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. 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 f = (v5 – 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 f 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 f ˜ 0.618034. So if this period ratio is trying to avoid rational numbers by being equal to f, 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. Venus and the pentagram LUCIFER | PENTAGRAM | SATAN’s SYMBOL | ÉLIPHAS LEVI | ALEISTER CROWLEY The pentagram (pentacle or pentalpha) is considered by some western occultists to trace its esoteric significance to an astronomical observance of the pattern of Venus' conjunctions with the Sun. This is not possible. A superior conjunction occurs when Venus is behind the Sun and the inferior conjunction is when Venus is between Earth and the Sun; in both cases Venus is not visible. But it is true that plotting the recurrence of Venus' westward elongation from the Sun, over five consecutive synodic periods, will create the points of a pentagram. This period is approximately 584 days long, each period determining a different point of the observed pentagram—taking approximately eight years, five days to complete the figure. One would get a pentagram by picking any sunrise date on which the morning star is prominent and then repeating the observation at 584 day intervals following that date. The synodic period of Venus, the time required for it to return to the same position relative to the Sun as seen by an observer on Earth, is 583.9211 days. There is a 0.0789 day slippage every 584 days, totalling a one day slippage each 12.67 synodic periods. This means that the pentagram figure is slowly revolving within an oval in a clockwise direction, alternating either one or two points ascendant roughly every 160 years. There is no observation point on Earth that would present a regular pentagram. Moving further north elongates the figure while on the equator the figure is an irregular pentagon. Compiled by Trevor W. McKeown. Calculations and text reviewed by Dr. Leigh Hunt Palmer.