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
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
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
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
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
Step 5: Making a Ring of Scissors and Hubs
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
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
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.