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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.

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

Jul 14, 2003, 6:56:18 PM7/14/03

to

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.

> 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.

--

| Jim Ferry | Center for Simulation |

+------------------------------------+ of Advanced Rockets |

| http://www.uiuc.edu/ph/www/jferry/ +------------------------+

| jferry@[delete_this]uiuc.edu | University of Illinois |

Jul 17, 2003, 4:57:44 PM7/17/03

to

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

Jul 27, 2003, 12:51:43 PM7/27/03

to

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:

2 1

+------o---------o The sidelength is

| | c = 0.647111423

| |

| |

| |

3 ' . 5 The 5-gon is equilateral

| | but *not* regular.

| |

| |

+-------o--------+

4

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

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