Soddy's Hexlet Soddy's Hexlet Imagine 2 spheres touching each other and the inside of a large sphere. In the applet the small spheres are blue and silver the outersphere is red. Now imagine the contact point between the 2 small spheres is surrounded by a necklace of spheres where each sphere in the necklace touches 5 spheres the 2 inner spheres the outer sphere and the necklace member ahead and the necklace member behind. The necklace spheres are green in the applet.
There are always 6 necklace spheres. The 2 neck spheres can be any size. They can be inside or outside the third sphere. The third sphere can be so large that it appears as a flat plane. Still exactly 6 necklace spheres. At the top of the applet controls is a choice box labelled presets selecting each choice except the last gives examples. The choice labelled ext. symmetric is interesting, the spheres actually invert and surround the entire setup but still exactly 6 spheres in that necklace.
You can examine the spheres by using the ROTATE and MOVE controls. The 45 in the ROTATE controls is angle in degrees it can be changed. The hide color control can be used to make spheres of a particular color invisible.

Why might there be exactly 6 spheres in the necklace?
First I need to explain what a geometric inversion is. Most people know what a geometric transformation is examples are shrinking, expanding, skewing. A geometric inversion is a transformation in which the original bit of geometry is pretty well unrecognisable. basically everything at infinity is reduced to a single point and everything at the point is expanded to infinity, the stuff inbetween is graded sort of evenly. Planes become spheres(usually) and spheres change size and position. Points of contact between sheres and between planes and spheres are always conserved.
 It turns out that all the cases we examined can be generated by doing geometric inversions of the same starting object and the 6 sphere necklace is in this object. The starting object is 6 same sized green spheres surrounding a samesized red sphere in a sort of sandwich with a blue and a pearl plane on either side. Shown here with the blue plane transparent.(planes are represented as hexagons)
To see this object click on the button at the bottom left labelled setup The button label changes to invert and the object changes to something very tiny. If you click on the resize button on the bottom left of the panel the object gets bigger. You may need to click resize a couple of times. resize centers and resizes the object you often need it to find your object after a change.
 When you invert and resize the blue and pearl planes turn into the blue and pearl spheres the large red sphere is descended from the small red sphere and the 6 green necklace spheres are descended from the regular spheres. Every different figure is the result of different inversion conditions, but the fact that there are 6 green spheres each in contact with 2 green spheres and the red pearl and blue spheres is an invariant condition of a geometric inversion of the starting object
Since you probably still do not have much of an understanding of how a geometric inversion works this may not mean a lot. If you want to learn more about inversions and Soddy's Hexlet try the following links
Eric Weisstein's World of Mathematics(hexlet page)
Edward Crane's Soddy's Hexlet
Geometric inversion is an extremely powerful geometric tool, proofs that take several pages can be reduced to a few lines. It is discussed in lots of tertiary level geometry texts. Excursions in geometry by C.Stanley Ogilvy is beautifully accessible to recreational geometers.

The buttons and fields under Initialise are part of a Soddy Generator you can use it to explore the possible ways of altering the inversion conditions and making new soddy geometries. All fields are reloaded when you select a new presets choice. The field green rotation controls the speed the green spheres rotate. There are 2 variables in the inversion. The distance to the center of inversion which is controlled by the field radial offset.The angle the blue and pearl planes make to the line to the center of inversion, this is controlled by the field "radial rotation".
The line ended with the save button can be used to save a hexlet you have made. It will appear in the presets choice box with any text you enter.
Click setup and the button text changes to invert and the object returns to the starting object. Often the object will disappear click resize a few times to find it. ("resize" centers the object and makes it about screen size.)
To get a visual feel for the Initialise controlls put the setup/invert button in the "invert" position, alter radial offset , radial rotation or balls diameter and then use the ROTATE buttons to see what you have changed. in relation to the black circle The black circle represents the sphere of inversion. You can see it's relationship to the startup object by using the ROTATE keys
The last of the presets structures is the 3d Arbelos I generated it when I realised that the arbelos system called Pappus's chain has a very similar inversion proof to soddy's Hexlet. The main difference is the number of balls in the initial object. The usual startup object has 7 balls 6 greens surrounding a red it has a balls diameter of 3 if you expand this to say 23 you can end up with the 3d arbelos. Even numbers can give quite strange results.