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

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[Daily puzzle] Diameter of circle

2154 sats \ 13 comments \ @south_korea_ln 14 Oct science

The two circles

and

with radii of 1 are tangent with each other and with the horizontal line at the bottom. Circle

is inscribed with a maximum possible radius inside

, and the horizontal line.

,... are successively inscribed following the same rule maximizing their radius.

Find the diameter of a circle

Previous iteration: #720410 (quite a few different valid solutions were found for this one, congrats~~)

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0 sats \ 12 replies \ @Scroogey 14 Oct

For

looking at the right triangle

Pythagoras says

which means the diameter of

This can be repeated of

giving

and

giving

etc.

I don't yet have the generic formula for

1 sat \ 11 replies \ @south_korea_ln OP 14 Oct

Looks good. I'm not sure about your value for

though.

0 sats \ 10 replies \ @Scroogey 14 Oct

For

Diameter is twice that, so

No?

106 sats \ 1 reply \ @Scroogey 14 Oct

That would give diameters

For the sequence of divisors there are multiple candidates

1 sat \ 0 replies \ @south_korea_ln OP 14 Oct

This is a great website. Didn't know about it. Made me check out the infamous https://oeis.org/A000127 sequence, for which there seem to be many more cases showing this unexpected change from 32 to 31.

1 sat \ 6 replies \ @south_korea_ln OP 14 Oct

Aren't you using the same symbol

to denote the different radii of

(as in the drawing) and

(as in the right-hand side of the formula below)? And same comment for the subsequent

equation?

For

For

I used a different approach to solve this problem (I'll post it as a solution with the next iteration), but that one gives me

for

102 sats \ 5 replies \ @Scroogey 14 Oct

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

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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.

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1 sat \ 0 replies \ @south_korea_ln OP 14 Oct

I'll check my notes tomorrow.