heres a question

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4 total messages Started by Brian Mcdonald Fri, 22 Jan 1993 02:19

#506

Author: Brian Mcdonald
Date: Fri, 22 Jan 1993 02:19

9 lines
801 bytes

how many polygons are there that all its sides are longer than any of its 
diagonals
?????????????????????????peace and i'm outa here



--  
uucp: uunet!m2xenix!puddle!128!18.4!Brian.Mcdonald
Internet: Brian.Mcdonald@p4.f18.n128.z1.fidonet.org

Re: heres a question

#515

Author: I am a terminator.
Date: Mon, 25 Jan 1993 18:10

32 lines
1683 bytes

In article <41295.2B62BCAE@puddle.fidonet.org>, Brian.Mcdonald@p4.f18.n128.z1.fidonet.org (Brian Mcdonald) writes:
|> how many polygons are there that all its sides are longer than any of its 
|> diagonals

All right! A COUNTING problem. Counting and categorizing are said to be the
two hardest topics in math. So let the count begin! Ha Ha Ha Ha (thunder
and lightning)

I submit the triangle to the count. For all x, if x is a diagonal of a
triangle, then every side of the triangle is longer than x. This is

                                  true

because triangles have no diagonals. 1 Ha Ha Ha Ha (thunder and lightning)

I ask
  (1) Prove that every polygon has a convex corner.
  (2) What comes after the triangle? (Can I say 2 Ha Ha Ha Ha
      3 Ha Ha Ha Ha?)

-- 

Henry Choy
choy@cs.usask.ca

What rolls down stairs alone or in pairs            This has been brought to 
Rolls over your neighbor's dog?                     you by the numbers 4
What's great for a snack and fits on your back?     and 9 and the letter P.
It's Log, Log, Log!  -- "The Log Song", from            -- Big Bird
                         Ren & Stimpy

Math is tough!       -- Barbie

Re: heres a question

#518

Author: Alan Kaufman
Date: Tue, 26 Jan 1993 20:24

5 lines
766 bytes

Every polygon has a "convex corner?"  Err, what is a convex corner?

[The intersection of a finite set of convex regions is convex, and a planar
pologon is the intersection of a finite number of half-planes
(which are convex), so "polygon" implies a convex figure. ]

Re: heres a question

#519

Author: David G Radcliffe
Date: Tue, 26 Jan 1993 22:31

21 lines
1412 bytes

Brian Mcdonald (Brian.Mcdonald@p4.f18.n128.z1.fidonet.org) wrote:
: how many polygons are there that all its sides are longer than any of its 
: diagonals
: ?????????????????????????peace and i'm outa here

There are infinitely many.  Consider the polygon ABCD, with
A = (-2,0), B = (x,1), C = (0,0), and D = (x,-1).  If x > sqrt(3)
then all the sides are longer than the diagonals.

If you allow polygons to intersect themselves, then you can draw an
n-pointed star with this property provided n is odd and greater than 4.

Is it possible to find a pentagon which does not intersect itself
with this property?  Is there an equilateral polygon with 4 or more
sides having this property?  Can you prove that no convex polygon
with 4 or more sides can have this property?  Are there other conjectures
about this problem which you find interesting?

--
David Radcliffe                                
radcliff@csd4.csd.uwm.edu

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