Our historical commentator Francisco Montesinos, who was banned from the comments section for a while due to a technical issue, returns with a proof of Napoleon’s theorem that we talked about last week:
“Let ABC be any triangle, D, E, F be the third vertices of the equilateral triangles constructed on each side, and M, N, P be their respective midpoints. Since M = A + D + B/3, N = B + E + C/3 and P = A + C + F/3, with a little patience it is easy to see that (P – M)e^i( pi/ 3 ) = N – M, which proves the theorem. To prove the second part, it is enough to observe that M + N + P = A + B + C and divide both terms by 3, which is a direct consequence of the fact that E + F + D = A + B + C I placed myself in an affine plane, which is known to be a space of points of the plane associated with a vector space. Although the beauty of the geometric proof of the theorem is unsurpassed, I believe that the one I propose is not far behind because of its scope and simplicity.”
More information
The Monge system
Adelaida López appropriately pointed out that among the mathematicians with whom Napoleon interacted, we must definitely mention Gaspard Monge, one of the fathers of descriptive geometry and creator of the dihedral system (also known as the Monge system in his honor). he developed it in his influential book Géometrie descriptive, published in 1799. Monge is also known, particularly among economists, for his important contributions to solving optimization problems (but that is another article).
The dihedral system, fundamental in technical drawing, consists of representing a three-dimensional object through its orthogonal projections on planes that intersect perpendicularly. Usually the elevation or front view, the plan view or top view and the profile or side view are shown, although sometimes two views are sufficient (the dreaded “monkeys” of engineering exams). I invite my attentive readers to reconstruct in mind (or with pencil and paper) the objects whose orthogonal projections are shown below (the dashed lines represent hidden edges).
The dihedral system. Carlo Frabetti
And as a note, an “Oral Monkey” (the simple description makes the drawing superfluous) from an exam sixty years ago at the School of Industrial Engineers in Madrid:
The height (front view) of an object is a circle, and its plan (top view) is a square with a side equal to the diameter of the circle with its two diagonals. What object is it?
Napoleon’s retreat
A chess set is known to represent the outline of a battle; But in the case of last week’s artistic finale, the author goes one step further because it is a real conflict. The black king is Napoleon. Box b1 is Moscow. Diagonal h1-a8 is the Berezina River. Box h8 is Paris. The white horses represent the Cossack cavalry. The queen on h1 is Marshal Mikhail Ilariónovich Golenishchev-Kutúzov. The white king is Tsar Alexander I. And there is a checkmate in 14 moves, corresponding to the 14 days of Napoleon’s retreat to Paris:
1. Cd2+, Ra2
2. Sc3+, Ra3
3. Cdb1+, Tb4
4. Ca2+, Rb5
5. Ca3+, Ra6
6. Nb4+, Ra7
7. Nb5+, Rb8
8. Ca6+, Rc8
9. Ca7+, Rd7
10. Nb8+, Ke7
11. Nc8+, Kf8
12. Nd7+, Kg8
13. Ne7+, Kh8
14. Rg2++
White can checkmate in fewer moves (how?), but in this way he only uses cavalry, paying homage to the memorable Russian victory of 1812 in 14 rounds.
You can follow THEME on Facebook, X and Instagram, or sign up here to receive our weekly newsletter.
Subscribe to continue reading
Read without limits
_