2024-01-26 10:24:25
Regarding the “hidden treasures” in the Pascal-Tartaglia-Jayam triangle, which we dealt with, once again, last week, Francisco Vicente rules, in relation to the Fibonacci sequence (see footnote). image), the following graph of my own creation:
And in relation to the presence of the number e in the triangle, Luca Tanganelli says: “A relationship of the Jayam triangle with e that occurs to me is the following. The triangle is drawn in its centered form, making the distance between two adjacent numbers equal to the distance between lines. Then a parabola is drawn that passes through the apex and the ends of the 1-2-1 row. The relationship between the value of the triangle on the vertical axis with respect to the value of the triangle on the parabola, at the same height, tends to e”. Brilliant, but can anyone think of a simpler relationship?
And Salva Fuster raises an interesting question that I submit to the consideration of my sagacious readers: “Looking for the cat’s three feet, it occurs to me to ask whether decimal numbers formed as 0,… in which we replace the ellipsis with concatenation of figures of the numbers that form the diagonals of the triangle, are always irrational (except the first):
0,1111…
0,1234…
0,13610…”
More information
Eulerian and Hamiltonian tours
A couple of weeks ago (see comments on 2024 and the tetrahedral numbers) a momentary confusion arose between Eulerian and Hamiltonian paths, which is a good pretext to point out the difference between both paths, which are often considered equivalent, although they are not.
An Eulerian path runs through all the edges of a figure (a graph, mathematically speaking) passing through each of them only once. The typical pastime of drawing a figure (for example, an open envelope) without lifting the pencil from the paper and without going back over a line already drawn, is solved by an Eulerian path. If the path is closed (that is, if it ends at the same point where it begins), it is an Eulerian cycle. Note that in the well-known open envelope pastime it is possible (and in fact it is inevitable) to pass through the same vertex again, but not through the same line.
Drawing of an envelope, which constitutes the classic example of an Eulerian path
In the Hamiltonian path, however, it is about passing through all the vertices only once. As in the previous case, if the path is closed it is called a Hamiltonian cycle. Of course, a path can be both Eulerian and Hamiltonian (what condition must a path meet to be both Eulerian and Hamiltonian?).
Like the recently revisited Pascal-Tartaglia triangle, what we know as Hamiltonian paths had already been studied long before by Eastern mathematicians. Already in the 9th century, the Indian poet Rudrata speaks of the “path of the horse”: a journey of the jumping jig around the entire board, passing each square only once (can you make that journey? Do you see why it is a Hamiltonian path? ?).
Hamilton studied the paths that bear his name in the Platonic solids, and in 1857 he allowed a puzzle based on the Hamiltonian paths to be marketed, consisting of finding a path along the edges of a dodecahedron that passed only once through all its vertices (it seems be that the 25 pounds he was paid on that occasion was all the money Hamilton ever received for his mathematical discoveries). You can have fun solving it without having to use a dodecahedron itself, that is, three-dimensional: its projection on the plane works just as well (so the first part of the problem consists of drawing a two-dimensional topological equivalent of the dodecahedron).
You can follow MATERIA on Facebook, X and Instagram, or sign up here to receive our weekly newsletter.
#paths #Euler #Hamilton #game #science