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

pull down to refresh

1 sat \ 6 replies \ @south_korea_ln OP 14 Oct 2024 \ parent \ on: [Daily puzzle] Diameter of circle science

Aren't you using the same symbol

r

to denote the different radii of

O_1

(as in the drawing) and

O_2

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

O_3

equation?

For $O_2$

$(1 - \frac{1}{2} - r)^2 + 1 = 1 + 2*r + r^2$

For $O_3$

$(1 - \frac{1}{2} - \frac{1}{6} - r)^2 + 1 = 1 + 2*r + r^2$

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

1/18

for

O_3

102 sats \ 5 replies \ @Scroogey 14 Oct 2024

For

O_3

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

O

)

The hypotenuse is 1 +

r

, where

r

denotes the radius of

O_3

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

O_n

The vertical leg is

1

minus the sum of all diameters

O_1

and

O_2

minus

r

O_3

So, Pythagoras gives

r

O_3

1 sat \ 4 replies \ @south_korea_ln OP 14 Oct 2024

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 2024

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

0 sats \ 2 replies \ @Scroogey 14 Oct 2024

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

So you're calculating

 $k_4 = k_1 + k_2 + k_3 + 2*\sqrt{k_1*k_2 + k_2*k_3 + k_3*k_1}$

where curvature

k_n

\frac{1}{r_n}

(the inverse of the radius).

If I pick

O

O'

and

O_1

to calculate

O_2

 $1 + 1 + 4 + 2*\sqrt{1*1 + 1*4 + 4*1} = 12$

Now we can simply repeat with

O

O'

and

O_2

to calculate

O_3

 $1 + 1 + 12 + 2*\sqrt{1*1 + 1*12 + 12*1} = 24$

So radius

\frac{1}{24}

or diameter

\frac{1}{12}

1 sat \ 0 replies \ @south_korea_ln OP 15 Oct 2024

And I guess a direct formula for circle

O_n

could be

 $d(n) = \frac{1}{n(n+1)}.$

1 sat \ 0 replies \ @south_korea_ln OP 14 Oct 2024

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.