In this series of “Impossible Objects” articles we will use the magic of 3D printing to explore strange objects that are (maybe) totally useless but at the same time absolutely essential.

We will start with something that defy intuition: **constant width shapes**. It’s is a rather boring name for something very cool. Like circles/spheres you can roll them and they will always have the same width even if they are not “round”:

The 3 sided shape called Reuleaux Triangle has a constant width when rotated.

Every odd number of side regular polygon can be transform to fit this property by drawing arcs of constant radius as shown in the next figure:

This video shows multiple of these strange “wheels” in action:

These shapes make a perfect tutorial to start modeling in Inventor for example and play with the constraints. I’ve made a set of wheels (IPT Inventor file) and converted them to STL for your printer.

- Start with a standard part and create a new 2D sketch
- Create all the wheels shapes with parametric dimensions:
- Create a circle and add a constraint with diameter name for the diameter
- Create with the polygon tool the triangle/pentagon/Heptagon and fix their dimension using diameter name

- Switch the initial shape to construction lines and add all center-point arcs. Make sure all the construction points are ‘snapped’ to the initial geometry (in blue). If they are not snapped, they will appears in green (under-constrained)

- Exit the sketch and extrude the circle with parametric distance call thickness. The sketch will be “consumed” by the extrusion, to right click on the sketch in the model try and select “share sketch”. Now extrude all the shapes. If the full profile cannot be selected it’s probably because there is an open loop in the sketch. To solve it edit the sketch and in the right click menu select “sketch doctor”, search for open loop and let inventor resolve the issue. You should be able to extrude all the shape now.

- We are nearly there! Add a small fillet on the front and back edges for the beauty. Save your part, then select export to “CAD format” in the File menu and create the .stl file. The preview let you adjust the precision of the triangle approximation.
- Print your new wheels! I’m eager to get the Form1 to test…

The files for this tutorial are [here for the inventor file] and [here for the STL]

**So are Reuleaux polygons really useless?**

The answer seems to be mostly… Beside a few coins, guitar picks and some signage the shape is not used in many applications. The Wankel engine uses a shape that is similar but with flatter sides. Maybe the fact that you can drill a square hole with them will finish to convince you?

In a coming post we’ll see the extension of these Reuleaux polygons in volume.

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