[sci.math] pentagons within squares

jml@cs.strath.ac.uk (Joseph McLean) (11/04/86)

I think Dave desJardins has hit the nail on the head.The largest pentagon
inscribed in a square is larger than the largest pentagon inscribed in
the largest circle inscribed in the square.

(How about that for a mouthful).

Simply conceive of placing one of the corners of the circled-pentagon 
at the middle of one of the square's sides.Since the distance from the
centre of this pentagon to the corner is the radius of the circle(which
is half the length of a side of the square),and this distance is the
smallest distance from the centre of the square to one of its sides
(varying from r to sqrt(2)*r),the other 4 corners of the pentagon will
not touch the square.Hence if we magnify the pentagon from the point in
contact with the square,we eventually obtain a strictly larger one
that touches the square in 3 places.Hence result.

What we can say is that the largest polygon inscribable inside a square
is "at least as large as" the largest polygon inscribable inside the
largest circle inscribable inside the square.

     jml,the circumlocutive mathematician.

jml@cs.strath.ac.uk (Joseph McLean) (11/11/86)

>
>Joseph McLean writes:
>> A pentagon can't be inscribed in a square ? Absurd.Whoever said that
>> all 5 corners of the pentagon need to touch the square ?
>
>My definition of "inscribe", from Webster's, states that all vertices
>of the inscribed polygon must touch a boundary of the polygon in
>which it is inscribed.
>
>Makes sense, when you think about it. Or else, here's two inscribed
>squares for you:
>
>			+--+
>			|  |
>		     +--+--+
>		     |  |
>		     +--+
>
>Who said that all 4 corners needed to touch the square? :-)
>
>Rick
>
>
If the dictionary says that all vertices must touch the square then
of course I accede.I presumed that inscribed meant lying totally within,
so that the above sketch would not be concidered legal by me either (before
I was corrected of course).
Mind you,if,for argument,there was a word that meant "lying totally within"
(subscribed?) then my little argument would hopefully provide the largest
such polygon lying within a square (which would then be at least as large as
the largest inscribed polygon).

Thanks for correcting me Rick,

   jml.