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:

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* – *k* steps 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 160 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.

You would love books written by Richard Anthony Proctor:

http://en.wikipedia.org/wiki/Richard_Anthony_Proctor

Pychon’s novel, Mason and Dixon, is about the

“transits of Venus” and also makes some nice

remarks about observing solar eclipses via

the shadows cast by the leaves of trees.

Kepler (who first recognized that the planetary

orbits are elliptical, rather than circular) was

quite enamored of the fact that the ratios of

the various planetary orbital periods are very

close to low-order rational fractions (like 13/8).

Hmm, I know Kepler’s attempts to fit planetary orbit radii to Platonic solids, but not his interest in planetary orbital period ratios! Thanks!

Giving away some of the answer to one puzzle, it seems the 13/8 Venus/Earth period ratio is a ‘coincidence’ rather than a resonance locked into place for dynamical reasons. Wikipedia has a nice table of these ‘coincidental’ ratios, and the Venus/Earth one comes the closest to being exact. But apparently the 3/2 Neptune/Pluto period ratio is an actual resonance.

“Hmm, I know Kepler’s attempts to fit planetary orbit radii to Platonic solids, but not his interest in planetary orbital period ratios!”The two are related via his third law: “radius” cubed is proportional to period squared.

Do you think that Kepler thought that if you have the right perspective, life seems a bit less complicated?

I seem to remember that it’s resonance *overlap* that leads to instabilities. I too lack intuition for this and need to do the math (that’s how to build intuition, right?). Does anybody have a clear review article on this?

I don’t know a review article, but there could be something good in the references here:

• Orbital resonance, Wikipedia.

(The article itself doesn’t answer my main questions, but it’s still fun to read… it talks about cool things like

retrograde damocloids!)I don’t have the energy to lead an attack on this subject, but if someone wants to tackle it and post things here, I’d be very happy to join in. Understanding the math of stable vs. unstable resonance is definitely on my list of things to do before I die.

This led me to discover something I should have suspected: Google Scholar can be fooled by the end of article on the first page of article . That’s how a GS search for “orbital resonance” can turn up “A Plant Leucine Zipper Protein that Recognizes an Abscisic Acid Response Element” [

Science250,4978: 267–71 (1990)]. It’s picking up the footnotes from the previous article.I wonder if this ever screws up their citation data. Spot-checking this particular instance, I didn’t see that happening, but who knows?

Many automatic bibliographic databases are screwed up like this. Usually, you have to send an email to a person to get it corrected. I do this for my own stuff, even if it reduces my citation count!

I’ve sent this article on to several friends of mine who use and study pentagrams in various occult and esoteric practices; if you observe anything out of the ordinary, let me know and I’ll tell them to ease up a bit. This: http://arxiv.org/pdf/1307.6731v1.pdf is just as much magic to me as what they’re doing, by the way.

I’m sort of disappointed that nobody tried puzzles 1 and 2, since these are fairly straightforward math puzzles. Puzzle 3 is much harder, since it involves celestial mechanics of a nontrivial sort.

I thought, the planet Venus was said in the Da Vinci code, to trace a perfect pentacle across the sky every FOUR years…..

I haven’t read

The Da Vinci Code, but it sounds like your recollection is correct:Just one mistake: the golden ratio isn’t 0,6180…, it is 1,6180…, so there is apparently no relationship between earth-Venus lobes and the golden ratio.

Actually both and its reciprocal are called the golden ratio: different people use this term differently! But since they are reciprocals, if the Earth-Venus lobes are related to one they’re related to the other. 8/13 is close to 0.6180… and 13/8 is close to 1.6180…

I’m more familiar with the triangle of Venus. :-)

“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? “Sounds like the plot for an Eco novel.

EXCELLENT WORK here.

I’m James Gyre, I run the Naked Geometry page on facebook and I’m always trying to bring lucidity and critical eyes to geometry claims from kind but gullible people.

Well done, totally sharing with my followers. I’m a fan.

What hurts my brain is how IO and GANYMEDE have a 4:1 resonance and meet up a three points in their orbit. I mean, “4” makes me think of not . I mean, I get it, I can grasp the math, but I still sets off a buzzing in my brain to get a triangle from anything associated with the number 4. I just think of the number 12 to stop the buzzing.

It’s fun! But it doesn’t have much to do with the number 12. It’s just because 3 = 4 – 1.

This is similar to how the Earth turns around its axis 364.25 times each year relative to the stars, but we see the Sun go around us 365.25 times a year. (See discussion here.)

Or take coins the same size and roll one around another—rolling without slipping. The rolling one will turn around

twice:(Pay attention to the little line segment on the rolling coin.)

If we roll a coin on another coin that’s twice as big, it turns around three times:

And so on.

Here’s another nice animation:

It’s from a blog called Under the SymmeTree.