July 2013 edit: I finaly printed the spheres see pictures [here]

Here we are back to our Impossible object series, and I promise this is the last time we are going to cover the Reuleaux polygons! Those constant witdth polygons can be extended to 3 dimensions to build non-regular spheres (I.E. spheroforms).

Here is a video of one ‘sphere’ based on a revolve of the Reuleau triangle:

Here is how to build these Reuleaux spheres:

Build half of the Reuleau polygons (see here how), and define the vertical segment as a center line (optional, the revolve command will let you select it anyway). Before closing the sketch, make sure that all the loops are closed with the sketch doctor tool.

Use full revolve for all the polygons and the circle (you will need to share the sketch). You can now export your spheroforms in STL and print.

These spheroforms are relatively easy to build using Matlab. Here is a parametric script that:

Build the regular polygon

Build the Reuleaux polygon

Rotate it through one axe of symmetry to get the cloud of points

Tessellate and save the result as a STL file

Note: The Matlab scripts are available HERE, Launch it with the start.m script. all code are copyrighted, only usable for non-commercial purpose and provided as is with no guaranty of any sort!

As a final note there are other spheroform like Meissner’s tetrahedron but I’ve covered enough the constant width solids for the moment. Maybe in the future…