/sci/ - Science & Math » Thread #14002639

91KiB, 1920x1806, circles.png

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Problem

Anonymous Fri 24 Dec 00:46:40 2021 No.14002639 View Reply Original Report

Quoted By: >>14003435

In the gap enclosed by three large circles with equal radii, there are infinite circles arranged in the way showed in the picture. If the area of the central circle is 1, what is the total area of all the circles inside the enclosed area?

Anonymous

Anonymous Fri 24 Dec 2021 01:17:20 No.14002723 Report

Quoted By: >>14002812

There is a theorem by Descartes that says if 3 circles are mutually tangent at 3 distinct points with curvatures k1, k2, k3 then if a 4th circles is tangent to all 3 and has curvature k4 then

This gives a radius for the next smaller circles touching the center circle a radius of 1/(3 + 2Sqrt(3))

I tried going further but I couldn't figure out an easy way to get the entire series to sum out and I ran out of time.

Anonymous

Anonymous Fri 24 Dec 2021 01:18:50 No.14002731 Report

Quoted By:

I dunno what happened to the formula. It was

(k1 + k2 + k3 + k4)^2 = 2(k1^2 + k2^2 + k3^2 + k4^2)

Where k is curvature; k = 1/r

Anonymous

View Same Google iqdb SauceNAO soddy circle formula.png, 109KiB, 1920x1556

Anonymous Fri 24 Dec 2021 01:36:09 No.14002812 Report

Quoted By:

>>14002723

Anonymous

Anonymous Fri 24 Dec 2021 04:10:29 No.14003435 Report

Quoted By: >>14003456

>>14002639
how are you defining a 'circle' and 'infinite'?

Anonymous

Anonymous Fri 24 Dec 2021 04:20:20 No.14003456 Report

Quoted By:

>>14003435
https://en.wikipedia.org/wiki/Circle
https://en.wikipedia.org/wiki/Infinity

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