The Earth and Venus Dance
You are looking down on the Sun from above, with the orbits of Venus and the Earth around it. Every day a straight line joins the two planets where they really were on that day, and after eight years the lines weave a flower with five-fold symmetry. The pattern is not drawn to look pretty: it comes from Venus going round the Sun 13 times while the Earth goes round 8.
| Venus orbits in 8 Earth years | 13 |
|---|---|
| Inferior conjunctions in 8 years | 5 |
| Mean synodic period | 583.9 days |
| Flower turns backwards | 2.4° every 8 years |
What you are looking at
The glowing point in the middle is the Sun. The faint dotted rings are the orbits of Venus (inside) and the Earth (outside), drawn from the planets' own positions, so they are the real ellipses with the Sun very slightly off centre.
Each thin line joins the two planets on one day. A line is shortest, about 0.26 AU, when Venus passes between the Earth and the Sun (an inferior conjunction), and longest, about 1.74 AU, when Venus is on the far side of the Sun. The colour runs from violet to pale gold over the years, so you can see which lines came first. The small gold marks on the Earth's orbit show where the Earth was at each inferior conjunction.
Turn on Trace the midpoints and the middle of every line is followed through time. That point draws a figure of its own, with five small loops close to the Sun, one for each time Venus passes behind the Sun.
The math
Venus takes 224.7 days to go round the Sun and the Earth takes 365.26. Eight Earth years come to 2,922 days and thirteen Venus years to 2,921 days, so after eight years both planets are back almost exactly where they started and the lines begin to retrace themselves.
The time from one inferior conjunction to the next is the synodic period, about 583.9 days on average. Each one runs between 580 and 588 days, because both orbits are slightly elliptical. Five of them make 2,920 days, so there are five conjunctions in eight years. Between one conjunction and the next the Earth goes round the Sun 1.6 times, so each conjunction falls about 144 degrees further back than the last, two fifths of a circle. That is where the five-fold symmetry comes from. Five is also 13 minus 8: the number of times Venus laps the Earth in eight years.
The fit is not perfect. Five synodic periods fall about 2.3 days short of eight years, so the whole flower turns about 2.4 degrees backwards every eight years. Choose 32 years to see four flowers laid on top of each other, each slightly turned.
How it is computed
The positions come from astronomy-engine, an open-source library built on the VSOP87 planetary theory, which agrees with the Swiss Ephemeris to well under an arcminute. For every day of the span the piece asks where the Earth and Venus are relative to the Sun, in the fixed J2000 ecliptic frame, so decades of lines are not smeared by the slow turning of the equinoxes.
A line is drawn every day, or every few days if you move the slider. Inferior conjunctions are found by scanning for the moments the two planets share the same heliocentric longitude and narrowing each one down to about a minute. The orbits are the first full lap of each planet from the same daily positions, and the turn counts in the caption come from following each planet's angle round the Sun, day by day.
Questions
Is this pattern real?
Yes. Every line joins the two planets where they really were on that day. The five-fold symmetry follows from the ratio of the two orbital periods, 13 Venus years to 8 Earth years. It is a close match rather than an exact one, which is why the flower slowly turns from one eight-year cycle to the next.
How is this related to the Rose of Venus?
It is the same cycle seen from a different place. The Rose of Venus follows Venus as seen from the Earth; here you look down on both planets from above the Sun. In both, five inferior conjunctions in eight years make the five-fold pattern.
Why five petals and not eight or thirteen?
The pattern repeats each time Venus laps the Earth, and in eight years Venus goes round the Sun 13 times to the Earth's 8, so it laps the Earth 13 minus 8, which is five times.
Does the start date change the picture?
Only its angle. A different start date turns the whole flower, because the conjunctions fall in different directions, but the shape and the five-fold symmetry stay the same.