My Math Forum  

Go Back   My Math Forum > Science Forums > Art

Art Art and Architecture Forum - Math in our world, culture, buildings, natural wonders, nature, movies, music


Reply
 
LinkBack Thread Tools Display Modes
February 6th, 2013, 06:00 PM   #1
Senior Member
 
Joined: Dec 2012

Posts: 450
Thanks: 0

drawing a closed figure

What ensures if you can draw a closed figure without lifting your hand off and without repeating through a line?

I tried different conjectures but all of them failed for some figures.

Please help me out...
mathmaniac is offline  
 
February 6th, 2013, 08:51 PM   #2
Math Team
 
Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 407

Re: drawing a closed figure

Hello, mathmaniac!

Quote:
What ensures if you can draw a closed figure without lifting your hand off
and without repeating through a line?

An even vertex has an even number of paths leading out of it.
An odd vertex has an odd number of paths leading out of it.

If all the vertices are even, the figure is traceable.

If there are any odd vertices, there must be exactly two of them.
Otherwise, the figure is not traceable.

Code:
              B
    A o * * * o * * * o C
      *     *   *     *
      *   *       *   *
      * *           * *
    H o               o D
      * *           * *
      *   *       *   *
      *     *   *     *
    G o * * * o * * * o E
              F



Code:
            A
            o
          *   *
        *       *
    E o * * * * * o B
      * *       * *
      *   *   *   *
      *     o     *
      *   * F *   *
      * *       * *
    D o * * * * * o C


Code:
    A o * * * * * o B
      *           *
      *     H     *
    G o * * o * * o C
      *     *     *
      *     *     *
    F o * * o * * o D
            E


soroban is offline  
February 7th, 2013, 04:08 PM   #3
Global Moderator
 
Joined: May 2007

Posts: 6,275
Thanks: 516

Re: drawing a closed figure

Quote:
An even vertex has an even number of paths leading out of it.
An odd vertex has an odd number of paths leading out of it.

If all the vertices are even, the figure is traceable.

If there are any odd vertices, there must be exactly two of them.
Otherwise, the figure is not traceable.
You need also that the figure is connected.
mathman is offline  
February 8th, 2013, 12:31 AM   #4
Math Team
 
Joined: Apr 2010

Posts: 2,770
Thanks: 356

Re: drawing a closed figure

What is connected? All vertices >= 2? If so, do you have an example that fails [color=#105287]soroban[/color]'s conditions for traceability?
Hoempa is offline  
February 8th, 2013, 12:22 PM   #5
Global Moderator
 
Joined: May 2007

Posts: 6,275
Thanks: 516

Re: drawing a closed figure

Two triangles not connected to each other. You can draw either one, but you need to lift the pencil to do both.
mathman is offline  
February 8th, 2013, 04:41 PM   #6
Senior Member
 
Joined: Dec 2012

Posts: 450
Thanks: 0

Re: drawing a closed figure

Thank you soroban,but i need time to check your conjecture.
mathmaniac is offline  
February 9th, 2013, 12:48 AM   #7
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: drawing a closed figure

Quote:
Originally Posted by mathmaniac
Thank you soroban,but i need time to check your conjecture.
It's no conjecture. It's the basics of graph theory.
mathbalarka is offline  
February 9th, 2013, 04:32 PM   #8
Senior Member
 
Joined: Dec 2012

Posts: 450
Thanks: 0

Re: drawing a closed figure

Have you got any proof for it?
mathmaniac is offline  
February 9th, 2013, 08:27 PM   #9
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: drawing a closed figure

Quote:
Originally Posted by mathmaniac
Have you got any proof for it?
Of course.

Quote:
Originally Posted by soroban
If there are any odd vertices, there must be exactly two of them.
Um, soroban, you are talking about Eulerian graphs, right? But not every traceable graph is Hamiltonian.

Shouldn't we have even number of odd vertices for Hamiltonian ones? i.e. GP(5,2) is the Peterson graph. Where the number of odd nodes are 10 (not 2).
mathbalarka is offline  
February 13th, 2013, 08:10 PM   #10
Senior Member
 
Joined: Dec 2012

Posts: 450
Thanks: 0

Re: drawing a closed figure

Quote:
Originally Posted by mathbalarka
Quote:
Originally Posted by mathmaniac
Have you got any proof for it?
Of course.
Then can you post it?
mathmaniac is offline  
Reply

  My Math Forum > Science Forums > Art

Tags
closed, drawing, figure



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Drawing xyz graph. ohc4r3 Linear Algebra 1 March 17th, 2013 05:25 AM
Closed set and closed ball Vasily Real Analysis 3 February 4th, 2013 12:49 PM
Is the union of infinite disjoint closed sets closed? 03sqq Real Analysis 4 November 13th, 2012 03:40 AM
Drawing balls with repetition butabi Advanced Statistics 1 November 27th, 2011 08:19 AM
How To Figure Gross With a Certain Net Figure Al_Ch Elementary Math 2 May 21st, 2009 05:53 PM





Copyright © 2017 My Math Forum. All rights reserved.