Black pixel-art hoodie; equation, symbols and sources in the physics panel below.

Color
Size

Size guideS–5XL · inches

Free US shipping$10 flat shipping outside the USMade to order, ships in 2–5 business days

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

Two harmonic motions at right angles draw a curve. When their frequencies form a small integer ratio, the curve closes into a knot.

With frequency ratio 3:2 and phase δ = π/2 the path is x = A sin(3ωt), y = B sin(2ωt + π/2). After one common period T = 2π/ω both oscillators return together, so the figure repeats.

If the ratio is irrational (for example 1:√2) the path never closes: it densely fills a rectangle. The front print is the closed 3:2 knot.

On the hoodie the amber bead marks the centre crossing of the filled silhouette.

Jules Antoine Lissajous studied these figures optically in the mid-1800s; they are still the quick check that two oscillators share a rational frequency ratio.

Equations

\[x = A\sin(3\omega t),\qquad y = B\sin(2\omega t + \pi/2)\]
\[\text{closed iff }\omega_x/\omega_y\in\mathbb{Q}\]
SymbolMeaningUnit
\(x, y\) coordinates of the tracing point \(\mathrm{m}\)
\(A, B\) amplitudes of the two drives \(\mathrm{m}\)
\(\omega\) common angular-frequency unit \(\mathrm{rad/s}\)
\(3\omega, 2\omega\) drive frequencies (ratio 3:2) \(\mathrm{rad/s}\)
\(\pi/2\) relative phase \delta \(\mathrm{rad}\)

Sources

  1. Lissajous, J. A. (1857) Mémoire sur l’étude optique des mouvements vibratoires, Annales de chimie et de physique, 3e série, 51, 147
  2. French, A. P., Vibrations and Waves (MIT Introductory Physics), Ch. on coupled / perpendicular oscillations
  3. Halliday, Resnick & Walker, Fundamentals of Physics, Ch. on simple harmonic motion (perpendicular oscillators)
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