The dodecahedral system Science game

Our historical commentator Francisco Montesinos, who was sidelined from the comments section for a while due to a technical issue, returns with proof of the Napoleon theory we talked about last week:

“Let ABC be any triangle, D, E, F being the third vertices of equilateral triangles built respectively on each…

Our historical commentator Francisco Montesinos, who was sidelined from the comments section for a while due to a technical issue, returns with proof of the Napoleon theory we talked about last week:

“Let ABC be any triangle, D, E, F being the third vertices of equilateral triangles built up successively on each side and M, N, P their respective centres. Since M = A + D + B/3, N = B + E + C/3 and P = A + C + F/3 It is easy to see with some patience that (P – M)e^i(pi/ 3 ) = N – M, which proves the theorem. To prove the second part, it is enough to note that M + N + P = A + B + C and dividing both terms by 3, which is a direct consequence of the fact that E + F + D = A + B + C I have placed myself in an asymptotic plane, which, as is known, is a space of plane points associated with a space Although the beauty of the geometric proof of the theorem is unparalleled, I believe that the one I am proposing is not far behind due to its scope within its simplicity.

Monge system

Adelaide Lopez pointed out in time that among the mathematicians with whom Napoleon interacted, we cannot fail to mention Gaspard Monge, one of the fathers of descriptive geometry and creator of the dodecahedral system (also known, in his honor, as the Monge system), which he developed in his influential book Descriptive geometrypublished in 1799. Monge is also known, especially among economists, for his important contributions to solving optimization problems (but that is another article).

The dodecahedral system, a basic system in technical drawing, consists of representing a three-dimensional object through its perpendicular projections onto planes that intersect perpendicularly. Typically, a height or line view, a plan or top view, and a profile or side view are displayed, although sometimes two views are sufficient (“monkeys” of scary geometry tests). I invite astute readers to mentally reconstruct (or with pencil and paper) the objects whose orthogonal projections appear below (dotted lines represent hidden edges).

As a note, “Verbal Monkey” (the simple description makes the drawing unnecessary) is taken from an examination sixty years ago at the School of Industrial Engineers in Madrid:

The elevation (front view) of an object is a circle, and its outline (top view) is a square with side length equal to the diameter of the circle times its diameters. What is an object?

Napoleon retreated

It is known that chess is a form of battle; But in the case of the artistic finale that took place last week, the author goes further, because it represents 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 Illarionovich Golenishev-Kutuzov. The White King is Tsar Alexander I. There is a companion in 14 movements that correspond to the 14 days from Napoleon’s retreat until his arrival in Paris:

1. CD2+, Ra2

2. NC 3+, RA 3

3. CDP1+, RP4

4. Ca2+, Rb5

5. Ca3+, Ra6

6.Nb4+, Ra7

7. NB 5+, RB 8

8. Ca6+, Rc8

9. Ca7+, Rd7

10.NB8+, Ke7

11. NC8+, KF8

12.Nd7+, Kg8

13.Ni7+, Kh8

14.Rg2++

White can mate in fewer moves (how?), but in this way he only uses cavalry and pays tribute to the memorable Russian victory of 1812 in 14 rounds.

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