J
John Baez
math.ucr.edu · Personal blog · on HN since 2007
John Baez, a mathematical physicist, writes expository pages on math and physics, including relativity, topology, and the Crackpot Index.
What HN saysCommenters value his explanations, citing the Relativistic Rocket page and his root-plot pictures, and say they love his blogs.
344HN people shared it
234linked it in comments
139stories submitted
3.1Kstory points
2 months agolast shared
2007people per year2026
Most-shared posts
- 1The Crackpot Index60 people · 60 points · first shared 2007 · 18 comments on HN
- 2Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]19 people · 153 points · first shared 2010 · 10 comments on HN
- 3How to Learn Math and Physics16 people · 16 points · first shared 2010 · 1 comments on HN
- 4The Beauty of Roots (2011)13 people · 524 points · first shared 2010 · 19 comments on HN
- 5Torsors (2010)13 people · 52 points · first shared 2010 · 14 comments on HN
- 6Can a Human See a Single Photon? (1996)11 people · 248 points · first shared 2008 · 91 comments on HN
- 7Is Energy Conserved in General Relativity?10 people · 6 points · first shared 2009 · discussion on HN
- 8The Relativistic Rocket10 people · 1 points · first shared 2016 · discussion on HN
- 9Baez9 people · first shared 2011
- 10Does Gravity Travel at the Speed of Light? (1998)6 people · 219 points · first shared 2010 · 132 comments on HN
- 11The Octonions (2001)6 people · 36 points · first shared 2013 · 1 comments on HN
- 12Hot water freezes faster than cold water (Mpemba Effect)6 people · 33 points · first shared 2008 · 15 comments on HN
- 13This Week's Finds in Mathematical Physics6 people · 3 points · first shared 2009 · discussion on HN
- 14Information Geometry5 people · 265 points · first shared 2015 · 20 comments on HN
- 15Why is C the symbol for the speed of light? (2004)5 people · 146 points · first shared 2014 · 121 comments on HN
What HN people wrote when they linked it
bananaflag · 2026-06-27
I am a mathematician, and I was never the kind to like to struggle by working on problems, but I developed a lot of intuition by 1) thinking deeply about definitions and proofs and why are they this way and not another 2) reading a lot of blogs and expository papers by great mathematicians, even (the more philosophically minded) mathoverflow q&a's (so I absorbed their way of thinking unconsciously). For example, I tell my students to read all 300 of John Baez's This Week's Finds posts [1] and they will deeply understand more math than 99% of their peers.
This is not the "standard" advice that usually gets peddled but for me it "worked".
[1] https://math.ucr.edu/home/baez/twf.html
antonvs · 2026-02-07
Spoken like someone who’s never read the Relativistic Rocket: https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.htm…
I really find it hard to understand how people confuse science fiction with reality. I love Tau Zero - I first read it over 40 years ago - but it’s fiction, ffs.
awanderingmind · 2024-12-09
This is cute, but I don't understand why it's on the arXiv. It's only two pages long, of which almost half is a picture and the bibliography, and as far as I can see contains nothing novel. I love Baez's blog(s) (https://johncarlosbaez.wordpress.com/ and https://math.ucr.edu/home/baez/), and it seems this would have been better served posted there as a post. Am I missing something?
ykonstant · 2024-06-19
This is related to the distribution of algebraic numbers of bounded height and degree; similar pictures have been presented e.g. in the classic post by Baez: https://math.ucr.edu/home/baez/roots/ I have also drawn some pictures: https://drive.google.com/file/d/1vx-U28OgGIs1mfrgBb7OtgVsPWAL0DSr/view…
The discipline that deals with these distributions is called Diophantine geometry and involves tons of important and deep mathematics.
mathgenius · 2023-04-26
The answer to these questions is the Riemann hypothesis. John Baez just wrote a related paper about "motives" for beginners (undergrads?) [1]. It's worth a read even if you squint at the equations like i do, there's plenty of interesting commentary as well.
[1] https://math.ucr.edu/home/baez/motives.pdf
sleepyams · 2023-03-19
One interesting result which you may find interesting: https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem
I also highly recommend this survey paper by John Baez and Mike Stay: https://math.ucr.edu/home/baez/rosetta.pdf
There are plenty of interesting results in category theory, in fact your comparison to abstract algebra is apt. There is only so much you can say about an arbitrary group in general, or an arbitrary topological space, just like there is only so much you can say about an arbitrary category.
wwalker3 · 2022-11-20
That's a great article! 1) I'd seen a lot of other cool stuff from the same author over the years (see https://math.ucr.edu/home/baez/README.html), but had somehow missed this one.
1) "Struggles with the Continuum" by John Baez
amznbyebyebye · 2021-09-17
Wow. Love this.
Searched for “Ramanujan”, one of my heros.
Found this gem- https://math.ucr.edu/home/baez/ramanujan/
Ramanujan’s “easiest” formula.
Awesome!!
Thorondor · 2021-05-03
Life, including spaceflight, would be pretty much the same as it is for us. Since the entire solar system would be moving, the relative velocities of the planets and moons would be the same as in any other solar system.
The rest of the universe would look quite different, though. Stars "ahead" of the solar system - in the path of its motion - would be noticeably blue-tinted and clumped together. Stars behind the system would be red-shifted. There are some great visualizations at [0].
[0] https://math.ucr.edu/home/baez/physics/Relativity/SR/Spaceship/spacesh…
cambalache · 2021-03-26
John is fantastic, not only a nice person with absolutely zero ego, he has brain and heart and he has such a deep love for science, a funny writer too. Anything that he writes (TWIMP,his diary,his blog Azimuth, n-category cafe, his book ) is highly recommended.
This is a must read for the aspiring mathematician or physicist:
https://math.ucr.edu/home/baez/books.html
Bonus points for being the nephew of Joan Baez.
Tossrock · 2020-12-17
Sometimes known as the Crackpot Index, wherein naming a new effect after oneself is worth 20 points: https://math.ucr.edu/home/baez/crackpot.html
lidHanteyk · 2020-02-11
Maybe. Another possibility for a "Rosetta Stone", or perhaps the Lovecraftian perversion of said stone, comes from the ADE situation [0][1]. Even category theory, that great unifier with its own "Rosetta" tables [2][3], cannot escape; quivers have an ADE property.
Either way, we still don't know what's up with 1728. That will truly unlock everything, I think. Understanding 2, 3, or 5 would be nice; understanding 8 or 12 or 24 would be groundbreaking; but I think understanding 1728 will also be understanding Langlands' programme entire.
[0] https://en.wikipedia.org/wiki/ADE_classification
[1] http://www-groups.mcs.st-andrews.ac.uk/~pjc/talks/boundaries/ade2.pdf
[2] http://math.ucr.edu/home/baez/rosetta.pdf
[3] https://ncatlab.org/nlab/show/computational+trinitarianism
dsr_ · 2018-11-05
Mass is energy, energy is mass. What's the difference? They are at different points in a curve that relates space to time.
If you trade away all the time, you get a maximum velocity: c. So the things moving at c don't experience time, and have momentum but not mass.
http://www.math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_ma…
In the same direction, if you take a mass and accelerate it, it experiences less time. There you have relativistic time dilation, as beloved by science fiction stories.
https://en.wikipedia.org/wiki/Time_dilation
And you simultaneously get length contraction in the direction of each velocity:
https://en.wikipedia.org/wiki/Length_contraction
These are all aspects of the same thing.
Really not a dumb question at all.
empath75 · 2018-10-22
http://math.ucr.edu/home/baez/rosetta.pdf
It's worth reading this if you've got a math background at all. It's a nice summary of what Category Theory is and potential applications for it.
JadeNB · 2018-08-07
This is a pleasant but intentionally very elementary summary of the relevant concepts. If you're interested in much more on this, one place to start is MO (https://mathoverflow.net/questions/4841/what-precisely-is-categorifica… or the n-Category Café (https://golem.ph.utexas.edu/category/2008/10/what_is_categorification.… or Baez's TWF 121 (http://math.ucr.edu/home/baez/week121.html).
Warning: all are much math-ier than Riehl's very nice slides, although the first one has enough content that you may find an answer that you like. If your math itch still isn't scratched, there's Baez and Dolan's paper "Categorification" (https://arxiv.org/abs/math/9802029). I don't know a programmer-focussed view on categorification; can anyone recommend one?
dsacco · 2018-02-26
I host over 30TB of excellent climate date here: https://mirrors.sorengard.com/azimuth.
These datasets were collected from various government agencies as part of a larger collaboration with John Baez and a couple of other scientists from the Azimuth Climate Data Project: http://math.ucr.edu/home/baez/azimuth_backup_project/
My server has a 1Gbps connection and can support sustained downloads of very large files. Unfortunately, the data isn’t well documented, but I’m working on improving that in the future. It’s organized roughly according to the agency it’s sourced from on a directory basis.
The archive includes datasets from NASA, NOAA, EPA, USGS, NCDC, CDC, EIA, FDA, USDA, DOE, and various subdepartments thereof.
aq3cn · 2017-10-16
There is similar website for physics textbook recommendation if anyone interested. The list was last updated in 1997.
http://math.ucr.edu/home/baez/physics/Administrivia/booklist.html
FredrikMeyer · 2017-10-07
John Baez is a fantastic writer. His home page is full of wonderful illustrations and expository stuff (physics, mathematics, climate science, ...) http://math.ucr.edu/home/baez/
mikhailfranco · 2017-04-12
For the parallel historical development of 'the continuum' in physics, I recommend this readable survey:
Paper:
https://arxiv.org/abs/1609.01421
https://math.ucr.edu/home/baez/continuum.pdf
Blog summary with discussion:
https://johncarlosbaez.wordpress.com/2016/09/08/struggles-with-the-con…
https://johncarlosbaez.wordpress.com/2016/09/09/struggles-with-the-con…
fantispug · 2016-09-06
Learning advanced mathematics without going to university would take an extreme amount of dedication, focus, and effort, but it's certainly possible. It's much easier with the resources available on the internet, and being able to connect with people through forums and stack exchange.
John Baez's recommendations: http://math.ucr.edu/home/baez/books.html
For theoretical physics 't Hooft's recommendations: http://www.staff.science.uu.nl/~gadda001/goodtheorist/
ttctciyf · 2016-07-07
John Baez's Prehistory of n-Categorial Physics [1] has some interesting nuggets about Feynman diagrams, in the "Feynman (1947)" section.
In particular, "The mathematics necessary for [interesting Feynman diagrams] was formalized later, in Mac Lane's 1963 paper on monoidal categories (see below) and Joyal and Street's 1980s work on 'string diagrams.'"
In other words, Feynman diagrams were (or at least can be taken as) an early precursor of diagrammatic approaches in category-theoretic mathematical physics.
The paper is really great for getting some context around these diagrams and a sense of the underlying mathematics, even for a category-theory ignoramus like myself, with a little "suspension of comprehension."
1: http://math.ucr.edu/home/baez/history.pdf
tzs · 2016-06-30
> I do know quite a bit about physics. But if I just said the EmDrive is nonsense due to conservation of momentum (which is the 100% correct answer with no caveats), people would shut me down because "NASA scientists know what they're doing".
I wouldn't. However, I would ask why you think violation of conservation of momentum is completely dispositive, rather than just a strong reason for great skepticism?
From Noether's Theorem [1] we know that conservation of momentum is a consequence of physics being invariant under space translation. As far as I've been able to find, we don't know that a universe that results from a big bang must be isotropic. Observationally, ours appears to be, but there are some hints that it might not be, and scientists are taking those seriously.
[1] http://math.ucr.edu/home/baez/noether.html
Steuard · 2015-05-02
Conservation of momentum can be derived as a necessary result in any theory whose basic laws are translation invariant (that is, if the defining equations are unchanged if you move everything in the universe 6.2 cubits to the left). This is true in classical physics, relativistic dynamics, quantum mechanics, quantum field theory, or even string theory). You can prove it in various ways depending on which formalism you're working in, but the most elegant proofs I know are based on Noether's Theorem (http://math.ucr.edu/home/baez/noether.html). It's beautifully general, and it applies equally strongly to classical or quantum systems.
spott · 2015-05-01
So, it is worth bringing up: GR doesn't require global conservation of momentum or energy.[0,1]
I'm not saying anything about the experiment, but I think your statement that "conservation of momentum is more deeply embedded..." isn't quite so accurate.
[0] http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-cons…
[1] http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
orbifold · 2014-11-15
For someone who is more of a mathematician than physicist, the following article might be of interest in this context: http://math.ucr.edu/home/baez/quantum/node4.html
The gist is that the state space of a multicomponent quantum mechanical system (think electrons in a metal) is given by the tensor product of the single component state spaces. As might be familiar from linear algebra, not every element in the tensor product of two vector spaces is simply a tensor product of two vectors. That's really all there is to entanglement, but unfortunately you can't explain that to a lay person.
gtani · 2013-08-21
Baez has published some great booklists and a FAQ for self study
http://math.ucr.edu/home/baez/physics/index.html
http://math.ucr.edu/home/baez/books.html
http://math.ucr.edu/home/baez/physics/Administrivia/booklist.html
carstimon · 2013-06-06
I prefer that point of view if you want to understand Euler's identity, and I find John Baez does it even better:
http://math.ucr.edu/home/baez/trig.html
The Baez article does leave it to the reader to convince himself that exp(i*theta) is good notation for a point on a unit circle.
I have problems with the attitude of the article you linked, though. Especially
"Therefore, I’d like to complain to the thousands of people who find Euler’s identity stunning and beautiful."
followed by a snide list of reasons why someone might find it beautiful. It's very common when doing math that something amazing is obvious an hour later. I believe that we are better served by reminding ourselves that (a) nobody knows everything, and (b) the basics facts are actually very beautiful.
btilly · 2013-05-26
Yes.
See http://en.wikipedia.org/wiki/John_C._Baez for some basic background. See http://math.ucr.edu/home/baez/ for more articles from him if you enjoyed this one.
One of my favorites is http://johncarlosbaez.wordpress.com/2012/12/27/our-galactic-environmen… which describes our galactic neighborhood. Everyone knows that Alpha Centauri is our nearest star. But I bet you don't know a lot more about it. It is a much more complex place than I had realized.
sbi · 2013-04-12
If you believe that Lie theory is "pratical," then yes. See John Baez's excellent survey: http://math.ucr.edu/home/baez/octonions/
mrcactu5 · 2013-01-06
john baez has been blogging for years on math and physics
* http://math.ucr.edu/home/baez/TWF.html
math is separated from the other disciplines in a very artificial way. but I am also skeptical of any one book who makes as bold claims this. Math (even freshman calculus) is very deep and takes years to master
these notes rough around the edges, but great for self-teaching.
Harvard's Math 55 tries to accomplish similar goals. Not as user friendly, but more traditional:
* http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/i…
* http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/i…