This instructable shows how to make an expanding and contracting sphere out of K'nex construction toys. The design is similar to that of the toy sold as the "Hoberman Sphere". (Note: the original US patent on the Hoberman sphere was issued in 1991 and is now expired.) In addition to being fun to build and fun to play with, it offers interesting lessons in mathematics.

Watch the K'nex sphere in action:

To make the sphere, you will need the following K'nex parts:

96 grey 2-slotted connectors
96 red 3-slotted connectors
96 green 4-slotted connectors
192 yellow rods
96 tan clilps
96 short white rods
96 blue spacer rings
48 short green rods
12 white 8-slotted hubs

Yep, that's a lot of K'nex. If you don't have everything on the list you can make some substitutions.

You can use any color rods instead of the yellow rods, just as long as they're all the same length. I've made the model using blue rods -- it's smaller and not as pretty but it's sturdier because it weighs less. I wouldn't recommend longer rods, though -- it might fall apart under the extra weight.

You can also substitute red connectors for some of the grey ones by ignoring the extra slot, but you have to be careful how you orient them or they can limit the range of motion of the model.

The white connectors and green rods can be left out, too, but you'll have to do something else to stabilize the "hubs" in Step 4 -- like inserting a blue rod as a diagonal brace (you'll see when you get there).

The blue spacers are also optional, but they really help keep the model moving smoothly and I strongly recommend using them if you can.

Step 1: Making Rhombuses

Remember rhombuses from math class? A rhombus is a quadrilateral with four equal length sides. Use all the yellow rods and the grey and green connectors to build 48 of them. The acute (sharp) angle in these rhombuses is 45 degrees and the obtuse (shallow) angle is 135 degrees.

After you've made all the rhombuses (Latin snobs call them "rhombi"), link them together in pairs by adding a clip to the end of a white rod, slipping the rod through the center of a green hub, add a blue spacer, then another rhombus (through the green hub again) and topping it off with another clip.

Step 2: Linking the Scissor Joints

Take two pairs of linked rhombuses from the previous step and arrange them as shown. From underneath, insert a white rod with clip through the grey connectors on the two lower rhombuses and add a blue spacer. Put the grey hubs on top over the white pivot rods and finish them off with clips (see closeup below).

Make 12 of these double scissor joints, but be careful -- it is easy to make mistakes. The assembly in the fourth picture is subtly different from the third -- and wrong! Notice in the third picture that if you examine the rhombuses from left to right you will see that they alternate being on top, on the bottom, on top, on the bottom. In the fourth picture, from left to right they are on the bottom, the top, the top, on the bottom. That joint won't flex like it needs to.

Also, if you look at the fifth picture you'll see two scissor joints that are mirror images of each other. Both will work, but not together. You need to make all your scissor joints alike! (Ok, ok, they can be made to work together, but it's complicated and I'm OCD, so humor me, ok?)

Step 3: Scissor Joint Theory

Note: this page is pure theory. Feel free to skip it if you really don't care about the foundations of the universe.

This figure shows the mathematical theorem behind the operation of this model. It states that if two triangles are connected together at a pivot as shown, the dotted lines connecting their vertices (the plural of vertex -- math talk for "corner") intersect at a constant angle, regardless of how the triangles are rotated. The proof is left as an exercise for the reader.

Because our triangles (which are really half of the rhombuses) have an obtuse angle of 135 degrees, the angle between the dotted lines is 45 degrees. So 8 of these scissor joints can be linked together in a ring (8x45=360) and slide in and out smoothly. Three of these rings can be made to cross each other at right angles in the xy, yz and zx planes and voila -- an expanding and contracting sphere!

These pictures show pairs of scissor joints flexing through their full range of motion. Cool, huh?

Step 4: Making the Hubs

Make 12 "hubs" as shown.

Step 5: Making a Ring of Scissors and Hubs

Link two scissor joint pairs together with a pair of hubs as shown in the first two pictures. Notice the position of the blue spacer. If a rhombus is below its neighbor the spacer goes above its grey connector. If a rhombus is above its neighbor, the blue spacer goes below its grey connector.

Link two more scissor joint pairs together. Then link the two pairs of linked joints together to form a a ring of eight scissor joints as show in the last picture.

Step 6: Making a Cross of Scissors and Hubs

In the last step we made a ring out of 8 scissors and 8 hubs. In this step we will make a cross out of 4 scissor pairs and 2 hubs. And then we'll make another one out of the remaining scissors and hubs.

Set a scissor joint pair down as shown and attach a hub to the ends of two rhombuses (first picture). Note the position of the blue spacers: if a rhombus is to the left of its neighbor, the spacer is added on the right of the grey hub. If a rhombus is to the right of its neighbor, the spacer is added to the left of the grey hub.

Add three more scissor pairs around the edges of the "bottom" hub, being careful to connect the bottom rhombus, not the top (second picture). Then flip the top hub down and connect it to the top rhombuses of all the scissors (third picture).

The fourth and fifth pictures show the whole assembly with the two hubs pulled apart, from the side and from above.

Step 7: Linking the Crosses to the Ring

Take the ring of scissor joints that you made in Step 5 and lay it flat in the middle of your work surface. Take one of the crosses you made in Step 6 and position it on top of the ring as shown below. Add the white rods and blue spacers to connect the ring and the cross. The first two pictures are almost identical. If you look carefully at the first you will that the connections haven't been made yet, but in the second they are there.

Then, flip the whole thing over and add the second cross to the other side. The sphere is complete!

Press or pull gently on two opposite hubs to make it expand and contract.

Thanks for sticking with me through this complex instructable. I hope you enjoyed it.

I would like to thank Sam MacInnes for demonstrating the model in the video and Margaret Minsky for lending me some extra K'nex pieces. I would also like to thank Mark Nahabedian for suggesting the use of the blue spacers to make its motion smoother.