system
Given the following post, respond as if you're commenting on Reddit:
human
Why is it that when you cramp flexible circles together you get hexagons (honeycomb) specifically, and not any other shape?

What is the mathematics going on? I'm expecting some equation that links pi to the circumference  of a circle and a constant, 6, or something like that.
gpt
The hexagonal lattice is the most structurally efficient tessellation - meaning that it is the lowest possible energy shape for a given area.  You can demonstrate this using nested soap bubbles, because the bubble walls always pull themselves into the lowest energy configuration. This is why bees evolved to build hexagonal honeycombs. It uses the least amount of wax in its construction. 

If you think about it, the most structurally efficient shape is the circle, because it requires the least amount of perimeter wall to enclose the greatest possible volume. When you nest circles together, however, you end up with unusable gaps between them. The most structurally efficient nesting pattern should then logically be some shape which tesselates perfectly without gaps, but which is otherwise as close to being a circle as possible. That is the hexagon.

You can construct a hexagon using only a drawing compass. First, construct a circle. Then, without changing the radius setting on the compass, place the point anywhere on the circumference of that circle, and make a mark on the circumference where the compass falls. Move the point of the compass to the new intersection and repeat. As you move around the circle, you are defining the vertices of a perfect hexagon inscribed within it, with a side length equal to the radius of the circle.