Fascinating 3-D shapes -

The upper left shows a classic Gömböc; the lower left is a paper mobius strip; the upper right is a Reuleaux triangle and below it is a Reuleaux pyramid.

Illustration by Scott Hamilton

By Scott Hamilton

Senior Expert Emerging Technologies

Over the last week I have been researching unique things to print on my 3-D printer, which led me down the track of searching for shapes. I found three shapes that are very interesting. One of the three I had heard of before, but learned some very new things.

The first shape was the Reuleaux triangle, a curved triangle pyramid with equal sides that has all the points connected with an arc of constant radius. A guitar pic is a Reuleaux triangle. The end result is a pyramid shape that, when rolled like a wheel under a flat surface, will roll smooth. The Reuleaux triangle is the first of a sequence of similar shapes all named after Franz Reuleaux, who used such shapes to build machines for translating motion from one type to another. The Reulaux triangle easily converts circular motion to square motion; as the object rolls the center point traces the path of square. As you increase the number of sides on a Reuleaux shape, the volume of the shape increases and the surface becomes smoother; once you reach infinitely small sides, you have a sphere. In fact, when you view a circle or sphere on a computer screen, it is created by drawing the smallest possible triangles and joining them together, so you are really seeing a very small-sided Reuleaux shape.

The second shape is called a gömböc, a convex three-dimensional homogenous body that, when resting on a flat surface, has just one stable and one unstable point of equilibrium. It was designed by a Russian mathematician, Vladmir Arnold, in 1995 and proven by Hungarian scientists Gabor Domokos and Peter Varkonyi. The shape is not entirely unique; just like the Reuleaux shapes, there are many varieties of gömböc, most of which are close to a sphere and all with a very tight shape tolerance of about one in a thousand. They are easier to prove mathematically than they are to produce due to the strict tolerances. The most famous solution to creating gömböc is capitalized to set it apart and has a sharpened top. The shape is close to the shape of some tortoises, allowing them to more easily turn over onto their feet.

The final shape is the only one I had heard of before this week and is the mobius strip discovered by August Ferdinand Mobius and Johann Benedict Listing in 1858. The mobius strip is a unique shape in that it is a 3-D shape with only one surface. You can make a mobius strip by taking a rectangular piece of paper, making a one-half twist in the paper and gluing the ends together. This is where it gets interesting; you took the two-sided piece of paper and made it one sided. If you draw a line down the center of the strip, you will find that you cross under your line and all the way back around the paper to where you started, proving there is only one side. I knew this about mobius strips for years, but the new thing that I learned is what happens if you split it down the center line you just traced. You turn the one-sided strip into a longer two-sided strip, and it remains in one piece. However, if you try to cut 1/3 of the edge off the strip, you will get two pieces, one long double twisted strip passing through the middle of a new, smaller mobius strip.

I suggest you take some time and play with the last one, unless you have access to a 3-D printer; then, by all means, download a model and print the others for a little fun with geometry.

Until next week, stay safe and learn something new.

Scott Hamilton is a Senior Expert in Emerging Technologies at ATOS and can be reached with questions and comments via email to scott.hamilton@atos.net or through his website at https://www.techshepherd.org.