I was looking over some ancient posts of mine and can’t find one of my earliest threads.
Take a circle, and draw an equilateral triangle around it.
Now circumscribe another circle, and a square.
Circle, pentagon, circle, hexagon, ad infinitum.
Something along the lines of this:
The ratio of the outermost circle to the innermost is around 8:1
Does this geometric progression have a particular name?
Reminds me of Johannes Kepler’s model of the solar system (bolding mine):
Beyond that: the ratio of the radii of the circles inscribed in and circumscribing a n-sided polygon can be shown to be cos(π/n) (where the angle is measured in radians.) So if you continue this product out to infinity, you will get a ratio of
cos (π/3) * cos(π/4) * cos(π/5) * …
which converges (!) to approximately 0.11494204 ≈ 1/8.70003663 according to Wolfram Alpha.
As to whether this infinite product has a name: I’m unaware of one. It seems like an interesting quantity, and I wouldn’t be surprised if some mathematician studied it back in the 19th century, but I wasn’t able to find any mention of it via a quick Google.
Looking up the long version of that number in Wolfram Alpha indicates in the “Possible closed forms” section that it’s fancy-K-sub-in, which they call the polygon inscription constant. Mousing over that gives a link to MathWorld, which says it’s also called the Kepler-Bouwkamp constant. The “Bouwkamp” part comes from this article published by him in 1965. Computers were (obviously) still in their infancy and Bouwkamp was an applied mathematician that worked for Philips, so there’s a necessary emphasis in his paper on getting values for that number that converge quickly.
I don’t see a name for the actual arrangement of circumscribed polygons other than “polygon circumscribing”.
Nitpick: while it is geometric and a progression, it is not what is ordinarily called a geometric progression, which is one in which has a constant ratio between successive terms, e.g 1, 2/3, 4/9, 8/27, 16/81,… and can “converge” only to 0, 1, or infinity, depending on the ratio.
Yeah, I was thrown by that, too. But really, what the OP is talking about has a better claim on the phrase “geometric progression” than the usual usage does, and how else would you describe it in the limited space of a title line?
Just so I got it straight…
looking at the figures:
Kepler-Bouwkamp on the left, Polygonic Circumscribing on the right, the outermost and innermost circles of both figures look the same size.
Is that the case?
Do the two geometric progressions describe the same size ratio?
(BTW, if somebody can find a more appropriate figure size, be my guest;))
Yes, both of those diagrams represent the same multiplications, just with one starting in the middle and working out, and the other starting on the outside and working in.