The connection between mathematics and football

First, the football surface "black" and "white"

Many people love football, but it seems that few people pay attention to the geometric shape and quantity of black and white leather blocks that make up the football surface.

There are two regular polygons on the surface of a standard football, one is a black regular Pentagon and the other is a white regular hexagon. It can be found that the edge of each black leather block has a common edge with the white leather blocks around it, while only three edges of each white leather block have a common edge with the black leather block. If the number of black patches is x and the number of white patches is y, 5x=3y= the total number of adjacent edges of black patches, so x: y = 3: 5. Using this relationship, we can know the total number of football leather pieces by counting the number of black leather pieces. For example, when we know that the black leather is 12, then the total number of leather is 8/3x12 = 32.

Second, the study of football "incident angle" α

The ultimate goal of using skills and tactics in football matches is to score goals, so whether you can shoot the ball into the opponent's goal at the end or shoot the ball into the opponent's goal with your head is the key to the success or failure of the game, that is, we often say whether you can do it with one foot. Because shooting is often done during running, the requirements for angle and distance from the door are very high. If we can add appropriate strength and direction at a certain angle and distance, the ball will definitely break into the door. Shooting can be divided into: close range-11meter; Shoot at a distance of 20 meters from a long distance; Mid-range shooting is between the two; According to the height of the incoming ball, it is divided into: groundball, rebound ball and volley ball; According to the flight path of the ball, it can be divided into straight ball and curved ball. Because the shooting distance is relatively close and the strength is very strong, it can be regarded as a straight ball.

Now let's take groundball as an example.

For example, Party A's winger passes the ball near the goal guarded by Party B and moves along a straight line. The straight line in the known direction is perpendicular to the bottom line, and the bottom line intersects with the point d on the extension line of the target AB. So what is the incident angle α?

Distance between edge and baseline (x) Tangent value of incident angle α (tanα) Incident angle α

10 0.366 20 degrees

15 0.244 13.7 degrees

20 0. 183 10.4 degrees

25 0. 1464 8.3 degrees

If the ball flies in the air after kicking.

ABCD is the vertical plane of the target, O is the starting point, O and AB determine the plane γ, the horizontal plane is β, the plane angle of dihedral angle γ-l-β is θ, tanO=2.44/x, tan∠BOC=2.44/OC, tan∠AOD=2.44/OD, OD > x,.

∴∠AOD〈θ,∠BOC〈θ。 Therefore, in order to get the ball into the net, the height angle and oblique angle of the central shot; The angle must be less than θ, if it is greater than the possible shooting height or mistake, if it is equal to the possible impact on the door frame.