reply on: [Daily puzzle] Diameter of circle \ stacker news ~science

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102 sats \ 5 replies \ @Scroogey 14 Oct \ parent \ on: [Daily puzzle] Diameter of circle science

For

(the top-left corner of the triangle is supposed to be the center of

)

The hypotenuse is 1 +

, where

denotes the radius of

The horizontal leg is simply 1 (this is true for all

The vertical leg is

minus the sum of all diameters

and

minus

So, Pythagoras gives

1 sat \ 4 replies \ @south_korea_ln OP 14 Oct

Ok got it. This seems correct. I'll have to figure out why my approach is giving a different result. I likely must have applied it incorrectly. Please don't spend more time on this, I'm assuming the error must be on my side at this point.

1 sat \ 0 replies \ @south_korea_ln OP 14 Oct

You're right, I incorrectly applied Descartes. 1/12 is correct. My bad...

0 sats \ 2 replies \ @Scroogey 14 Oct

Oh, neat, I didn't know about Descartes' theorem!

So you're calculating

where curvature

(the inverse of the radius).

If I pick

and

to calculate

Now we can simply repeat with

and

to calculate

So radius

or diameter

1 sat \ 0 replies \ @south_korea_ln OP 15 Oct

And I guess a direct formula for circle

could be

1 sat \ 0 replies \ @south_korea_ln OP 14 Oct

Yes, exactly.

Related to Apollonian gaskets

a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three.