For a unit cube, we place a plane on each edge such that the plane forms a $45^{\circ}$ angle with both faces that the edge belongs to and does not intersect the cube. What is the volume of the convex body bounded by these 12 planes?
I. solution. The body described in the problem can be constructed by placing a pyramid on each face of the starting cube (Figure 1). The base of these pyramids is the corresponding face of the cube, and their side faces are determined by the planes placed on the four edges of the corresponding face of the cube. Due to the symmetry of the cube, these pyramids are congruent.

![](https://cdn.mathpix.com/cropped/2024_05_02_681b69b66d59b1782e1dg-1.jpg?height=499&width=456&top_left_y=306&top_left_x=821)

Figure 1

To determine the height of the pyramids, consider a cross-section of the body by a plane that is the perpendicular bisector of one of the cube's edges. Such a plane bisects three other edges of the cube that are parallel to the selected edge and contains the apex of four of the six pyramids (Figure 2). Since the planes placed on the cube's edges form a $45^{\circ}$ angle with the faces of the cube, the angles marked $\alpha$ in the figure are $45^{\circ}$. Therefore, each of the four small triangles is isosceles and right-angled, so the height marked $m$ in the figure, which is also the height of the pyramids, is exactly half the hypotenuse, or $\frac{1}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_681b69b66d59b1782e1dg-1.jpg?height=507&width=510&top_left_y=1248&top_left_x=794)

Figure 2

Thus, the volume of the small pyramids is $\frac{1 \cdot 1 \cdot \frac{1}{2}}{3}=\frac{1}{6}$, and therefore the volume of the body is $1+6 \cdot \frac{1}{6}=2$ cubic units.

II. solution. Consider the six planes determined by the pairs of body diagonals of the cube. Each of these planes intersects two edges of the cube and forms a $45^{\circ}$ angle with the faces of the cube containing these edges (the part of such a plane inside the cube is shown in Figure 3). These planes divide the cube into six square-based pyramids, with the center of the cube being the common apex of the six pyramids. Reflect each of these six pyramids across the corresponding face of the cube. Since the side faces of the pyramids form a $45^{\circ}$ angle with the corresponding faces of the cube, their reflections form a $90^{\circ}-45^{\circ}=45^{\circ}$ angle with the corresponding faces. Thus, they form the planes and the body bounded by these planes described in the problem.

![](https://cdn.mathpix.com/cropped/2024_05_02_681b69b66d59b1782e1dg-1.jpg?height=320&width=309&top_left_y=2285&top_left_x=889)

Figure 3

Therefore, the convex body bounded by the planes consists of 12 congruent pyramids, and thus its volume is twice the volume of the original cube, or 2 cubic units.

Remark. The body examined in the problem is a rhombic dodecahedron (because it has 12 faces, and each is a rhombus). The body has many interesting properties, which can be found in detail, for example, in H. S. M. Coxeter's book "Introduction to Geometry" in Chapter 22.4.