I studied several of Hestenes' papers around ten years ago. Most of them are computational in nature and do not proceed to do geometry in an axiomatic way. This isn't to say he didn't write such a paper but I never found one and I read nearly all his extent papers. The thing I liked most was his derivation of the Kerr metric for a rotating black hole. I couldn't find anything wrong with it and I could actual understand every step, unlike Kerr's original paper. Other than that GA has done absolutely nothing to help gain a better understanding of QFT, GR, string theory, Lie groups etc. (for that an in depth understanding of differential forms is best. Physicists, even mathematicians who truly understand the magic of differential forms are even more rare than programmers with a deep understanding javascript!!!) IMO Geometric algebra is a useful collection of interesting computation tools. It reminds me of Pedrag Cvitanovic's "bird track" diagrams to calculate representations of Lie algebras. Nothing unique or fundamental is gained but it is remarkable that a completely orthogonal viewpoint to some very traditional topics exist. Other examples (from mathematics) of surprisingly novel viewpoints of traditional topics include Kuratowski who has a very unique approach to general topology and F. Riesz came up with the notion of "nearness" which simplifies difficult theorems in functional analysis but it could be used to recreate all of basic mathematical analysis.
It seems like the point isn't to gain anything unique or fundamental in physics or mathematics directly, but rather that, since it has the potential to unify the language used across a number of fields in mathematics and physics, that the adoption of a common language could eventually lead to significant progress. So (if it is in fact an effective unifier), it is something unique and fundamental in the realm of tools—though not within the fields the tools would be applied to. Or do you not think it would be an effective unifier (in the sense of communication) after all?
From experience I would have to say 'differential forms' seem to fit the bill for a unifying mathematical language as applied to geometry and physics. GA seems to me as more of a computation tool. Differential forms are pretty standard in many maths and physics texts. Many paper on the preprint Arxiv use differential forms. The only thing about DFs is that they are very efficient for communicating ideas and doing proofs. For practical calculations they don't really simplify anything (actually get in the way) but it's easy to transform them into the usual vector tensor notation. GA seems less flexible in this regard, you take the product it uses and you either like or lump it.
There's a bit of discussion of how differential forms fit into the geometric calculus framework in [1], which is probably the most concise and readable introduction to the geometric calculus approach to differential geometry. All the machinery of forms is available as part of geometric calculus.
Geometric algebra/calculus has a more direct way to deal with metrical information using the dot product. Forms only use the wedge product: in problems where the dot product would be useful, forms simulate it by applying the hodge dual twice, which is a less intuitive and less direct way to get the job done.
I cannot thank you enough for introducing me to Riesz. While finishing up my math degree, I came up with the idea that “it's all about distance/proximity/nearness”, and it looks like the same thing Riesz was thinking according to this: http://www.emis.de/journals/SEMR/v6/a1-10.pdf
Here are some old comments where I used the concept:
One interesting thing that comes to mind is spinor calculus.
Spinor calculus is a bit like geometric algebra; there is a mention to them in the paper (more precisely, to "twistors"). The central idea of spinor calculus relates like this: take your 4-vector (v^0, v^1, v^2, v^3) and form a 2x2 Hermitian matrix by:
V = v^0 I + v^1 s_1 + v^2 s_2 + v^3 s_3
where s_1, x_2, and s_3 are the Pauli matrices. Then it turns out that det V is the 4-norm of v^\mu:
det V = v^0 v^0 − v^1 v^1 − v^2 v^2 − v^3 v^3.
The Lorentz transforms must preserve the 4-norm and hence det V, but they must also be linear and map Hermitian matrices to Hermitian matrices, so that given a lorentz transform t, there is a Lorentz matrix L such that:
matrix (t v) = L (matrix v) L†
(that's not 100% accurate because it can't do PT flips; I think P is something like V → V^-1 while a 4-flip is V → -V; combine them together to get T). The Lorentz transforms are just the group det L = 1 -- the Möbius transformations SL(2, C).
The elegance of this comes when you look at null vectors, where det V = 0, making V a projection -- so V = u ⊗ u† for some u. The action of a Lorentz transform on u is then just u → L u, where L is the Lorentz matrix. Moreover when you work out what the ratio of the components of u are, tracing back through the mathematics, you get varios stereographic projections (x + i y) / (R - z), depending whether it's future-pointing or past-pointing.
So all the light that is coming in towards you is a bunch of null vectors that you can paint on a celestial sphere, projected to the complex plane by a stereographic projection, with Lorentz boosts as Möbius transformations of those points.
Immediate freebies: when a marble is speeding past you it still "looks like" a marble to you; it just seems "rotated" in a strange way, because Möbius transformations map circles to circles. Yes if you try to "work backwards" in your coordinates you'll construct a warped model of the system which is Lorentz-contracted, but that's not what you'll see.
Another freebie: as you accelerate faster and faster, the stars all "tilt" in the direction that you're going, crowding around the point you're travelling to. This is in sharp contrast to all those spacey TV shows where the stars "streak away." One can imagine that for a photon's timeless life, the event of its origin is the only thing behind it; and the entire rest of the universe is in front of it.
It actually gets even better; it turns out that you get to unify the spinor equations for the massless neutrino ∇_{AA'} u^A = 0; the photon ∇_{AA'} u^{AB} = 0, and the weak-field limit for gravity is something like ∇_{AA'} u^{ABCD} = 0 for the graviton. (That may not be 100% correct; I am working from memory here.)
All of that comes from something which is basically a quaternion/geometric algebra application to spacetime.
Unless the plane crash was due to a software bug, this story, however important it is, has no place here. I appreciate your interest and concern but this isn't the right forum.
This is a systems bug based on an underestimation of a technical bug, i.e. that flying 1,000 feet over a 32,000 feet restricted altitude was considered "safe" given the hardware used. Such an assumption ended up being wrong. OTOH, the OP makes the point that diverting flights isn't as simply as moving the joystick in a different direction, given the nature of crowded airspace and routes. It's also interesting to note, as the OP does, that flights at lower altitudes were routinely conducted over Afghanistan and Iraq.
I actually think the OP is overstating his case though...but I posted it here because I know there are flight enthusiasts who care about the details, such as whether 1,000 ft. above a restricted altitude is not much more reckless than speeding 63 mph over 65.
HN will be a sad place if it concerns itself only with bugs in pure logic, and not all the bugs and misunderstandings of details and mechanics that exist outside of software code.
The point is this has nothing to do with software or anything of interest to Hacker news. Why bother reading HN if it's going to be cluttered with posts and comments that clearly belong elsewhere (Huffingtonpost.com, Disqus, Twitter whatever...) then some sour grapes crybaby who seeks attention voted down your Karma to prove a point. I'll still read HN but some participants are just trollish. I'll never hire or work with anyone with the usernames above because in my world civility and good behavior counts, even with the "little things" like respecting HN guidelines (see below) but it's your life and do what you want with it. Sooner or later you'll put your foot in your mouth in the very worst way and panic all day over your mistake.
On-Topic: Anything that good hackers would find interesting. That includes more than hacking and startups. If you had to reduce it to a sentence, the answer might be: anything that gratifies one's intellectual curiosity.
Off-Topic: Most stories about politics, or crime, or sports, unless they're evidence of some interesting new phenomenon. Videos of pratfalls or disasters, or cute animal pictures. If they'd cover it on TV news, it's probably off-topic.
Teenagers act crazy bc biologically they are adults. If we were doing our job raising kids and not depriving them of real life experiences and honest learning opportunities, they would be better prepared to channel that amazing youthful energy, the risk taking, the ability to stay awake longer, learn faster, compete. They would be starting businesses of their own perhaps even competing against well established players and occasionally beating them at their own game. Most would simply become (find out) who they are and find their place in the world. They would be thinking independently, writing their own life script. Parents don't like that bc it doesn't validate the choices they made. Their child might fail in their endeavors thereby making the parents look bad, God forbid! Universities don't like it bc success is supposed to depend entirely on whether or not they get paid lots of money. Governments don't like it bc it would force them to change and adapt. Corporations hate it bc those kids are supposed to work for THEM for cheap and buy their CRAP. Did I mention clergy? They don't like it bc some of those kids might become something they aren't and usher in a new form of spirituality they don't 'agree with'. Kids (especially teenagers) are DANGEROUS. It's far better to undereducate them, setup barriers to keep them out of the real world, and deprive them of all the respect and responsibility accorded to 'read adults'. Then when they become neurotic and act out, claim that's the real reason kids need to be watched so closely and excluded from participating in the real world.
This was an amazing and somewhat sad story. How's Gembe doing today? Based on his near adoration of the Valve developers, it would be fitting if he worked in the gaming industry. Maybe he could set things right by helping to make Half-Life 3?
I worked programming fire alarm systems for a while, embedded device programming and test automation. Now I started doing realtime graphics in Bejing. Working on using the Kinect to do gesture input on presentations on really big screens. Like Powerpoint on steroids.
The article didn't have any information on the person you passed the source code to. I assume they tried to figure out who it was? What happened to them?
I think I already confirmed it on reddit after someone else said it. I shared it with SourceX from myg0t and he shared it with other people inside myg0t. It was really stupid to do ... yay hindsight :(
"This was actually one of the interview questions, don't know why they didn't use the answer. I work as a software developer and a bit of a system administrator. I work in a company that does physical security, like fire alarms and such. Most of the work I do is programming PC control software for our systems and also quite some firmware development for various uCs. I know quite a bit of different assemblers. Measurement and automation is another field that I'm currently learning more and more."
Sugar is the stuff life runs on but at our modern level of consumption it's undoubtedly the number one food toxin. Sugar is the secret to addicting the consumer and peddling high profit mush. It doesn't really matter much what form it's in sugar, evaporated cane juice (euphemistic term for what sugar actually IS), corn syrup a.ka. HFC (vilified but not substantively different than table sugar), molasses, wheat, corn starch, rice flour, etc. Modern wheat actually has a higher glycemic index than table sugar! Most of our food isn't toxic, just barely acceptable and sometimes poisonous e.g. food born illnesses. Our food is great at killing us slowly, lowering the quality of life and overall health and most of all attracting huge government $ubsidies! It's all about the money after all. Anyhow, I blathered on about this because it's probably the single most effective thing anyone could do for their health, keep their blood sugar stable and avoid cheap high glycemic addictive foods. BPA in tomato cans and other 'problems' deserve very low priority in relation to the sugar epidemic.
Pascual Jordan, one of the founding fathers of quantum mechanics along with Dirac, Heisenberg, Pauli and Born had his name essentially written out of the history books because of his association with the Nazi party which really amounted to overlooking what the Nazis were doing and saying some positive comments about the Nazis. I don't like what Jordan did. Heisenberg is a less controversial figure which some people claimed had skeletons in his closet regarding the Nazi party. Nothing really came of it and Heisenberg's reputation has been untouched except a few still hold questions regarding his conduct. I'd like to know if this issue was ever cleared up. Heisenberg is still (rightly) remembered as a chief architect but Pascual Jordan whose contributions equal or exceed Pauli is almost entirely written out.
I was really surprised one day to find all my youtube comments on Google+ for all my friends to see. Try explaining away comments about a fart video, drunken bums, cock fighting or some other unfortunate mistake. Oh well (shrugs shoulders)
That's a shame really. Recording history shouldn't be motivated by politics and other peoples' sensibilities being offended. Regarding Aleister Crowley being 'The Wickedest Man on Earth', not even close. If you've ever read M. Scott Peck's 'People of the Lie', then you know what I mean.
That was a title Crowley took for himself, as he enjoyed being a notorious figure. His actual behavior, though, amounted to little more than being a sex maniac and saying blasphemous things during a much more socially conservative era. His 'autohagiography' makes for entertaining reading if you can get past the turgid Victorian prose.
As for Parsons, the Wired article is being a bit clickbaity. In the last paragraph: Wired.co.uk contacted JPL and we asked whether Parsons had been written out of the history books. Historian Erik Conway said: "Jack Parsons is included in history books and other venues, and in fact, his role is discussed in the JPL-involved standard history, JPL and the American Space Program by Clayton R. Koppes. Parsons was one of the original founders of JPL. He was the team's chemist and developed the first castable solid propellant used to power aircraft."
I agree, and I'd even upgrade your "a bit clickbaity" to "very clickbaity". I happen to work at JPL, and elements of the Parsons story are generally known to people at the lab. I just don't think they're that interesting. Certainly, complaining that the oddball philosophy of one of the four founders is not featured in the lab tour is ridiculous. People come on the lab tour to see robots and rockets, not to talk about occult side-interests.
If you want interesting side stories, how about Qian Xuesen, who was also one of the original handful of JPL founders, and who was hounded from his position and went on to found the Chinese rocket program. (Discussed on HN previously, https://news.ycombinator.com/item?id=6905862). That's a historical figure who has actual significance, not just odd/occult tendencies.
The other interesting back story, besides Wehrner von Braun, is that of the principal founder of JPL, Theodore von Karman, who was a Hungarian Jew who left Europe in 1930. Von Karman was the first Director of JPL. Another founder with a back-story of considerable significance.
As context for future readers, the submitted HN title was the same as the actual title of the article: "Occultist Father of Rocketry 'Written Out' of NASA'S History". Definitely link-bait, although unedited primary-source link-bait.
(dang: Leaving notes when changing titles is great, thanks!)
