Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md
The most beautiful things in mathematics are results that are:
- simple to state but hard to prove:
- Fermat's Last Theorem
- number of unknown rationality, e.g. is rational?
- transcendental number conjectures, e.g. is transcendental?
- basically any conjecture involving prime numbers:
- many combinatorial game questions, e.g.:
- surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees, sometimes via pathological counter-examples. General surprise themes include:Lists:
- classification of potentially infinite sets like: compact manifolds, etc.
- problems that are more complicated in low dimensions than high like:
- generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
- classification of regular polytopes
- unpredictable magic constants:
- why is the lowest dimension for an exotic sphere 7?
- why is 4 the largest degree of an equation with explicit solution? Abel-Ruffini theorem
- undecidable problems, especially simple to state ones:
- mortal matrix problem
- sharp frontiers between solvable and unsolvable are also cool:
- attempts at determining specific values of the Busy beaver function for Turing machines with a given number of states and symbols
- related to Diophantine equations:
- applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson
Good lists of such problems Lists of mathematical problems.
Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.
Am I achieving insight, or am I just memorizing definitions?
As of 2019, the Standard Model and general relativity are incompatible. Once those are unified, we will have one equation to describe the entirety of physics.
There are also however also unsolved problems in electroweak interaction + strong interaction, which if achieved is referred to as a Grand Unified Theory. Reaching a GUT is considered a sensible intermediate step before TOE.
The current state of Physics has been the result of several previous unifications as shown at: en.wikipedia.org/wiki/Theory_of_everything#Conventional_sequence_of_theories so it is expected that this last missing unification is likely to happen one day, potentially conditional on humanity having enough energy to observe new phenomena.
It is unbelievable that you can't find easily on YouTube recreations of many of the key physics/chemistry experiments and of common laboratory techniques.
Experiments, the techniques required to to them, and the history of how they were first achieved, are the heart of the natural sciences. Without them, there is no motivation, no beauty, no nothing.
School gives too much emphasis on the formulas. This is bad. Much more important is to understand how the experiments are done in greater detail.
The videos must be completely reproducible, indicating the exact model of every experimental element used, and how the experiment is setup.
A bit like what Ciro Santilli does in his Stack Overflow contributions but with computers, by indicating precise versions of his operating system, software stack, and hardware whenever they may matter.
It is understandable that some experiments are just to complex and expensive to re-create. As an extreme example, say, a precise description of the Large Hadron Collider anyone? But experiments up to the mid-20th century before "big science"? We should have all of those nailed down.
We should strive to achieve the cheapest most reproducible setup possible with currently available materials: recreating the original historic setup is cute, but not a priority.
Furthermore, it is also desirable to reproduce the original setups whenever possible in addition to having the most convenient modern setup.
Lists of good experiments to cover be found at: the most important physics experiments.
This project is to a large extent a political endeavour.
Someone with enough access to labs has to step up and make a name for themselves through the huge effort of creating a baseline of amazing content without yet being famous.
Until it reaches a point that this person is actively sought to create new material for others, and things snowball out of control. Maybe, if the Gods allow it, that person could be Ciro.
Tutorials with a gazillion photos and short videos are also equally good or even better than videos, see for example Ciro's How to use an Oxford Nanopore MinION to extract DNA from river water and determine which bacteria live in its for an example that goes toward that level of perfection.
The Applied Science does well in that direction.
This project is one step that could be taken towards improving the replication crisis of science. It's a bit what Hackster.io wants to do really. But that website is useless, just use OurBigBook.com and create videos instead :-)
We're maintaining a list of experiments for which we could not find decent videos at: Section "Physics experiment without a decent modern video".
Ciro Santilli visited the teaching labs of a large European university in the early 2020's. They had a few large rooms filled with mostly ready to run versions of several key experiments, many/most from "modern physics", e.g. Stern-Gerlach experiment, Quantum Hall effect, etc.. These included booklets with detailed descriptions of how to operate the apparatus, what you'd expect to see, and the theory behind them. With a fat copyright notice at the bottom. If only such universities aimed to actually serve the public for free rather than hoarding resources to get more tuition fees, university level education would already have been solved a long time ago!
One thing we can more or less easily do is to search for existing freely licensed videos and add them to the corresponding Wikipedia page where missing. This requires knowing how to search for freely licensed videos:
- Wikimedia Commons video search, e.g.: commons.wikimedia.org/w/index.php?search=spectophotometry&title=Special:MediaSearch&go=Go&type=video
- YouTube creative commons video search
Related:
- relevant University YouTube channels:
- K-12 demo projects:
- books:
- Practical approach series by Oxford University Press: global.oup.com/academic/content/series/p/practical-approach-series-pas