Set of soccer ball

According to the sentence about football, there are two points on a football (here marked in red) that are in the same place in the room at the beginning of the first and second halves.

The theorem of soccer is a mathematical theorem from linear algebra and geometry , which clearly illustrates the properties of the rotating group . The theorem indicates the existence of two fixed points on a spherical surface after the sphere has been rotated in place any number of times . The basic mathematical statement of the theorem was first proven in 1776 by the Swiss mathematician Leonhard Euler with the help of elementary geometric arguments . ${\displaystyle \mathrm {SO} (3)}$

statement

The sentence of football is as follows:

"In every football game there are two points on the surface of the ball, which are at the same point in the surrounding area at the beginning of the first and second half, when the ball is exactly on the point of impact."

proof

Proof idea

Rotation of a sphere around an axis of rotation

In the following, football is idealized as a ball . During the first half, a soccer ball performs a series of movements in space. Since the soccer ball is placed back on the kick-off point at the beginning of the second half, the movements of the ball can be disregarded and only the rotations of the ball need to be considered. Each of these rotations can be described by an axis of rotation and a rotation angle . Points in space that are on the axis of rotation do not change their position when rotated.

An important property of three-dimensional space is that every successive execution of two or more rotations can be described by a single rotation. The axis of rotation of this rotation penetrates the surface of the soccer ball at two diametrically opposite points ( antipodes ). These two points must therefore be in the same place in the room at the beginning of the first and second half.

proof

After choosing a Cartesian coordinate system with the center of the sphere as the origin of the coordinates , every rotation in space can be described by a rotation matrix . A rotation matrix is ​​an orthogonal matrix with a determinant . If a sphere performs a total of rotations, then these can be specified by rotation matrices . The successive execution of these rotations then corresponds to the die product ${\displaystyle D\in \mathbb {R} ^{3\times 3}}$ ${\displaystyle \det D=1}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle D_{1},\ldots ,D_{n}}$

${\displaystyle Q=D_{n}\cdot \ldots \cdot D_{1}}$

of the rotary dies. Because the product of two orthogonal matrices is again orthogonal (see orthogonal group ) and the determinant of the product of two matrices is equal to the product of the determinants ( determinant product theorem ), the matrix is again an orthogonal matrix with determinants . If now and are the three (generally complex ) eigenvalues of , then holds ${\displaystyle Q}$ ${\displaystyle \det Q=1}$ ${\displaystyle \lambda _{1},\lambda _{2}}$ ${\displaystyle \lambda _{3}}$ ${\displaystyle Q}$

${\displaystyle 1=\det Q=\lambda _{1}\cdot \lambda _{2}\cdot \lambda _{3}}$.

Since the eigenvalues ​​of an orthogonal matrix apply and complex eigenvalues occur in complex conjugate pairs , at least one eigenvalue must be real and equal . This in turn means that there must be an eigenvector for which ${\displaystyle |\lambda |=1}$ ${\displaystyle Q}$ ${\displaystyle 1}$ ${\displaystyle x\in \mathbb {R} ^{3}\setminus \{0\}}$

${\displaystyle Q\,x=1\,x}$

applies. Such a vector and every scalar multiple of this vector is accordingly mapped onto itself by the matrix . The linear envelope of this vector defines a straight line through the origin that intersects the surface of the sphere at two points. These are the two sought-after points that are held during the entire rotation. ${\displaystyle x}$ ${\displaystyle Q}$ ${\displaystyle \langle x\rangle }$

use

In modern mathematical literature , the theorem about football is often given as a corollary , that is, as a direct consequence of previously proven theorems. In such a case, the proof of the theorem is usually quite simple. Gerd Fischer writes in his textbook on linear algebra , for example , that the theorem about football is easier to prove than vividly understood, and then proves it in one line.

The football theorem is a special case of a more general statement, according to which the orthogonal endomorphisms with positive determinants form a group , the so-called special orthogonal group , in a finite-dimensional real scalar product space. If the dimension of the underlying vector space is odd, then every mapping in this group has the eigenvalue . ${\displaystyle \mathrm {SO} (n)}$ ${\displaystyle n}$ ${\displaystyle 1}$

See also

• Euler's angle
• Momentary pole
• Borsuk-Ulam's Theorem
• Sentence of the hedgehog

literature

• Gerd Fischer : Linear Algebra: An Introduction for New Students . 18th edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-03944-8 .
• Michael Merz , Mario V. Wüthrich: Mathematics for economists . Vahlen, Munich 2013, ISBN 978-3-8006-4483-4 .

Individual evidence

1. ^ Leonhard Euler: Novi Commentarii academiae scientiarum Petropolitanae . tape 20 , 1776, pp. 189-207 ( online ).
2. ↑ ^{a b} Gerd Fischer: Lineare Algebra: An introduction for first-year students . Springer, 2008, p. 307 .
3. ^ ^{A b c} Michael Merz, Mario V. Wüthrich: Mathematics for economists . Vahlen, 2013, p. 244-245 .

Web links

Wikiversity: Plato and Cube Symmetry  - Course Materials

• Bernhard Elsner: Why the linking of rotations is one more thing. September 8, 2014, accessed April 18, 2015 (geometric proof).