November 10

triangulations

If you draw non-intersecting straight lines connecting the dots, so that no more lines can be added, your polygon is now split into triangles. We call this partition a triangulation... There are many triangulations of a polygon with the dots inside, but they all seem to have the same number of straight lines there. Why?

triangular and square numbers

triangular numbers: how many dots in a triangular pile? If we count empty triangle and start with 0, we get

0, 1, 3, 6, 10, 15, 21...

What happens if you take the differences between the neighboring triangular numbers? Try it:

1, 2, 3, 4, 5, 6,...

Seems like a pattern. Can one explain it?

One can also form square numbers:

0, 1, 4, 9, 16, 25,...

What happens if we take sums of the neighboring triangular numbers? Here's the beginning of the sequence:

0+1, 1+3, 3+6, 6+10, 10+15,...

Looks familiar? can one explain this?

cutting rectangles and finding g.c.d.

did not really work well. Postpone till later.

ball rotations

experiments: 

tying shoelaces in one sleek movement!