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i move away from the mike to breath in
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learning maths update: its stupid and i hate it, but i can differntiate and integrate now;
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hi this is my first post. i signed up on this strange place to post about math. i'm also self-studying math, taking a break from calculus to learn linear algebra. being a software person it seems like the applications of linear algebra are endless, unlike calculus which is fascinating but not obviously applicable to my sorts of work. i made a ray tracer:

the level of detail is nuts but it was surprisingly straightforward to program. just a couple hundred lines of code! a few simple vector operations drive the whole thing. the trickiest part was the dielectric material simulation because glass balls are tricky to visualize. i may keep working on this thing and add other primitive shapes (currently it only renders spheres), or i may move on to other types of 3d renderers. but eventually i want to tackle cutting edge fields (i.e. bullshit fields?) like machine learning. anyway, learning math through application like this is proving to be much more engaging than working through endless exercises like i did with calc.

question for math academics though: in your typical math program, what is linear algebra typically a prerequisite for? do you just go straight into the applied classes, or is there a "linear algebra 2" of sorts?
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thank you! yes currently i'm simulating lambertian surfaces (ideal matte), metallic surfaces (with adjustable diffusion for brushed metals), and dielectric (with adjustable refractive index). i'm blown away by how realistic the simple models are for these simulations.

the poole text looks much more approachable than other sources i've looked over. at first i tried strang's lectures on MIT OCW and his textbook... the guy seems really passionate and his lectures are apparently revered, but the online materials are so disorganized (a common issue with MIT OCW in my experience) that i couldn't get started.

this is the first time i've heard of exact constraint design, no surprise since i've never done much engineering. what's the purpose of the work you're doing? is it part of a bigger project? edit: just read this entire thread -- lots of pretty notes but i still have no idea what you're doing

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ahh interesting. i'm starting to see it now, although i'm getting lost in the theory. perhaps this is obvious, but, for kinematics, videos seem more intuitive than text/images: https://www.youtube.com/watch?v=fRo3FUWs4Sg&t=140 .. if i'm not mistaken, this mechanism is driven by a maxwell coupling built out of some standard fittings and 3d printed material? this demonstrates "picking up the slack" pretty clearly i think. this sort of function is really intuitive and i'm sure i've witnessed it in action plenty of times without knowing there was a developed theory behind it. engineering is cool

the way you're describing it makes it seem like kinematic coupling is a silver bullet solution to a lot of engineering problems. i saw blanding's book on amazon last night and it kind of comes across this way too: "... a unique and powerful set of rules and techniques to facilitate the design of any machine-of every type and size." but this is always what impassioned work sounds like i think.

it's fascinating to hear about hydraulic pump efficiency. i too have been struggling with the political implications of these sorts of technological advances. i can't comment on iron mining vs. copper mining with regards to imperialism (not familiar with the production numbers here). but along similar lines i've been preoccupied for several months with the implications of the next generation of robotics finally being profitable enough to replace, for example, the 75 million third world garment workers.

one of my loftier reasons for studying math is to help develop these new machines in some capacity. i'm not sure what the politics look like around this sort of effort... i'm unsure how exactly this sort of scientific work can be useful for communism. but, somehow, i feel like one of the only tangible responsibilities of the petty bourgeoisie is to advance the sciences, advance productive technology, and thus help lay the material conditions for socialism. (but at the same time, this self-important idea may be the result of several years of meaningless political struggle and is just my justification for the retreat into intellectual life... ahem)

anyway, these sorts of conversations are why i signed up here in the first place. thanks for the warm welcome! here's a picture you might enjoy that i put together for my bookshelf: https://i.imgur.com/eR1bJvx.jpg

e: oh and btw, as far as contributing to openscad's renderer.. the simplicity and realism of the renderer that produced the picture i posted comes at a cost: performance. ray tracing (which can render spectacular shadows and surface details) is often used for "offline" rendering, e.g. in the production of animated films. the tiny picture i posted took a whopping 3 minutes to render on my desktop (it would take the same computer 10,000+ years to render a pixar movie). i assume openscad currently uses a hardware rasterizer, which is much faster and used for realtime graphics, e.g. for video games. still, video games look beautiful, whereas openscad... is the result of overworked open source programmers. but i'm sure there are a few easy rendering tricks that would go a long way in making openscad's output look prettier :)

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yea, i agree with everything here. i'm shouldn't be so delusional to expect that i'll ever have to worry about these questions personally... besides, what you say:

so the material pre-conditions of socialism in textile-making countries probably aren't CNC garment machines, it's probably the destruction of the US navy, or at least that'd be a good start.

is tragically true. the technologies of socialism are clearly overripe, and they're actually rotting in under-utilization and misuse. so i have to apologize because my questions were selfish. but that was a fascinating article and your comments were very helpful so not all is lost. you've also reminded me i should read john smith's imperialism. i never finished it because i got so lost in the methodology but i bet i could tackle it now

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nearlyoctober posted:

read john smith's imperialism

you might consider rhizzone favorite "divided world divided class" by zak cope if you have not already

Y'all are so cool, I'm fascinated by the work and study everyone is doing.

I really want to do interesting technical work but I'm having trouble breaking in to the field. I got a machinists apprenticeship last year but I lived downtown and the factory was way out in suburbia. Since I don't drive, I had a really hard time making it work and I couldn't keep it up.

Now I'm in a technicians program in college. My challenge now is the paranoia and anxiety of being in this massive crowded school. Classes are trivial so far but I keep skipping them to avoid the bus/the halls.

I don't want to stink up the math thread with my gripes, I just needed to share these woes with people who will understand .. thank u
As someone who fucks up various things regularly due to weird anxieties about class attendance, here is what I would need to hear: get over your shit and go. It doesn't matter how awkward or stupid you think you are or what bad encounter you're envisioning, it will only last a few minutes at most and learning something you want to learn is way more important to your entire life than one little bullshit moment where you'll feel weird and bad for a bit. Then you can just get on to the learning part. Please do it.
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toyotathon posted:

!!! if you can learn calc on your own you can learn anything

as a complete idiot who self-taught my senior year of high school, i can't agree

The effect of automation on workers isn't just a crisis that leads to potentially revolutionary conditions. It's probably also one of the bigger reasons people start looking into Marxism well before then based on what they've learned here and there, even if their concern doesn't start at the level of conscious thought about that specific issue.

A lot of people, especially younger people, realize it's a dumb idea to try to suppress a lot of newer stuff that threatens people's livelihoods within the current system. They also realize that making life easier shouldn't carry the cost of making life harder. If you push hard on that contradiction and find yourself unable to accept the usual apologias, it doesn't lead to very many places besides Marxism.

Chasing that down doesn't necessarily make someone into an effective Marxist, and it won't create a revolutionary class consciousness directly, or even contribute indirectly to it if the person's class background prevents that. But Marx & Engels, and revolutionary socialism in general, make a lot more sense on that problem than liberalism/social-democracy does, or Luddism and medieval-fetish play-acting.

So, the problem has a lot of potential to start to change people's minds if they're not inclined toward wonky thinking about policy, since it's less an issue of one approach promising improved results over another, and more one where the usual alternatives can't really address the problem at all. It starts to make reorganizing society along different principles look like the more sensible approach, which isn't nothing.

I don't think anyone should underestimate how much of a big deal the contradiction is for pundits and mainstream economists, either, given the vast amounts of time and money that were spent very recently trying to convince everyone that when established industry jobs fled the West, its workers would all be trained to do computers and conjure value out of thin air. It's also one of the big reasons, maybe even the biggest one, that Donald Trump's umpteenth campaign for president found mass appeal. Right around the time that even the most stubborn politicians and talking heads had to publicly accept all that stuff had fallen apart if they wanted to maintain any credibility at all, Trump started promising everyone that he'd use tariffs and threats to make water flow uphill, and he was a lot better at mustering team spirit around that promise than Pat Buchanan or whatever other cranks had been carrying the torch.

Anyway, looking into the automation problem gives people raised on bourgeois propaganda a chance to think harder about socialism. Probably just as important, it tips people off that it won't do much if the ultimate goal of "socialism" is just higher top-bracket taxes funding social programs in Western countries, if the working class seizing the means of production is still treated as scary and profane. Remaking society to address the issue isn't as easy as reading a book and knowing it makes sense, of course.
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toyotathon posted:

kinda stoked on this (it's at the end)... studying exact constraint... had a 50hr train ride to read marx and think w/o any internet... remember from exact constraint that the universe has 6 degrees of freedom, 3 translation, 3 rotation.

what always weirded me was how in 3D you get 3 rotational DOF, but in 2D, it's not 2, it's only 1. and in 1D it's zero. so: 1D/0R, 2D/1R, 3D/3R. like why the jump up by 2. in 2D you can only spin shapes around your piece of paper. and 1D there's nothing to spin into. but 3D you get 3 axes of rotation.

well 0, 1, 3, those are the first 3 triangular numbers. and you can get triangular numbers by looking at all combinations of linearly independent vectors that span the space. so say the 3D basis is {1,0,0}, {0,1,0}, {0,0,1}. (meaning you can create any point in 3D, by adding those 3 and multiplying them by another #) so vector combinations are: {{1,0,0},{0,1,0}},{{1,0,0},{0,0,1}},{{0,1,0},{0,0,1}} = 3 total. these make planes... in 2D, say your basis is {{1,0},{0,1}}, well that's the only combo = 1 total. and in 1D, there are no combos.

so if you believe me then 4D should be the next triangular number. 0,1,3, then 6. i got internet again and looked it up and IT FUCKING IS. so that means 5D will be 10, in 6D 15 ways to spin something, etc. hella stoked that i'm not nuts thinkin bout 4D or whatever. i think these planes are actually the nullspaces of the standard rotation vectors that everybody uses to represent rotations.

is there a linearly-independent-vector-combo equivalent for 3 vectors, like a degree of freedom which requires 3D space (like how a rotation requires 2D space)? seems like reflections work in 2D too.. like you can make a 2D reflection matrix. hell they work in 1D too by changing the sign. so not reflections...

This is actually a topic that has been studied in great detail. What youre describing is the representation theory of rotation groups. The result that the number of independent rotations is a triangular number can be understood as a consequence of the fact that rotations in dimension n can be represented by antisymmetric nxn matrices. The number of degrees of freedom for such matrices is exactly n(n-1)/2, which is the triangular number result.

Representation theory is really cool to me but the way its most commonly formulated and presented is really dull and tedious which is a real shame (how its formulated and presented varies a little depending on whether youre talking to a mathematician or a physicist but theyre both sort of blah in different ways)
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toyotathon posted:

then i counted how many ways to make planes in each of the known (1D/2D/3D), ie how many linearly-independent 2D subspaces in each space, and saw that it matched my knowledge (0,1,3). i remembered from a prev post how the triangular numbers are related to binomials (binomials cuz i'm counting # of pairs), and made my guess for 4D/5D/6D

I think thats probably a sensible way to understand it. What youve done, i think, is count the number of independent ways to construct an axis of rotation by constructing the number of planes normal to the axis of rotation

I actually mentioned that derivation to a friend of mine and he said that apparently a similar argument is made in the e&m textbook by landau and lifschitz. They're really really stellar textbooks on physics and if youre really interested you might find them interesting. Theyre a little formal but they also give a lot of intuition.
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and all Newton had to do to figure that out, was to never have sex even once. *clearing throat & elbowing everybody with many goro arms*
*nervously* guess that rules me out
no you check out fine.

toyotathon posted:

another thing i like about the book is that, it treats mathematics as the science of distance and quantity, which it is. you can observe the dot product in nature using string. and it recognizes the development of sciences, as coming out of the observations of new proletarian labors, and where our labors blossom, so do the scientific rationalizations of labor:

“...the description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn... To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics and by geometry the use of them, when so solved, is shown; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring...” (from Newton's principia)

Youre correct to look into the use of the wedge product, since its the map that combines vectors in the ways that you are describing. Actually you might be interested in a book called "nonholonomic mechanics and control", which is really all about how to understand various kinds of mechanical constraints in the context of differential geometry, which naturally includes concepts like the wedge product and integrable and nonintegrable constraints.


actually you might be interested in a book called "nonholonomic mechanics and control"


I was re-reading Rudin and I found a great explication of one of his derivations here: http://mathforum.org/kb/message.jspa?messageID=7128174
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toyotathon posted:

nice... i picked up Introductory Real Analysis by kolmogorov but i'm v intimidated. definitely could use a book more like that link that goes slow and demystifies the process. struggling w/ Linear Algebra via Exterior Products now

check out "understanding analysis" by stephen abbott

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toyotathon posted:

good to see the tears school of note-taking in action