The Institute of Mathematics & its Applications tells us that applications of the Reuleaux triangle include “the cross-section of some pencils that are thought to be more ergonomic than traditional hexagonal ones”. It it true?

It’s a bit difficult to tell, because there are many brands of triangular pencil and they don’t all come with easy-to-interpret cross-sections or dimensions. A true Reuleaux triangle pencil would have a cross section with rounded sides and 120° corners, making the same angle as the corners of a hexagonal pencil, as shown in the upper left cross-section of the figure below.

Instead, all the ones I found online appear to fall into two other classes of shapes: flat-sided triangles with rounded-off corners (lower left of the figure) or smooth three-lobed shapes with no corners at all (lower right). Here are some:

• The best image I could find of the Dixon Tri-Conderoga is this brief post by Woodclinched. It appears to be a smooth three-lobed shape. Dixon also produce a “Tri-Write” triangular pencil, aimed at a lower-level market than the Tri-Conderoga; the 2021 Dixon product catalog shows both it and the Tri-Conderoga as having a cross-section with slightly rounded sides (flatter than a Reuleaux triangle but not completely flat) and rounded corners.

• The Faber-Castell Grip 2001 is another three-sided pencil, with added bumps on its sides to make it grippier. If we ignore the bumps for the purpose of determining its shape, Pencil Talk shows it as a flat triangle with rounded corners.

• The Koh-I-Noor Triograph, as shown in a review by Pencil Talk, has rounded corners, and flat-enough sides to clearly print the brand name in large letters on it.

• The Lyra Ferby is a German children’s brand with a smooth three-lobed shape.

• The Marco Grip-Rite appears, from its review by Polar Pencil Pusher, to have a smooth three-lobed cross-section, even though the review calls it a Reuleaux triangle.

• The Micador ColouRush Jumbo Triangular Pencils (an Asutralian branch of children’s pencils) have a cross-section image on the manufacturer website clearly showing a flat triangle with rounded corners.

• PaperMate make several triangular writing tools under the “Handwriting” brand, including PaperMate Handwriting woodcase pencils, aimed at children. I couldn’t find a good image of a cross-section of one, but their mechanical pencil in the same branch is clearly a smooth three-lobed shape. Pencil Revolution appears to show a similar shape for the wood pencils.

• I don’t know what brand of pencils the IMA used for the photo of their example, but it appears to be one with flat sides and rounded corners.

I also didn’t find pencils with flat-sided triangles (and non-rounded or only very lightly rounded corners) as their cross-sections. But if you search for “triangle pencil” you will find plenty of colorful plastic things with that shape that you can slide over your pencil to make it easier to hold.

I can’t tell you whether any of these are any good to write or draw with; for that, I prefer fountain pens. (I don’t have an Omas 360, but that one really does appear to be based on a Reuleaux triangle.) Beyond writing, if you’re hoping for a pencil that doubles as a constant-width roller for rolling your books smoothly across your desk, and don’t want boring circular pencils, the smooth three-lobed ones may be your best bet. The one I drew above really is a curve of constant width. It’s just not a Reuleaux triangle.

This has been yet another episode of “things that are not Reuleaux triangles”. See previously a triangular bottle, the Kölntriangle, rotors within rotors, a Polish football logo, Wankel engines, and many more.