### The Mathematics of Honeycomb: Part III

In the first two parts of the article, I have given a historical review of the honeycomb problem and a calculus-based technique to compute the honeycomb rhombic angle. In this article, I will try to give a crude analysis on the results we obtained in Part II.

For a closed curve in two-dimensional space, there exists an inequality to govern its perimeter

This inequality is called the isoperimetric inequality. It simply means that the ratio of

This shows that the theoretical maximum value of the ratio of

This value is approximately 0.08 for

This result is no…

For a closed curve in two-dimensional space, there exists an inequality to govern its perimeter

*L*and area*A*.This inequality is called the isoperimetric inequality. It simply means that the ratio of

*A/L2*cannot be greater than 1/4π. This result can be generalized to three-dimensional space to give:This shows that the theoretical maximum value of the ratio of

*V*to*S*3/2 (when you have the least surface for a given volume) is 1/6π1/2 or 0.0940316. For honeycomb cells, the ratio of*V*to*S*3/2 is:This value is approximately 0.08 for

*a*= 3.7 mm,*b*= 13.6 mm, which is pretty close to the theoretical maximum. On the other hand, the same ratio is approximately 0.078 for a prism with hexagonal top and hexagonal bottom.This result is no…