What would happen if there was only one type of figure in the world, for example, a shape such as a rectangle? Some things would not have changed at all: doors, cargo trailers, soccer fields - they all look the same. But what about door handles? They would be a little weird. What about car wheels? That would be inefficient. What about football? It’s hard to imagine. Fortunately, the world is full of many different forms. Are there regular polyhedrons 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. Firstly, there should be many sides and angles. In addition, it must be a closed form. A regular polygon is a shape with all equal sides and angles. Accordingly, in the wrong one they can be slightly deformed.
Types of Regular Polygons
What is the minimum number of sides a regular polygon can have? One line cannot have many sides. The two sides also cannot meet and form a closed form. And three sides can - this will turn out to be 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 shape will be called a rectangle. If you add the fifth side, you get a pentagon. Accordingly, there are hexagons, heptagons, octagons, and so on to infinity.
Elementary geometry
Polygons come in many forms: open, closed, and self-intersecting. In elementary geometry, a polygon is a planar 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 vertices and angles. The inside of a polygon is sometimes called its body.
Polyhedra in nature and human life
While many living forms abound in pentagonal patterns, the mineral world prefers double, triple, fourfold, and sixfold symmetry. The hexagon is a dense shape that provides maximum structural efficiency. It is very common in the field of molecules and crystals, in which pentagonal forms almost never occur. Steroids, cholesterol, benzene, vitamins C and D, aspirin, sugar, graphite - these are all manifestations of six-fold symmetry. Where in nature are regular polyhedrons found? The most famous hexagonal architecture is created by bees, wasps and hornets.
Six water molecules form the core of each snow crystal. So it turns out a snowflake. The edges of the eye of the fly form a tightly packed hexagonal arrangement. What other regular polyhedrons are there in nature? These are water and diamond crystals, basalt columns, epithelial cells in the eye, some plant cells, and much more. Thus, polyhedra created by nature, both living and nonliving, are present in human life in huge numbers and diversity.
What determines the popularity of hexagons?
Snowflakes, organic molecules, quartz crystals and columnar basalts are hexagons. The reason for this is their inherent symmetry. The most striking example is the honeycomb, whose hexagonal structure minimizes spatial imperfection, since the entire surface is consumed very rationally. Why divide into identical cells? Bees create the right polyhedra in nature in order to use them for their needs, including for storing honey and laying eggs. Why does nature prefer hexagons? The answer to this question can give 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 Take 428 squares with a side of 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 the hexagons is obvious. It is this form that helps to minimize space as much as possible and allows you to place as many figures as possible on a smaller territory. The honeycombs in which the bees store their amber nectar are wonders of precision engineering, an array of prismatic cells with an ideally hexagonal cross section. Wax walls are made with very precise thicknesses, the cells are carefully tilted to prevent viscous honey from falling out, and the entire structure is aligned with the earth's magnetic field. In an amazing way, the bees work simultaneously, coordinating their efforts.
Why hexagons? This is simple geometry.
If you want to put together the same cells in shape and size to fill the entire plane, then only three regular shapes will work (with all sides and with the same angles): equilateral triangles, squares and hexagons. Of these, hexagonal cells require the smallest total wall length compared to triangles or squares of the same area.
Therefore, the choice of hexagons by bees makes sense. Back in the eighteenth century, scientist Charles Darwin stated that hexagonal honeycombs "are absolutely perfect in saving labor and wax." He believed that natural selection endowed the bees with instincts to create these wax chambers, which had the advantage of providing less energy and time than with other forms.
Examples of polyhedra in nature
The compound eyes of some insects are packed in a hexagonal, where each face is a lens connected to a long thin retinal cell. Structures that are formed by clusters of biological cells often have shapes that are controlled by the same rules as bubbles in soapy water. 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 vesicles.
What defines these rules for soap films and bubble shapes? Nature is even more concerned about saving than bees. Bubbles and soap films are made of water (with the addition of soap), and surface tension pulls the surface of the liquid in such a way as to give it the smallest possible area. This is why droplets are spherical (more or less) when they fall: a sphere has a smaller surface area than any other shape with the same volume. On a wax sheet, water droplets are drawn into small beads for the same reason.
This surface tension explains the model of bubble rafts and foams. The foam will look for a structure that has the lowest overall surface tension, which will provide 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 various shapes and sizes. If you look closely, the correct polyhedrons in nature are not so correct. Their edges are rarely perfectly straight.
Regular bubbles
Suppose you can make an “ideal” foam in which all the bubbles are the same size. What is the perfect shape of the cell, which makes the total area of the wall of the bubble 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-sided polyhedron with square and hexagonal sides.
In 1993, a more economical, albeit less ordered, structure was discovered, consisting of a repeating group of eight different cell shapes. This more sophisticated model was used as inspiration for the foamy design of the swimming stadium during the 2008 Beijing Olympics.
The rules for the formation of cells in the foam also control some patterns observed in living cells. Not only the composite eye of the flies shows the same hexagonal packing of the facets as the flat bubble. The photosensitive cells inside each of the individual lenses also join in groups that look like soap bubbles.
The world of polyhedrons in nature
The 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 work plane, since many biochemical processes take place on the surface of the membranes, where enzymes and other active molecules can be embedded. Whatever the function of polyhedra in nature, do not bother creating complex genetic instructions, because the laws of physics will do it for you.
Some butterflies have winged flakes containing an ordered labyrinth made of a durable material called chitin. Exposure to light waves bouncing off the usual ridges and other structures on the wing surface causes some wavelengths (i.e. some colors) to disappear, while others reinforce each other. Thus, the polygonal structure offers an excellent means for producing animal color.
To make ordered networks from a hard mineral, some organisms seem to form from flexible flexible membranes and then crystallize solid material inside one of the interpenetrating networks. The honeycomb structure of the hollow microscopic channels inside the chitin spikes of an unusual marine worm, known as the sea mouse, turns these wave-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 may serve to deter predators.
Nature knows better
Flora and fauna abound 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 a leader in optimizing energy consumption. In addition to the 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 information spreads much faster. Why are regular hexagonal and irregular star-shaped polyhedra so common in nature? Probably necessary. Nature knows better, she knows better.