Thursday, November 08, 2012

Some hints for why the pyramid of hexagons is cubic

In my previous two posts I noted that a pyramid built of hexagonal boards contains a cubic number of hexagons and I provided an algebraic proof of this.

My goal is to provide a visual explanation of why this is so.

For those who would like to tackle this challenge themselves, here are a few visual hints...

Is the following shape a hexagon?

Or the silhouette of a cube?

Or the silhouette of the base and two adjacent sides of the same cube?

It's also possible to view this last figure from below, like this:
Looking from below makes it easier to see the cornerstone and edge pieces which aren't visible in the previous figure. So you can count the blocks more easily.
At this point you might want to count the number of blocks adjacent to the white cornerstone (including diagonally). And the number of blocks 2 steps away. And the number of blocks 3 steps away.
Are you seeing a pattern yet?

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