# Reuleaux polygon

Mark A. Mandel mamandel at UNAGI.CIS.UPENN.EDU
Wed Apr 30 17:59:37 UTC 2003

```On a University of Pennsylvania local newsgroup I mentioned the
"Rouleaux triangle". That elicited the following response (prefixed with
':'), and my appended followup:

>>>

: Yes! Though I believed it's spelled Reuleaux. There is a nice page about
: them here,
: http://mathworld.wolfram.com/ReuleauxTriangle.html . I should mention that
: any Reuleaux Polygon http://mathworld.wolfram.com/ReuleauxPolygon.html
: will work too.

Fascinating!

I first recall hearing of these in a story by Poul Anderson, who I think
used the "ou" spelling. I always thought it was "Rouleaux", and I ran 4
Google searches to start checking (after failing to find either spelling
in OED Online):

triangle        polygon
Reuleaux         606             214
Rouleaux        1330              30

I found the opposite skews for the two nouns interesting. And some of
the hits led to the evident origin of the term. As Amazon.com presents
it:

Kinematics of Machinery: Outlines of a Theory of Machines
by Franz Reuleaux

He lived 1829-1905, according to
http://www.seiflow.co.uk/Franz%20Reuleaux.htm, and was quite an expert
on kinematics and small machines (also
http://www.mae.cornell.edu/Reuleauxcoll/Sp.feat5.html).

I gather that he developed the theory of the polygons that are known by
his name. Each edge is an arc of a circle, and as the polygon rolls
along a flat surface, its upper limb maintains a constant distance from
the surface, just as with a circle, so that a Reuleaux-polygonal
cylinder can be used as a roller. And, I would guess, the spelling
"Rouleaux" arose from association with the verb "roll" ("rouler" in
French).

<<<

-- Mark A. Mandel
Linguistic Data Consortium, University of Pennsylvania

```