Jakob Steiner1796–1863
Steiner's porism
Swiss geometer who showed that a ring of circles, each touching its neighbours and two fixed circles, closes either for every starting circle or for none.
Steiner Chain
A ring of circles, each touching its neighbours and both of the circles it lies between. Either it closes exactly or it never does.
About this pattern
Jakob Steiner, 1826
Steiner's porism: if one such chain closes after a whole number of circles, every chain drawn between the same two circles closes after that same number, wherever it is started. So the ring can be turned freely and will always meet itself. The relation is exact: for a chain of n circles between concentric bounds, the inner radius is fixed by the sine of half the angle each circle occupies, and being out by any amount leaves a gap.
Behind the mathematics