32. Geodesics. This problem does not require knowledge of mathematics. Let's place a rubber band (so-called "prescription", used in pharmacies for packaging medicines) on a stationary cube in such a way that it holds on the cube and does not cross itself.

The line along which this rubber band will lie is called a geodesic line.

1) How many times will all geodesic lines cover the surface of the cube (i.e., how many geodesic lines will pass through each point on the surface of the cube)?
2) How many different families of geodesic lines cover the surface of the cube?
32. We will prove that through each point on the surface of a cube, there pass four different geodesics, and in total, we have seven families of geodesic lines.

If we assume that the cube is smooth, then a rubber band wrapped around it will be arranged in such a way that the perimeter of the polygon it forms will reach a minimum.
![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-083.jpg?height=538&width=1268&top_left_y=1645&top_left_x=384)

Fig. 42.

Three types of such positions, and therefore, three families of such geodesics, are shown in Fig. 43; they lie in planes parallel to the faces of the cube. To convince ourselves that there are other families of geodesics, let's cut the cube with a plane parallel to the diagonal of the base (Fig. 44),

Then, using the notations indicated in Fig. 44, we will have:

$$
\begin{gathered}
x+y=a \\
P K=a \sqrt{2}-2 x \operatorname{tg} \alpha, \quad K L=x \sqrt{1+2 \operatorname{tg}^{2} \alpha} \\
M N=a \sqrt{2}-2 y \operatorname{tg} \alpha, \quad L M=y \sqrt{1+2 \operatorname{tg}^{2} \alpha}
\end{gathered}
$$

and for the perimeter \( p \) of the hexagon \( K L M N O P \), we get:

$$
p=2 a \sqrt{2}-2 a \operatorname{tg} \alpha+2 a \sqrt{1+2 \operatorname{tg}^{2} \alpha}
$$

Thus, the perimeter \( p \) depends only on the angle \( \alpha \) and remains the same in all parallel planes; this perimeter
![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-084.jpg?height=366&width=1272&top_left_y=902&top_left_x=381)

Fig. 43.
![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-084.jpg?height=582&width=1220&top_left_y=1388&top_left_x=407)

Fig. 44.

reaches a minimum if \( \operatorname{tg} \alpha=1 / \sqrt{2} \). The sides of the hexagon \( K L M N O P \) will be parallel to the diagonals of the faces of the cube, and this hexagon represents a geodesic. There are four families of such geodesics, as indicated in Fig. 45, and therefore, together with the previous three, there are a total of seven families.