Why is a manhole cover round?

Additional material

Problem 25 in the July 1971 "Mathematical Games" column [Editor: Martin
Gardner's, in Scientific American] asks "Why are manhole covers circular
instead of square?"  (No!  Please do NOT start this thread up
again...)  The August answer is that a square manhole cover could fall
through the hole into the sewer.  The November column notes that many
readers gave the very reasonable answer "Because manholes are circular."
Another answer given was that it reduces the number of decisions sewer
workers putting the lids on have to make.

And reader John Bush "told of a recent explosion near his home in
Brooklyn that blew off a Consolidated Edison square manhole cover.
After the smoke cleared the cover was found at the bottom of the
manhole.  'Geometria invincibilis est', Bush concluded."

-- Mark Brader <msb@sq.com>

[ Editor: from a November 1993 sci.math/rec.puzzles thread. ]

>tycchow@cayley.mit.edu (Timothy Y. Chow) writes:
>>For a simple example, begin with an equilateral triangle ABC, and
>>replace edge AB with an arc AB centered at C; do the same trick with
>>each edge.
>
>Is there a name for this shape?

Yes.  It's called a Reuleaux triangle.

>  Ideally, a good name would be a single word with a unique ending so
>the word can be generalized for others of higher order (as in n-gon
>or n-tuple).

How about Reuleaux polygon?  (But I just made that up, so I have no
idea what the standard term, if any, really is.)

-- David A. Karr <karr@cs.cornell.edu>

>>A similar procedure can be used with any convex polygon.
>
>With odd number of sides, no?  I have a hard time visualizing doing
>this to a square.

Yes.

Martin Gardner's writing on the subject has already been mentioned in
this thread. but shouldn't we also mention Rademacher & Toeplitz "The
Enjoyment of Mathematics" (originally "Von Zahlen und Figuren")?  I
certainly hope this is still in print: it would be a great pity if it
were not.

-- Chris Thompson <cet1@phx.cam.ac.uk>

> In article <1993Nov17.180726.23764@cs.cornell.edu> karr@cs.cornell.edu
> (David Karr) writes:
> >ObPuzzle:  The rec.puzzles archive gives five alternative answers to
> >"why are manhole covers round?" in addition to the usual one.
> >The first alternative is:
> >
> >   It gives maximal area for a given amount of material.
> >
> >What is wrong with this answer?

Nothing.  The cover and the lip of manholes are made of cast iron and a
round shape certainly minimizes the amount of material needed to make
them both.

-- Alex Lopez-Ortiz <alopez-o@neumann.uwaterloo.ca>

*Applause*

-- Warwick Allison <warwick@cs.uq.oz.au>

In article <2cec2o$etj@vixen.cso.uiuc.edu> cburian@ux4.cso.uiuc.edu
(Christopher J Burian) writes:
>There's no reason to assume you need to maximize area.  Most pieces of
>heavy equipment aren't round.
>
>And, of course, all flat plates of equal thickness and area will use
>the same amount of material, no matter what their shape.

How 'bout this:  Maximum area to weight ratio.

You want the largest area possible for the guys to get through, but
you want to minimize the weight for that area because the things are
typically lifted out manually.

-- Jeff Shearer <jshearer@beta.tricity.wsu.edu>

[ Editor: the November 1993 posting that started the "ObPuzzle" quoted
in the material above, and another reply to it. ]

In article <1993Nov17.153655.4988@cee.hw.ac.uk> jack@cee.hw.ac.uk (Jack
Campin) writes:
>
>ristol@xerver.icl.fi (Risto Lankinen) wrote:
>> In other words, can you find an(y) other flat shape(s) such that a lid
>> of that shape can't fall thru a hole of the same shape.
>
>Fattened up variants of
>
>                  |
>                  |
>                  |
>                 / \
>                /   \
>               /     \

That depends on how you fatten them up.  The obvious shape generated by
taking all points within some small distance d of this skeleton *can*
fall through a hole of the same shape without touching the sides.  (It
has to rotate on the way down, but the designers of manhole covers have
to allow for such things.)

Of course you can always "fatten up" any shape into a circle, that's
(partly) why I say "it depends."

>shape be convex, in which case i suspect the circle is unique but can't
>prove it.

You can't because it's not.  Read Chapter 18 of Martin Gardner's book,
"The Unexpected Hanging," on curves of constant width.

Perhaps this problem should be in the rec.puzzles archive.  I couldn't
find it there, though I did find "why are manhole covers round?"

ObPuzzle:  The rec.puzzles archive gives five alternative answers to
"why are manhole covers round?" in addition to the usual one.  The first
alternative is:

   It gives maximal area for a given amount of material.

What is wrong with this answer?

-- David A. Karr <karr@cs.cornell.edu>

That any manhole cover of uniform fixed thickness and of the same area
will obviously use the same amount of steel if it is being cast in a
mould (and not cut by a giant cookie-cutter, in which case only shapes
which tile the plane should be used).

Unless the lip is made of a very expensive material, in which case it
does optimize it.

-- Alexander Pruss <pruss@math.ubc.ca>