The Kiss Precise For pairs of lips to kiss maybe Involves no trigonometry. 'Tis not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. If one in three, beyond a doubt Each gets three kisses from without. If three in one, then is that one Thrice kissed internally. 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. To spy out spherical affairs An oscular surveyor Might find the task laborious, The sphere is much the gayer, And now besides the pair of pairs A fifth sphere in the kissing shares. Yet, signs and zero as before, For each to kiss the other four, The square of the sum of all five bends Is thrice the sum of their squares. Frederick Soddy (1936)     The Complex Kiss Precise Yet more is true: if all four discs Are sited in the complex plane, Then centers over radii Obey the self-same rule again. Wilks, Mallows and Laguna (2001)     Index The Challenges Jump to Model Share Your Results Jump to GXWeb

Drag the centers to rearrange the circles - or use the table below to control location and curvature/radius. Try some Apollonian Gaskets of your own - and try the Quiz! JIGSAW Reverse QUIZ Gaskets EDIT MODE ⇓ ♬ RESET This document requires an HTML5-compliant browser. Gasket ▶ 0 0 50 Depth ▶ 0 100 100 Bends: [-1, 2, 2, 3, 3, 6, 6, 6, 6, 11, 11, 11, 11, 14, 14, 14, 14, 15, 15, 18, 18, 18, 18, 23, 23, 23, 23, 26, 26, 26, 26, 27, 27, 27, 27, 30, 30, 30, 30, 35, ...] Bend Radius Center(x) Center(y) App generated by GXWeb Show/Hide Fill Axis Labels

 

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Gaskets, Kisses and Mathematical Bubbles

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Some two thousand years after Apollonius of Perga played with his gaskets and some three hundred years after Pappus with his boot-maker's knife and chains, Rene Descartes developed his Kissing Circles Theorem: if four circles have curvature a, b, c and x - and hence radii $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ and $\frac{1}{x}$ - then $$(a+b+c+x)^2$$ $$= 2\cdot (a^2 + b^2 + c^2 + x^2)$$ Solving for x gives two solutions for the curvature of the fourth circle: $$x = a + b + c$$ $$\pm$$ $$2\cdot\sqrt{(a\cdot b + a\cdot c + b\cdot c)}$$

It is much less obvious that the very same equation can be used to compute the location of the fourth circle as well, and thus completely solve the drawing problem. This fact was discovered in the late 1990s by Allan Wilks and Colin Mallows of AT&T Labs, and Wilks used it to write a very efficient computer program for plotting Apollonian gaskets.

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It took the application of complex numbers to the problem (which were not recognised by mathematicians until some 150 years after Descartes) to compute the centres of all those circles! Perhaps most amazingly, it turned out to be Descartes Kissing Circle Theorem once again which lay at the heart of this solution! In 2001, when Allan Wilks, Colin Mallows and Jeff Lagarias published a long article in the American Mathematical Monthly, they ended it with a continuation of Soddy's poem entitled The Complex Kiss Precise (cited above).

You might notice in the last verse that Soddy generalizes the theorem to five spheres. The extended theorem becomes:

$$a^2+b^2+c^2+d^2+x^2 = 3\cdot (a+b+c+d+x)^2$$

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Enter your name:	? Abel, Nils Agnesi, Maria Bernoulli, Daniel Cantor, Georg Descartes, Rene Euler, Leonhard Fermat, Pierre Galois, Evariste Germain, Sophie Hypatia Johnson, Katherine Liebniz, Gottfried Lovelace, Ada Monge, Gaspard Newton, Isaac Noether, Emmy Pythagoras Ramanujan, Srinivasa Riemann, Bernhard Sylvester, James Tao, Terence Weierstrass, Karl' Zeno
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More to Explore: the Kissing Circles Collection

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If you have found something of interest in these challenges, then it might be time for you to try to create your own kissing circles using GXWeb, and then move on to where more answers may be found, in the GXWeb Kissing Circles Collection:

The Arbelos (the Shoe Maker's Knife) and the Pappus Chain Arbelos is an old Greek name for a shoemaker's knife, and is made up of three kissing semi-circles. The Pappus chain is a ring of circles between two tangent circles (in this case, the Arbelos) investigated by Pappus of Alexandria in the 3rd century AD. How many Pappus Chains can you recognise as you scroll through the many gasket configurations above?

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Farey Numbers and Ford Circles Not all Apollonian Gaskets lie within circles (unless you consider a straight line as just a circle with 0 curvature and infinite radius!) One lovely application of kissing circles are Ford Circles, which represent Farey Numbers. Suppose you wanted to make a list of all the rational numbers between 0 and 1. How would you start? One approach might be to begin with the denominators of the fractions: first, list those with denominator 1 - zero and 1. Next, add those with denominator 2: $\frac{0}{1},\frac{1}{2},\frac{1}{1}$. Add those with denominator 3 ($\frac{0}{1},\frac{2}{3},\frac{1}{2},\frac{2}{3},\frac{1}{1}$) and you are on your way to the Farey Sequence! Here we see our Apollonian Gasket bounded, not by a circle, but by two lines, which may be thought of as circles of infinite radius and curvature 0!

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Descartes' Kissing Circles Theorem Descartes' (Kissing Circles) Theorem describes a beautiful and perhaps surprising relationship between tangential (kissing) circles: in general, for any three circles tangent to each other, two other circles exist (one inner and the other bounding the three) and all are related by a quadratic equation.

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Apollonian Gaskets and Mathematical Bubbles! Kissing Circles take on a whole new look when they become bubbles in an Apollonian Gasket! Apollonios of Perga studied these mathematical bubbles in the third century BC, but it took a couple of modern mathematicians in the 1990s to show how, once again, Descartes Theorem can help to produce the coolest screensavers.

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