#then it has a chapter on topology and another on some famous constants

[thisisnotmyname]

no worries. I, too, find it extremely impressive when I see people solving hard oly geo problems, especially when they reduce those to 2-3 claims that seems approachable, sometimes even trivial. like, how did you even notice this thing in a diagram so cluttered?

that being said, I hate it when someone posts on an AoPS thread something that goes along the line of "this is a well-known property of X(57)". this is why I said I understand the hatred.

it's acceptable to expect your readers to know a few well-known results or to look them up, but looking at solutions that bring out the ETC or some very specific property of the sharkydevil point (cool name tho), I find, is discouraging. it's worse when you find it's almost impossible to solve a problem in reasonable time without having previous knowledge about stuff like this or resorting to almost mindless bashing.

I know there are olympiad geometry connoisseurs who enjoy working with these obscure configurations. it's just not my cup of tea.

Actually the ugliest mathematical objects is every olympiad geometry problem but they’re not even mathematical objects to me

[thesupergamercorpus]

Today I wanted to write about one of my ‘heroes’ or inspirations. The way I look at them is somewhat like this quote:

“Trophies and great men are not only to be gazed upon, but also inspire to do the same.”

I will write down one of the best stories I have read about John von Neumann. But first of all, this is the list he is known for. I will make the ones I think are most popular bold:

Abelian von Neumann algebra, Affiliated operator, Amenable group, Arithmetic logic unit, Artificial viscosity, Axiom of regularity, Axiom of limitation of size, Backward induction, Blast wave (fluid dynamics), Bounded set (topological vector space), Carry-save adder, Cellular automata, Class (set theory), Computer virus, Commutation theorem, Continuous geometry, Coupling constants, Decoherence theory (quantum mechanics), Density matrix, Direct integral, Doubly stochastic matrix, Duality Theorem, Durbin–Watson statistic, EDVAC, Ergodic theory, Explosive lenses, Game theory, Hilbert’s fifth problem, Hyperfinite type II factor, Inner model, Inner model theory, Interior point method, Koopman–von Neumann classical mechanics, Lattice theory, Lifting theory, Merge sort, Middle-square method, Minimax theorem, Monte, Carlo method, Mutual assured destruction, Normal-form game, Operation Greenhouse, Operator theory, Pointless topology, Polarization identity, Pseudorandomness, Pseudorandom number generator, Quantum logic, Quantum mutual information, Quantum statistical mechanics, Radiation implosion, Rank ring, Self-replication, Software whitening, Sorted array, Spectral theory, Standard probability space, Stochastic computing, Stone–von Neumann theorem, Subfactor, Ultrastrong topology, Von Neumann algebra, Von Neumann architecture, Von Neumann bicommutant theorem, Von Neumann cardinal assignment, Von Neumann cellular automaton, Von Neumann interpretation, Von Neumann measurement scheme, Von Neumann ordinals, Von Neumann universal constructor, Von Neumann entropy, Von Neumann Equation, Von Neumann neighborhood, Von Neumann paradox, Von Neumann regular ring, Von Neumann–Bernays–Gödel set theory, Von Neumann universe, Von Neumann spectral theorem, Von Neumann conjecture, Von Neumann ordinal, Von Neumann’s inequality, Von Neumann’s trace inequality, Von Neumann stability analysis, Von Neumann extractor, Von Neumann ergodic theorem, Von Neumann–Morgenstern utility theorem, ZND detonation model

Cellular automata → this one appears to function and replicate like DNA. Cellular automata preceded the discovery of the structure of DNA.

Decoherence theory (quantum mechanics) → quantum states get continuously ‘pushed around’ by external influences (like being observed e.g. the double-slit experiment), which can change their original state. A quantum state resides in a ‘superposition’. Superposition simply means a state where two or more ‘states’ are combined, like an up and down state simultaneously. When that is the case, a quantum system resides in coherence. When observing that system, decoherence, or wave function collapse happens e.g. the original quantum system both had an up and down state simultaneously, but after being observed, now only has either an up state or a down state.

Merge sort → see the chapter 08/31/2019—Top-down, bottom-up thinking, sorting algorithms, and working memory where I discuss this computer sorting algorithm and combine it with top-down and bottom-up thinking.

Self-replication → a machine replicating itself. If machines are also able to upgrade themselves with each replication, a so-called technological singularity can occur (Google it).

Von Neumann architecture → essentially how our computers are built.

Now onto some stories of him. Most information is taken from Wikipedia.

Examination and Ph.D.

He graduated as a chemical engineer from ETH Zurich in 1926 (although Wigner says that von Neumann was never very attached to the subject of chemistry), and passed his final examinations for his Ph.D. in mathematics simultaneously with his chemical engineering degree, of which Wigner wrote, “Evidently a Ph.D. thesis and examination did not constitute an appreciable effort.”

Mastery of mathematics

Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: “Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods.” He went on to explain that the three methods were:

A facility with the symbolic manipulation of linear operators;

An intuitive feeling for the logical structure of any new mathematical theory;

An intuitive feeling for the combinatorial superstructure of new theories.

Edward Teller wrote that “Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique.”

Cognitive abilities

As a six-year-old, he could divide two eight-digit numbers in his head and converse in Ancient Greek. When he was sent at the age of 15 to study advanced calculus under analyst Gábor Szegő, Szegő was so astounded with the boy’s talent in mathematics that he was brought to tears on their first meeting.

Hans Bethe on von Neumann

Nobel Laureate Hans Bethe said “I have sometimes wondered whether a brain like von Neumann’s does not indicate a species superior to that of man”, and later Bethe wrote that “von Neumann’s brain indicated a new species, an evolution beyond man”.

Edward Teller

Edward Teller admitted that he “never could keep up with John von Neumann.”

Teller also said “von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us.”

George Dantzig

George Dantzig is the mathematician who thought that two problems on the blackboard were homework. He solved them and handed them, albeit a bit later, so he thought they were overdue.

Here’s the plot twist: They were two famous unsolved problems in statistics with which the mathematics community struggled for decades.

When George Dantzig brought von Neumann an unsolved problem in linear programming “as I would to an ordinary mortal”, on which there had been no published literature, he was astonished when von Neumann said “Oh, that!” before offhandedly giving a lecture of over an hour, explaining how to solve the problem using the hitherto unconceived theory of duality.

Johnny as a student

George Pólya, whose lectures at ETH Zürich von Neumann attended as a student, said “Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he’d come to me at the end of the lecture with the complete solution scribbled on a slip of paper.”

Nobel Prizes

Peter Lax wrote, “To gain a measure of von Neumann’s achievements, consider that had he lived a normal span of years, he would certainly have been a recipient of a Nobel Prize in economics. And if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too. So the writer of these letters should be thought of as a triple Nobel laureate or, possibly, a ​3 1⁄2-fold winner, for his work in physics, in particular, quantum mechanics”.

von Neumann as a teacher

Von Neumann was the subject of many dotty professor stories. He supposedly had the habit of simply writing answers to homework assignments on the board (the method of solution being, of course, obvious). One time one of his students tried to get more helpful information by asking if there was another way to solve the problem. Von Neumann looked blank for a moment, thought, and then answered, “Yes.”

Henry Ford

Henry Ford had ordered a dynamo for one of his plants. The dynamo didn’t work, and not even the manufacturers could figure out why. A Ford employee told his boss that von Neumann was “the smartest man in America,” so Ford called von Neumann and asked him to come out and take a look at the dynamo.

Von Neumann came, looked at the schematics, walked around the dynamo, then took out a pencil. He marked a line on the outside casing and said, “If you’ll go in and cut the coil here, the dynamo will work fine.”

They cut the coil, and the dynamo did work fine. Ford then told von Neumann to send him a bill for the work. Von Neumann sent Ford a bill for $5,000. Ford was astounded – $5,000 was a lot in the 1950s – and asked von Neumann for an itemised account. Here’s what he submitted:

Drawing a line with the pencil: $1

Knowing where to draw the line with the pencil: $4,999

Ford paid the bill.

David Blackwell

Blackwell did a year of postdoctoral research as a fellow at the Institute for Advanced Study in 1941 after receiving a Rosenwald Fellowship. There he met John von Neumann, who asked Blackwell to discuss his Ph.D. thesis with him. Blackwell, who believed that von Neumann was just being polite and not genuinely interested in his work, did not approach him until von Neumann himself asked him again a few months later. According to Blackwell, “He (von Neumann) listened to me talk about this rather obscure subject and in ten minutes he knew more about it than I did.”

von Neumann was the only genius

Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1911. This was one of the best schools in Budapest, part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. Despite being run by the Lutheran Church, the majority of its pupils were Jewish. The school system produced a generation noted for intellectual achievement. Wigner was a year ahead of von Neumann at the Lutheran School. When asked why the Hungary of his generation had produced so many geniuses, Wigner, who won the Nobel Prize in Physics in 1963, replied that von Neumann was the only genius.”