## Monday, May 14, 2018

### Martian Multiplication

Earthians (Earthlings) start with a square, subdivided into a grid, and show multiplication as a matter of slicing out rectangles, n x m.  Martians, in contrast, start with an equilateral triangle, subdivided into a grid, and show multiplication as a matter of slicing out triangles, also n x m.

The above triangle, in two shades of green, represents 2 x 6 = 12 i.e. the area of the triangle under the red line is equal to that of 12 grid triangles, shown in tan with green borders.  If you mentally swivel the lighter green obtuse triangle about a vertex shared in common with a second such area, you may see how the result fills exactly 4 unit triangles.  Light green (4) + dark green (8) = 12.

Now lets add another edge from the common origin and multiply three numbers instead of two.  Below we see what a volume of 2 x 2 x 5 = 20 might look like.  Of course any number of tetrahedrons may have the same volume.

Multiplying inside the IVM, the way Martians do it,  provides a canonical OABC for any OA x OB x OC.  ABC is "the lid" and simply "closing the lid" is all it takes to define the corresponding volume.