reading i like though, some borges shortstories are beautiful narrative explorations of mathematical concepts which makes glimpsing at that mystery more accessible i think?

anyway, in case youve not read them.. smoke a spliff, have a cup of tea, and sit down with one. they are all short. suggested...

- garden of forking paths: quantum mechanics / infinity

- library of babel: infinity / topology

- lottery of babylon: probability / chaos

- the aleph: infinite sets

- blue tigers: cardinality

- book of sand: dense set

dimashq posted:u put the subtracted value into a small box and keep it secret, keep it safe

...thats division not subtraction

tears posted:trying to learn maths but having some problems when u do a "subtraction" where does the bit u take away go to? example: 13 - 3, where does the 3 go to?

Google "number line"

toyotathon posted:kitchen math

experimenting adding gluten to my flour for bread. turned 10% gluten AP flour into 15%. so far, the bread texture is a lot better, going to keep upping the gluten 1% each batch and see what happens.

i hope this 'gluten-free' fad keeps going cuz, right now there's a gluten glut. my pet theory for the mass gluten allergy is, since white people have been eating bread for millenia, to just now diagnose a mass allergy, that's probably not real. BUT-- maybe the chemical processing step that removes gluten (dissolving the starch in water, removing sediment incl gluten, then drying the starch) might be removing the actual causative agent, like a new pesticide or something. someone was tellin me that their friend with celiacs was fine eating european-wheat breads, maybe cuz of different pesticide regimes in europe and US.

the formula to figure out how much of something to add is:

x = b*(c - a) / (1 - c)

where,

x = the amount to add (weight)

a = the original %/100 (from 0 to 1)

b = total original weight

c = target %/100 (from 0 to 1)

if you want to stick with percents you can turn that '1' into '100'.

so if you have 51oz of 10% gluten AP flour, and want to turn it into 15% gluten bread flour,

x = 51oz * (0.15 - 0.10) / (1 - 0.15) =3ozif you don't believe the formula: 10% of 51oz is 5.1oz gluten, 5.1+3=8.1, 8.1/(51+3)=15%

started with semigroups/monoids/groups in this number theory book that's supposed to get to the integers eventually. i am very interested in group theory if anybody wants to PM chat cuz of the euclidean group and its relation to kinematics/exact constraint, and why point-symmetry in solid bodies makes 3-DOF, line-symmetry makes 5-DOF, and plane-symmetries and higher give 6-DOF (?). or honestly geometry. vvvvvv shit TG that gives me an idea.

Classic suggestion for group theory, that isn't too hard to find a pdf of online if you want, is Algebra Chapter 0 by Aluffi. The intent is a basic algebra book that builds naturally into category theory needed for algebraic geometry. The way your book seems to go about it, by starting at monoids and working backwards, makes me think that either there is a lot of hand waving or it is perhaps too advanced to be a good introduction to groups. Sorry if I misunderstood and an intro level book is not what you're looking for.

the quaternions are for cad but another cool thing in this cad. i'm writing polyhedra code to render pipes, and re-writing it to render an outer pipe diameter, from s=0 to 1, then it loops back on the inside, s=1 to 0, to render the inner pipe diameter. to stitch it up, the last points all connect to the first points. i was going to write a second case, for when a solid filled pipe (no inner diameter), links back on itself, like an ouroboros, or: a torus. but the code's identical, don't gotta make a special case for it, because they're both toruses, they both stitch the first points to the last.

i also realized the code's going to be the same for any "spheres" where the first set of points in a circle, all make triangles connected to the same pole, on the first and last sets -- two poles in a sphere. wish i understood more about spheres and toruses and etc cuz there are maybe more general shapes the code could do w/o much addl effort.

edit: reading about quaternions and the multiplication rules and yo, this defines the cross product! that is cool. cross product from out of algebra. friendship ended with Josiah Willard Gibbs, now William Rowan Hamilton is my best friend.

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the way this regular tetrahedra is balanced on an edge, if you take a cross section level to the sea, it forms rectangles:

it's a cube where every rectangle is braced with a crossbar, in a certain way. and every bar is held in at least 6 places (but many are over-constrained)

edit i knew i fucked up the stairs... it's like a double mobius and in the middle you're on em upside down oh well

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Edited by toyotathon ()

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probably the longest function description i've written in my life:

Description:

Determines how to mesh two sets of points. The mesh is

a set of triangles between the points. This finds all

possible triangle combinations, and finds the set with

the smallest sum of triangle perimeters, which is closest

to the analytical model.

This is an interesting math problem -- given two number lines,

with n and m points, how many ways are there to connect the

points to each other, such that every point is connected at

least once, and no lines cross? It makes a ribbon of triangles.

It helps to draw two lines, dot them irregularly with points,

and try out some solutions. Remember, no lines cross, and first

connects to first, and last to last. Number the points 0, 1, 2...

Like this: (ASCII's no good for drawing the triangle lines tho)

0----1--2--------3----4-------5---6-----------7 n=7

lines = n+m+1

0--1------2---3-----4-----5--6-----7---8------9 m=9

One trick is to note that, one side of each triangle is on the

number line, and the other two sides are between. So that, when

the points are numbered 0 to n or m, as you draw triangles from

one side to the other, each triangle drawn advances one of the

number lines by one.

If you then draw an ordered list of how the points connect, like

[n,m] = [0,0], [0,1], [1,1], [1,2], etc, you'll notice that the sum

of both point ID's increases by one (since one side of each triangle

lies on one or the other number line). The sum goes 0, 1, 2, 3...

It is the set of natural numbers (with zero), up to n + m,

the number of triangles between the number lines.

There are many ways to draw the triangles, so we're trying to figure

out which, and how many. You can draw the first line, between 0,0

exactly one way, [0,0]. The second line (first triangle) can be

drawn [0,1] or [1,0]. The third [0,2], [2,0], [1,1]. See the pattern?

Given a natural number c, and two natural numbers a and b (<= n), there

are c+1 ways to sum to c.

Does that mean we've figured out how many ways there are to draw

these triangles? That'd be nice if this were the formula: adding up

all combinations to c, you get 1 + 2 + 3 + ... + c+1, which is always

a triangular number. Then the product of the triangular numbers, up

to c, would be the number of ways to draw the triangles. But because

we have the condition that no lines can cross, and because a and b

max out before c, the actual solution set is smaller.

Let's visualize all combinations in a triangular pyramid:

[0,0]

[0,1] [1,0]

[0,2] [1,1] [2,0]

[0,3] [1,2] [2,1] [3,0]

[0,4] [1,3] [2,2] [3,1] [4,0]

[0,5] [1,4] [2,3] [3,2] [4,1] [5,0] etc

And let's restate the condition that no lines can cross: no number

in a set can go backwards, only forwards by 1, to draw the next line.

So you can't go [0,1], and draw the next line at [2,0]. That means

the path, from top of this pyramid to the bottom, must be either down

and right, or down and left, zigzagging or hugging an edge.

When you add up how many ways you can reach each number in the pyramid,

it becomes the Staircase of Mount Maru (or, Pascal's triangle). That's

useful for knowing the rough probability of reaching each number.

To apply the other condition (n and m maximums), we can truncate the

pyramid into a diamond shape. So for [n,m], if n_max=4, m_max=3,

[0,0]

[0,1] [1,0]

[0,2] [1,1] [2,0]

[0,3] [1,2] [2,1] [3,0]

[1,3] [2,2] [3,1] [4,0]

[2,3] [3,2] [4,1]

[3,3] [4,2]

[4,3]

Now we know the full solution space, and all paths thru it, and the

rough probability of reaching each one. And using the Staircase of

Mount Maru, we can calculate the number of ways to draw the triangles:

1

1 1

1 2 1

1 3 3 1

4 6 4 1

10 10 5

20 15

35 <---ways to draw the triangles

i dunno had fun going thru it, thought it was fun how much triangle shit came out of a triangle-drawing problem. triangular numbers, pascal's triangle. altho like so much else in white culture, pascal didn't discover it first, indian and chinese mathematicians did, by hundreds of years. some white just claimed the territory. one of its first names is The Staircase of Mount Maru.

edit: really glad i saw this was a binomial problem b/c the internet's filled with pseudocode that goes, 'for a traveling salesman starting in the top left corner of a grid, who can only move right and down, here are all his possible paths to get him to the bottom-right corner'. like this https://xlinux.nist.gov/dads/HTML/allPairsShortestPath.html flip the diamond on its side and this problem becomes that problem. math: it's great

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tears posted:can sum physicist explain to me why interger spin particles dont follow the pauli exclusion principle. basically why can multiple bosuns occupy the same quantum state while multiple femions cant. do i need to know Bose–Einstein and Fermi–Dirac statistics?

Bose-einstein and fermi-dirac statistics are consequences of what you're describing. The reason half-integer spin particles cant occupy the same states has to do with symmetry properties of the wave function. Essentially, half-spin particles require (for reasons that are somewhat formal) that the wavefunction is antisymmetric under exchange of particles. That means that when you swap an electron for a "different" one the wavefunction picks up a minus sign. A consequence of this is if the two swapped particles were in the same state the wavefunction there would be equal to minus itself, meaning it must be zero, so the probability of that happening is exactly zero.

with femions where its +1 + -1, which is impossible for particles, though possible for particle anti-particle interactions? idk, i feel like im missing something

i dont understand this bit though:

c_man posted:symmetry properties of the wave function. Essentially, half-spin particles require (for reasons that are somewhat formal) that the wavefunction is antisymmetric under exchange of particles.

why do half spin particles require that the wavefunction is asymetric under exchange of particles? is it a maths thing?

tryna get my head round the higgs mechanism next, wtf is mass?

i'm sure that math teaching is part of the bourgeoisie's big school con. nobody leaves school feeling the way they do about math, discouraged, from their gentleperson's C in art class. nobody sees that C and gets scared to ever pick a marker up again. after a con's done and the money's gone, and the victim starts getting wise, there's always a cool-down where the con-artist makes em feel like it was a game that they lost fair and square. it helps prevent blowback or revenge. was math class a fair game? there are only so many well-paid technical jobs. math class is more for sorting than learning. not a lot of people are turned off by reading forever from english class. maybe its closest analog is phys ed, another naked sorting system. i had a facebook data scientist one time tell me math was like harry potter magic. really embarrassing thing to say. but that's part of the harry potter ideology, that there's a special privileged race of man that's destined for advantage. math's not like that but the ideology around it is, so don't let all that turn you off.

The Higgs mechanism is therefore also called the Brout-Englert-Higgs mechanism, or Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism,[8] Anderson-Higgs mechanism,[9] Anderson-Higgs-Kibble mechanism,[10] Higgs-Kibble mechanism by Abdus Salam[11] and ABEGHHK'tH mechanism [for Anderson, Brout, Englert, Guralnik, Hagen, Higgs, Kibble, and 't Hooft] by Peter Higgs.[11]

physicists are fucking morons

i went thru school intimidated by dy/dx and limped thru the courses, but have since changed my mindset during self-study and am finally allowing myself to learn it. calculus isn't that hard, it's how one thing relates to another thing, like space and time or energy and distance or posts per hour. i was over at my buddy's recently, in the last few months he's filled notebooks w/ math problem sets in self-study. film major cuz he got intimidated in college by the weed-out calc courses. we're gonna co-study linear algebra next month out of Halmos. it's what i'm saying. there's a whole ideology around STEM built by the profiteering class, standing in our way, existing for the mass-acceptance of high-paid, male mental laborers. 'of course they get paid more, they know magic and teacher told me i don't'

firstly, from both myself, and from talking to my colleagues, none of us imagine we have some special innate talent in mathematics, and probably more than any other discipline, it's mastery is the product of time, patience, and focus, but little else. during my time in graduate school, i had found that most of my friends were not terribly successful in math during their regular schooling; in fact, a couple of them had failed math classes in highschool. i have had many friends that showed much more promise than myself when i was younger who ended up burning out. i didn't finish my phd because i had a special talent for math, i finished it because i didn't want to do anything else.

secondly, i am suspicious of the individuality of learning styles. i have been teaching college math for about 7 years now, and have read a bit of the empirical research on pedagogy. my understanding is that individualizing teaching for students has not shown to be effective. we don't really learn concepts in ways that are that much different than each other on aggregate. obviously, there are certain concepts that take longer to click for some students than others, and as a teacher you try to show different contexts to the students, so perhaps one way of approaching a problem will click faster than the other; however, ultimately, i do not think this is a difference in types of learning, but rather the specific educational histories of various students. if a student has never learned to add fractions (as many of my college intro students struggle with), teaching them about rational polynomial functions becomes a problem. furthermore, memory can be a stupid thing, most students only remember pieces of their studies, and they try to bridge those pieces of knowledge into a piecemeal system of understanding, which can cause particular difficulties for mathematics due to the very cumulative and unforgiving nature of it.

thirdly, i do think the "intellectual class" nature of it is more pronounced in highly capitalist countries. i do not buy into the complete cynicism some of you may have towards "western" education; however, i do think that schools everywhere serve as sorting system in addition to a education system. this is not necessarily a terrible thing in and of itself. even in communist countries, students were tracked to what they showed interest and "talent" in. however, in communist states, there was generally a much more even playing field that children started off with. most elementary schools were relatively uniform in quality, and with a (more) leveled class structure, you didn't have students in one school coming from asbestos apartments competing against students in another school who all had private tutors and full stomachs. my mother, a social academic from yugoslavia, went on a social sciences and languages track during highschool, and never considered herself good at math; however, she never had this fear of mathematics i've seen in the united states. talking to friends and family from eastern europe, they might be amazed that i am a mathematician, but they're always eager to ask me what i'm studying and aren't afraid to have me sit down and show them at least the schematics of my research topics. americans on the other hand just say "dude, i always sucked math, i don't understand how you do that shit" and back off.

finally, i am not a physicist, but when understanding anything, it's important to not mistake the metaphor from the propositions themselves. there's a difference between understanding that gravity is a force that pushes things together, and actually being able to get actual scientific statements from it. the mathematics

**is**the concept you are trying to learn.

tears posted:thanks, so let me see if ive got this straight - they cant occupy the same state because they would cancel each other out - like wave interferance - and thats impossible right, two particles cant just disapear each other. But when bosuns occupy the same quantum state its like additive interferance which is just +1 + +1. they would appear industinguisahble/identicle to each other just +n - hence the bose-enstein condensate where you get them to all do it at the same time and bring quantum wave shit into real life

with femions where its +1 + -1, which is impossible for particles, though possible for particle anti-particle interactions? idk, i feel like im missing something

i dont understand this bit though:

c_man posted:symmetry properties of the wave function. Essentially, half-spin particles require (for reasons that are somewhat formal) that the wavefunction is antisymmetric under exchange of particles.

why do half spin particles require that the wavefunction is asymetric under exchange of particles? is it a maths thing?

tryna get my head round the higgs mechanism next, wtf is mass?

The first bit is exactly right, interference is precisely the concept to be invoking here. As for why fermions require this antisymmetry, im not of a way of understanding this that doesnt go into the guts of quantum field theory which requires a decent amount of sort of esoteric math, at least as far as i understand it. There could be a more straightforward explanation of why that should be the case but i dont know it. Thats really what math can help with the most, precision and dealing carefully with abstract concepts in regular ways.

I also agree with most of what elemenop said. I was never particularly gifted in mathematics, i just spent a lot of time on it and i found it more or less enjoyable, and eventually i developed some kind of fluency. Imo the process is similar to learning a language or musical instrument, and it can be easier or harder for various people for various reasons. Institutional preparation is a big one and any one those is a fairly daunting task to tackle without the active pedagogical action of some institution, which leaves a lot of people behind when they dont match the institutional vision of the desired students.

tears posted:try teaching a class of 30 14 year olds who hate school and maths

i have...it's difficult, and i certainly empathize with your struggle

tears posted:if i learned the maths what proportion would i find that was simply descriptive rather than explainatory?

Iirc he describes the processes of doing the calculations in detail either in detail in simple situations where the answer can be understood in the correct way or heuristically in terms of the procedure you would carry out to arrive at the quantitatively correct answer. So i think it would be both descriptive and explanatory.