# Forged lightning | Science game

Last week’s “oral monkey” sparked a wide and interesting discussion (see comments by The dodecahedral system). This is the projective representation of a two-cylinder Steinmetz solid: the intersection of two cylinders with the same radius and whose axes are perpendicular to each other (there is also a three-cylinder Steinmetz solid, which is the intersection of three cylinders of equal radius whose axes are perpendicular and intersect at the same point). This shape is well known to architects, because when two half-barrel aisles intersect perpendicularly, they lead into a vault, very common in Romanesque churches, called the cloister vault, which is a double barrel of the Steinmetz type.

These solids are named in honor of the prolific German mathematician and engineer Charles Proteus Steinmetz (1865-1923), who determined their volume. Although he was not the first: the genius Archimedes, who anticipated calculus by two thousand years, had already determined it. Can you simulate Archimedes and calculate the volume of the intersection of two cylinders per unit radius without resorting to integrals? And its surface area? Do you see any relationship with the size and area of ​​the ball?

We’ve been talking recently about applications of complex numbers (for example, to discover buried treasure or prove Napoleon’s theorem), and it should be noted that Steinmetz applied them effectively to the study of alternating current circuits, and his works, both theoretical and experimental, played an essential role in replacing direct current with alternating current. And thus in the industrial development of the United States at the end of the nineteenth century and the beginning of the twentieth century. In addition, he created a new, very safe type of lightning rod, earning him the nickname Lightning Forger.

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As for the three calendar projections that appeared last week, there is an error in one of them (can you spot which one?). This, in perspective, is the solid thing that leads to another prediction:

### Napoleon III theory

Returning to Napoleon’s theory, we wondered at the time whether it could be extrapolated to three-dimensional space (hence Napoleon III: in this case III means three-dimensional). That is by saying:

If, given any tetrahedron, we construct on each of its faces two isosahedral tetrahedrons (with four equal faces), their centers (and also the surrounding centers and centers) will be the vertices of the new tetrahedron, which, by analogy with Napoleon’s triangle, we will call “Napoleonic tetrahedron”. . What would its shape be: regular, similar to the elementary tetrahedron…? What happens to the individual vertices of the four tetrahedrons, that is, those corresponding to the faces of the initial tetrahedron? But before tackling the complex and multifaceted theory of Napoleon III, there is a simpler task:

Obviously, if we start from a regular tetrahedron, the centers of the four tetrahedrons built on its faces will be the vertices of a regular Napoleon tetrahedron. Can you calculate its size? Just as Little Red Riding Hood goes to her grandmother’s house, you can reach the solution by the longest or shortest path.

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