This image, made by Greg Egan
With the passage of each year, you will see the Sun go around the Earth. As the Sun goes around the Earth 8 times, Venus goes around the Sun 13 times, and traces out the pretty curve shown here.
It’s called the pentagram of Venus, because it has 5 ‘lobes’ where Venus makes its closest approach to Earth. At each closest approach, Venus move backwards compared to its usual motion across the sky: this is called retrograde motion.
Now, I’m sure you’ve noticed that these numbers:
3, 5, 8, 13
are consecutive Fibonacci numbers.
As you may have heard, ratios of consecutive Fibonacci numbers give the best approximations to the golden ratio φ = (√5 – 1)/2. This number actually plays a role in celestial mechanics: the Kolmogorov–Arnol’d–Moser theorem says two systems vibrating with frequencies having a ratio equal to φ are especially stable against disruption by resonances, because this number is hard to approximate well by rationals. But the Venus/Earth period ratio 0.615187 is actually closer to the rational number 8/13 ≈ 0.615385 than φ ≈ 0.618034. So if this period ratio is trying to avoid rational numbers by being equal to φ.
It’s all rather tricky, because sometimes rational numbers cause destabilizing resonances, as we see in the gaps of Saturn’s rings: Here’s a figure from James Ferguson’s 1799 book Astronomy Explained Upon Sir Isaac Newton’s Principles:
Naturally, some people get too excited about all this stuff—the combination of Venus, Fibonacci numbers, the golden ratio, and a ‘pentagram’ overloads their tiny brains. Some claim the pentagram got its origin from this astronomical phenomenon. I doubt we’ll ever know. Some get excited about the fact that a Latin name for the planet Venus is Lucifer. Lucifer, pentagrams… get it?
In The Da Vinci Code, someone claims that Venus traces “a perfect pentacle across the ecliptic sky every 8 years.”
Every 8 years, Venus goes around the Sun 13.004 times. So the whole pattern keeps shifting. It makes a full turn about once every 1920 years.