# Equation of the Day #16: The Pentagram of Venus

**The above image is known as the Pentagram of Venus; it is the shape of Venus' orbit as viewed from a geocentric perspective. ****This animation**** shows the orbit unfold, while ****this one**** shows the same process from a heliocentric perspective. There are five places in Venus' orbit where it comes closest to the Earth (known as perigee), and this is due to the coincidence that**

**When two orbital periods can be expressed as a ratio of integers it is known as an ****orbital resonance**** (similar to how a string has resonances equal to integer multiples of its fundamental frequency). The reason that there are five lobes in Venus' geocentric orbit is that 13–8=5. Coincidentally, these numbers are all part of the ****Fibonacci sequence****, and as a result many people associate the Earth-Venus resonance with the golden ratio. (Indeed, ****pentagrams themselves**** harbor the golden ratio in spades.) However, Venus and Earth do not exhibit a true resonance, as the ratio of their orbital periods is about 0.032% off of the nice fraction 8/13. This causes the above pattern to precess, or drift in alignment. Using the slightly more accurate fraction of orbital periods, 243/395, we can see this precession.**

**This is the precession after five cycles (40 Earth years). As you can see, the pattern slowly slides around without the curve closing itself, but the original 13:8 resonance pattern is still visible. If we assume that 243/395 is indeed the perfect relationship between Venus and Earth's orbital periods (it's not; it precesses 0.8° per cycle), the resulting pattern after one full cycle (1944 years) is**

**Which is beautiful. The parametric formulas I used to plot these beauties are**

**Where ***t*** is time in years, ***r*** is the ratio of orbital periods (less than one), and ***τ* = 2*π*** is the circle constant.**

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