# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Sun, 19 May 2013 17:45:26 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 4 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: paradoxPosted: Wed, 8 Feb 2012 16:24:54 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 28 Dec 2007 12:01:53 UTC
Posts: 1261
Make a equilateral triangle ABC of side 10cm BC AS BASE
Now for going from b to c there are two ways, one b-c and another b-a-c.
b-a-c has length of 20 cm
if we divide AB, BC and AC into halves say D, E & F respectively, then way b-d-e-f-c has length 20 cm.
now we divide BD, BE, DE, EF, FC& EC into 4 halves and join those points in similar manner. The way still has length 20 cm
we keep on dividing it infinitely and the path has length of 20 cm every time.
But BC has length of 10 cm and while infinite divisions it is near to BC.

_________________
There is no god in this world except PARENTS and i have lost ONE

Top

 Post subject: Re: paradoxPosted: Wed, 8 Feb 2012 19:05:59 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7949
Location: Lexington, MA
Hello, mun!

Quote:
Make a equilateral triangle ABC of side 10cm; BC as base.
Now, for going from B to C, there are two ways: one B-C and another B-A-C.
B-A-C has length of 20cm.

if we divide AB, BC and AC into halves say D, E & F respectively,
. . then path B-D-E-F-C has length 20 cm.

Now we divide BD, BE, DE, EF, FC & EC into 4 halves and join those points in similar manner.
The path still has length 20 cm.

We keep on dividing it infinitely and the path has length of 20 cm every time.
But BC has length of 10 cm and while infinite divisions it is near to BC.

No matter how many times we compute the zig-zag path, it looks like this:

Code:
*       *       *                   *
/   \   /   \   /   \               /   \
B *-------*-------*-------*-- . . . --*-------* C
: - - - - - - - -  10 - - - - - - - - - - - :

Even at the molecular (or even atomic) level,
. . the length of the zig-zag path is always 20 cm;
. . the length of BC is always 10 cm.

Years ago, I saw a similar problem involving the diagonal of a square.
Suppose we have a square with side 4.
Code:
A o---------------o B
| *             |
|   *           |
|     *         |
4 |       *       |
|         *     |
|           *   |
|             * |
D o---------------o C
: - - - 4 - - - :

We see that

Bisect and and form this zig-zag path.
Code:
D
A o-------o
| *     |
|   *   |
|     * |
4 |       o-------o F
|      E  *     |
|           *   |
|             * |
D *---------------o C
: - - - 4 - - - :

We see that the zig-zag path has length 8.

Divide the horizontal and vertical segments again and we have:
Code:
G
A o---*
| * |
|   *---* I
|  H  * |
4 |       o---* K
|      J  * |
|           *---* M
|          L  * |
D *---------------o C
: - - - 4 - - - :

We see that the zig-zag path has length 8.

As we continue this process, the zig-zag path approaches the diagonal.

Therefore: .

Top

 Post subject: Re: paradoxPosted: Wed, 8 Feb 2012 20:07:19 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sun, 24 Jul 2005 20:12:39 UTC
Posts: 3688
Location: Ottawa Ontario
Nope Soroban: a paradox is 2 doctors.

_________________
I'm not prejudiced...I hate everybody equally!

Top

 Post subject: Re: paradoxPosted: Wed, 8 Feb 2012 22:13:59 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7949
Location: Lexington, MA

You're thinking of paramedics, Denis.

. . (Or is that paramour?)

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 4 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous