largest 5-gon in a square

Rainer Rosenthal

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Jul 14, 2003, 5:34:46 PM7/14/03

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There are many equilateral 5-gons, which
can be inscribed in the unit square.

It seems as if the largest of them have
a corner point in the middle of a side
of the square.

Is there an elegant proof for this symmetry?

Regards,
Rainer Rosenthal
r.ros...@web.de

Jim Ferry

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Jul 14, 2003, 6:56:18 PM7/14/03

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Rainer Rosenthal wrote:
> There are many equilateral 5-gons, which
> can be inscribed in the unit square.
>
> It seems as if the largest of them have
> a corner point in the middle of a side
> of the square.

I disagree. I find that the largest of them is symmetric
about the diagonal of the square. This configuration does
not yield a vertex at a midpoint of a side.

To see this, consider the minimum separation of two parallel
lines (enclosing the pentagon) at a given angle t. This distance
is proportional to f(t) = cos(mod(t,pi/5)), with the understanding
that the range of mod(t,pi/5) is -pi/10 to pi/10. The minimum of
max(f(t),f(t+pi/2)) is obtained for t = pi/20, yielding a
configuration symmetric about the square's diagonal.

Rainer Rosenthal

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Jul 17, 2003, 4:57:44 PM7/17/03

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Jim Ferry wrote

> Rainer Rosenthal wrote:
> > There are many equilateral 5-gons, which
> > can be inscribed in the unit square.
> >
> > It seems as if the largest of them have
> > a corner point in the middle of a side
> > of the square.
>
> I disagree. I find that the largest of them is symmetric
> about the diagonal of the square.

Oh well, you are right and I am sorry for my error.
What I was looking for, is an elegant proof for

the smallest equilateral 5-gon

inscribed in the unit square.

This is a nice problem, when you look at all the possible
deformations of maximal enscribed equilateral 5-gons.

My stupid mistitling comes from a switch in the view of
the original question, which asked for the largest circum-
scribing square araund an equilateral 5-gon, whose sides
are all equal to 1. In our discussion in de.sci.mathematik,
Klaus Nagel provided this "dual" view.
See Message-ID: <3F117B0D...@t-online.de>

Best regards,
Rainer Rosenthal
r.ros...@web.de

Rainer Rosenthal

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Jul 27, 2003, 12:51:43 PM7/27/03

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Jim Ferry wrote

> Rainer Rosenthal wrote:
> > There are many equilateral 5-gons, which
> > can be inscribed in the unit square.

Hello Jim,

a nice proof has been provided by Klaus Nagel for the problem
of finding the smallest such 5-gon. Because I asked for the
largest one (which was not what I really wanted), I did not
carefully read your reply until now, sorry.

The largest equilateral 5-gon in the unit-square is indeed
symmetric about the square's diagonal. But your proof
seems to assume that I asked for a *regular* 5-gon.

That is not the case. Here is, what you wrote:

> To see this, consider the minimum separation of two parallel
> lines (enclosing the pentagon) at a given angle t. This distance
> is proportional to f(t) = cos(mod(t,pi/5)), with the understanding
> that the range of mod(t,pi/5) is -pi/10 to pi/10. The minimum of
> max(f(t),f(t+pi/2)) is obtained for t = pi/20, yielding a
> configuration symmetric about the square's diagonal.

The largest equilateral 5-gon in the unit square is situated
as follows:

Thanks for your interest. I will wait for possible replies here in
this thread and present the solution of Klaus Nagel, regarding the
smallest equilateral 5-gon within the unit square.

Rainer Rosenthal
r.ros...@web.de