The Pentagram of Venus


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.




days. Venus orbits the Sun once every


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


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 = mn lobes? (The pentagram of Venus has 5 = 13 – 8 lobes.)

Puzzle 2: Under what conditions does the planet move forward j = nk 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 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.


45 Responses to The Pentagram of Venus

  1. JhM says:

    You would love books written by Richard Anthony Proctor:

  2. Paul Kainen says:

    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).

    • John Baez says:

      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.


      The two are related via his third law: “radius” cubed is proportional to period squared.

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

  4. Leo Stein says:

    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?

    • John Baez says:

      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.

    • Blake Stacey says:

      This led me to discover something I should have suspected: Google Scholar can be fooled by the end of article N on the first page of article N+1. That’s how a GS search for “orbital resonance” can turn up “A Plant Leucine Zipper Protein that Recognizes an Abscisic Acid Response Element” [Science 250, 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!

  5. 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.

  6. John Baez says:

    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.

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

    • John Baez says:

      I haven’t read The Da Vinci Code, but it sounds like your recollection is correct:

      In chapter six of The Da Vinci Code, Dan Brown’s character, Langdon, was amazed to learn that the planet Venus traces a perfect pentagram (also known as a pentacle or pentalpha) in the sky over a four-year period. This isn’t quite correct.

      The claim that Venus traces a perfect pentagram appears to have originated with western occultists. In reality, Venus traces an imperfect pentacle every eight years…

  8. Maria says:

    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.

    • John Baez says:

      Actually both (\sqrt{5} - 1)/2 = 0.6180... and its reciprocal (\sqrt{5} + 1)/2 = 1.6180... 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…

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


    Sounds like the plot for an Eco novel.

    • Klaus Schilling says:

      Venus may only be called Lucifer (carrier of light) if it is visible (from earth, due to reflecting sunlight) just before sunrise (it is a morning star); whereas, as an evening star, it is named Hespera.

  11. James Gyre says:


    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.

  12. Craig says:

    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 \Box not \bigtriangleup . 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.

    • John Baez says:

      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.

  13. John Baez says:

    Here’s another nice animation:

    It’s from a blog called Under the SymmeTree.

  14. […] a near perfect pentagram and is really cool. (A more detailed and scientific explanation in this blog post. The discussion afterwards is also […]

  15. love this info !!
    im obsesionate with this geometrics patterns and the meanings that is inside, im glad that im not alone, keep the good job.
    Btw, i feel a strong connection with this symbol, near perfect, near fibonacci, near lucifer (the light of the morning) and beautiful geometrics of a rose.
    Also venus, the planet, the mythology goddess and astrology around( yes i said astrology and im virgo lol) im very prone to make this a tattoo to remember me that im not perfect but a beautiful human being, just a tiny brain in this universe full of harmony and well organized chaos.

  16. John Baez says:

    Another nice image of the pentagram of Venus, created by Matthew Henderson using Mathematica:

    • Randall Wetmore says:

      Except the orbits aren’t perfect spheres, they’re elliptical.

      • John Baez says:

        Right, there are all sorts of ‘imperfections’. The frequency ratio is not exactly 13:8, etcetera. I believe Venus has the closest to circular orbit of all the planets, with an eccentricity of just 0.0067. The eccentricity of Earth’s orbit is 0.0167, and for Mars it’s 0.0934. I seem to recall it was problems with understanding the orbit of Mars that first led Kepler to guess that planetary orbits were ellipses.

  17. clifton says:

    Hey John, I would love to get an exact vector of the rose of venus. How would I go about getting or creating one?

    • John Baez says:

      I don’t know what you mean by “an exact vector”. You can draw the Rose of Venus by looking up the period of Earth’s orbit (1 year), the period of the orbit of Venus (0.616 years), the distance from Earth to the Sun (1 AU) and the distance from Venus to the Sun (0.723 AU).

      Then draw the Sun orbiting the Earth in a circle of radius 1, and draw Venus orbiting the moving Sun in a circle of radius 0.723, making sure that Venus takes 0.616 times as long to go around the Sun as it takes for the Sun to go around the Earth. This is more or less what Greg Egan did, though he adjusted the number 0.616 to make it equal 5/8 = 0.625. This makes the pattern simpler and more beautiful, though not astronomically accurate.

  18. Tamfang says:

    Using your numbers of days, I get this ratio:
    I hope that worked! Too bad there’s no preview here.

  19. Tamfang says:

    To put it another way, 8\frac{1}{15} \,: \,13\frac{1}{6}

  20. sunspotlover says:

    I found à connection between the orbit of Venus, the Earth and a circle of 360 degrees… Made a PDF file of it… It is the most strange theory concerning the fact why the ancients choose 360 degrees… To whom can I send it?

  21. Tamfang says:

    Seeing this thread again provoked me to create a diagram to add to the WP article on the Hilda asteroids:

  22. Nathan says:

    So what would it look like if the period ratio was exactly equal to φ?

  23. Herbert Knuettel says:

    Quote: “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:”

    What do I not understand here? If the entire pentagram makes a full turn of 360 degrees in 160 years, the slippage must be 18 degrees for every 8 years. While we are at that, how many degrees are Venus and Earth off the mark made by the first conjunction during the sixth conjunction?

    Bear with me, I am not an astronomer or mathematician but an interested layman ;-)

    Cheers, Herbert

    • John Baez says:

      I probably didn’t check the arithmetic—I probably just blindly copied down stuff I read here and there, not checking it for consistency. Let’s see. Check my work!

      According to NASA the sidereal orbital period, or ‘year’ for short, of Venus is 224.701 days, while that of the Earth is 365.256 days. We want the sidereal rather than tropical years here. It would be nice to have more decimal places, but this will do.

      So, 8 Earth years is 2922.048 days, while 13 Venus years is 2921.113 days.

      More importantly for us, 8 Earth years is 8 × 365.256 / 224.701 = 13.00416 Venus years. So that matches what I said. I vaguely remember doing this calculation myself before. Maybe it’s even on the blog article! I haven’t reread it.

      So, every 8 years the points of the Pentagram of Venus ‘slip forward’ by .00416 × 360 degrees. That’s about 1.5 degrees. So, for the pentagram to rotate 360 degrees takes about 360 / 1.5 = 240 of these 8-year periods, meaning about 240 × 8 = 1920 years.

      Or I could avoid using the number 360 twice by saying: each 8 years the pentagram makes .00416 of a full turn, so to make a full turn it takes 8 / .00416 years, or about 1923 years. Good! It’s nice to do a calculation twice and get the same answer. (I don’t really have good enough numbers to start with, to settle the last decimal place here.)

      I’ll fix my article! Thanks, I guess nobody had checked this out.

      • Herbert Knuettel says:

        Your calculation is of great help! Gives me once more affirmation that ancient astronomers used Venus as time giver for very long periods. They must have understood the long Venus cycle of 1920 years! Thanks for the help!



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