05/28/2020
Solving a Handshaking Problem using Recursion
https://medium.com/cantors-paradise/solving-a-handshaking-problem-using-recursion-650b1df31d57
Let’s solve a nice problem brought to my attention by Lawrence Bryan. Thank you Lawrence :)
A weekly Medium publication of math-, science- and technology-related essays.
(24)
https://medium.com/cantors-paradise/solving-a-handshaking-problem-using-recursion-650b1df31d57
Let’s solve a nice problem brought to my attention by Lawrence Bryan. Thank you Lawrence :)
“Oh that place. It’s so crowded nobody goes there anymore.” — Yogi Berra
https://medium.com/cantors-paradise/the-el-farol-bar-problem-a60205dd3f86
“Oh that place. It’s so crowded nobody goes there anymore.” — Yogi Berra
https://medium.com/cantors-paradise/proving-the-cauchy-goursat-theorem-cddcd0962273
Let’s prove a beautiful Theorem from complex analysis!! For the determined amateur with some knowledge of 12th grade math and calculus.
https://medium.com/cantors-paradise/the-powers-of-two-leading-digits-and-benfords-law-af6bb216cfc1
An intuitive way to understand Benford’s Law and how it applies to the powers of two
https://medium.com/cantors-paradise/solving-the-gaussian-integral-f8dc9b178fa
Lord Kelvin wrote of this integral: “A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville…
“Sir: Some recent work […] leads me to expect that the element of uranium may be turned into a new and important source of energy in the immediate future.”
https://medium.com/cantors-paradise/the-einstein-szil%C3%A1rd-letter-ec7f1f525704
The now famous Einstein-Szilárd letter was written at the initiative of Hungarian nuclear physicist Leó Szilárd with help from Edward…
https://medium.com/cantors-paradise/supplementary-angle-identities-9329de0f03b8
Whenever two angles add up to π.
Sometimes the simplest proofs give the best insights into how mathematicians think 👌
https://medium.com/cantors-paradise/a-simple-proof-of-fermats-little-theorem-7d3a406fdec6
Sometimes the simplest proofs give the best insights into how Pure Mathematicians think
https://medium.com/cantors-paradise/platonist-roger-penrose-sees-mathematical-truths-61a45840fe00
What does Roger Penrose mean when he claims to “see” mathematical truths?
Albert Einstein has been one of the most celebrated scientists of all time. It is funny, that his ideas are also, among the most difficult…
https://medium.com/cantors-paradise/powers-of-two-ones-tens-and-hundreds-of-digits-6ebcb24ebf1a
An examination of repeating digits in the first n trailing digits of the powers of two
https://medium.com/cantors-paradise/laplaces-equation-62a4e758f0ef
Flows and fields.
https://medium.com/cantors-paradise/how-control-allocation-for-multirotor-systems-works-f87aff1794a2
A mathematical explanation of how drones function.
https://medium.com/cantors-paradise/how-aristarchus-found-the-size-of-the-moon-c21ff0d0ee0e
How big are you, my precious Moon?
https://medium.com/cantors-paradise/the-powers-of-two-an-introduction-ff661ebf5638
An introduction and index to an article series on the properties of and patterns in the powers of two
Probably the longest entry in the Online Encyclopaedia of Integer Sequences, and rightly so!
https://medium.com/cantors-paradise/the-catalan-numbers-3c591c5765e3
Probably the longest entry in the OEIS, and rightly so.
“There’s a tradition of scientists approaching senility to come up with grand, improbable theories. Wolfram is unusual in that he’s doing this in his 40s” — Freeman Dyson
https://medium.com/cantors-paradise/richard-feynmans-advice-to-a-young-stephen-wolfram-1985-d572dc360c18
«You don’t understand “ordinary people”. To you they are “stupid fools”. »
Do Prime Numbers Contain a Deep Pattern?
https://medium.com/cantors-paradise/the-prime-numbers-cross-hint-of-a-deeper-pattern-2110c348f03b
I have always been fascinated with “origins”: the origin of life on Earth, the origin of human beings, the origin of the Universe, and of…
https://medium.com/cantors-paradise/six-proofs-that-there-are-infinitely-many-primes-33037bc2c54e
If you only know Euclid’s proof, here are a few others
Why can’t the cube be doubled or the circle be squared using a compass, a straight edge and a finite number of steps?
https://medium.com/cantors-paradise/why-cant-the-cube-be-doubled-or-the-circle-be-squared-eb4b7d1d96bc
It took over 2000 years for algebra to expose the limitations of the straight edge and compass
This is a gentle introduction to an important concept in combinatorics that plays a major role in algebra and geometry — root systems
https://medium.com/cantors-paradise/all-about-root-systems-20c651b19d25
A visual introduction
The Nobel Prize in Physics 1965 is awarded jointly to Sin-Itiro Tomonaga, Julian Schwinger and Richard P. Feynman “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.”
https://medium.com/cantors-paradise/qed-the-strange-theory-of-light-and-matter-df50782b1651
The Nobel Prize in Physics 1965: Sin-Itiro Tomonaga, Julian Schwinger, and Richard P. Feynman
https://medium.com/cantors-paradise/a-method-of-counting-the-number-of-solutions-61afdcdefd87
Linear algebraic equations are one of the simplest equations that we can solve. If there is only one variable, then the solutions are…
https://medium.com/cantors-paradise/what-is-probability-for-90e5dbc8f7e8
To start with an analogy, what is money? Textbooks and Wikipedia say that money is a medium of exchange, and a store of value, and a unit…
"Task 1: To attend von Neumann’s lectures and take copious notes, complete unfinished proofs, see them through the secretarial jungle, and promptly circulate them to all American university libraries.
These notes ran to over 600 pages. Weekly.”
https://medium.com/cantors-paradise/the-duties-of-john-von-neumanns-assistant-in-the-1930s-8d08e538fe2b
“What, I asked, are the duties of an assistant to Professor von Neumann?”
We discuss a proof of the celebrated theorem of Turan. In the process, we utilize the Probabilistic Method and the Cauchy-Schwarz…
https://medium.com/cantors-paradise/the-rubiks-icosahedron-and-the-mathieu-group-m12-444ae64d1ac8
Modelling this construction using GAP
Various ways to prove the infinity of prime numbers:
https://medium.com/cantors-paradise/how-to-prove-the-infinity-of-primes-9ccfbe9bdd6a
Two simple proofs about the infinity of primes
https://medium.com/cantors-paradise/pythagoras-theorem-with-proof-d8b57fccf115
Where the sum of two squares meets the third.
"If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn’s lemma may well be able to help you" - Tim Gowers
https://medium.com/cantors-paradise/what-is-zorns-lemma-83980becef80
The quick and dirty guide to one of the stranger mathematical curiosities.
We discuss the concept of mathematical induction on the natural numbers and give some examples.
“I see now with horror, what I did: My letter developed into a monster of 21 pages! But pp. 7-20 are mathematical, so please do not read them seriously unless you are very bored.”
https://medium.com/cantors-paradise/john-von-neumanns-1935-letter-to-oswald-veblen-3acbe1b69098
Mathematical Gossip
We present the beautiful Erdős-Szekeres theorem about monotone subsequences of finite sequences of distinct real numbers.
https://medium.com/cantors-paradise/faltingss-theorem-and-the-mordell-conjecture-59e1917f5cdb
On the number of rational solutions to polynomials.
New York, NY
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Einstein, Bohr, Dirac, Heisenberg, Born, Schrödinger, Curie, Lorentz and many more. The list of attendees at the 1927 Solvay Conference on Physics and Chemistry was and is unparalleled. Amazingly, video exists, narrated by Nancy Thorndike Greenspan. Read more: https://medium.com/cantors-paradise/the-golden-age-of-quantum-physics-9a3ddac68413
Cantor's paradise was originally an expression used by David Hilbert (1926, page 170) in describing set theory and the infinite cardinal numbers developed by Georg Cantor.
Cantor’s Paradise is a weekly Medium publication of math-, science- and technology-related essays
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