The correct hexagon: how interesting it is and how to build it

Is there a pencil around you? Look at its section - it is a regular hexagon or, as it is also called, a hex. This shape also has a cross-section of the nut, a hexagonal chess field, a crystal lattice of some complex carbon molecules (eg, graphite), a snowflake, honeycombs and other objects. A giant regular hexagon was recently discovered in the atmosphere of Saturn. Does not it seem strange to use so often the nature of this form for its creations? Let's look at this figure in more detail.

A regular hexagon is a polygon with six identical sides and equal angles. From the school course we know that it has the following properties:

• The length of its sides corresponds to the radius of the circumscribed circle. Of all the geometric figures, this property has only a regular hexagon.
• Angles are equal to each other, and the value of each is 120 °.
• The perimeter of the hexagon can be found by the formula P = 6 * R, if the radius of the circumscribed circle is known, or P = 4 * √ (3) * r, if the circle is inscribed in it. R and r are the radii of the circumscribed and inscribed circle.
• The area occupied by a regular hexagon is defined as follows: S = (3 * √ (3) * R ² ) / 2. If the radius is unknown, substitute for it the length of one of the sides - as is known, it corresponds to the length of the radius of the circumscribed circle.

A regular hexagon has one interesting feature, thanks to which it has received such a wide distribution in nature - it is able to fill any surface of the plane without overlaps and spaces. There is even the so-called Lemma Pala, according to which the right hexagon, whose side is 1 / √ (3), is a universal tire, that is, it can cover any set with a diameter of one unit.

Now consider the construction of a regular hexagon. There are several ways, the simplest of which involves the use of a compass, a pencil and a ruler. First, draw an arbitrary circle with a compass, then in an arbitrary place on this circle make a point. Without changing the solution of the compass, put the point at this point, mark the next incision on the circle, continue until until we get all 6 points. Now it remains only to connect them with each other by straight lines, and the desired figure will be obtained.

In practice, there are times when you want to draw a large hexagon. For example, on a two-level gipsokartonnom ceiling, around the attachment of the central chandelier, you need to install on the lower level of six small lamps. The compasses of such sizes will be very, very difficult to find. What should I do in this case? How do I draw a large circle? Very simple. It is necessary to take a strong thread of the necessary length and tie one of its ends opposite the pencil. Now it remains only to find an assistant who would press the second end of the thread to the ceiling at the desired point. Of course, in this case, minor errors are possible, but they are unlikely to be noticeable to an outsider.