Interactive
Two planets orbit the sun. Join them with a straight line every few days and never erase it. Because their periods sit close to a small whole-number ratio, the lines don't smear into mush — they settle into a rose. Venus + Earth is the Qalarc mark.
Periods are sidereal, in days. Ratio is the best convergent of the period
ratio with denominator ≤ 12 — the same continued-fraction test astronomers
use to spot a near-resonance. Orbit radii from Kepler III
(a ∝ T^(2/3)), circles not ellipses.
The maths
Venus takes about 224.7 days to go round the sun; Earth takes 365.26. Divide one by the other and you get 1.6255 — which is within 0.034% of 13/8 = 1.625. In eight Earth years Venus completes almost exactly thirteen of its own. That "almost" is the whole trick: exact would be sterile, wildly off would be noise. Nearly is what draws.
Put both planets on circular orbits and sample time in fixed steps of N days. At each step, work out where each planet is and draw one straight line between them. Never erase. That's the entire algorithm:
// angle of a body at time t (days), starting from the top of its orbit const ang = -Math.PI/2 + 2*Math.PI * t / period; const x = cx + r * Math.cos(ang), y = cy + r * Math.sin(ang); // one chord per N days, drawn at very low alpha so the weave accumulates for (let t = 0; t < totalDays; t += N) { line( posOf(planetA, t), posOf(planetB, t) ); }
Each individual chord is uninteresting. The picture lives in the envelope — the curve the whole family of chords is tangent to. That envelope is a hypotrochoid, and its petal count falls straight out of the resonance.
Take a p:q resonance — the inner planet does p orbits while the outer does q. Now watch from a frame that rotates with the outer planet. In that frame the inner planet appears to loop around p − q times before the whole configuration repeats. The pattern therefore has |p − q|-fold rotational symmetry, and you get |p − q| petals:
| Pair | Ratio | Petals | Error | Cycle closes |
|---|---|---|---|---|
| Venus · Earth (the logo) | 13:8 | 5 | 0.034% | 8.00 yr |
| Earth · Mars | 15:8 | 7 | 0.308% | 15.05 yr |
| Venus · Mars | 3:1 | 2 | 1.875% | 1.88 yr |
| Mercury · Venus | 23:9 | 14 | 0.050% | 5.54 yr |
| Jupiter · Saturn (Great Inequality) | 5:2 | 3 | 0.674% | 58.91 yr |
For Venus and Earth: |13 − 8| = 5. Five petals, five cusps, 72° apart. Those cusps are the five inferior conjunctions — the moments Venus passes between us and the sun — and they're why the figure is known as the Pentagram of Venus.
Venus·Earth is off true 13:8 by only 0.034%, so successive 8-year cycles land almost on top of each other. The rose stays crisp for decades before it visibly drifts.
Jupiter·Saturn misses 5:2 by 0.674%. The triangle slowly rotates instead of closing — the drift Laplace called the Great Inequality. Crank the years slider up to watch it smear.
Mercury·Venus is 23:9 — fourteen petals. High-order resonances need a longer span and a smaller chord interval before the structure resolves.
The mark
Of every pair in the inner solar system, Venus·Earth is the one that draws best — and we didn't choose it for the astronomy, we chose it because of what it looks like when it's finished.
The full build — the density analysis that places the five eyes, the canvas spinner, and the Python pipeline that bakes the favicons — is written up on the logo story page.
Caveats
This is a circular-orbit approximation, and deliberately so. Orbits are drawn as perfect coplanar circles with radii from Kepler's third law (a ∝ T^(2/3)), every planet starting at the top of its orbit at t = 0. Real orbits are ellipses, mutually inclined, with their own precession.
For the resonance geometry that hardly matters — the petal count and symmetry depend only on the period ratio, which is accurate here. What you lose is second-order detail: Mercury's eccentricity of 0.206 would visibly squash its rose, and true conjunction points wander a little. The shape is honest; the ephemeris is not.
