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Equation of the Day #16: The Pentagram of Venus

Akano

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khqm1OK.png

 

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

 

cE8zi1M.png

 

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.

 

vDC1pTb.png

 

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

 

ZfPPxMj.png

 

Click for hi-res image.

 

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

 

huhDFO7.png

 

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

 

TmFf04p.png

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Oh no, the pentagram summoned the spirit of a dork! And he's talking about equations!

 

Egassem sdrawkcab! Egassem sdrawkcab! Egassem sdrawkcab! Egassem sdrawkcab!

 

That's gorgeous. :o

 

I was taken aback by it the first time I saw it. 'Tis quite pretty. ^_^

 

akanohi.png

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