Surfaces of Revolution

March 19, 2021
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A surface of revolution is a hollow 3D shape that is created by rotating the outline of a 2D shape around a line. Watch this video to learn more about them and see some examples.

There is a really neat ornament papercraft based on this principle. Here’s a video showing how to make the surface of revolution of half a circle with that technique:

You can adapt this to make any surface of revolution. Instead of half a circle, draw the shape of your choice on the copy paper. Just make sure one of its sides is the long edge of the rectangle. For example, if you choose a bell, instead of the full thing, do half of it so that it has a straight edge.

If you find it hard to follow the video, here are the step-by-step written directions.

Materials:

  • Color tissue paper
  • Glue
  • Ruled notebook sheet
  • Copy paper
  • Pencil
  • Needle
  • Thread
  1. Cut a rectangle on the notebook sheet, short edge aligned with the sheet’s lines. The size you choose for the rectangle will roughly be the size of your craft.
  2. Use that rectangle as a stencil to cut 25 or more equal rectangles on tissue paper and two on copy paper.
  3. Mark six lines in the notebook sheet to divide the rectangle into 7 roughly equal sections. You can use the notebook sheet’s lines to help you with that.
  4. Mark the first line A, then the next one B, then A again, and so on.
  5. Take one tissue rectangle and align it on the table below the notebook rectangle. Put glue continuing the A lines.
  6. Place another tissue paper rectangle on top of the first one and press to paste them.
  7. Now put glue on the B lines and paste another rectangle.
  8. Continue in this fashion, putting glue alternatingly in the A and B lines, until you use all the tissue rectangles.
  9. Glue one of the copy paper rectangle on top of the stack. This time put glue all over the rectangle.
  10. On the other copy paper rectangle, draw whatever shape you want to rotate. You can choose anything, but one of its sides must be the long edge of the rectangle. So for example, if you choose a bell, instead of the full thing do half of it, so that it has a straight edge.
  11. Cut your shape and glue it to your stack of tissue rectangles, on top of the tissue paper side. Make sure to align the straight edge of your shape with one of the long edges of the rectangle.
  12. Cut the stack along the outline of your chosen shape.
  13. Use the needle to pass a piece of thread on the upper corner of the stack and tie a loop. Do the same on the lower corner.
  14. Open up your piece. Notice how when you do that, you are rotating the chosen shape around the straight edge of your stack.
  15. To finish the craft, you can glue one side to the other, but to me, it is more fun to keep it like that and be able to rotate it and close it back up at will.

If you found this topic interesting, take a look at this series of videos explaining a math object that comes from surfaces of revolution.

 

This post was contributed by Buckeye Aha! Math Moments, the outreach program from OSU’s Department of Mathematics. Check out their website for more fun math activities.