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In mathematics, an Apollonian gasket or Apollonian net is 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. It is named after Greek mathematician Apollonius of Perga (Ἀπολλώνιος ὁ Περγαῖος).

The algorithm uses René Descartes' formula aka Descartes theorem to calculate the curvatures (bends) of the generated circles along with complex Descartes theorem to calculate the position of the newly created circles.

• Press Space or Left Mouse Click to generate new gaskets!
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Apollonian circles experienced perhaps their most glorious rediscovery in 1936, when the Nobel laureate (in chemistry, not mathematics) Frederick Soddy became mesmerized by their charm. He published in Nature a poetic version of Descartes’ theorem, which he called “The Kiss Precise”:

Four circles to the kissing come
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb,
There’s now no need for rule of thumb.
Since zero bend’s a dead straight line,
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

Yet more is true: if all four discs
Are sited in the complex plane,
Then centers over radii
Obey the self-same rule again.

Helpful resources on Apollonian Gasket fractals and the Descartes theorem

• https://en.wikipedia.org/wiki/Apollonian_gasket
• https://mathlesstraveled.com/2016/04/27/apollonian-gaskets/
• https://www.americanscientist.org/article/a-tisket-a-tasket-an-apollonian-gasket