Regular polyhedra in nature

What would happen if there was only one type of figure in the world, for example, a form such as a rectangle? Some things would not change at all: doors, freight trailers, football fields - they all look the same. But what about the door handles? They would be a little strange. And wheels of cars? That would be ineffective. And football? It's hard to imagine. Fortunately, the world is full of many different forms. Are there regular polyhedra in nature? Yes, and there are a lot of them.

What is a polygon?

In order for a figure to be a polygon, certain conditions are necessary. First, there must be many sides and angles. In addition, it must be a closed form. A regular polygon is a figure with all equal sides and angles. Accordingly, in the wrong they can be slightly deformed.

Types of regular polygons

What is the minimum number of sides that a regular polygon can have? One line can not have many sides. The two sides also can not meet and form a closed form. And three sides can - so it turns out a triangle. And since we are talking about regular polygons, where all sides and angles are equal, we mean an equilateral triangle.

If you add one more side, you get a square. Can a rectangle, where the sides are not equal, be a regular polygon? No, this figure will be called a rectangle. If you add the fifth side, you get a pentagon. Accordingly, there are hexagons, heptagons, octagons and so on ad infinitum.

Elementary geometry

Polygons are of different types: open, closed and self-intersecting. In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of rectilinear segments in the form of a closed polyline or contour. These segments are its edges or sides, and the points where two edges meet are peaks and corners. The inner part of a polygon is sometimes called its body.

Polyhedra in nature and human life

While pentagonal patterns abound in many living forms, the mineral world prefers double, triple, quadruple and sixfold symmetry. Hexagon is a dense shape, which provides maximum structural efficiency. It is very common in the field of molecules and crystals, in which pentagonal forms almost do not occur. Steroids, cholesterol, benzene, vitamins C and D, aspirin, sugar, graphite are all manifestations of sixfold symmetry. Where are the right polyhedra in nature? The most famous hexagonal architecture is created by bees, wasps and hornets.

Six molecules of water form the core of each snow crystal. So it turns out a snowflake. The flank of the fly forms a tightly packed hexagonal arrangement. What are the correct polyhedra in nature? They are crystals of water and diamond, basalt columns, epithelial cells in the eye, some plant cells and much more. Thus, polyhedra created by nature, both living and non-living, are present in a person's life in a huge number and variety.

What determines the popularity of hexagons?

Snowflakes, organic molecules, quartz crystals and columnar basalts are hexagons. The reason for this is the inherent symmetry. The most striking example is the honeycomb, the hexagonal structure of which minimizes the spatial defect, since the entire surface is consumed very rationally. Why divide into identical cells? Bees create regular polyhedra in nature in order to use them for their own needs, including for storing honey and laying eggs. Why does nature prefer hexagons? The answer to this question can be given by elementary mathematics.

• Triangles. Take 428 equilateral triangles with a side of about 7.35 mm. Their total length is 3 * 7.35 mm * 428/2 = 47.2 cm.
• Rectangles. We take 428 squares with a side about 4.84 mm, their total length is 4 * 4.84 m * 428/2 = 41.4 cm.
• Hexagons. And, finally, take 428 hexagons with a side of 3 mm, their total length is 6 * 3 mm * 428/2 = 38.5 cm.

The victory of hexagons is obvious. It is this form that helps to minimize space and allows you to place as many figures as possible in a smaller area. Cells in which bees store their amber nectar are wonders of precise engineering, an array of prismoid cells with perfectly hexagonal cross-section. Wax walls are made with a very precise thickness, the cells are carefully tilted to prevent the loss of viscous honey, and the entire structure is aligned in accordance with the magnetic field of the Earth. Surprisingly, bees work simultaneously, coordinating their efforts.

Why are hexagons? This is a simple geometry

If you want to collect together in the same shape and size cells, so that they fill the entire plane, only three regular figures (with all sides and with the same angles) will work: equilateral triangles, squares and hexagons. Of these, hexagonal cells require the least total length of the wall as compared to triangles or squares of the same area.

Therefore, the choice of hexagons by bees makes sense. As early as the eighteenth century, the scientist Charles Darwin stated that hexagonal honeycombs "are absolutely ideal in saving labor and wax." He believed that natural selection gave bees instincts to create these wax chambers, which had the advantage of providing less energy and time than creating other forms.

Examples of polyhedra in nature

The compound eyes of some insects are packed in a hexagonal, where each facet is a lens connected to a long thin cell of the retina. Structures that are formed by clusters of biological cells often have shapes controlled by the same rules as bubbles in a soap solution. The microscopic structure of the face of the eye is one of the best examples. Each facet contains a cluster of four photosensitive cells that have the same shape as a cluster of four ordinary bubbles.

What determines these rules of soap films and the shape of the bubbles? Nature is even more concerned with saving than bees. Bubbles and soap films are made of water (with the addition of soap), and surface tension draws the surface of the liquid in such a way as to give it the smallest possible area. This is why drops are spherical (more or less) when they fall: the sphere has a smaller surface area than any other shape with the same volume. On the wax sheet, water droplets are drawn into small beads for the same reason.

This surface tension explains the model of bubble rafts and foams. Foam will look for a structure that has the lowest total surface tension, which will ensure the smallest wall area. Although the geometry of soap films is dictated by the interaction of mechanical forces, it does not tell us what the shape of the foam will be. A typical foam contains polyhedral cells of different shapes and sizes. If you look closely, the correct polyhedra in nature are not so right. Their edges are rarely perfectly straight.

Correct Bubbles

Suppose that you can make an "ideal" foam, in which all the bubbles have the same size. What is the perfect shape of the cell, which makes the total area of the bubble wall as small as possible. This has been discussed for many years, and for a long time it was believed that the ideal cell shape is a 14-faceted polyhedron with square and hexagonal sides.

In 1993, a more economical, albeit less ordered structure was found, consisting of a repeating group of eight different cell forms. This more sophisticated model was used as inspiration for the foamy design of the swimming stadium during the 2008 Olympic Games in Beijing.

The rules for the formation of cells in the foam also control certain patterns observed in living cells. Not only the composite eye of flies shows the same hexagonal packing of facets as a flat bubble. Light-sensitive cells inside each of the individual lenses also join in groups that look just like soap bubbles.

The world of polyhedra in nature

Cells of many different types of organisms, from plants to rats, contain membranes with such microscopic structures. Nobody knows what they are for, but they are so widespread that it is fair to assume that they have some useful role. Perhaps they isolate one biochemical process from another, avoiding cross-interference.

Or maybe this is just an effective way to create a large working plane, since many biochemical processes occur on the surface of membranes, where enzymes and other active molecules can be embedded. Whatever the function of polyhedra in nature, do not bother to create complex genetic instructions, because the laws of physics will do it for you.

Some butterflies have winged flakes containing an ordered labyrinth of a durable material called chitin. The impact of light waves bouncing off from ordinary ridges and other structures on the surface of the wing leads to the fact that some wavelengths (that is, some colors) disappear, while others reinforce each other. Thus, the polygonal structure offers an excellent means for producing animal color.

To make ordered networks of a hard mineral, some organisms seem to form a form of soft flexible membranes, and then crystallize solid material within one of the interpenetrating networks. The honeycomb structure of hollow microscopic channels inside the chitinous thorns of an unusual marine worm known as a marine mouse turns these hair-like structures into natural optical fibers that can direct light, changing it from red to bluish-green depending on the direction of illumination. This discoloration can serve to deter predators.

Nature is more visible

The plant and animal world is replete with examples of polyhedra in living nature, as well as the inanimate world of stones and minerals. From a purely evolutionary point of view, the hexagonal structure is the leader in optimizing energy consumption. In addition to obvious advantages (space saving), polyhedral grids provide a large number of faces, therefore, the number of neighbors increases, which has a beneficial effect on the entire structure. The end result of this is that the information spreads much faster. Why are the right hexagonal and irregular stellate polyhedra in nature so common? Probably, so it is necessary. Nature knows best, she knows best.