Steiner's chain

One circle inside another, and between them circles placed one by one, each touching both and the one before; the one marked 1 is the first. Drag the slider Where the first circle goes, or press play: the chain slides round between the two circles, its circles growing and shrinking, and the last always just touches the first; the first-to-last gap in the corner stays at 0. Drag the centre of the inner circle and the chain changes but still closes; the inner circle shrinks as it moves off centre, because the two circles have to match in size and position for exactly one ring to fit. With Inner circle enlarged or shrunk at 0, change n and the inner circle takes the size at which n circles just close. Move that slider away from 0 (the percentage is of the inner radius in the concentric view, top right) and the last circle misses the first: larger leaves a gap (coral), smaller makes them overlap, and the corner goes negative. Move the start again and the gap widens and narrows, but no starting place closes it. Top right is the concentric view: a map that takes circles to circles and keeps every touching circle touching can make any two circles, one inside the other, concentric. There the circles are all the same size and a new start only turns the whole ring, so whether it closes is plain to see and cannot depend on the start; the map carries that back to the pair you see. Jakob Steiner found this.