Imagine two circles - kissing!
Now add a third and a fourth circle to kiss both of these - each can only be placed in one of THREE positions, but could have many different sizes.
Once the fourth kissing circle is added - and their sizes locked - then everything changes! Any further kissing circles are fully determined - they may only lie in certain positions and their sizes (radii OR their reciprocals, curvatures or bends) are fixed.
⇑ Tap for more...In fact, given any THREE kissing circles, there are only TWO possible subsequent circles - a feature which Descartes recognised as quadratic!
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. This relationship is described most easily in terms of the curvature of each circle, where curvature is defined as the inverse of the radius.
If the 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
\[(a+b+c+x)^2 \]\[= 2\cdot (a^2 + b^2 + c^2 + x^2)\]A little rearrangement(!) allows us to find the curvature (and hence, radius) of the inner and outer tangent circles given any three (here the greater of the two solutions describes the curvature of the inner circle and the lesser value (possibly negative) relates to the outer bounding circle):
\[x = a + b + c \]\[\pm\]\[2\cdot\sqrt{(a\cdot b + a\cdot c + b\cdot c)}\]
Of particular interest here is the special case where one of the three circles has infinite radius (and hence zero curvature) - i.e. one of the circles is a straight line, with the other two circles tangent to it and to each other.
This case (shown below) may serve to define Ford Circles and the next Ford Circle between two tangent circles with curvatures a and b will have curvature (derived from the above equation where \(c = 0\)):
\[x = a + b + 2\cdot \sqrt{(a\cdot b)}\]Here it is interesting to note that, when used to represent the rational numbers in the Farey Sequence, the radius of the Ford Circle standing on the rational value \(\frac{a}{b}\) is \(\frac{1}{2\cdot b^2}\). The centre of such a Ford Circle is \((\frac{a}{b},\frac{1}{2\cdot b^2})\).
Farey(3) is modelled below when both outer circles have radius \(\frac{1}{2}\). You may explore other Farey Numbers by entering farey(a) in the input box below, press the Input button, and use the a-slider. The same method can be used to explore arbelos(a) and pappus(a)!
You might check that this holds for the model below and then explore the values for other circle groupings!
radius A | ||||
0 | 0.5 | 20 | ||
a = | curvature of A | \(\frac{1}{radius(A)} \) | 2 | |
radius B | ||||
0 | 0.5 | 20 | ||
b = | curvature of B | \(\frac{1}{radius(B)} \) | 2 |
c (curvature C) = \(a + b + 2\cdot \sqrt{(a\cdot b)} = \) 8
radius C = \(\frac{1}{c} = \) 0.125
d (curvature D) = \(a + c + 2\cdot \sqrt{(a\cdot c)} = \) 18
radius D = \(\frac{1}{d} = \) 0.056
e (curvature E) = \(b + c + 2\cdot \sqrt{(b\cdot c)} = \) 18
radius E = \(\frac{1}{e} = \) 0.056