CTK Exchange
CTK Wiki Math
Front Page
Movie shortcuts
Personal info
Awards
Terms of use
Privacy Policy

Interactive Activities

Cut The Knot!
MSET99 Talk
Games & Puzzles
Arithmetic/Algebra
Geometry
Probability
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
My Logo
Math Poll
Other Math sit's
Guest book
News sit's

Recommend this site

Manifesto: what CTK is about Search CTK Buying a book is a commitment to learning Table of content Products to download and subscription Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

CTK Exchange

Subject: "wilson's theorem - geometric proof"     Previous Topic | Next Topic
Printer-friendly copy     Email this topic to a friend    
Conferences The CTK Exchange This and that Topic #906
Reading Topic #906
bryan Huh
guest
Aug-19-09, 08:20 AM (EST)
 
"wilson's theorem - geometric proof"
 
   Found this proof on this website: https://www.cut-the-knot.org/blue/GeometricWilson.shtml

An important part of this proof is this point:

These polygons, too, are congruent in pairs since the vertices of a polygon could be traversed in two directions. IN ANY EVENT, WE SEE THAT THE NUMBER OF SUCH POLYGONS IS DIVISIBLE BY P.

This is not clear to me. I see that there are p! permutations using the p points, but if you take out polygons which are congruent by mere direction of traversal, it becomes p!/2. Then if you take out polygons which are congruent by mere rotations (referred to as "shifting of indices" in the proof) we get (p-1)!/2 polygons.

Then it looks like we subtract the star polygons which there are (p-1)/2 of to get the total number of "irregular polygons."

This is my understanding. But I'm not getting how there are supposed to be "p" of these irregular polygons.

Thank you,
Bryan


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexbadmin
Charter Member
2427 posts
Aug-19-09, 08:39 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
1. "RE: wilson's theorem - geometric proof"
In response to message #0
 
   >Found this proof on this website:
>https://www.cut-the-knot.org/blue/GeometricWilson.shtml
>
>An important part of this proof is this point:
>
>These polygons, too, are congruent in pairs since the
>vertices of a polygon could be traversed in two directions.
>IN ANY EVENT, WE SEE THAT THE NUMBER OF SUCH POLYGONS IS
>DIVISIBLE BY P.

Think of a permutation as a sequence of jumps from a vertex to another vertex so that, for example, 1 3 6 5 ... starts with one and then makes jumps of 2, 3, -1, ... vertices.

Now, the same series of jumps can start with any of the p vertices producing (congruent, but distinct for the sake of enumeration) polygons. This is why the number of such polygons is divisible by p. This is truly a simple geometric argument. The sentence about them too coming in pairs is unnecessary and misleading.

I am on vacation. Will change the wording when back.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top

Conferences | Forums | Topics | Previous Topic | Next Topic

You may be curious to have a look at the old CTK Exchange archive.
Please do not post there.

Copyright © 1996-2018 Alexander Bogomolny

Search:
Keywords:

Google
Web CTK