Pick’s formula, as stated last week, allows us to find the surface of a lattice polygon based on its interior and perimeter points, this couldn’t be easier:
S = I + P/2 -1
where i is the number of interior points and p is the number of circumferential points. The various methods of proving this theorem have led to a lively discussion among readers, including whether Pick’s theorem is a true theorem or simply “a formula that works”, as by Francisco Montesino. The argument was made (see the first twenty comments in the previous installment). The question is not trivial, because the difference between what we call a “theorem” and what we call a “property” is not qualitative, but of degree, since all its implications exist on any mathematical basis. Is it a theorem that the sum of the angles of a triangle is 180º? And the equality of angles opposite to the vertex? It could be argued that the latter is obvious (suffice it to realize that they have the same supplementary angle, i.e. they lack the same amount to reach 180º); But the concept of “obvious” is very subjective, and it makes us refer once again to the old contradictions or “heaps of”, which we have already dealt with on more than one occasion. Manuel Amoros expresses it in a humorous way that is no less eloquent: “If it is easy for me to understand, it is a formula; If I’m having a hard time understanding it, here’s a motto: If I don’t understand a big deal, it’s a theorem”. (Might add: if I get it, but can’t prove it, it’s conjecture.)
To the temptation of extending Pick’s theorem to three-dimensional space, John Reeve showed in 1957 that this is not possible, by the following counter-example:
Let us consider a quadrilateral whose vertices are the points of the spatial coordinate axes (0, 0, 0), (0, 1, 0), (1, 0, 0) and (1, 1, r), where r is a Natural numbers (integer and positive). In the attached figure, we see Reeve’s tetrahedron with a fixed base and variable height, when r takes on the values 2, 3 and 4 respectively. How does this quaternion prove the inapplicability of Pick’s theorem in 3D space? And after this question of medium difficulty, one easy and one difficult: What is the volume of Reeve’s quadrilateral? Does the conclusion of the inapplicability of Pick’s theorem in 3D apply to other dimensions?
for questions related to osteotomyIt seems to be They haven’t aroused the interest of my astute readers yet (I can’t believe they oppose them), so they’re pending.
Returning from the three-dimensional grid to the quintessential two-dimensional grid, the 8×8 board, and taking on the topic of safe strategy games discussed in previous weeks, Ignacio Alonso takes a problem from Peter Winkler’s highly recommended collection of mathematical puzzles. proposed:
Adam begins his game of snakes (nothing to do with Google) by marking any square on a chessboard. Eva then marks a square adjacent to the square marked by her partner. And so, in turn, they each mark a square next to the marked last, forming a snake on the board. The game ends when one of the two cannot mark the box, and thus loses the game.
Is there a square that guarantees Adam a win if he marks it on his first move? In other words, is Winkler Snake a safe strategy game?
Carlo Frabatti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and youth, among them ‘Maldita Physics’, ‘Malditas Matématicas’ or ‘El Gran Juego’. He was the screenwriter of ‘The Crystal Ball’.