It turns out that the answer to this question is closely related to the regular polyhedra. Ancient Greek mathematicians and philosophers were fascinated by the regular polyhedra, also known as Platonic solids , attributing to them many mystical properties. The Platonic solids are polyhedra with the greatest possible degree of symmetry: All their faces are equilateral polygons with the same number of sides, and the same number of faces meet at every vertex.
Euclid proved in his Elements that there are only five such polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron see Figure 7. Figure 7. The five basic Platonic solids shown here have been known since antiquity. Examples of all generalized soccer-ball patterns can be generated by altering Platonic solids. Although Euclid used the geometric definition of Platonic solids, assuming all the polygons to be regular, modern mathematicians know that the argument does not depend on the geometry.
Each Platonic solid can be described by two numbers: the number K of vertices in each face and the number M of faces meeting at each vertex. The possible solutions can be determined quite easily. The complete list of possible values for the pairs K, M is:. Strictly speaking, this is only the list of genuine polyhedra satisfying the above equation.
The equation does have other solutions in positive integers. These solutions correspond to so-called degenerate Platonic solid s, which are not bona fide polyhedra. The first case can be thought of as a beach ball that is a sphere divided into M sections in the manner of a citrus fruit. The Platonic solids give rise to generalized soccer balls by a procedure known as truncation.
Suppose we take a sharp knife and slice off each of the corners of an icosahedron. At each of the 12 vertices of the icosahedron, five faces come together at a point.
When we slice off each vertex, we get a small pentagon, with one side bordering each of the faces that used to meet at that vertex. At the same time, we change the shape of the 20 triangles that make up the faces of the icosahedron. By cutting off the corners of the triangles, we turn them into hexagons.
The sides of the hexagons are of two kinds, which occur alternately: the remnants of the sides of the original triangular faces of the icosahedron, and the new sides produced by lopping off the corners. The first kind of side borders another hexagon, and the second kind touches a pentagon. In fact, the polyhedron we have obtained is nothing but the standard soccer ball. Mathematicians call it the truncated icosahedron. Figure 8. Chopping off corners, or truncation, converts any Platonic solid into a generalized soccer ball.
In particular, the standard soccer ball is a truncated icosahedron. After truncation, the 20 triangular faces of the icosahedron become hexagons; the 12 vertices, as shown here, turn into pentagons. The same truncation procedure can be applied to the other Platonic solids.
For example, the truncated tetrahedron consists of triangles and hexagons, such that the sides of the triangles meet only hexagons, while the sides of the hexagons alternately meet triangles and hexagons. The truncated icosahedron gives values for k, m and n of 5, 3 and 2. Figure 9. Generalized soccer balls fall into 20 types. In this table, k represents the number of sides in any black face; the product m x n is the number of sides in any white face.
Every side of a black face meets a white face. Every n th side of a white face meets a black face. The columns b and w represent the number of black and white faces in the simplest representative of each type. However, this is not true for other values of n. The minimal realization of type 8 is combinatorially the same as the World Cup ball shown in Figure 2, whereas type 10 is the standard soccer ball.
Are these the only possibilities for generalized soccer ball patterns, or are there others? Just as we did for the Platonic solids, we can express the number of faces, edges and vertices in terms of our basic data.
Here this is the number b of black faces, the number w of white faces, and the parameters k , m and n. Now, because the number of faces meeting at a vertex is not fixed, we do not obtain an equation, but an inequality expressing the fact that the number of faces meeting at each vertex is at least 3. The result is a constraint on k, m and n that can be put in the following form :.
This may look complicated, but it can easily be analyzed, just like the equation leading to the Platonic solids.
It is not hard to show that n can be at most equal to 6, because otherwise the left-hand side would be greater than the right-hand side. With a little more effort, it is possible to compile a complete list of all the possible solutions in integers k, m and n. Alas, the story does not end there. However, Braungardt and I were able to determine the values of k, m, n that do have realizations as soccer balls; these are shown in the table in Figure 9, where we also illustrate the smallest realizations for a few types.
The numbers of hexagons in these examples are 30, 60 and 2, respectively. Note that in the latter case the color scheme is reversed, so the hexagons are black rather than white. The numbers of carbon atoms are 80, and 24, respectively.
The last of these is the only fullerene with 24 atoms. In the case of 80 atoms, there are 7 different fullerenes with disjoint pentagons, but only one occurs in our table of generalized soccer balls. For atoms, there are , fullerenes with disjoint pentagons. Figure Braungardt and I discovered something very intriguing when we tried to see whether every generalized soccer ball comes from a branched covering of one of the entries in our table.
However, it is not true for other values of n! The minimal example is just a tetrahedron with one face painted black Figure 10a. Another realization is an octahedron with two opposite faces painted black Figure 10b.
This is not a branched covering of the painted tetrahedron! A branched covering of the tetrahedron would have 3, 6, 9, … faces meeting at every vertex—but the octahedron has 4. In the tetrahedron example, there are two different kinds of vertices: a vertex at which only white faces meet, and three vertices where one black and two white faces meet. Moreover, the painted octahedron has yet another kind of vertex.
Every vertex has the same sequence of colors, which goes black, white, white, black, white, white, …, with only the length of the sequence left open. Because the definition of soccer balls through conditions 1 , 2 and 3 does not specify that soccer-ball polyhedra should be spherical, there is a possibility that they might also exist in other shapes.
Louis, MO We are open Saturday and Sunday! Subject optional. Email address: Your name:. Calculate the approximate area a regular hexagon with the following side length:. Possible Answers:. Correct answer:. Explanation : How do you find the area of a hexagon? There are several ways to find the area of a hexagon. In a regular hexagon, split the figure into triangles. Find the area of one triangle.
Multiply this value by six. Regular hexagons: Regular hexagons are interesting polygons. Hexagons are six sided figures and possess the following shape: In a regular hexagon, all sides equal the same length and all interior angles have the same measure; therefore, we can write the following expression. There are in a circle and the hexagon in our image has separated it into six equal parts; therefore, we can write the following: We also know the following: Now, let's look at each of the triangles in the hexagon.
Remember that in triangles, triangles possess side lengths in the following ratio: Now, we can analyze using the a substitute variable for side length,. Now, we need to multiply this by six in order to find the area of the entire hexagon. This is denoted by the variable in the following figure: Alternative method: If we are given the variables and , then we can solve for the area of the hexagon through the following formula: In this equation, is the area, is the perimeter, and is the apothem.
Solution: In the given problem we know that the side length of a regular hexagon is the following: Let's substitute this value into the area formula for a regular hexagon and solve. Round the answer to the nearest whole number. Report an Error. A single hexagonal cell of a honeycomb is two centimeters in diameter.
Solution: In the problem we are told that the honeycomb is two centimeters in diameter. Substitute and solve. We know the following information. Each one takes another one away from the pile, so there are 90 pairs. Her goal is to make math as playful for kids as it was for her when she was a child.
Her mom had Laura baking before she could walk, and her dad had her using power tools at a very unsafe age, measuring lengths, widths and angles in the process. Search for:. Hint if needed: Every shape edge is shared with 1 other shape… Answers: Wee ones: 5 sides. But for a regular hexagon things are not so easy since we have to make sure all the sides are of the same length.
To get the perfect result you will need a drawing compass. Draw a circle, and, with the same radius, start making marks along it. Starting at a random point and then making the next mark using the previous one as the anchor point draw a circle with the compass. You will end up with 6 marks, and if you join them with the straight lines , you will have yourself a regular hexagon. You can see a similar process on the animation above.
The hexagon calculator allows you to calculate several interesting parameters of the 6-sided shape that we usually call a hexagon. Using this calculator is as simple as it can possibly get with only one of the parameters needed to calculate all others, as well as including a built-in length conversion tool for each of them.
We have discussed all the parameters of the calculator, but for the sake of clarity and completeness we will now go over them briefly:. If you like the simplicity of this calculator we invite you to try our other polygon calculators such as the regular pentagon calculator or even 3-D calculators such as the pyramid calculator , triangular prism calculator , or the rectangular prism calculator.
Everyone loves a good real-world application , and hexagons are definitely one of the most used polygons in the world. Starting with human usages, the easiest and probably least interesting use is hexagon tiles for flooring purposes. The hexagon is an excellent shape because it perfectly fits with one another to cover any desired area. If you're interested in such a use, we recommend the flooring calculator and the square footage calculator as they are very good tools for this purpose. The next case is common to all polygons, but it is still interesting to see.
In photography, the opening of the sensor almost always has a polygonal shape. This part of the camera called the aperture , and dictates many properties and features of the pictures taken by the camera.
The most unexpected one is the shape of very bright point-like objects due to the effect called diffraction grating , and it is illustrated in the picture above. One of the most important uses of hexagons in the modern era, closely related to the one we've talked about in photography, is in astronomy. One of the biggest problems we experience when trying to observe distant stars is how faint they are in the night sky. That is because despite being very bright objects, they are so very far away that only a tiny fraction of their light reaches us; you can learn more about that in our luminosity calculator.
On top of that, due to relativistic effects similar to time dilation and length contraction , their light arrives on the Earth with less energy than it was emitted. This effect is called the red shift. The result is that we get a tiny amount of energy with a bigger wavelength than we would like. The best way to counteract this, is to build telescopes as big as possible. The problem is that making a one-piece lens or mirror bigger than a couple meters is almost impossible, not to talk about the issues with logistics.
The solution is to build a modular mirror using hexagonal tiles like the ones you can see in the pictures above. Making such a big mirrors improves the angular resolution of the telescope as well as the magnification factor due to the geometrical properties of a "Cassegrain telescope". So we can say that thanks to regular hexagons we can see better, further and more clearly than we could have ever done with only one-piece lenses or mirrors.
Did you know that hexagon quilts are also a thing?? The honeycomb pattern is composed of regular hexagons arranged side by side. They completely fill the entire surface they span, so there aren't any holes in between them. This honeycomb pattern appears not only in honeycombs surprise! In fact, it is so popular that one could say it is the default shape when conflicting forces are at play, and spheres are not possible due to the nature of the problem.
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