In 1929, a Swiss/German mathematician/sculptor named Paul Schatz discovered an intriguing new 3 dimensional shape, formed from a vesica piscis by rotating one of its circles 90 degrees & shrinkwrapping the result*.

An oloid (“*oh*-loh-*weed*”) is a peculiar solid. Check out how it rolls. It’s awesome:

Notice that as it rolls, every point on the surface touches the ground. The surface can be flattened without distortion onto the S shaped path it traces. That’s called a developable surface, meaning a surface that can be unrolled onto a plane.

Very few solids have this property. Spheres don’t – that why mapmakers have to compromise with cartographic voodoo to represent the surface of the earth on a flat paper map, distorting the shape, size or distance. If the earth had the shape of an oloid, a world map could be perfect.

I know what you’re thinking. You’re thinking, How would *I* make an oloid? Let us count the ways:

1. 3d printer. I made my first oloid by writing some Ruby code that created a 3d model of the solid (seen at top) in Google SketchUp, exporting the model into an STL file, uploading this to Shapeways, a 3d printer service, paying like $35 for a 40mm radius (length ~2.5 inches) white nylon oloid that showed up in the mail a week later, which I love more than life itself.

2. Carve it from wood. Arie Brederode has detailed instructions for precisely carving an oloid.

3. Fold the surface from the path. Since the oloid’s developable surface can be unrolled onto a plane, we can do the opposite: draw the path on paper & roll it up into the solid.

Here’s a great site produced by Gijs Korthals Altes with printable patterns for several hundred papercraft polyhedra. Print it up, cut it out, tape it together, & you have a paper oloid. Fill it up with foam or plaster of paris or a polymer & you have a solid.

4. Wireframe. Connect the 2 circles with struts or spokes or something. This fellow at some sort of oloid conference (!?) seems to be having a good time. The other is from the artist Uwe Prolingheurer.

* More precisely, taking the convex hull of the 2 circles. A convex hull is the least surface enclosing a set of points. & as long as we’re being nerdy…

Here’s a fun fact you can bring up the next time you’re having trouble making small talk, like on a first date or at a funeral (or both!): The surface area of an oloid is equal to that of a sphere with the same radius: *A* = 4π*r* ^{2}

That’s surprising. That means that if you take one of those circles we started with & make a sphere, or take both of them & make an oloid, you’d need exactly as much paint to cover either solid. Is that cool, or what?

But wait, there’s more! Archimedes worked out upper & lower bounds for √3:

Remember how he admired the vesica piscis for the √3 it displays? Well, guess who was the first person to work out the surface area of a sphere? Yup. Archimedes. I swear, what a guy! If he could only have seen an oloid, he would have had a eureka moment, for sure.