Thursday, June 23, 2016

Playing with silly hexagons and cubes for fun

Today I was bored, so I decided to see if I could make a hexagon with some dowels and an elastic band. Then I got wondering what numbers you could make hexagons with. I drew a hexagon to figure out "Hexagonal Numbers", which are just numbers of circles (or mini hexagons) of equal size that form hexagons. Some numbers are: 1, 7, 19, 37, and 61. Then, because I was still bored, I decided to figure out a formula to figure out what numbers are hexagonal.

I decided to make a 'T Table' to figure out the formula.

Here is my attempt to recreate it on the computer:
                   x                    y
0 1
1 7
2 19
3 37
4 61
5 91
6 127
7 169
 I tried to figure out a formula, but I didn't get any further then to figure out that the previous Y value + 6 * X is equal to Y. For example, 7 + 6 * 2 = 19. This method was good enough for me, even though it is strange.

 I started thinking about how hexagons looked like cubes viewed at an angle (See right image). Then I was thinking about how they have the same about of sides as cubes (six).

Then, after thinking about cubes and hexagons and such for a while, I figured out that if you add up some adjacent hexagonal numbers (starting at 1) you get a cubic number! An example is 1+7+19 = 27, which can form a cube, such as the Rubik's cube on the right.


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