Picture a single bright dot on a dark screen. Now imagine two springs tugging it at the same time — one pulling it left-and-right, the other up-and-down. If both springs wiggle at exactly the same speed, the dot just slides along a slanted line or traces a tidy oval. But let one spring wiggle faster than the other and the dot begins to loop back on itself, weaving bows, figure-eights, and pretzels that seem to hang frozen in the air. Those frozen weaves are Lissajous figures — the picture you get when two simple back-and-forth motions happen at right angles.
Build: two motions, two dials
Each pull is just simple harmonic motion. Sideways the dot follows $x = A_x\cos(\omega_x t)$; up and down it follows $y = A_y\cos(\omega_y t + \varphi)$. Two things decide the whole shape. The first is the frequency ratio $\omega_x : \omega_y$ — how many side-to-side wiggles happen for every up-and-down wiggle. The second is the phase $\varphi$ — the head start one motion has over the other. Try the simplest case, equal frequencies (a $1{:}1$ ratio): with $\varphi = 0$ the dot just runs along a diagonal line, but give the vertical motion a quarter-cycle head start ($\varphi = \pi/2$) with equal reach, and that line fattens into a perfect circle. Same two motions — only the timing changed.
Deepen: when the figure locks shut
The pattern settles into one closed, repeating figure only when the ratio is a ratio of whole numbers, $\omega_x/\omega_y = p/q$ (in lowest terms). Then the whole curve repeats after a period
— the moment both wiggles line back up to their starting positions. You can read the ratio straight off the picture: count how many times the curve kisses a vertical edge (that gives $p$) and a horizontal edge (that gives $q$). A $3{:}2$ figure brushes the sides three times and the top twice. If instead the ratio is irrational — say $\omega_y = \sqrt{2}\,\omega_x$ — the two motions never re-sync, so the curve never closes; it slowly shades in the entire rectangle, a first glimpse of "ergodic" motion. In the simulation, the sliders $\omega_x$ and $\omega_y$ set the two wiggle rates, $A_x$ and $A_y$ their reach, and $\varphi$ the timing offset between them.
Try this in the sim above
Set $\omega_x = \omega_y$ and sweep $\varphi$ from $0$ toward $\pi$ — watch the diagonal line bloom into a circle and collapse back into the opposite diagonal. Next, set the ratio to $3{:}2$ and count the edge-touches to confirm $p$ and $q$ for yourself. Finally, switch to the Irrational Ratios mode and watch the curve stubbornly refuse to close, gradually filling the box with ever-finer detail.
Section 03
Equations & Derivation
Parametric Equations
$$x(t) = A_x \cos(\omega_x t + \varphi_x),\qquad y(t) = A_y \cos(\omega_y t + \varphi_y)$$
Frequency Ratio & Closed Curves
$$\frac{\omega_x}{\omega_y} = \frac{p}{q} \in \mathbb{Q} \;\Longleftrightarrow\; \text{curve closes after period } T = \frac{2\pi p}{\omega_x} = \frac{2\pi q}{\omega_y} = \frac{2\pi}{\gcd(\omega_x,\omega_y)}$$
Cartesian Form (special case $\omega_x = \omega_y = \omega$)
Smallest integers such that $\omega_x/\omega_y = p/q$
dimensionless
$T$
Period of the closed curve
s
$N_x, N_y$
Touches on the vertical ($x$-extrema, $=p$) and horizontal ($y$-extrema, $=q$) edges
integer
Step 1Two perpendicular SHMs. Take $x = A_x\cos(\omega_x t)$ and $y = A_y\cos(\omega_y t + \varphi)$. The point $(x,y)$ traces a curve as $t$ varies — a Lissajous figure.
Step 21:1 ratio. If $\omega_x = \omega_y = \omega$, eliminate $t$. From $\cos\omega t = x/A_x$, then $\sin\omega t = \pm\sqrt{1 - x^2/A_x^2}$. Substitute into $y = A_y\cos(\omega t + \varphi) = A_y(\cos\omega t\cos\varphi - \sin\omega t\sin\varphi)$, square and rearrange — you get the boxed Cartesian form, which is an ellipse oriented at angle determined by $\varphi$.
Step 3Phase $\varphi = 0$. Curve degenerates to line $y = (A_y/A_x)x$. Phase $\varphi = \pi/2$: ellipse aligned with axes. Phase $\varphi = \pi$: line $y = -(A_y/A_x)x$. The figure rotates as $\varphi$ varies.
Step 4Rational ratio. If $\omega_x/\omega_y = p/q$ (with $\gcd(p,q) = 1$), the curve closes after time $T = 2\pi p/\omega_x = 2\pi q/\omega_y$. The figure has $p$ lobes touching the $x$-extrema and $q$ lobes touching the $y$-extrema. Counting lobes is the classic way to measure a frequency ratio on an oscilloscope.
Step 5Irrational ratio. If $\omega_x/\omega_y$ is irrational, the curve never closes. It densely fills the rectangle $[-A_x, A_x] \times [-A_y, A_y]$ — a precursor of ergodic behaviour and quasi-periodic motion. This is mathematically equivalent to the two-frequency torus in nonlinear dynamics.
How simulation variables map to the equations
Sliders ωₓ and ω_y set the two frequencies; Aₓ, A_y set amplitudes; φ sets the relative phase. Notice that integer ratios produce closed figures (1:1 ellipse, 1:2 figure-of-eight, 3:2 trefoil-like, etc.). The "Source Oscillations" mode shows the underlying $x(t)$ and $y(t)$ signals separately so you can see how perpendicular projection produces the 2D pattern.
Reference: A. P. French — Vibrations and Waves (MIT Introductory Physics Series), Ch. 1 §1-9: "Combination of Two Vibrations at Right Angles"; Halliday, Resnick & Walker — Fundamentals of Physics, 10th Ed., §16-7.
Section 04
Frequently Asked Questions
They are the visual signature of an oscilloscope in $x$-$y$ mode — drive the $x$ and $y$ deflection plates with two signals, and the resulting pattern reveals their frequency ratio and phase. They're used to calibrate frequency standards (NIST has used them since the 1950s), to measure phase delay in audio gear, in laser-based displays, and even in the logo of the Australian Broadcasting Corporation (a 3:2 Lissajous).
💡 An oscilloscope in XY mode is a Lissajous machine.
A point oscillating with frequency $\omega_x$ along the horizontal and $\omega_y$ along the vertical, simultaneously. The trail traces the resulting curve. Change the ratio of frequencies to switch between ellipses, figure-eights, hourglasses, and complex multi-lobed shapes. Adjusting the phase $\varphi$ rotates and reshapes the figure smoothly.
💡 You're watching what an oscilloscope draws when fed two sine waves into XY inputs.
If $\omega_x/\omega_y = p/q$ then after time $T = 2\pi p/\omega_x = 2\pi q/\omega_y$, both $\omega_x t$ and $\omega_y t$ have advanced by integer multiples of $2\pi$. So $(x(t+T), y(t+T)) = (x(t), y(t))$ — the trajectory has returned to its starting state. Conversely, if the ratio is irrational no such common period exists.
💡 Closed curves require a common period — the LCM of the two periods.
For irrational $\omega_x/\omega_y$, the trajectory in 2D is dense — given any point in the bounding rectangle, the curve will eventually pass arbitrarily close to it. This is a consequence of the Weyl equidistribution theorem: $(\omega_x t \mod 2\pi, \omega_y t \mod 2\pi)$ is uniformly distributed on the torus $\mathbb T^2$ when the ratio is irrational. Watch the simulation with $\omega_y = \sqrt{2} \,\omega_x$ and you'll see ever-finer detail emerge.
💡 Irrational ratios produce ergodic motion on a 2D torus.
For a closed Lissajous with ratio $\omega_x/\omega_y = p/q$ (lowest terms), count tangent points on a vertical edge — that gives $p$. Count tangent points on a horizontal edge — that gives $q$. So a figure with 3 horizontal touches and 2 vertical touches has ratio 2:3.
💡 Lobe count = one frequency; bounce count = the other.
Although Nathaniel Bowditch had drawn similar curves in 1815, Jules Antoine Lissajous (1857) made the systematic study using two tuning forks with mirrors. He bounced light off them and projected the resulting curves onto a screen — the first mechanical x-y oscilloscope. He used these patterns to measure tuning fork frequencies with great precision, predating electronic frequency counters by a century.
💡 Lissajous invented the analog frequency meter in 1857.
For $\omega_x = \omega_y$ and equal amplitudes, $\varphi = 0$ gives $y = x$ (line at 45°). $\varphi = \pi/2$ gives a perfect circle. As $\varphi$ increases from 0, the curve opens up into an ellipse, becomes a circle at $\pi/2$, closes back into the perpendicular line at $\pi$, and so on. The Cartesian equation $x^2 + y^2 - 2xy\cos\varphi = \sin^2\varphi$ shows this continuous transformation algebraically.
💡 Phase variation rotates and squashes the ellipse — same conic, different orientation.
❌ Lissajous figures only exist for sine waves — square waves can't make them.
✅ Any two periodic signals fed into orthogonal axes will produce a closed pattern if their fundamental frequencies are rationally related. Square waves produce a Lissajous-like rectangular pattern; triangles produce diamond-shaped figures. The classic smooth curves come from sinusoidal sources but the principle is general.
📖 A. P. French — Vibrations and Waves, MIT, §1-9.
❌ A figure-of-eight Lissajous means $\omega_y = 2\omega_x$.
✅ A figure-of-eight with the loop on its side ("∞" oriented horizontally) has $\omega_y/\omega_x = 2$ — but only at certain phases. At other phases of the same ratio you get an open hourglass. The "shape" of a Lissajous depends on both the frequency ratio and the phase, not just the ratio.
📖 HRW 10th Ed., §16-7.
❌ Lissajous figures are just a curiosity — they have no real physics application.
✅ They are the canonical visualisation of any two-frequency system: planetary orbital resonances (Earth-Venus dance traces a Lissajous-like rosette), atomic orbitals in 2D harmonic traps, beam coupling in particle accelerators (betatron oscillations), and the Pythagoras tree of musical intervals (octave = 1:2 = ∞ symbol; fifth = 2:3, etc.). They're a fundamental tool, not a curiosity.
📖 Symon — Mechanics, 3rd Ed., §3.10.
❌ The number of lobes equals the frequency ratio.
✅ The number of lobes is the numerator and denominator separately, not the ratio. A 3:2 Lissajous has 3 lobes touching one pair of edges and 2 lobes touching the other — a total of 5 visual "points," not 1.5 of anything. Learn to count edges separately.
📖 French — Vibrations and Waves, §1-9, Fig. 1-21.
❌ Lissajous patterns are unique to mechanical / electrical systems.
✅ They appear in any 2D system with two oscillation modes: a particle in a 2D harmonic trap (cold atom experiments), Foucault pendulum precession (a slow Lissajous), polarisation states of light (Stokes parameters trace ellipses on the Poincaré sphere — an analogue), and even in molecular vibrations of triatomic molecules.
📖 Born & Wolf — Principles of Optics, 7th Ed., Ch. 1.4.
❌ A 1:1 Lissajous is always a circle.
✅ A 1:1 Lissajous is generally an ellipse, not a circle. It becomes a circle only when amplitudes are equal AND phase is exactly $\pi/2$ (or $3\pi/2$). At zero phase it degenerates to a straight line; at $\pi/4$ it's a tilted ellipse. The "orbit shape" is sensitive to phase — this is exploited in optical polarimetry to measure birefringence.
📖 Hecht — Optics, 5th Ed., §8.1: "The Polarization Ellipse".
Misconception research: Crouch (2005), "Lissajous figures and the misconception of period"; Cromer (1981), Am. J. Phys. 49, 30 — "Stable solutions using the Euler approximation."