# Mathematics and Soccer

Mathematics and Soccer

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Today, soccer is one of the most “big-time” sports games which is viewed and played by a lot if people. Along with soccer, basketball’s NBA, American Football’s Super Bowl and Major League Baseball is also one of the of the primetime games that we watch and talk about. The game of soccer is affected by a lot of factors. Some of these are: player skill, team play, strength of player, speed of player, technique, strategy, training and many more. All these factors affect how a player or a team can perform well when the game of soccer is being played. Unknown to some of us, the game of soccer is also largely related and affected by mathematics and physics.

Geometry in mathematics and physics, in particular, has a lot of effect in harnessing the capabilities of a player. Simple things like the right angle to kick a ball, the right amount of force to be applied and many different measurements and instant calculation are included and integrated with soccer which has a lot to do about mathematics. Geometry will be the main focus of this paper. How geometry affects soccer and what are the different interrelationships that can be found in the game will be discussed. An introduction to geometry and a brief discussion of some topics in geometry will also be included.

Some quick samples of how math is largely involved in soccer are enumerated as follows:

ü Reliance on numbers, dimension and graphs of coaches and players in order to help improve player statistics and training.

ü Coaches make use of framework and line references and other portions of geometry to inform the members of the team on what position in the field they should be.

ü The goalie relies on the knowledge of angles to know where to position himself when defending the team’s goal.

ü Algebra helps a coach a lot because it gives him the information about which team is more likely to decline its playing performance in the next half.

ü Each team utilizes data analysis and probabilities to help them improve their training and help each member of the team.

ü Each player can assess and have an idea of what angle and what amount of force should he kick the ball for him to score a goal or to pass it to his teammate.

Soccer and Geometry

Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry. (H.M.S Coxeter 1907-2003)

According to Coxeter, Geometry helps us to make more accurate and impressive deductions and calculations. As a branch of mathematics, geometry is involved in the manipulations and organization of different lines, shapes, angles and points. Geometry gives a lot of mathematical meaning to the meaningless shapes, lines, points and symmetry (Roberts 2006).

Knowledge of these angles, lines, points and symmetries will give rise to a deeper understanding of geometry (Birkhoff et al 2000). These different understandings will help a lot in visualizing and learning geometry and how to relate and use it in soccer. One of the common topics in geometry which is very much related to soccer is angle bisection.

In geometry, bisection is the division of something into two equal parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are segment bisectors and angle bisectors. (Hilbert 1999)

An angle bisector divides the angle into two equal angles. An angle only has one bisector. Each point of an angle bisector is equidistant from the sides of the angle. The interior bisector of an angle is the line or line segment that divides it into two equal angles on the same side as the angle. The exterior bisector of an angle is the line or line segment that divides it into two equal angles on the opposite side as the angle. (http://en.wikipedia.org/wiki/Bisection)

The following is a classic example of how bisection is applied in soccer:

The task of a goalkeeper also involves geometry: Think of the following problem: if an attacking player is approaching the goalkeeper, where does the goalkeeper need to stand to have the best chance of preventing a score?

A good goalkeeper does not stand on the goal line too much of the time. He stays a bit in front of his goal, because it makes no sense to dive behind the goal line to “save” a ball. When a single attacker is approaching him, he will try to be on the angle bisector of the lines from that player to the goal posts. He will turn his body toward the approaching player, so that when he dives to the side to stop a shot, he is as far from the left hand side as from the right hand side.

P———goal———P

/ /

/ ####GK /

/ # /

/ /

/ /

/ /

/ /

Attacker

The shortest distances from the goalkeeper (GK, on the not-drawn angle bisector) to the lines from the attacker to the goal posts are perpendicular to these lines (“drawn” as ***). So the goalkeeper will not dive to the side, but to maximize his reach he will always dive slightly forward.

P_________goal_________P

Goal Keeper ****

Zenadine Zidane *****

******** *******

************* **********

***********

This is a simplification, because in the sketch above the attacker finds more space to curve the ball on the right hand side of the goalkeeper (seen from the Attacker’s viewpoint) than he finds on the left hand side. There also are interesting dilemmas for the goalkeeper about how far he should stand in front of his goal: when he is far from his goal the goalkeeper can get a single player’s ball more easily, but the attacking player can also more easily lob the ball over the

goalkeeper. And when a second player is coming at the same time, that player has a free path to the goal.

The Soccer Ball’s Geometry

Here are a some trivial and great ideas stored in a soccer ball. First, look carefully at a soccer ball and you’ll see that it’s the intersection of two Platonic solids – the icosahedron and the dodecahedron. In fact, like the dodecahedron it has 12 5-sided faces, and like the icosahedron it has 20 6-sided faces. You might look at Euler’s formula, which relates the number of faces, edges, and vertices of a solid. For example, a cube has 6 faces, 12 edges, and 8 vertices (corners). The tetrahedron has 4 faces, 6 edges, and 4 vertices.

Conclusion

The use of geometry in soccer and the deeper understanding of it will further fuel the desires of students in learning mathematics. Different interrelationships between soccer and mathematics will help students to understand the importance and such subject. Teaching math will be much easier to professors if they combine topics like these to quickly get the attention of their students. Lastly, mathematics will always be renowned to be a part of soccer and its role in the strategy of the game will always be recognized and used.

Works Cited

G. D. Birkhoff and R. Beatley, Introduction to Geometry, AMS Chelsea Publ., 2000, 3rd edition

D. A. Brannan, M. F. Esplen, J. J. Gray, Angles and Bisection, Cambridge University Press, 2002

Changkyu Seol, and Kyungwhoon Cheun. “A low complexity Euclidean norm approximation.(Technical report).” IEEE Transactions on Signal Processing 56.4 (April 2008): 1721(1726). Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008

<http://find.galegroup.com.libproxy.csun.edu:2048/itx/start.do?prodId=EAIM>.D. Hilbert, Analyzing Geometry, Open Court, 1999

F. Klein, Uses of Geometry, Dover, 2004

D. Pedoe, Practical Geometry, Dover, 1988

Shillingsburg, Peter L. “A Primer of Textual Geometry.(Book review).” Papers of the Bibliographical Society of America 102.1 (March 2008): 113(3). Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008

<http://find.galegroup.com.libproxy.csun.edu:2048/itx/start.do?prodId=EAIM>.

S. Roberts, Sports and Math, Walker & Company, 2006

Munier, Valerie, Claude Devichi, and Helene Merle. “A physical situation as a way to teach angle.(teaching geometry).” Teaching Children Mathematics 14.7 (March 2008): 402(6). Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008

<http://find.galegroup.com.libproxy.csun.edu:2048/itx/start.do?prodId=EAIM>

Riley, Mark T. “Euclid.” Great Thinkers of the Western World. HarperCollins Publishers, 1992. 36(4). Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008

RILEY, MARK T. “ARCHIMEDES.” Great Thinkers of the Western World. HarperCollins Publishers, 1999. 44. Expanded Academic ASAP. Gale. California State Univ, Northridge. 22 Apr. 2008

OUTLINE

I. Introduction

a. What is Soccer

b. Role of Math in soccer

D. Pedoe

Riley, Mark

c. Subjets in Math that are needed in Soccer (Geometry)

D. Pedoe

F. Klein

d. Examples of how math is needed in Soccer

II. Soccer and Geometry

a. Role and importance of Geometry

P. Shillingsburg

F. Klein

b. Angle Bisection in Geometry

G. D. Birkhoff

D. A. Brannan

S. Roberts

c. Examples of Bisection in Soccer

III. Soccer Ball’s Geometry

a. The use of Geometry in the Soccer Ball

S. Roberts

IV. Conclusion

V. Works Cited