Four solids of the same volume, the regular tetrahedron with side [math] \frac 1 {\sqrt {2}} [/math], a quarter-octahedron tetrahedron made up of two equilateral triangles with side [math] \frac 1 {\sqrt {2}} [/math] and a square folded at right angles according to the unit hypotenuse, and the twenty-fourth cube, composed of a half square of unit hypotenuse, of a height [math] \frac 12 [/math] placed at the applomb of the corner, and the pyramid with a square base of side [math] \frac 12 [/math] and of the same length, the apex at the applomb of a corner.

Prove analytically, or using Cavalieri's principle, that these three solids have the same volume.