Mathematics 12.

In this lesson we shall see a couple of examples for applications of three-dimensional geometry and its relationships. We hope that these examples will validate that our knowledge about three-dimensional geometry is not autotelic, as the world we live in applies them in many fields.

We can find examples for deliberate applications, but maybe it is even more exciting to discover and understand how three-dimensional geometry appears in the laws of nature itself.

One of the oldest problems in three-dimensional geometry was proposed by Kepler in \latex{ 1611 }. He stated that congruent balls can best fill the space in a cubic close packing (Figure 70) or a hexagonal close packing, but he could not prove his conjecture.

Oranges, watermelons at the market or old cannonballs in a museum are usually piled in a cubic close packing. According to computations, this way they fill \latex{ 74.05 } percent of available space.

For a long time, those who worked on this problem only arrived at the conclusion that many mathematician believes and every physicist knows that this is the best possible way to fill the space.

This optimal way of filling the space is also behind the observation that for a short while after stepping on wet sand the ground right under our feet becomes dry. The explanation behind this is that due to the deforming forces the optimal structure of the dust particles is broken and water leaves through the resulting gaps. When the deforming force ceases to exist, the dust particles reform the structure to fill the space optimally, and the surplus of water ends up on the surface.

It also follows from this optimal structure that if we mix soap and petrol such that the ratio of soap is more than \latex{ 75 }%, then the petrol will not create a flammable mixture, thus it can be safely used in households. This way the petrol floats in soap, and not the other way around.

387 years have passed between Kepler's proposition and the proof of the theorem, as it was solved by mathematician Thomas Hales from the University of Michigan.

In his proof, there are hundreds of thousands of computational tasks with \latex{ 100-200 } variables and \latex{ 1,000-2,000 } constraints. After finishing with the computation, the rest of the proof is convincing, there was no reason to suppose that Kepler's conjecture is invalid. Yet for mathematicians, this result is only thought of as a starting point. One of the next steps could be to determine the most dense structure of congruent balls in more than three dimensions. One would be able to make conclusions concerning higher dimensions from the result.

During the work of bees, the creation of cells and the collection of honey are done simultaneously. It is clear that the more time the first task takes, the less time remains for collection; and honey is their food for winter. Therefore they had to solve the problem to determine the shape of the cells such that their capacity to store honey is as big as possible while the need for material and work remains as small as possible.

A bumblebee lives alone, thus for it the most convenient form is the sphere. The material needed for the surface is the smallest in this case if it needs to encase a certain volume.

On the other hand, bees form colonies. They need their winter honey storage rooms to be placed next to each other. This excludes, for example, the use of cylinder or pentagonal prism shaped cells. Triangular or square prisms, however, can be placed next to each other; but for a given volume it is more useful to increase the number of faces. Thus bees use the most useful solid, the hexagonal prism.

It is interesting that nature creates similar objects in many places. One can see hexagonal basalt columns at many places as hardened volcanic rock. The reason behind this is the flow of material during heat transfer.

It is not by chance that the fuel cells used in nuclear plants have the shape of a hexagonal prism, which enclose the capillaries containing the uranium pastilles.

Volume calculation is one of the oldest areas of mathematics. Ancient mathematicians had a huge amount of knowledge of this area. However, advancement of science created problems which required to determine the volume of higher dimensional objects to answer. 

As dimension increases, this problem seems to be a hopeless task even with today's technical background. Several researcher is trying to invent algorithms which could handle this type of problem. (Figure 71)

There were Hungarian mathematicians (László Lovász and Miklós Simonovits) participating in the invention of one of the best methods available. Since determining the volume of the ball is not a hard task even in higher dimensions, the concept of the algorithm is to enclose the object in question by a ball, and then pick points from this ball uniformly at random. Then they count the occurrence of the points being inside the object itself. If the number of the points is large enough, then the ratio of points inside the object and the total number of points is equal to the ratio of the volume of the object and the volume of the ball. This way one can approximate the volume in question by arbitrary precision.