15.34. In space, 4 points are given, not lying in the same plane. How many different parallelepipeds exist for which these points serve as vertices?
15.34. Consider a parallelepiped for which the given points are vertices, and mark its edges connecting these points. Let \( n \) be the maximum number of marked edges of this parallelepiped emanating from one vertex; the number \( n \) can vary from 0 to 3. A simple enumeration shows that only the variants depicted in Fig. 113 are possible. First, any of the four points can be the first, the second can be any of the remaining three, and so on, i.e., the 4 points can be numbered in 24 different ways. After numbering the given points, the parallelepiped in each case is uniquely restored, so we need to determine which numberings lead to the same parallelepiped.

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-271.jpg?height=243&width=262&top_left_y=398&top_left_x=154)

a) \( n=0 \)

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-271.jpg?height=243&width=256&top_left_y=398&top_left_x=403)

\( \delta \quad n=2 \)

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-271.jpg?height=244&width=264&top_left_y=398&top_left_x=648)

b) \( n=2 \)

![](https://cdn.mathpix.com/cropped/2024_05_21_3f2bc5f5a1af26514cfag-271.jpg?height=244&width=256&top_left_y=398&top_left_x=875)

\( 2 n=3 \)

Fig. 113

a) The parallelepiped in case a) does not depend on the numbering.

b) Numberings \( 1,2,3,4 \) and \( 4,3,2,1 \) lead to the same parallelepiped, i.e., there are 12 different parallelepipeds in total.

c) Numberings \( 1,2,3,4 \) and \( 1,4,3,2 \) lead to the same parallelepiped, i.e., there are 12 different parallelepipeds in total.

d) The parallelepiped depends only on the choice of the first point, i.e., there are 4 different parallelepipeds in total.

In the end, we get that there are \( 1 + 12 + 12 + 4 = 29 \) different parallelepipeds.

Let a sphere \( S \) with center \( O \) and radius \( R \) be given in space. Inversion with respect to the sphere \( S \) is a transformation that maps any point \( A \), different from \( O \), to a point \( A^{*} \) lying on the ray \( O A \) at a distance \( O A^{*} = R^{2} / O A \) from the point \( O \). Inversion with respect to \( S \) will also be called inversion with center \( O \) and power \( R^{2} \).

Throughout this chapter, the image of a point \( A \) under inversion is denoted by \( A^{*} \).