Rigel Kent

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 \(L\) and area \(A\): $$A \le \tfrac{1}{4\pi}L^2$$ This inequality is called the isoperimetric inequality . It simply means that the ratio of \(A/L^2\) cannot be greater than \(\frac{1}{4\pi}\). This result can be generalized to three-dimensional space to give: $$V \le \tfrac{1}{6\sqrt{\pi}}S^{3/2}$$ This shows that the theoretical maximum \(V/S^{3/2}\) (when you have the least surface for a given volume) is \(\frac{1}{6\sqrt{\pi}} = 0.0940\). For honeycomb cells, the ratio of \(V\) to \(S^{3/2}\) is: $$\frac{V_{\rm honeycomb}}{(S_{\rm honeycomb})^{3/2}} = \frac{\frac{3\sqrt{3}}{2}a^2 b}{\left(6ab + \frac{6}{\sqrt{8}}a^2...

As explained in Part I of the article, instead of building their house with the easy method of hexagonal top + hexagonal bottom , honeybees are building their cells with hexagonal top + rhombic bottom . In this article, I will try to give a calculus-based demonstration on how the honeybees achieve the economization of their wax resources with a rhombic angle of 70°32'. Suppose we start with a hexagonal prism with flat bottom like the one shown in the figure above. Let \( AB = a\), \(A''A = b\). Then it can be shown that the surface area \(S_0\) and the volume \(V\) of this prism are: $$S_0= 6ab + \frac{3\sqrt{3}}{2}a^2, \quad V = \frac{3\sqrt{3}}{2}a^2b$$ respectively. Mark a point \(B'\) on the prism so that \(B'B'' = x\) and slice a tetrahedron from the prism along the line \(A''C''\) through point \(B'\). Then, flip and rearrange the sliced tetrahedron on top of the prism, as shown below. If you repeat this proc...

The fact that honeycombs is hexagonal is rather well-known. However, this is only a two-dimensional description of the honeycomb. The three-dimensional description of the honeycomb structure is more interesting, but most people are unaware of it. The base of a honeycomb unit cell is a composite surface formed by three rhombuses A real honeycomb has many cells. Each cell is a special type of hexagonal prism. The usual hexagonal prism has a flat hexagonal base, but the bottom of the honeycomb cell is not flat. The base of a honeycomb unit cell is a composite surface formed by three rhombuses . When many of these unit cells are glued together side-by-side, a slab is formed. And when two of these slabs are glued back-to-back, a honeycomb is formed. When the bases of many honeycomb unit cells are glued together The first person who took the trouble to measure the angle of the honeycomb structure is a French astronomer named Giacomo Filippo Maraldi . In 1712, Marald...

Darwin did something very important for us some 150 years ago, when he published his "On the Origin of Species" in 1859. This famous book, as you might already know, is about evolution. However, contrary to popular belief, Darwin is the not the originator of the concept of evolution. Darwin's grandfather, for example, had also a similar idea. The contribution of Darwin is that he conceived a logical notion to explain the mechanism of evolution. Darwin's explanation is quite simple indeed: the evolutionary mechanism is by natural selection. The important thing to highlight here is that evolution is not a theory, it is an observable fact. For example, that human and other lifeforms living on earth is an observable fact. To explain this fact, we may invoke the Jewish notion that we are created by Elohim using only dust of the ground. The Chinese also have a similar notion for the origin of humankind, they postulated that we are created from soil by a goddess named Nǚwā ...