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 11-20-2006, 01:48 AM #1 Wise Young Administrator     Join Date: Jul 2001 Location: New Brunswick, NJ, USA Posts: 37,975 Geometric multiplication Here is an interesting video of a multiplication method that just requires counting and no knowledge of the multiplication table. http://zappinternet.com/index.php?video=zebQzoBhoR How does it work? Wise.
 11-20-2006, 09:41 AM #2 rfbdorf Senior Member     Join Date: Sep 2005 Location: Oregon Posts: 2,715 Very neat. You just count the crossings. Of course, 2 lines cross 3 lines 6 times (for example).- Richard
 11-20-2006, 09:59 AM #3 In the wind Senior Member     Join Date: Sep 2005 Location: Arkansas Posts: 274 Determinants were first discovered by Leibniz, forgotten and later worked on by Cramer and even later by Cayley. Normally the numbers are arranged in what we call a matrix or a pattern called a determinate today Much cleaner than the lines drawn in the video. The same system not only works with but is the basis for linear algebra
 11-20-2006, 10:27 AM #4 rfbdorf Senior Member     Join Date: Sep 2005 Location: Oregon Posts: 2,715 Is this really the same thing as a determinate? There is some similarity, but I don't see the subtraction of the other terms (sorry, forgot the terminology). This looks like straightforward multiplication to me, except replacing the multiplication of single place integers with the counting of crossings. - Richard (gotta get to work now)
 11-20-2006, 06:52 PM #5 XYNaPSE Senior Member     Join Date: Feb 2006 Posts: 1,121 Determinants have to do with multiplying 2X2 and 3X3 matrices I think. What this person is doing has nothing to do with matrices. Its just straight up multiplication. Very cool.
11-20-2006, 09:35 PM   #6
Wise Young

Join Date: Jul 2001
Location: New Brunswick, NJ, USA
Posts: 37,975
Quote:
 Originally Posted by rfbdorf Very neat. You just count the crossings. Of course, 2 lines cross 3 lines 6 times (for example).- Richard
I agree with rfbdorf. Counting the number of crossings is multiplication. By dealing with concentric layers of crossings, we are simply performing multiplication digit by digit and doing carryovers for each decimal position. It is simply a graphical depiction of the standard multiplication procedure, as illustrated below.
Attached Images

Last edited by Wise Young; 11-20-2006 at 09:37 PM.

11-22-2006, 09:05 AM   #7
In the wind
Senior Member

Join Date: Sep 2005
Location: Arkansas
Posts: 274
Quote:
 Originally Posted by XYNaPSE Determinants have to do with multiplying 2X2 and 3X3 matrices I think. What this person is doing has nothing to do with matrices. Its just straight up multiplication. Very cool.
Determinates (The matrices) get much larger than 3X3, there's not a limit, they tie the coefficients back to the geometry,

Penrose does an excellent job of showing the relationship (or in my mind, the “sameness”) of the number line to geometry, trigonometry, inverse, imaginary (i) and infinite numbers (and in the last case, demonstrating that there is no difference) with an abstract called the unit circle, a circle where r=1.

The video is a very neat example of this relationship, and shows that multiplication is just addition disguised, but if you really want to itch your brain ask yourself how far does this relationship between numbers and reality go.

A simple place to start, but off the subject of this thread, is to try to move from 1 to 2 without actually getting to 2 but just see how close you can get. These basic arguments (metaphysics) are 3000 years old…

 11-23-2006, 06:08 AM #8 Tufelhunden Senior Member     Join Date: Mar 2006 Location: Santa Cruz, California Posts: 1,961 Actually, In The Wind, you are correct in your first statement that determinants are linked to geometry. The absolute value of a determinant of an NxN matrix gives signed area and volumes of the geometric shapes in which the determinant's original matrix represents. For example, if you have a 2x2 matrix (NxN), the absolute value of the matrix equals the area of the parallelogram spanned by the row vectors. So, if you have a matrix: 51 34, you can create a cartesian graph (the one shaped like an L with the x-axis horizontal and the y-axis vertical, and mark the points (5,1) and (3,4) on your graph. conntect these points to the origin of the graph (the origin is the corner of the L). You now have two vectors in 2-dimensional space. You then take another vector similar in length and angle as your (3,4) vector, but instead of originating it from the origin of your cartesian graph, originate it from the point (5,1) so it looks like a "vector shooting off from another vector". You do the same thing to your original (3,4) vector, where you would replicate another (5,1) vector and "shoot it" from the point (3,4). Your two "new" vectors should now come to a single point (should be (8,5)) and the resulting geometric shape is a parallelogram. Well, the absolute value of the determinant from your original matrix is the area of this parallelogram. If you take the determinant of a 3x3 matrix, (this matrix gives three vectors in three-dimensional space), the resulting geometric shape is a parallelopiped (a parallelogram cube, if you want a visual), and the absolute value of the determinant of the 3x3 matrix is the volume of the resulting parallelopiped. I say absolute value because negative area or negative volume do not exist. __________________ End triangle inequality...triangle equality NOW!

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