The geodesic dome is a geometric curiosity that anyone can make in their own home. Popular as tents at the Burning Man event, touted by Buckminster Fuller as the housing design of the future, geodesic domes are interesting just as curious conversation pieces. You don't need to build a 30-foot diameter dome to appreciate the mathematical and architectural beauty of the geodesic dome. You can build one in your own home for just pennies.
Lay out the outer decagon. Create a regular decagon as the outer bottom base of your geodesic dome. A regular decagon is a 10-sided figure whose sides are all the same length. Think of a stop sign with ten sides instead of eight. Connect your toothpicks together with clay.
Create a layer of regular triangles. Moving upward from the outer decagon, create a series of regular triangles. The top point of your triangles should be in the middle of the side of the decagon that they come from. There should be a total of 10 triangles when you are done. Connect the triangles with your clay as you go. Lean them forward and in a little, but don't worry about getting the angle exactly right. The construction of the geodesic dome will force you to create the proper angle as you go, and the malleability of the clay makes creating minor changes easy.
Connect the triangle tops. Create a series of inverted triangles by connecting the top points of your triangles together with clay. The end result will look like a regular pentagon.
Create the "star" layer. Starting at the top point of the pentagon, stick a toothpick straight up from every other clay connection on the pentagon. Place a piece of clay on the end of the toothpick. Connect the piece of clay with a toothpick on either side to the nearest clay connections on the pentagon one layer down. The result should look like a star.
Lay the final pentagon layer. Find the clay connection point on the last layer with three toothpicks coming out of it. Connect all of these points with a toothpick to make your final pentagon layer.
Connect the roof of your geodesic dome. Take five more toothpicks and stick them in each corner of the last pentagon you created. Connect the toothpicks in the middle of the pentagon with a final piece of clay.
Don't get frustrated. This is not a process you will perfect on the first attempt. Try to remember that you may need to have a few practice runs before you have a perfect geodesic dome. Every time you make one that doesn't come out exactly the way you like, try and have an honest appraisal of what you can do to make the dome more to your liking.