When I visit the local hardware store, I run into boxes of carpenter's pencils. Why are they rectangular (i.e. skewed octagonal) in shape?
The answer lies in mathematics!
First, as any user of a carpenter's pencil knows, when you drop one, it stays right there and does not roll. This aspect proves helpful to anyone building structures of tall heights, on temporary or uneven flooring, or in an outside wind.
Second, the rectangular shape of the lead allows the user to draw either a thin or a wide line. All it takes is a 90-degree rotation of the pencil.
Are these mathematics-based reasons true? Yes, says Carl Reichenbach, product manager for Dixon -Ticonderoga--primary maker of carpenter's pencils.
Does any of this change when informed that many of the modern carpenter's pencils are elliptical in shape, not rectangular? Does mathematics again provide the reason for this modern shape?
Also, a new art form involves sculptures made using carpenter's pencils. This has nothing to do with math, but is interesting. Consider this example of a carved boot:
For more sculptures, check out Dalton Ghetti's "miniature masterpieces" created on tips of carpenter's pencils. Very unusual hobby!
Source: Adapted from D. Feldman's A World of Imponderables, 1992, p. 601