Geometry Problem involving packing Hexagons into Circles

[chriscots]

Hello All,

I have been give a particular task with packing hexagonal shapes with radius 0.105m, into different circular areas. This is not a 3D problem, and I have been trying to search for answers on the topic of "packing" but haven't seemed to find any that fit my requirements.

So the idea is I need the shapes to stay flat and they cannot be manipulated. I want to maximize the area covered in the circle with a given amount of hexagonal shapes. For instance in a 2 foot=0.6096m diameter circular shape I can fit around 7 hexagonal shapes. Even though the logical answer is around 21 the geometry behind it allows me to comfortably pack 7.

The same sort of experiment took place with a 4.64m diamater circlular shape. Mathematically from formulas I can fit 539 of these shapes comfortably. (Area hexagon =0.0286m^2 and Area of circle=16.97m^2, but 16.97/0.0286=593?! So a difference of 44 shapes not needed cause they don't fit.

I guess my overall question is their a relationship between this particular hexagons area and different areas of any circles? I don't want one hexagon inscribed in one circle, it's many little hexagons in any particular sized circle.

Thanks in advance for any help,

 

[conquest]

Hi

I don't have the answer right away, but to find something like this out I would start by taking a big patch of hexagon's of the size you want (like this http://blog.wuphonsreach.org/2010/09/civ5-blank-hexagon-grid-images-24x24.html) and drawing the circle on it.

Then if they are made to fit you can see the general "shape" the hexagon' s are in (a sort of zigzag lined hexagon) then work from there to find a relation between te area of such a shape that would fit into the area of a circle (i.e. if you add another ring of hexagon's you have to much area and it won't fit) in terms of the area of the hexagon (or radii of both of course).

Good luck!