You may not know it, but mathematics is all around you in the world today- from thebreakfast you eat in the morning, to the hobbies you enjoy, to the complex world of computersand games. In this paper, it’s going to be my goal to show you how math is related to the sportof soccer.
Soccer, in essence, is a fairly simplistic sport. The basic rules are simple, but some of themore particular ones can become slightly confusing. The MLS (Major League Soccer)recognizes seventeen basic rules which players and coaches must abide by. However, all ofthese are not entirely important to understand the game. First of all, you need a regulation sizeball and two netted goals, eighteen feet by eight feet. Each team consists of eleven players, oneof whom must be the goalkeeper. In addition, both teams are allowed to have a select number ofsubs. The number of subs varies depending on the level at which you are playing. No playersare allowed to touch the ball with their hands, besides the goalie, who is only given this privilegeif he is inside the eighteen yard box around the goal. Player uniforms must embody a shirt,socks, shorts, shin guards, and shoes. In addition the goalkeeper must wear colors thatdistinguish him from other players, the referee, and the referee assistants. The game is run by amain referee and two assistants. The main referee is responsible for control of the game andhis/her decisions regarding facts connected with play are final. The referee’s assistants aid thereferee by indicating offside, when the ball is out of play, and which team gets a corner kick, agoal kick, or a throw in. Furthermore, they denote when a substitution needs to be made. Gamelength for professional games is ninety minutes with two forty-five minute halves. Time iscontinuously running. Points (called goals) are awarded to the team that passes the ballcompletely over the goal line and into the other team’s goal. Another important part of the gameis the calling of fouls and penalties. “Fouls are called for any of the following six offenses in amatter that is considered by the referee to be careless, reckless, or using excessive force. A. KicksB. TripsC. Jumps atD. ChargeE. StrikesF. Pushes”(Major)Anything from free kicks for the opposing team, to game suspensions can be given out for theseoffenses. The game is won by the team that has scored the most goals after the entire ninetyminutes of play has expired. If the number of goals scored by each team is the same after theentire ninety minutes, the game goes into a ten minute overtime with two five minute halves. Ifthe score remains tied, the game goes into a shoot out until one team has kicked more goals thanthe other.
In brief, soccer is played in this way. Now you may look at all of that writing and saythat there is no way that any of this sport could have anything to do with math, but surprisinglyenough, it does.
Math is present in almost anything you do. All sports, games, hobbies, and more have anumber of ways in which they are involved in math. Soccer is not left out. Many mathematicaltheories apply to the sport of soccer. To start simple many geometrical shapes are on a soccerfield. The field is rectangular, the goal boxes are rectangular, and the center of the field is acircle. Even more difficult things can be calculated using math. For instance, the probability ofactually scoring a goal can be calculated by finding the angle to the goal (geometry) and byfinding the center of gravity (physics). More physics applies in calculating the distance and inwhat direction a ball will travel when kicked by using projectile motion and initial velocity. However, in this paper I am going to concentrate on one main focus, and this focus is the shapeof the actual soccer ball itself.
If you actually look at a soccer ball in depth, you will notice that it is an intricate patternof pentagons and hexagons covering a spherical surface. This shape is technically called atruncated icosahedron, a more complex version of a polyhedra. Basically there are five platonic solids, which are the cube, the tetrahedron, thedodecahedron, the octahedron, and the icosahedron. “Known to the Greeks, there are only fivesolids which can be constructed by choosing a regular convex polygon and having the samenumber of them meet at each corner. The cube has three squares at each corner; the tetrahedronhas three equilateral triangles at each corner; the dodecahedron has three equilateral triangles ateach corner; With four equilateral triangles, you get the octahedron, and with five equilateraltriangles, the icosahedron”(Hart). The number of faces, edges, and vertices can be related to each other using a fairlysimple formula called Euler’s formula. ” The Euler formula reveals a relationship among thethree elements of the polyhedron; vertices, edges, and faces. The Euler formula states that, V,the number of vertices minus, E, the number of edges plus, F, the number of faces of apolyhedron is always equal to two”(Koelm). This can be illustrated by looking at the number offaces, edges, and vertices of the five platonic solids. They are as follows: faces edges verticestetrahedron 4 6 4cube 6 12 8octahedron 8 12 6dodecahedron 12 30 20icosahedron 20 30 12To prove the theory correct, take an octahedron for example. If you plug the numbers intoEuler’s formula, you get an equation that looks like this:6 – 12 + 8 = 2To firmly establish this theory, use a tetrahedron. When it’s statistics are plugged into Euler’sformula the equation looks like this:4 – 6 + 4 = 2Again the equation is equal to two. If the math is done correctly, those answers should be thecorrect ones. This formula also works for the soccer ball, namely the truncated icosahedron. After carefully counting all the vertices, faces, and edges, I found that the truncated icosahedronhas these characteristics: faces edges verticestruncated icosahedron 32 90 60The soccer ball falls into the category of a archimedean semi-regular polyhedra “A keycharacteristic of the Archimedean solids is that each face is a regular polygon, and around everyvertex, the same polygons appear in the same sequence”(Hart). To more fully understand thesoccer ball is to begin with the regular icosahedron itself, which is one of the five platonic solidslisted earlier. This shape by definition has twenty faces, each being an equilateral triangle. Thefaces are arranged so that five triangles meet at a vertex at the top and bottom. There are fivesuch vertices throughout this particular shape. The truncated part of the soccer ball can beexplained best by the illustration that follows, but basically it is defined like this: Each vertex iscut off along a plane perpendicular to the radius at the vertex. Since the vertex is formed by theintersection of five triangles, the new facets created by this cutting are pentagons. Then theremaining portions of the triangle are converted into hexagons.
Now if you put that all together, you have to wonder how flat polygons can be puttogether to form a spherical ball. The answer to this question is fairly simple. The angles of thesides of a hexagon are 120 degrees, while the measures of the angles of the sides of a polygonare 108 degrees. Since the ratio of hexagons to polygons on a soccer ball is roughly 2:1, youwould add up the angles in this way: 120 + 120 + 108 = 348If you’ll notice the angles add up to be 348 degrees. 360 degrees is the measure of a flat surface. Because the angle measurement of the addition of the three angles equals 348, a number lessthan 360, which is that of a flat surface, the result when all thirty faces are fit together in thisform is a spherical shape. Overall, that is basically the way that a truncated icosahedron is created. Basically, itstarts off as a regular polyhedron, and then is cut at it’s vertices until the shape of the soccer ballis created. After reading this paper, I hope that you’ll realize the significance of mathematics insports. Most people, when told that there is a lot of math involved in daily activities, laugh at thewhole concept of it. Not just soccer, but numerous things can be applied to mathematics. Thispaper hopefully was a good example of just how much soccer was related to math. Furthermore,this was only one small aspect of the mathematical side of soccer. Subsequent theories and ideascould be applied to many other aspects of the game. So just remember, math is not just somepointless classes you take in school, it can be applied ANYWHERE!
