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
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 |
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
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