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This image, made by Greg Egan, shows the orbit of Venus.
Look down on the plane of the Solar System from above the Earth. Track the Earth so it always appears directly below you, but don’t turn along with it. 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.
Actually, what I just said is only approximately true. The Earth orbits the Sun once every
365.256
days. Venus orbits the Sun once every
224.701
days. So, Venus orbits the Sun in
224.701 / 365.256 ≈ 0.615187
Earth years. And here’s the cool coincidence:
8/13 ≈ 0.615385
That’s pretty close! So in 8 Earth years, Venus goes around the Sun almost 13 times. Actually, it goes around 13.004 times.
During this 8-year cycle, Venus gets as close as possible to the Earth about
13 – 8 = 5
times. And each time it does, Venus moves to a new lobe of the pentagram of Venus! This new lobe is
8 – 5 = 3
steps ahead of the last one. Check to make sure:
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That’s why they call it the pentagram of Venus!
When Venus gets as close as possible to us, we see it directly in front of the Sun. This is called an inferior conjunction. Astronomers have names for all of these things:
So, every 8 years there are about 5 inferior conjunctions of Venus.
Puzzle 1: Suppose the Earth orbits the Sun n times while another planet, closer to the Sun, orbits it m times. Under what conditions does the ‘generalized pentagram’ have k = m – n lobes? (The pentagram of Venus has 5 = 13 – 8 lobes.)
Puzzle 2: Under what conditions does the planet move forward j = n – ksteps each time it reaches a new lobe? (Venus moves ahead 3 = 8 – 5 steps each time.)
Now, I’m sure you’ve noticed that these numbers:
3, 5, 8, 13
are consecutive Fibonacci numbers.
Puzzle 3: Is this just a coincidence?
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 not doing a great job!
It’s all rather tricky, because sometimes rational numbers cause destabilizing resonances, as we see in the gaps of Saturn’s rings:
whereas other times rational numbers stabilize orbits, as with the moons of Jupiter:
I’ve never understood this, and I’m afraid no amount of words will help me: I’ll need to dig into the math.
Given my fascination with rolling circles and the number 5, I can’t believe that I learned about the pentagram of Venus only recently! It’s been known at least for centuries, perhaps millennia. 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?
I got the above picture from here:
• Venus and the pentagram, Grand Lodge of British Columbia and Yukon.
This website is defending the Freemasons against accusations of Satanism!
On a sweeter note, the pentagram of Venus is also called the rose of Venus. You can buy a pendant in this pattern:
It’s pretty—but according to the advertisement, that’s not all! It’s also “an energetic tool that creates a harmonising field of Negative Ion around our body to support and balance our own magnetic field and aura.” 
In The Da Vinci Code, someone claims that Venus traces “a perfect pentacle across the ecliptic sky every 8 years.”
But it’s not perfect! 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. You can see this slippage using this nice applet, especially if you crank up the speed:
• Steven Deutch, The (almost) Venus-Earth pentagram.
Also, the orbits of Earth and Venus aren’t perfect circles!
But still, it’s fun. The universe is full of mathematical beauty. It seems we need to get closer and closer to the fundamental laws of nature to make the math and the universe match more and more accurately. Maybe that’s what ‘fundamental laws’ means. But the universe is also richly packed with beautiful approximate mathematical patterns, stacked on top of each other in a dizzying way.
