Polygons. Visual Guide (2019). A new type of pentagons covering the plane has been discovered

A sensation in the world of mathematics. A new type of pentagons has been discovered that cover the plane without breaks and without overlaps.

This is only the 15th type of such pentagons and the first to be discovered in the last 30 years.

The plane is covered with triangles and quadrangles of any shape, but with pentagons everything is much more complicated and interesting. Regular pentagons cannot cover a plane, but some irregular pentagons can. The search for such figures has been one of the most interesting mathematical problems for a hundred years. The quest began in 1918, when mathematician Karl Reinhard discovered the first five suitable figures.

For a long time it was believed that Reinhard had calculated all possible formulas and that no more such pentagons existed, but in 1968 the mathematician R.B. Kershner found three more, and Richard James in 1975 brought their number to nine . That same year, 50-year-old American housewife and math enthusiast Marjorie Rice developed her own notation method and, within a few years, discovered four more pentagons. Finally, in 1985, Rolf Stein increased the number of figures to fourteen.

Pentagons remain the only figure about which uncertainty and mystery remain. In 1963, it was proven that there are only three types of hexagons covering the plane. There are no such triangles among convex heptagonal, octagonal, and so on. But with the Pentagons, not everything is completely clear yet.

Before today Only 14 types of such pentagons were known. They are shown in the illustration. The formulas for each of them are given at the link.

For 30 years no one could find anything new, and finally the long-awaited discovery! It was made by a group of scientists from the University of Washington: Casey Mann, Jennifer McLoud and David Von Derau. This is what the little handsome guy looks like.

“We discovered the shape by computer searching through a large but limited number of variations,” says Casey Mann. - Of course, we are very excited and a little surprised that we were able to open the new kind pentagon."

The discovery seems purely abstract, but in fact it can find practical use. For example, in the production of finishing tiles.

The search for new pentagons covering the plane will certainly continue.

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Ozhegov’s explanatory dictionary states that a pentagon is a bounded by five intersecting lines, forming five internal corners, as well as any object of a similar shape. If given polygon All sides and angles are the same, then it is called regular (pentagon).

What is interesting about the regular pentagon?

It was in this form that the well-known building of the United States Department of Defense was built. Of the three-dimensional regular polyhedra, only the dodecahedron has pentagon-shaped faces. And in nature there are absolutely no crystals whose faces would resemble a regular pentagon. In addition, this figure is a polygon with a minimum number of angles, which cannot be used to tile the area. Only a pentagon has the same number of diagonals as the number of sides. Agree, this is interesting!

Basic properties and formulas

Using formulas for an arbitrary regular polygon, you can determine all the necessary parameters that a pentagon has.

• Central angle α = 360 / n = 360/5 =72°.
• Internal angle β = 180° * (n-2)/n = 180° * 3/5 = 108°. Accordingly, the sum of the internal angles is 540°.
• The ratio of the diagonal to the side is (1+√5)/2, that is (approximately 1.618).
• The side length of a regular pentagon can be calculated using one of three formulas, depending on which parameter is already known:

• if a circle is described around it and its radius R is known, then a = 2*R*sin (α/2) = 2*R*sin(72°/2) ≈1.1756*R;
• in the case when a circle with radius r is inscribed in a regular pentagon, a = 2*r*tg(α/2) = 2*r*tg(α/2) ≈ 1.453*r;
• it happens that instead of radii, the value of the diagonal D is known, then the side is determined as follows: a ≈ D/1.618.

• The area of ​​a regular pentagon is determined, again, depending on what parameter we know:
• if there is an inscribed or circumscribed circle, then one of two formulas is used:

S = (n*a*r)/2 = 2.5*a*r or S = (n*R 2 *sin α)/2 ≈ 2.3776*R 2 ;

• The area can also be determined by knowing only the length of the side side a:

S = (5*a 2 *tg54°)/4 ≈ 1.7205* a 2 .

Regular pentagon: construction

This geometric figure can be built in different ways. For example, fit it into a circle with a given radius or build it on the basis of a given side. The sequence of actions was described in Euclid's Elements approximately 300 BC. In any case, we will need a compass and a ruler. Let's consider a method of construction using a given circle.

1. Select an arbitrary radius and draw a circle, marking its center with point O.

2. On the circle line, select a point that will serve as one of the vertices of our pentagon. Let this be point A. Connect points O and A with a straight line.

3. Draw a line through point O perpendicular to line OA. Designate the intersection of this straight line with the circle line as point B.

4. Halfway between points O and B, construct point C.

5. Now draw a circle whose center will be at point C and which will pass through point A. The place of its intersection with line OB (it will be inside the very first circle) will be point D.

6. Construct a circle passing through D, the center of which will be at A. The places of its intersection with the original circle should be marked with points E and F.

7. Now construct a circle whose center will be at E. This must be done so that it passes through A. Its other intersection of the original circle must be marked

8. Finally, construct a circle through A with its center at point F. Label the other intersection of the original circle with point H.

9. Now all that remains is to connect the vertices A, E, G, H, F. Our regular pentagon will be ready!