#201


this mechanism's formula is simple, it's pitch1 - pitch2. one side is a US 4-40 UNC thread and the other is a metric M4x0.5 fine thread. the pitches are 0.5mm and 25.4/40 = 0.64mm, so each rotation of the bolt, nuts on either end change distance by pitch1-pitch2 = 0.64-0.5 = 0.14mm. if i'd used a 4-48, it'd be about 30 microns per rotation, very very fine movement from hand-scale motion.

Edited by toyotathon ()

#202
what
#203
i was thinking like that one is tiny but it could be used for fine motion on leadscrews or positioning systems. like if i wanted to move the tool exactly 60.4 microns you could use a much larger version of that w/ one nut fixed, and the free nut connected to the tool, with the leadscrew connected to a stepper or servo positioning motor (which would unfortunately also have to move). then a 30-micron pitch difference means to move the tool for example 60.4 microns, there should be 2 full rotations + 0.4/30 rotations (or 2+0.4/30, or 2.0133 new degrees of rotation of the bolt), to move the free nut 60.4 microns. you could do 75-nanometer positioning with 400 steps of the a stepper motor: 30 micron pitch / 400 steps per rotation = 3/40 = 0.075 micron, or 7.5nm with the microstepping on my lathe. very cheap cuz that part costs like 3 cents of aluminum and i dunno how much wall power.

i was thinking another thing which could be done with them is to line up many of them like either (male1+female2)-(male2+female1)-(male1+female2)... or (male+male)-(female+female)-(male-male)-(fe... , with the one pictured being male+male. it would be a screw that telescopes in on itself, like shrinks in length, when you tighten it, and grows in length when you loosen it. it isn't really much different of a mechanism from a threaded rod (a bolt with no top) in a regular, but longer, nut, which moves one pitch per rotation. like just picture a nut on a bolt, a long nut, hanging off one end of a bolt, and then another bolt on the other side of the nut. just that repeating pattern of male-female-male-female bolts and nuts.

edit: ooo the above would actually have much more mechanical advantage than the turnbuckle mechanism, and would not require such a fine thread as the turnbuckle, so it could possibly bear an even higher load. a turnbuckle has a left-hand and a right-hand thread, so it's formula is pitch1+pitch2, and this is a right-hand and right-hand (or left and left), pitch1-pitch2. because the pitch is smaller, a moving load moves a shorter distance per rotation, but while traveling it is subject to a greater force for a given torque, on the dual-pitch bolt vs the turnbuckle. so you could get good mechanical advantage with this mechanism. this is a benefit of subtraction, but it has a tradeoff that the major diameter of the larger pitch side, must be smaller than the minor diameter of the finer pitch side.

Edited by toyotathon ()

#204
i move away from the mike to breath in
#205
devoting all residual crank energy towards a new system of degrees, where one degree is equal to one rotation around the circle

edit- hell yea i'm not a crank:
"One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer should be 150° + 270° = 420°, but when thinking in terms of the circle group, we need to "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360° which gives 420° = 60° (mod 360°).

Another description is in terms of ordinary addition, where only numbers between 0 and 1 are allowed (with 1 corresponding to a full rotation). To achieve this, we might need to throw away digits occurring before the decimal point. For example, when we work out 0.784 + 0.925 + 0.446, the answer should be 2.155, but we throw away the leading 2, so the answer (in the circle group) is just 0.155."

it's already used here https://en.wikipedia.org/wiki/Circle_group 1 big degree is just a better system!!!!!

Edited by toyotathon ()

#206
learning maths update: its stupid and i hate it, but i can differntiate and integrate now;
#207
!!! if you can learn calc on your own you can learn anything


edit: one thing you now have the power to do is to go out and apply calculus recklessly + check if it's true. you might misapply it (in which case you learn more calc) or the two things you're comparing might not be differentiable to each other (you learn about the world), but the first time i did where neither happened, me + 2 buddies hiked to the top of a falls and we were guessing at the height of it, so we threw a rock horizontally and timed the descent from release to splash. we were comparing distance to time. iirc 45'/15m tall! the next time i went alone to a lower falls and saw a newt the size of my forearm in the back of a shallow cave... 2 regular-sized newts were ambling towards her from the river... probably unrelated to calculus

Edited by toyotathon ()

#208
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?
#209
that is a beautiful work! you're doing a material simulation, like with a non-flat, rough surface? that's pretty awesome! the center one looks like a matte neptune.. i'm working on CAD stuff in OpenSCAD to understand & hopefully simulate exact constraint and our labor is probably overlapping if you want to PM. between Algebra Chapter 0 w/ a friend and now Linear Algebra- Modern Intro by Poole in class, Poole was much more straight-forward.. welcome to rhizzone

Edited by toyotathon ()

#210
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

Edited by nearlyoctober ()

#211
man i use OpenSCAD for modeling, and it makes the saddest renders... you'd make a lot of people happy if you either added, or let the devs know how easy it is to add surface lighting effects. that's really cool. you probably got spillover knowledge too into like light-matter interaction and surface roughness.

yeah poole is great, it's like larson's calculus, the goal is to teach, not make ya re-read proofs 10 times for a flash of insight. and racetrack, the vector addition game it starts w/, is actually really fun to bust out with friends (rule is, add the prev vector, then accel/decel/turn within 1 square, no going off the track, no collisions)... i think there are a few ppl here learning linear algebra right now if anybody wants to play it in-this-thread:


exact constraint is a theory that, like, solid objects in the universe have 6 degrees of freedom, 3 translational (xyz) and 3 rotational (pitch yaw roll). to subtract degrees of freedom from objects, they have to contact other objects... 6 minus N, N for the # of contact points. so in your pic the spheres each touch the plane in 1 place, so they have 5 DOF: 3 rotational, 2 translational. if a sphere were glued down, and another rolled into the glued sphere, as long as they're touching, the free sphere would have 1 translational DOF: the 1D circle around the glued sphere.

what i realized is that, if you take the "superposition", every possible position, of things, you can derive this same relationship, 6-N=DOF. so in the above, pick a point inside one of the spheres (pick the center... easier on the brain cuz the symmetry hides the 3 rot. DOFs), and imagine every possible position on the plane the sphere could be...... it makes a second plane, 1 radius above. it has 5 degrees of freedom-- a 2D plane plus 3 rotation = 5. if you take the 1-radius normal off that glued-down sphere (= a spherical shell, r1+r2) and find the intersection between the plane and this sphere shell, it's obvious the intersection between a sphere, and a plane that cuts thru the sphere, is a circle... and this corresponds to the 1D kinematic path (a circle can be described 1-dimensionally, like (cosine(t), sine(t)) ). introduce a 3rd contact, the intersection between the 1D circle and a new 2D shell, a line going thru a plane, this is a 0D point -- shoot a BB thru a sheet of paper, it leaves a tiny hole -- so the point inside the ball is now fixed in xyz. basically everywhere the superposition isn't differentiable, you find a kinematic path, so long as the vector of force is within a range made by the tangents of the superposition. wow i'm sounding mathy, this class must be working, or i'm full of shit lol

i read blanding's book and emailed profs who cited it, about the superposition-intersection way to see the theory, they said it was novel (altho blanding def suggests it in some diagrams) and confirmed my observations... but they're busy and i'm a rando guy so it's my burden/joy to figure it cuz who else. trying not to crowd out marxist study tho. math is huge so i'm hoping that i picked the correct subjects to study this with: linear algebra and differential geometry... actually any insight? 9 months ago i didn't know what math subjects there were... i started in group theory/topology and i didn't understand it or find much... but linear algebra so far shows why intersections drop dimensions. anyway it's a cool theory, b/c even w/o the benefit of predicting full kinematic paths, used in labs today, it's nanometer-accurate. it is just *true*! a classic kinematic mechanism is maxwell's coupling, check the # of points of contact:

it also makes it so i can be really sloppy w/ worksmanship and exact constraint will 'pick up my slack'... harder to describe this one tho. in my experience it reduces the # of really tight tolerances in parts. so it's a science that amateurs can use to make good parts on inexpensive/DIY machines.
or look at this probe that renishaw wants to charge like $12k for: https://slideplayer.com/slide/4888346/ that can be made on the lathe for $10!

it's part of a bigger project... i'm working on a pump design.. the current way the gerotor pump works is, the moving parts are not well-constrained, they should be 1-DOF but actually have 4-DOF which engineers try to precision-machine away ($$$). so stuff rattles around in its housing ---> chaotic pressure fluctuation. hoping exact constraint will give nice & smooth pressure output. in my fantasy we get hydraulic power up to the efficiency of electric power transmission (nearly all energy losses in hydraulic systems are in the pumps, viscous losses are minimal). steel and water for short-scale power transfer+storage is better than imperialism robbing andean copper imo, and is easier labor for DIY. for micro-chemistry, it's also good to be able to meter exact volumes of fluid continuously, rather than with syringes that need to be re-filled.

sorry for the long, technical post! welcome to rhizzone www.readsettlers.org www.readmarxeveryday.org posters here + marxist books can really help teach about science... sorry if you already know all this but w/ dialectics, the central insight imho, that the waves shape the shore and the shore shapes the waves, things have properties only in relation to others, everything changes in time (altho superpositions are "timeless"), great insights for any study, exact constraint in particular! amazing to see this marxist philosophy applied to shaft rotation, then class struggle history, without missing a beat. and if we fail to make land struggle, to seize & free the land from the settler landowners there won't be much math or people or animal life left, cuz topsoil's depleted in ~50 harvests. that's our burden/joy to solve too

Edited by toyotathon ()

#212
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 :)

Edited by nearlyoctober ()

#213

nearlyoctober posted:

i too have been struggling with the political implications of these sorts of technological advances... 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.



check out how garment-making CNC machines are used in practice, in india where high-tech meets low-tech:

The textile and garment industry is composed of the knitwear, fashion and textile (fabric-making) industries. In the knitwear enterprises, there is functional upgrading towards brand manufacturing. Additionally, there is also process upgrading through automation (i.e., using computer (or computerised) numerical control (CNC) machines). The former has enhanced demand for a small number of highly skilled workers in design and administration. The latter has reduced the number of manufacturing workers and deskilled them. A most fascinating finding with regard to the CNC machines is that even illiterate workers, guided by visual elements, can oversee their faultless operation; the manufacturers of these machines deliberately make them with a view to deskilling the machine operators.



fascinating the class parasite that owns the CNC machines saves rupees by keeping the production line illiterate. mass literacy is one of the pre-conditions of any technical society, socialist or capitalist, but at this front in the class war, owners are overturning the old wisdom. maybe we don't have to pay taxes for teachers after all, they think. and our workers can't read marx if they're illiterate! anyway it's worth looking into how CNC garment manufacturing is done in practice today and why it's done that way, this is one sample.

also i'd say, you should not automate anybody's job w/o their permission, and if you do, do NOT tell the boss, keep it a secret within the class. in my first job out of school i sized valves, and every valve i had to size, i had to produce a valve fact sheet, so that 20 yrs later when an operator would pull it during PM, they'd know what they were holding, & maybe order a like-for-like replacement. i'd size the valves, but we'd contract out the fact sheet to a guy who wrote valve fact sheets. unfortunately i wrote a spreadsheet so we could fill in these fact sheets ourselves. here's my fuckup: to be a good new-hire boy, i told my boss about my work. one of my coworkers took me aside and did NOT mince words about my error: the only value that the boss sees in a spreadsheet like that is to reduce headcount. which is what he did-- he cut the guy's contract. had i kept it a secret within the class, 1) it'd have been actually less work for us, cuz the spreadsheet took more than zero time for us to run, and 2) it've been less work for the guy, could've given it to him. learn from my error, don't think by helping make automated garment machines, that you're doing garment workers a favor -- once the parasite hears about it, it will be used to enrich him. they know this game better than you, cuz it's their whole livelihood, leeching labor.

there are 75 million garment workers b/c imperialism shaped that labor market to slave-like conditions, to its national preferences. like, taxes on a shirt sold in a US mall, are greater than the sum of all wages to the workers who stitched it, and the cotton-pickers who grew & processed the raw material. so the taxes that fund the US navy, that ensure that container ships leaving port in india arrive in LA, are greater than the sum of wages to all workers in primary production. this is how imperialism funds & reproduces itself from year to year. 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.

if you're petty-bourgeoisie learning about garment work second-hand, and have never been a garment worker yourself, how will you learn what these machines need to do? i couldn't understand kinematic theory until i took machining classes after work and learned the lathe w/ my hands.. i read blanding as penance cuz i took apart a lathe, then 'overconstrained' (>6 contacts) its z-axis when i put it back together... so it was knowledge flowing directly from a labor i did. maybe you are labor genius but you will probably not be able to understand garment manufacturing in detail, w/o first filling a closet with clothing you've made. but if it's something you want to do, might i suggest you proletarianize as a garment-worker? if you need to stay in your class you could find instructions build a loom and take apart a Singer, otherwise you could spend years fumbling like i did making crappy shit!!!

Edited by toyotathon ()

#214
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

Edited by nearlyoctober ()

#215

nearlyoctober posted:

read john smith's imperialism


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

#216
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
#217
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.
#218
it's good to have a good reason to go to class. like you're learning it not for an abstract obligation to be in school, but because you need to solve a specific problem or job. it's also good if that specific problem/job intersects w/ your ideology some way. like i took machining classes cuz i needed to make a part on the lathe then i left after i was done figuring out the lathe* & how to make my part. there were some gunsmith guys in there basically doing that too like rifling barrels or whatever

*i was not

Edited by toyotathon ()

#219
one interesting thing about the pythagorean theorem is that it works to describe distance in any dimension(s). so hypotenuse = sqrt(x^2 + y^2) or a^2 + b^2 = c^2, and this is a visual proof https://upload.wikimedia.org/wikipedia/commons/9/9e/Pythagoras-proof-anim.svg. so it describes the distance between two points, a and b. in 1D it also describes distance on the number line, by "erasing" the negative sign. so a = sqrt(a^2) = sqrt((-a)^2). it also works in 3D, like to take the string-distance between two opposite corners of a room, by measuring each length of wall and height to ceiling. this is string-distance = sqrt(L^2 + W^2 + H^2), or the 3D normal distance and it works for the identity elements of addition and multiplication, 0 = sqrt(0^2), and 1 = sqrt(1^2) for whatever that's worth.
#220

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

#221
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.
#222
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...

Edited by toyotathon ()

#223
combining 2 unique vectors, that's unity of opposites, rotation is the result of their contradiction
#224

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.

#225
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)
#226

c_man posted:

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.



okay thank you... i just learned symmetric/antisymmetric matricies in the linear algebra class so i'll reread that section and work some numbers.

was my reasoning physical? the reasoning began, well actually it began drawing a triangle on 2 rollers, and i noticed that when it's normal force vectors were parallel, its DOF was translational, and when not, rotational. then i was like well what's a rotation, it's modulo motion... like 359.99deg-->0deg. then that, modulo isn't possible on a physical 1D line (in R1 euclidean space... like i get that a circle is 1D). it's only possible to go in one direction, and return to where one starts, in 2+ dimensions (again in flat space). and when the two vectors AREN'T parallel (= are linearly independent), they can make a plane. 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. does that go against representation theory?

another interesting thing i learned is that, the dimension of a space A = null(A) + rank(A). and that's probably why exact constraint works, 6 = #DOF + #constraints, say, if A is a collection of normal force vectors, then that's the # of constraints. the nullspace should be the space of motion. i think i'm close, but i don't see how it relates to my previous insight yet, about the space of the submanifold, the intersections.

Edited by toyotathon ()

#227

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

#228
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.