Another roof, another proof.
bobshush said: Re: the gears typeface, sketch of a proof idea - find the shortest TSP cycle between the centers of all the gears, which is guaranteed to be non-crossing. Wrap a conveyor belt around this path, case analysis shows this can be done in a taut manner. Am I missing something?
I’ve been thinking about the conveyer belt problem as well for a while now, this kind of question really appeals to me. Especially since there appears to be a counterexample with seven disks of differing sizes (however I can’t see yet why it is a counterexample, I thought I have found a solution). Online information about the conveyer belt problem seems scarce but this and this papers give some nice basic explanations.
Your idea is a very good start, but your missing one small (but vital) detail: by moving the belt from the centers to the circumference of the disks, it is possible you introduce self-intersections to the belt. So you still need to prove that you can always find a Hamiltonian cycle between the centers of the disks which still allows to validly “puff up” the belt from centers to circumferences, and this is where the main difficulty lies, I’m afraid…
Good luck with your further efforts,
I don’t think that everyone should become a mathematician, but I do believe that many students don’t give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it.
I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.
The Rado graph
The Rado graph is the unique (up to isomorphism) countable graph R such that for every finite graph G and every vertex v of G, every embedding of G−v as an induced subgraph of R can be extended to an embedding of G into R. This implies R contains all finite and countable graphs as induced subgraphs.
Rado gave the following construction: identifiy the vertices of the graph with the natural numbers. For every x and y with x<y, an edge connects vertices x and y in the graph if the xth bit of y's binary representation is nonzero.
Thus, for instance, the neighbors of vertex 0 consist of all odd-numbered vertices, while the neighbors of vertex 1 consist of vertex 0 (the only vertex whose bit in the binary representation of 1 is nonzero) and all vertices with numbers congruent to 2 or 3 modulo 4.
Mathematician Maryam Mirzakhani is the first woman to win a Fields Medal. It had been an all-boys club since the prizes were established in 1936. Mirzakhani, a native of Iran, is a professor at Stanford University. She won for her work on “the dynamics and geometry of Riemann surfaces and their moduli spaces.” Here’s how Nature summed up her contributions:
"Perhaps Maryam’s most important achievement is her work on dynamics," says Curtis McMullen of Harvard University. Many natural problems in dynamics, such as the three-body problem of celestial mechanics (for example, interactions of the Sun, the Moon and Earth), have no exact mathematical solution. Mirzakhani found that in dynamical systems evolving in ways that twist and stretch their shape, the systems’ trajectories “are tightly constrained to follow algebraic laws”, says McMullen. He adds that Mirzakhani’s achievements "combine superb problem-solving ability, ambitious mathematical vision and fluency in many disciplines, which is unusual in the modern era, when considerable specialization is often required to reach the frontier".
Erica Klarreich wrote a wonderful summary of Dr. Mirzakhani for Quanta magazine, which is worth a read. She’s apparently quite the generalist—deriving intellectual satisfaction in “crossing the imaginary boundaries people set up between different fields.” Among her diverse body of work outside dynamics is her doctoral dissertation on geodesics of hyperbolic surfaces, which another researcher called “the kind of mathematics you immediately recognize belongs in a textbook.” Meanwhile, she’s an unassuming character herself, with a deep love of her work and a phenomenal work ethic: "You have to spend some energy and effort to see the beauty of math."
In 2007, students from the Dutch Film Academy graduated with the short “Gödel”, directed by Igor Kramer with Robert Stuc in the title role.
World-famous logician Kurt Gödel developed paranoid symptoms, including a fear of being poisoned, after the assassination of Moritz Schlick by a pro-Nazi student. It was Schlick who had aroused Gödel’s interest in logic during his seminars. This short depicts a retired Gödel, realizing his surroundings are a filmset, feeding his paranoia.—
This topological magic trick by Martin Gardner appeared in his Perplexing Puzzles and Tantalizing Teasers. Take a dollar bill (or a strip of paper) and a pair of paper clips. Attach the clips to the bill as in the picture below. Make sure everything’s positioned correctly and quickly pull the ends of the bill. This will make the paper clips slide together, interlock, and fly off the bill!
It is also possible to magically connect multiple paper clips, but in practice, if you got too many of them, they tend to bend rather than to interlock.
Differentiating is like squeezing toothpaste out of the tube. Integrating is like putting the toothpaste back into the tube.