#geometry-and-trigonometry | Mathematics | Page 177

opal blaze Jul 30, 2018, 2:42 PM

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ffs

mint sandal Jul 30, 2018, 2:42 PM

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Hence corresponding angles are equal too

opal blaze Jul 30, 2018, 2:42 PM

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then it is easy

mint sandal Jul 30, 2018, 2:42 PM

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Which solves the exercise

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No need for circles

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And whatever else you got there

opal blaze Jul 30, 2018, 2:47 PM

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ok, angle show a = b, when line g=h. then a/g = b/h so ah = bg and since h = g then a=b

opal blaze Jul 30, 2018, 4:00 PM

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"Chord EF is the perpendicular bisector of chord BC" does this mean that EF passes through the middle of BC

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or does it only mean that the are orthogonal.

night sluice Jul 30, 2018, 6:46 PM

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I've never done geometry in my life, but I self-studied up to real analysis and abstract algebra, AMA

mint sandal Jul 30, 2018, 6:57 PM

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In that case

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You should study geometry

clear haven Jul 30, 2018, 6:59 PM

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gottem

forest dove Jul 30, 2018, 8:25 PM

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Pfft geometry

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Do more algebra instead

rugged moat Jul 30, 2018, 8:47 PM

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Geometry is pretty much useless after computers

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Algebra though......

fallen ivy Jul 30, 2018, 9:32 PM

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Algebraic geometry

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Fite

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With Grothendieck meme

opal blaze Jul 30, 2018, 11:02 PM

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Is the answer 9?

📎 ec68df235250dd477f7e2929767c358b.png

hard gale Jul 30, 2018, 11:55 PM

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does RE cut the diagonals at their respective midpoint?

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cause if they do, I have no idea how to show it

errant ether Jul 30, 2018, 11:57 PM

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I think it does, you could show it by constructing similar right triangles with hypotenuses AI and AZ as well as DP and DT, respectively

hard gale Jul 30, 2018, 11:57 PM

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ah yeah well if zevenate is right the answer is 9 yes

opal blaze Jul 31, 2018, 12:29 AM

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nice, I can't look up any answers for the question. So I wanted to make sure. Thank you @hard gale @errant ether

hard gale Jul 31, 2018, 12:29 AM

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^^

eager pendant Jul 31, 2018, 7:27 AM

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@opal blaze that's from aops vol 1?

tiny sphinx Jul 31, 2018, 9:37 AM

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@rugged moat Why do you think geometry is useless after computers?

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I write graphics programs and I have to use geometry all the time.

rugged moat Jul 31, 2018, 9:38 AM

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Hard geometry is.

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Basic geometry is pretty useful

tiny sphinx Jul 31, 2018, 9:39 AM

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What is hard geometry, and why is it useless?

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Are you saying because it's isomorphic to algebraic theories?

rugged moat Jul 31, 2018, 9:39 AM

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I would explain, but I don't have the time rn

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nvm

tiny sphinx Jul 31, 2018, 9:40 AM

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I know of geometrical algebra for example.

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Are you just bullshitting? >:)

rugged moat Jul 31, 2018, 9:40 AM

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GWcmeisterPeepoShrug

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I will be back in ~an hour

tiny sphinx Jul 31, 2018, 9:41 AM

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No worries. You just got me curious

mint sandal Jul 31, 2018, 9:42 AM

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@rugged moat me too

rugged moat Jul 31, 2018, 9:42 AM

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GWsplBahtman GWmythicalHyperREEEE

tiny sphinx Jul 31, 2018, 9:42 AM

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Do you think most of the problems on this server are homework problems? 😃

thorn valley Jul 31, 2018, 9:42 AM

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yeah

tiny sphinx Jul 31, 2018, 9:42 AM

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A previous problem looked pretty homework-y

rugged moat Jul 31, 2018, 9:42 AM

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Many, yes

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Probably 99%

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I gotta run

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Fuuuuh

tiny sphinx Jul 31, 2018, 9:43 AM

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Go go go!

thin wasp Jul 31, 2018, 9:51 AM

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If you're referring to my previous question, it's not homework

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I just find contest maths especially fascinating, and I'm working through a relevant book with exercised

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And when a friend or I can't solve them, I'll ask this discord.

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@hollow edge My solution was something like: You let some the midpoint of PB be M', say, and reflect D about M' to D'. Then you angle chase, show that ABD'P is cyclic, and prove that D'DA is a straight line proving the condition

opal blaze Jul 31, 2018, 11:57 AM

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@eager pendant yes, do you have a answer sheet or something?

eager pendant Jul 31, 2018, 11:57 AM

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no but i did all the geometry problems

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ok

tiny sphinx Jul 31, 2018, 12:46 PM

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@thin wasp No, I was referring to https://cdn.discordapp.com/attachments/326138757474680852/473493630884118529/bd5151782586dc22057e972fc2531528.png

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It literally says "EXERCISE". I forgot.

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I thought maybe your problem was homework, but that would be pretty rough if you only had like two days

opal blaze Jul 31, 2018, 1:55 PM

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@tiny sphinx Yeah, that question was really simple. I just misunderstood what they were asking....

steep temple Aug 1, 2018, 10:58 AM

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What’s wrong with homework questions @tiny sphinx

opal blaze Aug 1, 2018, 5:45 PM

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Is the answer 6(sqrt(2) - 1)

📎 ad0be8e24bf0bf007cd4707b5fb03f68.png

rugged moat Aug 1, 2018, 5:49 PM

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$$π(\frac{90}{360}-\frac{60}{360})×6^2$$

charred spearBOT Aug 1, 2018, 5:49 PM

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📎 latex.png

rugged moat Aug 1, 2018, 5:50 PM

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$$3π$$

charred spearBOT Aug 1, 2018, 5:50 PM

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📎 latex.png

rugged moat Aug 1, 2018, 5:50 PM

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GWqlabsFacepalm

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I read that as area

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Nvm this

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Try 2:

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$$2π(\frac{90}{360}-\frac{60}{360})×6$$

charred spearBOT Aug 1, 2018, 5:51 PM

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📎 latex.png

hollow edge Aug 1, 2018, 5:51 PM

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~~you need to use special triangles~~

rugged moat Aug 1, 2018, 5:51 PM

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$$π$$

charred spearBOT Aug 1, 2018, 5:51 PM

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📎 latex.png

rugged moat Aug 1, 2018, 5:52 PM

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The answer is π

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Just a plain old π

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@opal blaze

hollow edge Aug 1, 2018, 5:52 PM

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for.... https://cdn.discordapp.com/attachments/326138757474680852/474271420822716419/ad0be8e24bf0bf007cd4707b5fb03f68.png ???

rugged moat Aug 1, 2018, 5:52 PM

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Yup

opal blaze Aug 1, 2018, 5:52 PM

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@rugged moat are you sure about that?

hollow edge Aug 1, 2018, 5:52 PM

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it asked for length....

rugged moat Aug 1, 2018, 5:53 PM

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Difference in legths

hollow edge Aug 1, 2018, 5:53 PM

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the answer is 6(sqrt(2) - 1)

rugged moat Aug 1, 2018, 5:53 PM

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The formula is 2πr×@/360

hollow edge Aug 1, 2018, 5:53 PM

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segments

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not arclength

rugged moat Aug 1, 2018, 5:53 PM

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Goddamnit

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Ffffffffff

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So apply trigonometry?

hollow edge Aug 1, 2018, 5:53 PM

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no need

rugged moat Aug 1, 2018, 5:54 PM

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Or just brute force it I guess

hollow edge Aug 1, 2018, 5:54 PM

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60 deg one is a equilateral triangle

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90 deg allows pythagorean

opal blaze Aug 1, 2018, 5:54 PM

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Yeah that is what I did

rugged moat Aug 1, 2018, 5:54 PM

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Guess that works

hollow edge Aug 1, 2018, 5:54 PM

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no need for fancy trig

rugged moat Aug 1, 2018, 5:55 PM

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π is such a pretty solution

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While that's ugly

hollow edge Aug 1, 2018, 5:55 PM

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tau/2 is an ugly solution

rugged moat Aug 1, 2018, 5:55 PM

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:/

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Tau is 2π, right?

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Or was it π/2?

hollow edge Aug 1, 2018, 5:55 PM

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yeah

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2pi

rugged moat Aug 1, 2018, 5:56 PM

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√2 is uglier

opal blaze Aug 1, 2018, 6:14 PM

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2(sqrt 2), I am asking because I don't have the option to check myself.

📎 4f9dab6fe0b14087d820544d1724ab81.png

hollow edge Aug 1, 2018, 6:17 PM

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that seems right

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you mean you want an explanation?

opal blaze Aug 1, 2018, 6:29 PM

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no, just confirmation

clear haven Aug 1, 2018, 11:05 PM

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https://www.youtube.com/watch?v=pQa_tWZmlGs @hollow edge

YouTube

3Blue1Brown

How to convert a non-math-lover (Dandelin spheres)

A beautiful proof of why slicing a cone gives an ellipse. Thoughts on going sponsor-free (for those who missed it): https://www.patreon.com/posts/going-spons...

▶ Play video

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conic section thingie

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ehhh

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you may have already seen it lel

clear haven Aug 2, 2018, 2:31 AM

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pls

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floating point boio

stuck jewel Aug 2, 2018, 10:26 PM

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https://m.youtube.com/watch?v=_60sKaoRmhU

YouTube

Khan Academy

Introduction to the line integral | Multivariable Calculus | Khan ...

Introduction to the Line Integral Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line_integrals_topic/line_integrals/v/line-i...

▶ Play video

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Why did he have to put the small chang of ds

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Into the pythag thm?

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Couldnt he just have put it as dx instead?

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Because if they are measuring infinitely small sections of tne curve, it woulnt matter in the end

frank frost Aug 2, 2018, 10:28 PM

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variable x was already taken

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and ds is not the same as dx

stuck jewel Aug 2, 2018, 10:30 PM

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Why wouldnt thet be the same?

frank frost Aug 2, 2018, 10:33 PM

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because we are not just changing x, we are also moving in the y axis

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and this entire thing is kinda informal

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in order to be able to properly talk about stuff like dS, with intuition about the curve

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one needs to do quite a lot of work

stuck jewel Aug 2, 2018, 10:35 PM

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What should I study before co tinuimg with this?

frank frost Aug 2, 2018, 10:36 PM

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it also can be done in different ways, some of those appeal more to intuition, but those are usually much, much harder

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I would just recommend to find a more formal exposition

stuck jewel Aug 2, 2018, 10:38 PM

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Ok

frank frost Aug 2, 2018, 10:38 PM

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for example

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there is an excellent textbook by Zorich, which has it formalised quite good

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http://instructor.sdu.edu.kz/~verbovskiy/Math Analysis 1/Zorich1_en.pdf

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as for stuff more appealing to intuition

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it really requires quite a number of prequisites to get right

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do you know what a category is?

stuck jewel Aug 2, 2018, 10:42 PM

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Oh shoot no

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And thx btw

frank frost Aug 2, 2018, 10:42 PM

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no problem)

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I don't really like informal expositions like that video, so keep that in mind

stuck jewel Aug 2, 2018, 10:44 PM

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Ok

fallen ivy Aug 2, 2018, 11:32 PM

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@frank frost were you about to use category theory to explain the line integral

frank frost Aug 2, 2018, 11:33 PM

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well, a) I don't understand it well myself
and b) I did not think somebody would ask :D
but I would try to paint a sketch and direct you to a book where you can read more

fallen ivy Aug 2, 2018, 11:33 PM

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Sure

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I am really good with category theory

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If this uses ends and coends I will die oml

frank frost Aug 2, 2018, 11:34 PM

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instead of building just one integral, one would want to build ALL the integrals, so we are building diff geometry as a whole basically

fallen ivy Aug 2, 2018, 11:34 PM

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Oh yeah lol

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Work in category of manifolds

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Use a dum category theoretical construct to get integrals?

frank frost Aug 2, 2018, 11:38 PM

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well, not quite, this one is more geometric in it's approach

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and also synthetic: one introduces multiple axioms for geometry, which make it possible to build infinitesimal analysis on "smooth objects"

fallen ivy Aug 2, 2018, 11:40 PM

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Alright

frank frost Aug 2, 2018, 11:40 PM

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the resulting geometry is non-euclidean (it's possible to have multiple distinct lines between 2 points)

fallen ivy Aug 2, 2018, 11:40 PM

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Link me the book?

frank frost Aug 2, 2018, 11:41 PM

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📎 sdg99.pdf

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I was too lazy to google the thing so I just uploaded it 😄

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the second part is more categorical, the first one thinks more in terms of "sets", but has more category theory then I could handle

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rigorously everything is formulated in terms of categories

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honestly I think it's the only book on diff geometry I might actually finish, just because of it's immense appeal to intuition

shut ridge Aug 3, 2018, 1:58 PM

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I found this definition on wiki: In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function

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what exactly should I imagine that inverse function to look like? I understand that the mapping needs to be continuous but how would two spaces have continuous mapping without the inverse function being continouos?

plain wigeon Aug 3, 2018, 2:32 PM

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Are you sure that last statement is correct? You contradicted the definition

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The inverse function is continuous

opal blaze Aug 3, 2018, 3:14 PM

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3/2 is the right answer?

📎 dc423afec613f173c77e73c514101e7c.png

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6 - sqrt(3) ?

📎 e188d645c89dc17452a11688e5a026b5.png

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📎 c431d8716d012a9da0a73fd662ac969e.png

opal blaze Aug 3, 2018, 4:25 PM

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8*sqrt(3)

📎 d53af2e498e8c2679349c4be9ab99480.png

fallow edge Aug 3, 2018, 4:30 PM

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📎 image.png

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That’s what I got for 178

eager pendant Aug 4, 2018, 10:01 AM

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@opal blaze I got 1/2 for 173, 3/2 for 174, and 32 for 178

dense holly Aug 4, 2018, 2:55 PM

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@shut ridge it's perfectly possible to have a mapping be continuous and have it's inverse not be continuous

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easiest way to see this is to look at the definition of continuity (preimages of open sets are open) and then let the domain have the discrete topology and the target have the chaotic topology. Then all maps will usually have this property

dense holly Aug 4, 2018, 3:16 PM

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as a more concrete example, consider the map from angles in [0,2pi) to the unit circle. this map is continuous, but it's inverse isn't (consider neighbourhoods of the image of 0)

ancient mesa Aug 4, 2018, 4:07 PM

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So at what point does geometry turn into topology? The line seems blurred to me

mint sandal Aug 4, 2018, 4:34 PM

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yh it's blurred

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i would say geometry turns into topology when you start talking about things which are invariant under homeomorphisms or something

plain wigeon Aug 4, 2018, 6:26 PM

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Yes. Usually less rigidity is an easier way of looking at it

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So invariance under homeomorphisms you can look at it from the fundamental group perspective or just look at that homotopically

dense holly Aug 4, 2018, 9:39 PM

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topology is just anything you can get with just topologies and topological spaces

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if you have an atlas, it's not really topology anymore

plain wigeon Aug 4, 2018, 9:41 PM

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Then that gets a bit circular for someone unfamiliar

shut ridge Aug 5, 2018, 6:06 AM

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thanks for explaining :)

hollow edge Aug 7, 2018, 3:45 PM

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You need to find the radius of the circumcircle

upper karma Aug 8, 2018, 12:08 AM

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I'm very slow with math pls explain

📎 Screen_Shot_2018-08-07_at_8.07.32_PM.png

stark ember Aug 8, 2018, 12:10 AM

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because x is the refence angle that would make 6 the opposite side and 19 would be the hypotenuse

upper karma Aug 8, 2018, 12:10 AM

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yes and how do i solve x then

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SOHCAHTOA

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What sides do you have?

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hypotenuse and op angle

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Hence, what relation should you use?

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sin

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Okie dokie

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So what's the relationship?

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uhhhhh

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wdym

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sin(x) = ?

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6sin19??

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Wat

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SOHCAHTOA

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Sine of the angle is equal to the ratio between the opposite and the hypotenuse.

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sin(6)=19?

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What's the angle in the diagram?

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90

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Is that the one we're trying to find?

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no

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So what angle are we trying to find?

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A lol

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okie dokie

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What other information do we have

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Relative to A?

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it's an approximation

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...?

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Of what

stark ember Aug 8, 2018, 12:16 AM

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that question looks like it is asking for the approximate measure of angle x

upper karma Aug 8, 2018, 12:16 AM

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of x

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^^

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Small-angle approximation smh

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But still, the ratio is important

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Since the relationship that you are intending to use is

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x ~ sin(x)

stark ember Aug 8, 2018, 12:17 AM

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pao think about it this way. S=O/H, C=A/H, T=O/A. that helped me

upper karma Aug 8, 2018, 12:17 AM

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so i would use sine then, right?

stark ember Aug 8, 2018, 12:17 AM

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yep

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whats the opposite

upper karma Aug 8, 2018, 12:17 AM

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6

stark ember Aug 8, 2018, 12:17 AM

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and whats the hypotenuse

upper karma Aug 8, 2018, 12:17 AM

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19

stark ember Aug 8, 2018, 12:18 AM

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So the sine of theta is 6/19

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do you know where the sin to the power of -1 is on your calculator?

upper karma Aug 8, 2018, 12:19 AM

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i do but im using a digital calculator cus i dont have one rn :/

stark ember Aug 8, 2018, 12:19 AM

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does your digital calculator have one?

upper karma Aug 8, 2018, 12:19 AM

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i'm using the google calc and it has sin

stark ember Aug 8, 2018, 12:20 AM

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does it have a 2nd button?

upper karma Aug 8, 2018, 12:20 AM

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Inv ? In?

stark ember Aug 8, 2018, 12:21 AM

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📎 image.jpg

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it would be the blue button

upper karma Aug 8, 2018, 12:21 AM

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this is it

📎 Screen_Shot_2018-08-07_at_8.20.59_PM.png

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yes i have that one but i dont think it's charged lmao

stark ember Aug 8, 2018, 12:21 AM

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charge it

upper karma Aug 8, 2018, 12:21 AM

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is there a way to do it without it tho

stark ember Aug 8, 2018, 12:21 AM

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and make sure to set your calculator to degree mode for trig

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so on a TI 84 you would press 2nd -> sin -> Opposite -> Division symbol -> hypotenuse -> ) -> Enter

upper karma Aug 8, 2018, 12:24 AM

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ok i'll be back

stark ember Aug 8, 2018, 12:24 AM

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take a screenshot of what i just said

upper karma Aug 8, 2018, 12:26 AM

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i got 18.40848017

stark ember Aug 8, 2018, 12:27 AM

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that is correct, and what is that simplified

upper karma Aug 8, 2018, 12:27 AM

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18

stark ember Aug 8, 2018, 12:27 AM

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it says round to the nearest degree

upper karma Aug 8, 2018, 12:27 AM

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yup i got it thx

stark ember Aug 8, 2018, 12:28 AM

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yeah, it seems to be asking for a whole number so yeah that is probably it

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and your welcome, always happy to explain concepts

valid hedge Aug 8, 2018, 10:25 PM

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Is anybody here interested in quantum topology/ khovanov homology/ categorification?

plain wigeon Aug 8, 2018, 11:45 PM

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Ask any question you may have

clear haven Aug 9, 2018, 2:26 AM

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i think that may have been the question lolol

neon fossil Aug 9, 2018, 3:32 AM

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no one is interested in any of those

keen aspen Aug 9, 2018, 5:26 AM

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Lolol

valid hedge Aug 9, 2018, 12:42 PM

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I'm a PhD student working in this area. Nobody else in my school is super interested in this stuff besides my advisor (and the rep theory people, but they take different approaches)

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So if anybody knows about this stuff, I'd love to chat

ornate reef Aug 9, 2018, 1:17 PM

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Quantum topology sounds interesting

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But I’m too stupid

valid hedge Aug 9, 2018, 1:22 PM

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It sounds way fancier than it actually is

hollow edge Aug 9, 2018, 5:32 PM

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~~these researchers are watching way too much tv~~

valid hedge Aug 9, 2018, 6:16 PM

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We really have too much time on our hands

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I wrote a short and incredibly vague overview here https://www.reddit.com/r/math/comments/95x6ve/khovanov_homology_and_related_topics/

Khovanov homology and related topics • r/math

Hello math community. Is there anybody here who is interested in Khovanov homology, categorification of skein modules, and things like this? I'm a...

narrow slate Aug 9, 2018, 7:47 PM

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I’m interested

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I suck though

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So as a first dumb question; why are they often called quantum invariants (the invariants for links and stuff)

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I think there was one that sprung up from path integrals in chern-simons theory or somethjng

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But apart from that I’m not sure what’s deserving of the ‘quantum’ tag

valid hedge Aug 9, 2018, 7:51 PM

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You know, I'm not sure. I think it's due to the connection with TQFT (topological quantum field theory) which started out in physics and then the math people came and fixed the physicists' mess

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Yep just like you said

narrow slate Aug 9, 2018, 7:52 PM

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Hmm ok; I know they often tag on ‘quantum’ in a lot of geometrical setting whenever you have noncommutative stuff going on I think (eg in noncommutative geometry) but I wasn’t sure where the ‘noncommutativity’ was arising in the study of knots

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Er another dumb question so what’s categorification got to do with knots lol

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I was thinking the other day of forming a category with knots as objects; and morphisms as (maybe) homotopies between the knots when they are embedded in 4D instead of the usual 3 (since nonequivalent knots can’t be homotoped to each other in 3 by def n.) but after a while that sounded like a dumb idea

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And I’m sure that this ‘categorification’ you speak of is completely different lol

valid hedge Aug 9, 2018, 7:55 PM

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There are plenty of "quantized" algebra things here where you force multiplication to commute up to a power of some indeterminate. Somehow this is important in hardcore algebra but I don't know much about this. There is a way to build noncommutative algebraic objects from link diagrams (cluster algebras) and this has caught some steam

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Nope it's pretty much what you suggest

narrow slate Aug 9, 2018, 7:56 PM

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Wait what

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But in this category... all the objects are isomorphic right? Waiiiit...

valid hedge Aug 9, 2018, 7:57 PM

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More realistically the maps are cobordisms. So no not everything is isomorphic

narrow slate Aug 9, 2018, 7:57 PM

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Oh

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That makes a lot more sense

valid hedge Aug 9, 2018, 7:58 PM

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Yea but your idea isn't crazy, this works but unfortunately it doesn't work super well

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A cobordism between links in 4-space will induce a map on homology but it's only defined up to +/- 1

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There are ways to fix this sign issue but there is subtle

narrow slate Aug 9, 2018, 8:00 PM

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Hmm nice at least it’s good to know about this category!

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Are there any problems you’re thinking about in this field?

valid hedge Aug 9, 2018, 8:01 PM

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Yep if you're interested Bar-Natan has an paper that's super readable and will probably answer a lot of your questions

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I'm looking at a few problems but one in particular is how do all these cluster algebra things fit into the older categorification models

forest dove Aug 9, 2018, 9:12 PM

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I'm still a little new to the general math game but this sounds quite interesting

valid hedge Aug 9, 2018, 9:16 PM

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It's fun!

forest dove Aug 9, 2018, 9:18 PM

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My immediate question would be, what is the Euler characteristic of a graded module? Is it (-1)^n rank of the nth guy?

valid hedge Aug 9, 2018, 9:20 PM

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It's a Laurent polynomial x^n dim(A_n)

forest dove Aug 9, 2018, 9:21 PM

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Ah so this is graded both sides

valid hedge Aug 9, 2018, 9:21 PM

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It's useful bc you need to shift the gradings around

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Yes it's graded in homology and then again each module is graded

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This is sometimes called Poincare polynomial

neon fossil Aug 9, 2018, 9:46 PM

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@forest dove

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that guy

upper karma Aug 10, 2018, 1:30 PM

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Yo anyone here

thorn valley Aug 10, 2018, 1:45 PM

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nope

tiny sphinx Aug 11, 2018, 4:59 PM

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Hello

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📎 gfx_01_p.jpg

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📎 gfx_for_03.png

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http://www.iquilezles.org/www/articles/sphereao/sphereao.htm

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d is the length of the entire left line. D is the length of the line from the tip of the cone to the disk. The author says "We can quickly find how R and D relate to r and d by applying Pythagoras theorem twice after noting that the radius of the disk line intersects the sphere position line in ninety degrees."

I don't follow. I think something else must be used like the law of cosines.

copper valve Aug 11, 2018, 5:49 PM

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(d-D)²+R²=r²
R²+D²=H²
H²+r²=d²
Where H is the length from the cone vertex, to the vertex between R and r

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then
R²+D²+r²=d²
R²=d²-D²-r²
R²=r²-(d-D)²=r²-d²-D²+2dD

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hmm

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if we subtract these, we get
0
=R²-R²
= (r²-d²-D²+2dD) - (d²-D²-r²)
= 2r²-2d²+2dD

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r²=d²-dD

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hmmm

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well what do the equations that are supposed to occur look like when we square em

#

R=(r/d)√(d²-r²)

#

R²=r²(d²-r²)/d² = r²-(r⁴/d²)

#

the thing is, this formula for R doesn't involve D, but mine does

#

so maybe I shoulda cancelled out D instead

#

D²=d²-r-R²
D²=2dD+r²-d²-R²

#

hmm

#

ya this is pretty weird

hard gale Aug 11, 2018, 6:00 PM

#

📎 a90.png

#

I also tried a ton of things, we're just as lost as you are...

tender hare Aug 11, 2018, 9:24 PM

#

Hi, i had a question about this, in the box it says that P is clearly Compact, How can we prove that?

📎 unknown.png

forest dove Aug 11, 2018, 9:25 PM

#

Intersection of compact sets

mint sandal Aug 11, 2018, 9:33 PM

#

is intersection of compact sets always compact, even in non haussdorf spaces?

forest dove Aug 11, 2018, 9:34 PM

#

Not automatically, it uses the fact that compact sets are closed

mint sandal Aug 11, 2018, 9:35 PM

#

yeah i figured thats what you had to assume

plain wigeon Aug 11, 2018, 9:36 PM

#

Yep, closedness

forest dove Aug 12, 2018, 6:32 AM

#

Topology isn't restricted to 3D, and I dunno how you make sense of an "infinitesimal shape"

#

Topology is the study of spaces where it makes sense to think about continuous functions

past mantle Aug 12, 2018, 6:37 AM

#

I thought so, that's why I asked xd

bleak rivet Aug 12, 2018, 9:40 PM

#

Im having some troubles with expressing trigonometric functions as a positive acute angle

reef inlet Aug 12, 2018, 9:40 PM

#

JustAsk

bleak rivet Aug 12, 2018, 9:41 PM

#

I have no idea where to even start with this. Just by fooling around I found if I subtract 180 it is often the answer but you have the negative sign. I dont know if there is a formula or what

#

but even when I google it I can seem to find a clear answer

reef inlet Aug 12, 2018, 9:42 PM

#

if it's in the second quadrant, substract it from 180

#

third: subtract 180 from it

#

fourth, substract 270 from it

bleak rivet Aug 12, 2018, 9:42 PM

#

and one is just itself then

reef inlet Aug 12, 2018, 9:43 PM

#

although you have to be careful of the signs

#

sin(x) if x is in the fourth quadrant is negative

#

same for third quadrant

bleak rivet Aug 12, 2018, 9:44 PM

#

right I remember quadrant 1 is all positive

#

2 sin is positive

#

3 tan is positive and 4 cos is positive

reef inlet Aug 12, 2018, 9:44 PM

#

top qudrants are positive ( because they're above the x axis) and bottom quadrants are neg

bleak rivet Aug 12, 2018, 9:44 PM

#

so cos 220 would be -40

reef inlet Aug 12, 2018, 9:45 PM

#

memorize it like this instead:
sin is the y coordinate, the y coordinate is how much up you are so if you're in the top quadrants then it's pos and if you're in the bottom then it's negative

bleak rivet Aug 12, 2018, 9:45 PM

#

right

#

cos is x

reef inlet Aug 12, 2018, 9:46 PM

#

yep

bleak rivet Aug 12, 2018, 9:46 PM

#

okay thanks

reef inlet Aug 12, 2018, 9:46 PM

#

no problem

bleak rivet Aug 12, 2018, 9:46 PM

#

I was confusing things and stuff lol

#

its clear now

reef inlet Aug 12, 2018, 9:46 PM

#

Ok360

eager pendant Aug 15, 2018, 6:44 AM

#

has anyone read "plane euclidean geometry" by a.d. gardiner and c.j. bradley?

#

is it good? I can't find a sample/pdf online

mossy vine Aug 15, 2018, 7:46 AM

#

I'm currently reading it, it's pretty good imo

#

it contains a lot of theory and problems

#

some of which are quite challenging

eager pendant Aug 15, 2018, 7:48 AM

#

are there any ukmt books you've been through in the past / you'd recommend?

mossy vine Aug 15, 2018, 7:49 AM

#

I've been through the olympiad primer, I don't know if I'd recommend it though

#

the theory in that is somewhat shallow, understandably, because it's aimed at BMO1

#

the problems in it can all be found online because they're past BMO1 problems, the best thing about it is the solutions for the problems which generally aren't available online

#

I'm not sure if I'd recommend it just for that though, because I'd say you tend to get more out of the problems the less help from something like that you use

#

I have the BMO2 equivalent but I haven't really been through it much

#

so I couldn't comment on it for that

#

the only other UKMT book I have is the new problems in euclidean geometry

#

they seem like very good problems, they're quite challenging

eager pendant Aug 15, 2018, 8:18 AM

#

alright, thank you

past mantle Aug 15, 2018, 4:48 PM

#

i was looking through a Geometry book i had, it says that its impossible to trisect an angle

#

by construcking

#

constructing*

upper karma Aug 15, 2018, 4:48 PM

#

that's true ^

past mantle Aug 15, 2018, 4:49 PM

#

how?

#

i want to know

upper karma Aug 15, 2018, 4:49 PM

#

here's a proof: https://terrytao.wordpress.com/2011/08/10/a-geometric-proof-of-the-impossibility-of-angle-trisection-by-straightedge-and-compass/

What's new

Terence Tao

A geometric proof of the impossibility of angle trisection by stra...

One of the most well known problems from ancient Greek mathematics was that of trisecting an angle by straightedge and compass, which was eventually proven impossible in 1837 by Pierre Wantzel, usi…

past mantle Aug 15, 2018, 4:50 PM

#

thx

#

there is a part of the proof i dont understand

#

📎 g3fwr.PNG

#

the start of this part

mint sandal Aug 15, 2018, 4:52 PM

#

which part exactly

past mantle Aug 15, 2018, 4:53 PM

#

all of that :/

mint sandal Aug 15, 2018, 4:53 PM

#

x^2+y^2=1

#

you don't recognize that?

upper karma Aug 15, 2018, 4:53 PM

#

he's basically just saying we'll draw a circle at the origin on euclidean 2 space

#

aka normal cartesian plane

past mantle Aug 15, 2018, 4:53 PM

#

i do

#

indicated by R^2?

upper karma Aug 15, 2018, 4:53 PM

#

that's euclidean 2 space

mint sandal Aug 15, 2018, 4:54 PM

#

set of all pairs of real numbers

past mantle Aug 15, 2018, 4:54 PM

#

oh

upper karma Aug 15, 2018, 4:54 PM

#

2 dimensions basically

past mantle Aug 15, 2018, 4:54 PM

#

alright

narrow haven Aug 18, 2018, 7:43 AM

#

Hi guys, can someone please explain quaternions to me?

#

if this is off topic, please tell me what topic they fall into

#

I mean, I can google the formulas, but I need really understand WHY they work. otherwise I'll never remember them or be able to use em effectively.

upper karma Aug 18, 2018, 7:46 AM

#

what are you using them for?

#

rotations in 3D space I assume?

narrow haven Aug 18, 2018, 7:48 AM

#

yep

#

so i have this definition

#

q = w + xi + yj + zk

#

i know i, j, k are the "imaginary" component

#

from knowing basic complex number like 4 + 5i

#

where 4 lies along the x axis and 5 is the magnitude of the perpendixular component, right?

#

so I have that basic 2d intuition from those complex numbers

#

but then the space in between is just empty

#

looking for some intuition here

upper karma Aug 18, 2018, 7:50 AM

#

here this is probably the best thing I've ever watched explaining quaternions

#

https://www.youtube.com/watch?v=FRD0PgsY3pU&t=0s&list=PLE211C8C41F1AFBAB&index=15

YouTube

Terumag

Humane Rigging: chapter 03, video 04

If you like this video please consider buying the DVD: http://www.blender3d.org/e-shop/product_info_n.php?products_id=146 Quaternion Rotation The entire cont...

▶ Play video

#

we can talk about it after that if you're still having questions since you probably will

narrow haven Aug 18, 2018, 7:51 AM

#

ok thank you

#

watching it right now

narrow haven Aug 18, 2018, 8:38 AM

#

k watched it

#

@upper karma so actually that brings up a question about euler angles

#

he says teh order of the axes doesnt matter

#

so in other words, if you compose a 3D rotation by multiplying 3 rotation matrices

#

it doesnt matter what order u multiply em in, correct?

upper karma Aug 18, 2018, 8:41 AM

#

no, generically speaking if you multiply 3D rotation matrices the order matters

narrow haven Aug 18, 2018, 8:41 AM

#

like the 3 classic rotation matrices

#

📎 rotMat.jpg

#

but then Im confused

upper karma Aug 18, 2018, 8:41 AM

#

take two dice or some kind of object

narrow haven Aug 18, 2018, 8:41 AM

#

bc order DOES matter in some cases

upper karma Aug 18, 2018, 8:41 AM

#

rotations just don't commute

#

yeah order matters

narrow haven Aug 18, 2018, 8:42 AM

#

like imagine z is up, x goes into teh screen, and y is right

#

and imagine a jar w the label facing you alligned w the Z acis

#

axis

upper karma Aug 18, 2018, 8:42 AM

#

yep, I'm agreeing with you that order matters

narrow haven Aug 18, 2018, 8:43 AM

#

but... it seems like tehre are cases where it DOESNT matter

#

I cant figure out which is true

upper karma Aug 18, 2018, 8:43 AM

#

for rotation matrices specifically

narrow haven Aug 18, 2018, 8:43 AM

#

I thought you just rotate em in any order and they concatenate to the same total rotation

upper karma Aug 18, 2018, 8:43 AM

#

rotate 180 degrees on x and rotate 180 degrees on y

#

order doesn't matter

#

haha idk if that's what you are asking

narrow haven Aug 18, 2018, 8:44 AM

#

ok, following my example

upper karma Aug 18, 2018, 8:44 AM

#

ahhh I see what you're saying

narrow haven Aug 18, 2018, 8:44 AM

#

rotate the jar 180 on Z so teh label faces away from you

upper karma Aug 18, 2018, 8:44 AM

#

yes what you're talking about is orientation

#

if you're rotating a vector

#

or moving a single point

narrow haven Aug 18, 2018, 8:44 AM

#

then tlt it along the x, the then "pitch" it toward you along the Y

upper karma Aug 18, 2018, 8:44 AM

#

you can end up at the same location

#

but spun around your own axis

#

in some cases

narrow haven Aug 18, 2018, 8:44 AM

#

so now it's kindof tilted toward you w teh label facing away

#

but if you rotated it around X and/or Y, THEN Z

#

the Z rotation will result in it "swinging"

#

instead of "rolling" around its own axis

#

which equals a totally different transformation

#

yes, if you see the bottom of the jar at the origin and the top of the jar as a point, then if oyu rotate it arond the z axis first, that rotation does nothing

#

but if oyu rotate it around the X or Y first, THEN the Z, then the Z will effectively translate it

#

so totally different

#

like, what am I missing here?

upper karma Aug 18, 2018, 8:48 AM

#

I'm not sure cause I am not entirely inside your mind but

narrow haven Aug 18, 2018, 8:48 AM

#

am I confusing an object centered at the origin w a VECTOR radiating out from the origin?

upper karma Aug 18, 2018, 8:48 AM

#

when you multiply matrices you're getting the end result of a lot of transformations combined in succession yeah

#

I don't know, possibly, I think you might be thinking about the orientation

#

vectors don't have an orientation themselves

#

like you can spin around in an office chair

#

a vector looks identical when you spin it on its axis

#

if a jar is in the same location as the vector though, it will get moved around right

#

so you need to think of all the vectors of all the positions of particles in your object being rotated instead of just one vector

#

that's my guess

#

as to what you're confused about, idk

#

if you have a monopoly game or yahtzee or somet dice lying around

#

get them out and rotate them in different ways or something

#

idk if that will help haha

#

like do the same transformations in opposite order or

narrow haven Aug 18, 2018, 8:51 AM

#

so you's saying a jar is not a discrete point, it's mesh of vertices, so it's a different situation

#

gotta think of each vertex in isolation

#

yeah thats what Ive been doing

upper karma Aug 18, 2018, 8:52 AM

#

well you might think of the jar as being just like

#

http://prntscr.com/kk2jbj

Lightshot

Screenshot

Captured with Lightshot

#

http://prntscr.com/kk2jfd

Lightshot

Screenshot

Captured with Lightshot

#

but it's more like this

#

so rotating it aorund the z-axis might appear to not be moving anything

#

but really all these points are, just I think a jar might not be the best example because it's spherically symmetric

narrow haven Aug 18, 2018, 8:53 AM

#

ya I see ur point

#

thats why I said it has a label that is facing you

#

so we can keep track of if it turns around

upper karma Aug 18, 2018, 8:54 AM

#

I said spherically I meant cylindrically symmetric sorry

#

oh ok

#

yeah so I guess since we can't label the side of a vector that's annoying but

narrow haven Aug 18, 2018, 8:55 AM

#

yeah so the crazy thing is, Pavel Grinfeld says that an euler matrix is teh combination of Z, X, and Z

#

NOT Z, X, and Y

#

which is like wtf

#

https://www.youtube.com/watch?v=A6lf8t9WXn8&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=91

YouTube

MathTheBeautiful

Linear Algebra 21g: Euler Angles and a Short Tribute to Leonhard Euler

https://bit.ly/PG_Patreon - Help me make these videos by supporting me on Patreon! https://lem.ma/LA - Linear Algebra on Lemma https://lem.ma/prep - Complete...

▶ Play video

upper karma Aug 18, 2018, 8:55 AM

#

haha at which time stamp

narrow haven Aug 18, 2018, 8:55 AM

#

hold on

#

wrong one

#

https://www.youtube.com/watch?v=n6yeak4FITs&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=92

YouTube

MathTheBeautiful

Linear Algebra 21h: The Matrix that Represents an Arbitrary Rotati...

https://bit.ly/PG_Patreon - Help me make these videos by supporting me on Patreon! https://lem.ma/LA - Linear Algebra on Lemma https://lem.ma/prep - Complete...

▶ Play video

upper karma Aug 18, 2018, 8:56 AM

#

on the board it looks like he has Rx,Ry, Rz

narrow haven Aug 18, 2018, 8:56 AM

#

skip to around 1:00

upper karma Aug 18, 2018, 8:56 AM

#

maybe he just misspoke

narrow haven Aug 18, 2018, 8:56 AM

#

i dont think so

#

he never uses the 3rd axis

#

he "twists" it around Z, then tilts it around Y, then after it gets tilted, rotating it around Z takes on a new meaning, so he "swings" it around Z once again

#

it's really fucking w me

upper karma Aug 18, 2018, 8:58 AM

#

oh ok

#

that's because after each transformation you're looking at everything from a new perspective

#

let me think of a nice way to explain this

#

probably best to draw a picture of some kind

#

actually here's something to help you visualize it

#

http://prntscr.com/kk2lmo

Lightshot

Screenshot

Captured with Lightshot

#

if you multiply this by the column vector (x,0,0)^T

#

that gets you exactly where the x axis ends up

#

so what you can do is now set all the angles = 0 except the first z rotation

#

then after you are comfy with that, maybe fix it and try to wiggle the next angle

#

and just sorta see what each part does on its own

#

then repeat for the y and z although you'll probably start to see the trend

narrow haven Aug 18, 2018, 9:07 AM

#

sorry, I dont know what this picture means

#

is this the concatenation of the three elementary rotation matrices?

#

my intuitive understanding is that w euler angles, each axis of rotation is "nested" inside another

#

so each subsequent one becomes relative to the other

#

yeah, he mustve made a mistake by using Z twice in that video

#

why would he discard a whole axis, when euler angles are clearly expressed in X,Y,Z values? that makes absolutely no sense

#

damn, when you think about it, his point does make sense

#

Im guessing maybe, it's because he started his "vector" pointing up along the vertical axis

#

In Unreal 4, I'm using to working w a coordinate system where x points into the screen, Y points right, and Z points up

#

and the forward vector of the actor point along that X axis

#

so naturally, you would first rotate around the X axis to get teh roll, then pitch around the Y axis, the yaw around the Z axis

#

@upper karma so I guess it really matters which axis your forward vector starts out alligned to

#

whichever axis it is alligned to to begin with, you'd better get your rolling out of teh way first

#

he makes it confusing by making talking about roll like "twist" and making it vertical

upper karma Aug 18, 2018, 9:40 AM

#

ahh sorry I was distracted and I still am but I'll come back later

narrow haven Aug 18, 2018, 9:40 AM

#

its okay

#

just read the message I sent when u get the chance

#

its a really subtle thing

#

basically it's about which axis you define to be the "roll" axis. to me, the convention Pavel presents is really bad, because what's the use of a "twist" if it's vertical. it makes much more sense to think of a forward vector lying down horizontally, rolling that first, then moving it into proper lattitude and longitude positions

upper karma Aug 18, 2018, 10:00 AM

#

ahhh I see what you're talking about, yeah

#

I guess he's more theoretical and a mathematician he doesn't really care since he just needs them to represent the objects in his mind or something like that

#

I went through his tensor calculus lectures several years ago, and I liked those but I guess at the end of the day it does just boil down to convention and he's just working on different stuff

narrow haven Aug 18, 2018, 10:24 AM

#

Nah he's amazing

#

In general for understanding linear algebra

#

But juat this one point us impractical hehe

#

Btw

#

📎 JPEG_20180818_182531.jpg

#

this formula for SLERP seems weird

#

I kinda understand why sin would need to be in there, because otherwise the interpolation btwen the 2 points would take a "shortcut", and not follow the path of the circle of rotation

#

however, look at the formula

#

dont the sin thetas on top and bottom just cancel out?

hard gale Aug 18, 2018, 10:30 AM

#

I guess it's sin(theta(1-t)), just look at the theta*t on the second part

narrow haven Aug 18, 2018, 10:30 AM

#

📎 JPEG_20180818_183017.jpg

#

ohh, he's decomposing the quaternion at time t into its component that lies along the starting quaternion q1 and the ending quaternion q2 and adding those components together

#

so that's basically how interpolation works I guess, blending between two "poles" with an alpha value, which in this case is time t

upper karma Aug 19, 2018, 10:31 PM

#

I was thinking about conics earlier today and had an idea

#

so an ellipse you can trace out with a loops of string and two pins in a board

#

what if you imagine a kind of material that has constant surface area constantly, and there are now 3 fixed "foci" instead of 2 and you use one extra free point to stretch it out tight and use that to define a surface

#

what would this shape be?

clear haven Aug 19, 2018, 10:55 PM

#

😮

umbral snow Aug 19, 2018, 11:20 PM

#

What, like you have three pins in a board, and a string around them?

clear haven Aug 19, 2018, 11:31 PM

#

i guess three pins in a space, and like a cloth i guess in between the three

#

and then the 4th spot is kind of pushed out of the plane the 3 points occupy

#

making three triangles (excluding the triangle formed in the plane the three pins are) whose areas have a constant sum i guess

upper karma Aug 20, 2018, 12:29 AM

#

yeah it's a fictional cloth

#

it's basically like a 2D string instead of 1D string in a sense

#

so you will always have a little tetrahedron

#

except in the cases when it happens to lie in the same plane as the three points

clear haven Aug 20, 2018, 12:55 AM

#

ye

#

im curious

#

my first idea would be to try to make the outside edge of this shape where the cloth is still on the same plane

#

like

#

ehh

#

how to word

upper karma Aug 20, 2018, 1:33 AM

#

I am trying to figure out how to make it in some 3D software I'm trying to learn

#

so it's a good exercise

#

there's a isosurface node, so I'm thinking I'll make it by defining the 3 points and then making the magnitude of the 4 sides equal a constant, and see what it graphs me

prisma marlin Aug 21, 2018, 12:01 PM

#

📎 JPEG_20180821_200059.jpg

#

I forgot how to do 5i

#

Can someone show me hoe

#

How*

#

My greatest issue is the cos 90-x part

hard gale Aug 21, 2018, 12:06 PM

#

do you see something on here?

📎 unknown.png

#

@prisma marlin

prisma marlin Aug 21, 2018, 12:44 PM

#

@hard gale explain it pls

#

Wait

#

OH

#

Nvm you don't need to explain it

#

I think I get it now

#

Wait can you explain for tangent

#

Which is 5 ii

hard gale Aug 21, 2018, 1:04 PM

#

in the same way sin(90-x) = cos(x)

#

$$\tan(90-x) = \frac{\sin(90-x)}{\cos(90-x)}$$

charred spearBOT Aug 21, 2018, 1:05 PM

#

📎 latex.png

prisma marlin Aug 21, 2018, 1:06 PM

#

Oh yea

hard gale Aug 21, 2018, 1:07 PM

#

~~that's more for iii but eh~~

prisma marlin Aug 21, 2018, 1:07 PM

#

$$\tan(x) = \frac{\sin(x)}{\cos(x)}$$

charred spearBOT Aug 21, 2018, 1:07 PM

#

📎 latex.png

prisma marlin Aug 21, 2018, 1:07 PM

#

Right

sonic wren Aug 21, 2018, 2:00 PM

#

I wondered if you can calculate line-sphere (as an example) intersection in a non-euclidean space as example on a sphere, without converting to euclidean space on the way? (I have no real knowledge of this topic, so sorry if that question is dumb)

upper karma Aug 21, 2018, 2:07 PM

#

Copied from questions-1: Is there anyone who is well versed in the topic of Geometry and Topology? When it comes to Vertices, Edges, Faces, Meshes, 3 Dimensional Coordinates and UV Mapping as in 3D Modelling? I need a very complex formula.

#

I need a formula that can create Seams on any Mesh so that the resulting UV Layout will have 1 Island and no overlapping parts.

#

I'll probably rather ask in the 3d modelling community

upper karma Aug 21, 2018, 2:36 PM

#

Hi can someone help me understand this problem because im having a bad time trying to understand this bc i suck at math sangry

📎 Screen_Shot_2018-08-21_at_9.15.27_PM.png

thorn valley Aug 21, 2018, 2:49 PM

#

is there a specific part you don't get? @upper karma

upper karma Aug 21, 2018, 2:51 PM

#

question 2 because i dont understand what they mean by " drawing diagonals from one vertex only"

thorn valley Aug 21, 2018, 2:52 PM

#

it's like the diagram

#

you pick one point

#

and draw all lines connecting to other points

#

notice how all the lines are coming from A in the diagram

#

(A = one vertex only)

upper karma Aug 21, 2018, 2:56 PM

#

Oh so technically when drawing the lines it just has to be all coming from one vertex? i think i got it

thorn valley Aug 21, 2018, 2:56 PM

#

yup

visual trail Aug 21, 2018, 4:40 PM

#

vector as = 2(vector sc). how to i write vector es as a product of only vector ad and vector ab

📎 20180821_183732.jpg

#

(this is just a part of my task, the full task i to find the ratio of de and ec)

thick portal Aug 21, 2018, 4:55 PM

#

ASB ~ CSE

#

BS=2SE

torn linden Aug 23, 2018, 12:15 AM

#

Can someone walk me through this problem? I know it seems very basic but this is just something extra.

📎 5f626546-d2d5-49aa-8765-d5622581946c.jpg

keen aspen Aug 23, 2018, 1:25 AM

#

Okay so notice that they are vertical angles

#

So you can set the two pairs equal to each other

#

4y=y+x+10

#

And then you also have the two supplementary angles

#

So 4y+2x=180

#

Now you can do your soe

upper karma Aug 23, 2018, 12:20 PM

#

yes

upper karma Aug 23, 2018, 2:44 PM

#

@upper karma Pls leik if u agree

keen bison Aug 23, 2018, 5:33 PM

#

Hey guys

#

I'm taking a class in Geometry and Algebraic Topology

#

and I was curious as to whether or not you could provide some intuition to what the 'Holomorphic 1-Forms' are on a Riemann surface

#

The class coordinator has been following Foster's lectures on Riemann surfaces but after the second week she skipped like 200 pages and jumped straight into the section on vector bundles

#

so I don't really have much intuition on the topics we've been covering

#

I only just read upward in this chat and well

#

I think this sort of question doesn't belong here sorry

dreamy wraith Aug 23, 2018, 7:26 PM

#

i'd say this belongs here, just because other questions are simpler, it doesn't mean that your question doesn't belong here

keen bison Aug 23, 2018, 7:27 PM

#

Oh well that's good to hear

sly crane Aug 24, 2018, 3:48 PM

#

@keen bison Do you have intuition for what 1-forms on an ordinary smooth manifold are? Or are you asking specifically for intuition for the holomorphic condition?

keen bison Aug 24, 2018, 3:49 PM

#

No I have no intuition at all regarding those things

sly crane Aug 24, 2018, 4:04 PM

#

Basically, a 1-form is something which allows you to do line integrals on a manifold. In general, you can think of k-forms as the things which allow us integrate over k-submanifolds.

#

So, since Riemann surfaces are locally modeled with a complex structure, it is natural to consider 1-forms which are holomorphic (as opposed to just smooth).

#

This means doing line integrals along such forms shares (at least locally) many of the wonderful properties of complex integration.

#

In general, differential forms are very interesting algebraically. They have an operator called the exterior derivative d which takes k-forms to (k+1)-forms. This geometric picture of integrating by Stokes' Theorem: if omega is a k-form and M is a orientable (k+1)-manifold, then we have that $$\int_M d\omega = \int_{\partial M} \omega$$

charred spearBOT Aug 24, 2018, 4:23 PM

#

📎 latex.png

sly crane Aug 24, 2018, 4:29 PM

#

By De Rham's theorem, algebraically studying how the of the vector space of k-forms relate by the exterior derivative is the actually the same as computing the singular cohomology (with real coefficients) of our space.

#

Holomorphic 1-forms \omega have some cool algebraic properties. First, is that they are always closed, i.e., d \omega = 0. This follows from the Cauchy-Riemann equations. The Poincare lemma then tells us that locally \omega is exact, i.e., \omega = dg for some function g locally. But moreover, one can show that his g must be holomorphic.

#

Another cool algebraic property is that the dimension of the complex vector space of of holomorphic 1-forms on a Riemann surface is equal to the genus of the surface. By contrast, the vector space of ordinary 1-forms is infinite dimensional.

#

@keen bison I have to go. Hopefully this helps. If you have any questions, I should be able to answer them later.

neon fossil Aug 24, 2018, 7:39 PM

#

nice

forest dove Aug 24, 2018, 7:45 PM

#

Wait what?

#

How many fucking things are equal to the genus?

upper karma Aug 24, 2018, 11:58 PM

#

woah

neon fossil Aug 25, 2018, 2:08 AM

#

the genus determines a riemann surface topologically so probably a lot

thin wasp Aug 25, 2018, 2:14 AM

#

Could someone help me with this?

📎 geohomework.PNG

#

@thick portal I think this might be from your AIMO as well

#

I know the answer but I don't know how to get there

thin wasp Aug 25, 2018, 2:57 AM

#

<@&286206848099549185>

thin wasp Aug 25, 2018, 6:57 AM

#

^ I got the answer, never mind.

night lotus Aug 25, 2018, 6:04 PM

#

📎 JPEG_20180824_212917.png

#

I have all the answers, but I don't know how to get them for angles 4, 6, 7, 8, 9, 10, 11, and 12

#

I've been stuck on this problem for so long and I need some help

hard gale Aug 25, 2018, 6:06 PM

#

so 4

#

let's workout with arc AC

#

do you know what inscribed/central angles are?

night lotus Aug 25, 2018, 6:07 PM

#

yes

hard gale Aug 25, 2018, 6:07 PM

#

it's exactly this to find 4

#

you know m (arcAC) (central angle)

night lotus Aug 25, 2018, 6:08 PM

#

yea

hard gale Aug 25, 2018, 6:08 PM

#

and you want to find m<AFC, with F on the circle

#

<AFC is the inscribed angle here

night lotus Aug 25, 2018, 6:08 PM

#

so that would make it 80 degrees

#

?

hard gale Aug 25, 2018, 6:09 PM

#

80/2 degrees

#

1/2 of m(arcAC)

night lotus Aug 25, 2018, 6:09 PM

#

oooh ok

#

so 40

hard gale Aug 25, 2018, 6:10 PM

#

ye

night lotus Aug 25, 2018, 6:10 PM

#

okay

#

and angle 6 would be 20?

hard gale Aug 25, 2018, 6:11 PM

#

yep

#

7 is the same thing

night lotus Aug 25, 2018, 6:13 PM

#

what about angle 9?

#

would i do 75 + 40 + 40 +30 + 70?

#

divided by 2?

night lotus Aug 25, 2018, 6:30 PM

#

<@&286206848099549185> How do I find angles 9 and 12?

#

📎 JPEG_20180824_212917.png

upper ether Aug 25, 2018, 10:48 PM

#

if you know arc GE 9 is half of that p sure

#

why is angle 5 not 75

#

i literally cant see whats written for real and whats not

vague berry Aug 25, 2018, 11:53 PM

#

Hey could you guys help me with this problem?

📎 image.jpg

upper karma Aug 25, 2018, 11:54 PM

#

what have you tried

vague berry Aug 25, 2018, 11:54 PM

#

I got several answers

upper karma Aug 25, 2018, 11:54 PM

#

good

vague berry Aug 25, 2018, 11:55 PM

#

So what I did was counted the cubes as like corners

#

So each side would have a 2x2

#

For the corner

#

And that would be 4 2x2’s

#

*6

#

So 24

#

And then I had another approach of counting the edges corners and the middle pieces

upper karma Aug 25, 2018, 11:55 PM

#

hmm the corner thing seems suspicious to me

vague berry Aug 25, 2018, 11:56 PM

#

Oh wait

#

I just realized that

#

Ok

#

So 2x2 tight

#

Right

#

So then

upper karma Aug 25, 2018, 11:56 PM

#

I don't get where you're getting the 2x2 thing from

vague berry Aug 25, 2018, 11:56 PM

#

There would be 8 total

#

Hmm

#

Oh so

upper karma Aug 25, 2018, 11:56 PM

#

there are 8 corners I agree

vague berry Aug 25, 2018, 11:56 PM

#

Like I think the problem is asking for how many cubes are there

upper karma Aug 25, 2018, 11:57 PM

#

yep

vague berry Aug 25, 2018, 11:57 PM

#

Like a trick type of problem

upper karma Aug 25, 2018, 11:57 PM

#

how do you normally find area of a grid?

vague berry Aug 25, 2018, 11:57 PM

#

Instead of just counting 9*6

upper karma Aug 25, 2018, 11:57 PM

#

https://topdrawer.aamt.edu.au/var/aamt/storage/images/media/tdt/patterns/p_cm_m2_fig2/277898-1-eng-AU/P_CM_M2_fig2.jpg

#

for instance how many squares are in this picture?

#

you can count them right, but there's a faster way yeah

vague berry Aug 25, 2018, 11:57 PM

#

17

#

?

upper karma Aug 25, 2018, 11:58 PM

#

I just mean single blocks alone

#

1x1 squares

vague berry Aug 25, 2018, 11:58 PM

#

Oh 12

upper karma Aug 25, 2018, 11:58 PM

#

not trying to be fancy here, just the 2D version of your problem basically

#

ok so how'd you solve that to get 12

vague berry Aug 25, 2018, 11:58 PM

#

That’s it

#

Oh I did 4*3

upper karma Aug 25, 2018, 11:59 PM

#

right so for your shape you have how many layers of 3x3 cubes?

vague berry Aug 25, 2018, 11:59 PM

#

6?

upper karma Aug 25, 2018, 11:59 PM

#

3

vague berry Aug 25, 2018, 11:59 PM

#

I didn’t understand

#

Rephrase that question please?

upper karma Aug 25, 2018, 11:59 PM

#

http://prntscr.com/kmwgf3

Lightshot

Screenshot

Captured with Lightshot

vague berry Aug 26, 2018, 12:00 AM

#

Oh ok

#

So that’s what you mean

#

3 then

upper karma Aug 26, 2018, 12:00 AM

#

http://prntscr.com/kmwglm

Lightshot

Screenshot

Captured with Lightshot

#

since you know how to find one layer, it's just like finding the area, 3x3

#

but now you have 3 of these stacked up

#

that's how I think of it

vague berry Aug 26, 2018, 12:01 AM

#

27

#

?

upper karma Aug 26, 2018, 12:01 AM

#

yep

vague berry Aug 26, 2018, 12:01 AM

#

That’s it wow cool

upper karma Aug 26, 2018, 12:01 AM

#

yeah not too bad

vague berry Aug 26, 2018, 12:01 AM

#

Thank you

upper karma Aug 26, 2018, 12:01 AM

#

in fact if each cube was measured to be a cubic inch of volume

#

then you'd have exactly 27 cubic inches

#

since the shape it came from was 3in x 3in x 3in

#

=27 in^3

vague berry Aug 26, 2018, 12:02 AM

#

So all you had to do was find the volume of the cube

upper karma Aug 26, 2018, 12:02 AM

#

I don't know if that makes it more clear or not but this is literally counting the volume yeah

#

yep yep

vague berry Aug 26, 2018, 12:03 AM

#

So there aren’t like hidden cubes within the cube xD

upper karma Aug 26, 2018, 12:03 AM

#

nah lol

#

there could be in more complicated problems but it's not possible to stack cubes to the top like that and leave a hole inside

vague berry Aug 26, 2018, 12:04 AM

#

Cool thanks

upper karma Aug 26, 2018, 12:05 AM

#

yeah np

keen bison Aug 26, 2018, 6:27 AM

#

@sly crane Hey thanks man! That was really helpful

upper karma Aug 27, 2018, 3:03 AM

#

📎 Screenshot_28.png

#

hey

#

sorry

#

for this kind of question

#

i should make 2 equations

#

and make them equal each other right

zenith ember Aug 27, 2018, 4:01 AM

#

Yes to writing two equations.

#

Probably no to setting them equal ...

kind ravine Aug 27, 2018, 7:58 AM

#

group

upper karma Aug 27, 2018, 11:36 PM

#

So, exercise from Hatcher 1.1. Show that Borsuk-Ulam applies to the 2-torus. Ie. any continuous function, pair of antipodal points, blah blah blah.

A brainlet answer here might be to just say "hey, this works for S1 and since S1^2 is just a product topology of S1 it follows that Borsuck-Ulam applies component wise to each of the S1s, so S1^2 has an extension of Borsuk-Ulam". Is there any more intelligent/rigorous of a way to do this without effectively recreating the proof of Borsuk-Ulam for S1 provided in Hatcher?

sly crane Aug 27, 2018, 11:57 PM

#

your argument doesn't work. maps T -> R^2 (where T is the torus) don't necessary decompose as a product of two maps S^1 -> R.

#

and actually, Borsuk-Ulam does not work for the torus

forest dove Aug 27, 2018, 11:59 PM

#

Turns out the exercise was asking whether it holds. He had his question answered elsewhere: it doesn't

upper karma Aug 29, 2018, 4:51 AM

#

Given that you can't cut a square up into an odd number of triangles of all equal area, can you cut up this 3x3 square without a 1x1 center square into an odd number of triangles of all equal area?

#

http://prntscr.com/ko2yd8

Lightshot

Screenshot

Captured with Lightshot

upper karma Aug 31, 2018, 4:21 AM

#

Can someone help me with his question?

📎 image0.jpg

umbral snow Aug 31, 2018, 4:23 AM

#

You have that
4y + 4x - 10 = 90
8x + 3 + 6y - 1 = 180

#

That's a system of two equations with two unknowns, know how to solve?

upper karma Aug 31, 2018, 4:24 AM

#

not really

umbral snow Aug 31, 2018, 4:29 AM

#

Simplfy:
4x + 4y = 100
8x + 6y = 178

Multiply the first equation by 2:
8x + 8y = 200
8x + 6y = 178

Subtract downward:
2y = 22
y = 11

Solve either for x:
x = 14

upper karma Aug 31, 2018, 4:30 AM

#

thx so much

umbral snow Aug 31, 2018, 4:31 AM

#

Note it does not bisect. Very close though

upper karma Aug 31, 2018, 4:31 AM

#

ok

upper karma Aug 31, 2018, 4:47 AM

#

How do I finish this?

📎 image0.jpg

umbral snow Aug 31, 2018, 4:51 AM

#

3y + 9x = 82
9x = 82 - 3y
x = (82 - 3y)/9

undone shell Aug 31, 2018, 5:19 AM

#

solve through substitution

#

so you see what he did was he moved all the x terms to the left

#

then simplified them and found x in terms of y

upper karma Aug 31, 2018, 5:30 AM

#

Ya I got it but I have another question

umbral snow Aug 31, 2018, 5:31 AM

#

Go for it

upper karma Aug 31, 2018, 5:32 AM

#

Do I divid by -1?

📎 image0.jpg

umbral snow Aug 31, 2018, 5:37 AM

#

The first line should actually read
90 < 22 - x < 180

You can also multiply by -1. Note that multiplying by a negative will flip inequalities, so you have:
-158 < x < - 68

hushed fulcrum Sep 1, 2018, 12:51 PM

#

anyone good with rectifying planes nonzero unit speed curves?

deft badger Sep 1, 2018, 4:14 PM

#

Hey, guys. I apologize if this isn't the appropriate channel for this problem. I'll give some more information on the specifics.

📎 33.PNG

#

I need to find the vector X

#

Only known information is Vector(A), Vector(B), and the radii of the circles.

upper karma Sep 1, 2018, 4:17 PM

#

X just appears to lie on your circle thonker

#

Is this moving or something?

deft badger Sep 1, 2018, 4:18 PM

#

Yes, it can move along the circle it lies on

#

So can B

upper karma Sep 1, 2018, 4:18 PM

#

Well in that case

#

This is pretty straight forward, OA+AX = OX

deft badger Sep 1, 2018, 4:20 PM

#

I don't know where X lies

#

That is just a visualization

upper karma Sep 1, 2018, 4:21 PM

#

hm

#

Well

#

The good news is that there are only two possible positions of X with just your information alone

deft badger Sep 1, 2018, 4:23 PM

#

Yes, I can eliminate the other one

upper karma Sep 1, 2018, 4:23 PM

#

So ye

deft badger Sep 1, 2018, 4:23 PM

#

I need the formula 😉

upper karma Sep 1, 2018, 4:24 PM

#

Heh, it'll be some function of the radii

#

You know ALL the circles radii?

deft badger Sep 1, 2018, 4:24 PM

#

Yes

upper karma Sep 1, 2018, 4:26 PM

#

Tbh, someone might pen it up for you

#

But the effective process is that you'd start with a center

#

You then know that you need a circle constrained between the inner most and outermost ring. It follows that only two possible circles with this constraint intersect B

#

From here, you get two centers

#

Idk how you can eliminate one of those as I haven't thought that far ahead yet.

deft badger Sep 1, 2018, 4:28 PM

#

I can check the signed angle

#

of both points

upper karma Sep 1, 2018, 4:28 PM

#

Then you have an algorithmic plan

deft badger Sep 1, 2018, 4:31 PM

#

I don't see how this translates into a function

upper karma Sep 1, 2018, 4:32 PM

#

Yeah, you're on your own there. But this'll be a process with a function

deft badger Sep 1, 2018, 4:34 PM

#

I mean, I have the answer, the straight formula. Just have to dig through some old code. I just want to understand the numbers rather than for the function to just work

upper karma Sep 1, 2018, 4:34 PM

#

Well that's the reasoning

deft badger Sep 1, 2018, 4:35 PM

#

Perhaps if I "rephrase" the question

#

📎 332.PNG

#

Imagine AB and BC as "limbs"

#

With A being the hip joint and B being the knee joint

#

How then, do you move C (Then being the foot in this euphamism) such that it reaches the position of the target?

#

We can see the first step being rotating B around A so that it's distance to the target is equal to BC's distance

#

By limiting the angle of ABC between 0 and 90, we can eliminate one of the two possible positions

#

Then we simply rotate C around B so it reaches it's target

#

Now

#

The problem is in the first step, how do we know by how much should we rotate B around A?

upper karma Sep 1, 2018, 4:46 PM

#

hi

mossy vine Sep 1, 2018, 4:47 PM

#

if you don't mind a geometrical solution

#

draw a circle centered at the target with the same radius as the circle centered at B

#

the intersection with the circle that is the locus of B is where B should be

#

if you like, you want the intersection of the points that are the length of the shin from the target and the points that the knee joint can be at

deft badger Sep 1, 2018, 4:51 PM

#

Might be useful to keep in mind, but I need a function

#

This is a programming task

mossy vine Sep 1, 2018, 4:52 PM

#

then create a function from that...

upper karma Sep 1, 2018, 4:52 PM

#

^

mossy vine Sep 1, 2018, 4:53 PM

#

you have two circles which you have the equation of, you can solve to find their intersection(s)

deft badger Sep 1, 2018, 5:03 PM

#

Alright, will do. Thanks

white linden Sep 2, 2018, 3:35 PM

#

📎 IMG_20180902_210235.jpg

#

Can some do the IIIrd one

upper karma Sep 2, 2018, 6:25 PM

#

which part is it that is giving you trouble, the area of the trapezoid on the bottom?

#

for anyone with topology experience, i have a noob question:

#

📎 Screen_Shot_2018-09-02_at_14.26.09.png

#

sorry for small picture, but my understanding is that deMorgan's laws would produce the opposite statement

#

so i'm having trouble figuring out what is special about R^n that would make this the case

umbral snow Sep 2, 2018, 7:37 PM

#

Let A and B be closed sets. Then A', B' (which I'll let be the complements) are open.

We know that A' ∩ B' is open by definition. So (A' ∩ B')' is closed

#

And I believe there's no "countable" condition on intersections

#

That means A ∪ B is closed by DeMorgan

#

@upper karma

#

Nothing special about Rⁿ here, this is generally true

upper karma Sep 2, 2018, 7:41 PM

#

yeah that's the problem I was having; because the question seems to imply the opposite case

#

i emailed him asking if its a type-o

umbral snow Sep 2, 2018, 7:42 PM

#

Nope! I got it backwards. You can only intersect finitely many open sets

#

So you can only union finitely many closed sets by the logic above

upper karma Sep 2, 2018, 7:42 PM

#

i understood what you meant as this was the logic I was using as well

#

i'm just going to wait for his reply for now

#

thanks for the help!

umbral snow Sep 2, 2018, 7:44 PM

#

Agreed, I think he got it backwards w
As well. Good luck!

serene dirge Sep 3, 2018, 12:38 AM

#

Hey can anyone prove that a set in which every infinite subset contains a limit point is compact. The converse is really easy, but I'm having a lot of trouble proving Weierstrass property => compactness

#

I'm thinking maybe I do a contradiction and assume that the set isn't compact, and thus has some open cover without a finite subcover, but i don't know how to connect that to the limit point concept in order to show that it conflicts.

forest dove Sep 3, 2018, 9:34 PM

#

@serene dirge have you figured it out yet or no?

#

I guess I'll just rant anyway and if it's useful it's useful. So, it'll be easier for me to phrase this in terms of sequences

#

I'll start by saying it should be more or less clear that "Every infinite subset has a limit point" <=> "Every sequence has a convergent subsequence"

serene dirge Sep 3, 2018, 9:36 PM

#

Yea, that's the equivalent statement of the B.W theorem'

#

Go on!

#

(In a nice way, not an aggressive one haha)

forest dove Sep 3, 2018, 9:38 PM

#

Yeah for sure

#

I'm thinking it might be worthwhile to throw in the stuff about complete + totally bounded?

#

That feels like a nice intermediate that could help prove it

#

(I've seen the proof some time ago but don't remember how it goes so I'm kinda feeling it out, do bear with me if I'm at all slow/scattered)

serene dirge Sep 3, 2018, 9:39 PM

#

It

#

is a hard proof

#

My real analysis professor actually wasn't sure either haha.

forest dove Sep 3, 2018, 9:40 PM

#

So, sequential compactness obviously implies completeness, because if you're Cauchy and have a convergent subsequence, you converge

serene dirge Sep 3, 2018, 9:40 PM

#

Oh btw, just to be clear, I'm using the equivalence class of cauchy sequence approach to our fields, and not the dedekind cut approach. Just in case that's relevant.

#

Right

forest dove Sep 3, 2018, 9:42 PM

#

It won't be relevant, as long as you have some field satisfying the axioms you're good

serene dirge Sep 3, 2018, 9:42 PM

#

Alrighty. Never hurts to clarify though.

forest dove Sep 3, 2018, 9:43 PM

#

Okay so assume you're not totally bounded, meaning there's some real number epsilon > 0 such that you can't cover your space with balls of radius epsilon

serene dirge Sep 3, 2018, 9:43 PM

#

Ok so we are doing contradiction then?

forest dove Sep 3, 2018, 9:43 PM

#

Uh, for this intermediate step yeah

#

I don't really tend to worry too much about whether a proof is by contradiction or contrapositive and all, I just kinda roll with the logic 😛

serene dirge Sep 3, 2018, 9:44 PM

#

Totally, I wasn't criticizing at all! Sorry I'm like, hypercaffenated right now, so I'm feeling very inquisitive.

forest dove Sep 3, 2018, 9:45 PM

#

So yeah, if you're not totally bounded, take an epsilon such that you can't cover the space with finitely many epsilon-balls. Take a point x_1, take a ball around it of radius epsilon. Take x_2 outside that ball, cover that in a ball of radius epsilon. Keep going

#

This process never terminates, since you can't cover the space with finitely many balls. So this gives us a sequence

#

But there's not even a Cauchy subsequence because for any i and j, d(x_i,x_j) > epsilon

#

So if you're sequentially compact, then you're complete and totally bounded

serene dirge Sep 3, 2018, 9:47 PM

#

Ooh right, and then just choose epsilon as the minimum ball we picked for our x's and we violate the cauchy criterion.

#

Hence contradicting the notion that there exists a convergent sequ... actually hold on

forest dove Sep 3, 2018, 9:47 PM

#

Well, I chose the same epsilon for each

#

Like, I fix epsilon. Then I take B_epsilon(x_1). Then choose x_2, then take B_epsilon(x_2). And so on

serene dirge Sep 3, 2018, 9:48 PM

#

Oh, ok I see.

#

So our neighborhoods aren't necessarily disjoint though right?

#

We're only ensuring that the x's lie outside of each others balls.

forest dove Sep 3, 2018, 9:49 PM

#

Yeah exactly, the neighborhoods aren't disjoint, but the x's themselves are at least epsilon from each other because the centers never overlap

#

Makes sense?

serene dirge Sep 3, 2018, 9:49 PM

#

Yep

forest dove Sep 3, 2018, 9:50 PM

#

So now it suffices to show that complete + totally bounded implies compact

#

For what it's worth this feels like it's gonna be by contradiction

#

Actually I'm not even 100% sure if this intermediate step is needed, we'll see

#

So let's say you have an open cover with no finite subcover

serene dirge Sep 3, 2018, 9:53 PM

#

And a sequence {x_n} in the set, presumably

#

We just need to show that for any generic sequence, the fact that we don't have a finite subcover is inhibitory to convergence.

forest dove Sep 3, 2018, 10:04 PM

#

Ohhhh

#

Okay so there are a bunch of problems in Rudin about this, I think you need to use a base for the space

#

Let me pull it up, I think I was assigned those problems in analysis

#

Okay so

#

Let's say you have a metric space

#

If you give me a bunch of open sets V_alpha, that's called a base if every open set is a union of some of them

#

Now, a metric space is separable if it has a countable dense subset

#

Separable metric spaces have countable bases, because you take the countable dense subset and take the epsilon balls for epsilon rational

#

Now, if every infinite subset of a metric space has a limit point, it's separable

#

The reason is that it's totally bounded, as proven above. Choose a (finite) cover of the space of balls of radius 1/n. Take the centers, keep going. That's a countable dense subset

#

So, you know spaces where infinite subsets have limit points have countable bases. So now you take an open cover, you can find a countable subcover. How? Take the countable base, G_1, G_2,...

#

Choose an open set in the cover containing G_1. Then choose an open set in the cover containing G_2. etc

#

So now assume your countable subcover, let's say {O_1,...}

#

Doesn't have a finite subcover

#

Let F_n be the complement of O_1 \cup ... \cup O_n

#

Then F_n is not empty for any n, but the intersection of the F_n is empty

#

But now choose x_i \in F_i

#

And I think the point is this is gonna be infinite, has a limit point, but that'd be in the intersection of the F_n, which is not empty, so rip

#

Anyway gotta go, but hope this helped @serene dirge. Sorry to rush out

upper karma Sep 4, 2018, 4:14 AM

#

Hello, I am looking for help with projecting 3D figures onto a 2D plane using a 2D game engine. I've done a fair bit of research and nothing really helps...

#

Figures as in, an array of vectors.

errant ether Sep 4, 2018, 4:33 AM

#

Projecting a vector onto the normal of the plane then subtracting the resultant projection from the original vector should give you the portion of the vector lying on the plane

upper karma Sep 4, 2018, 4:41 AM

#

(im trying to understand what you just said)

errant ether Sep 4, 2018, 4:48 AM

#

So you have a set of vectors, each of which you want to project onto a plane, right? Each plane has a unique normal vector determined by its coefficients; for a plane ax + by + cz = d, the normal vector n is <a, b, c>. For your vector r, the vector projection onto n is n*n·r/(|n|^2)

upper karma Sep 4, 2018, 5:11 AM

#

k ty

errant ether Sep 4, 2018, 5:12 AM

#

np, don't forget that the formula I wrote is for the projection, which you have to subtract from r

upper karma Sep 4, 2018, 5:13 AM

#

yeh

serene dirge Sep 4, 2018, 8:48 AM

#

@forest dove Hey, sorry I had to go. Tremendous thanks for your help. I think I'll go through those exercises myself and see if I can't figure it out fully. Huge thanks again!

median mirage Sep 4, 2018, 9:03 PM

#

Hi,
I move my question over to this server since I guess it's not really an advanced math question.

say I have a point A as well as a circle B. I now want to enclose the circle with a cone relative to A (i.e. the tangents, λr & λf, in the attached image) What's the approach to calculate said tangents ? I know the position of A, the center of B as well as the radius of B.

📎 Unbenannt.png

mossy vine Sep 4, 2018, 9:06 PM

#

the length of the tangents? or the lines specifically?

median mirage Sep 4, 2018, 9:08 PM

#

i don't care about the length

#

I pretty much want the directional vectors

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if that makes sense

mossy vine Sep 4, 2018, 9:08 PM

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well we can write any line in the plane as ax+by=c

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rearrange for y in terms of x

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substitute into (x-m)^2 + (y-n^2) = r^2

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should get a quadratic in x

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set discriminant = 0

median mirage Sep 4, 2018, 9:11 PM

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rearrange for y in terms of x what does that mean ?

mossy vine Sep 4, 2018, 9:11 PM

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y = (c-ax)/b

median mirage Sep 4, 2018, 9:13 PM

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ah, sorry I'm not a native english speaker. What does substitute intomean ? what do I substitute ?

humble ibex Sep 4, 2018, 9:14 PM

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So you replace all of your y values with (c-ax)/b

mossy vine Sep 4, 2018, 9:15 PM

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yeah, that

median mirage Sep 4, 2018, 9:15 PM

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oh, but what's m now ?

mossy vine Sep 4, 2018, 9:16 PM

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centre of the circle is (m,n)

median mirage Sep 4, 2018, 9:20 PM

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does that mean the resulting function is
(x-m)^2 + ( ((c-ax)/b) -n^2) = r^2
?

humble ibex Sep 4, 2018, 9:21 PM

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(x-m)^2 + ( ((c-ax)/b) -n)^2 = r^2
Should be this I think

median mirage Sep 4, 2018, 9:23 PM

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set discriminant = 0
and this means to get r^2 on the other side of the equals sign, and replace it with 0 ?

delicate swan Sep 5, 2018, 3:15 AM

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I feel like there has to be a word for this idea, does anyone know it?

Basically, to me it feels like length, area, and volume are just different specific cases of a broader category of "how much space does this take up."

Length is the one-dimensional version

Area is the two-dimensional version

Volume is the three-dimensional version

These things feel very strongly related. What is the abstract version, the word for the nth-dimensional form of this idea?

umbral snow Sep 5, 2018, 4:03 AM

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You can generalize each as the n-th iterated integral within an n-dimensional manifold. This allows one to define these ideas for upper dimensions.

delicate swan Sep 5, 2018, 4:10 AM

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Yep, that'll work, but I'm most interested in what this category of "thing" is called

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A sort of fill-in-the-blank:

"Length is the one-dimensional version of ____"

zenith ember Sep 5, 2018, 4:11 AM

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The term that comes to my mind for higher dimensions is n-volume

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But I doubt that's satisfying.

delicate swan Sep 5, 2018, 4:13 AM

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Ahhh, see, I suspected that from my quick wikipedia dive

But seeing as how "volume" is "the 3-dimensional version of <thing>," it is an unsatisfying word choice to use a specific case as the name of the general phenomenon

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I was hopeful there was something more flowery, even if it would never be used in any practical context.

zenith ember Sep 5, 2018, 4:14 AM

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Alternatively, you could think of volume as the general term, but we have unique other terms for the 2 and 1 dimensional versions.

errant ether Sep 5, 2018, 4:14 AM

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^ that's how I like to think of it

delicate swan Sep 5, 2018, 4:15 AM

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You know what, I like that --- I think I'll switch to that mentality. Thank you!

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Hm, but then "volume" becomes ambiguous with "3-volume"

Oh boy

upper karma Sep 6, 2018, 1:29 AM

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Does anybody here have an opinion on either of Spanier's or Rotman's textbooks on algebraic topology?

forest dove Sep 6, 2018, 1:59 AM

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I've started using Rotman, didn't yet dedicate enough time to get far but I really like it

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Spanier's apparently really hard

upper karma Sep 6, 2018, 5:34 AM

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Are the first 4 chapters of Munkres enough general topology to learn algebraic topology?

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Or must I also learn the metrization theorems, complete metric/function spaces, and baire spaces?

forest dove Sep 6, 2018, 5:55 AM

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Probably not. Function spaces meaning compact-open topology is relevant in a way but I think it can be black boxed

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Basically, you think of currying functions, like how a function from X x Y to Z is a function from X to the set of functions from Y to Z

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I think there's some weird point-set condition that practically everything satisfies (compactly generated and weak Hausdorff are words I've heard to this effect) where if you put the compact open topology on function spaces, this is a homeomorphism

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Otherwise I think you can go about life nicely without worrying about fancy point-set

sly crane Sep 6, 2018, 6:57 AM

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@upper karma first 4 chapters of munkres should definitely be enough to start learning alg top. you can always learn those other things later when you come across them later (which you probably won't for a while).

upper karma Sep 6, 2018, 7:00 AM

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@forest dove @sly crane thank you for the guidance

sly crane Sep 6, 2018, 7:04 AM

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also, i'd look into hatcher's book. it is really good.

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and if you are interested in a more differential point view, bredon's topology and geometry is great

upper karma Sep 6, 2018, 7:18 AM

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@sly crane I heard that Hatcher's use of Delta-complexes is uncommon and that he doesn't use as much rigor or category theory as other authors

sly crane Sep 6, 2018, 4:14 PM

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@upper karma hatcher is not lacking rigor at all. his book is just full of other things as well (examples, motivation, intuition). hatcher may not use as much category theory, but he definitely introduces when it is really necessary.

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also, i would not worry about the delta-complex thing. delta-complexes are just a small generalization of simplicial complexes which lead to much more efficient computations. for example, the most efficient simplicial complex structure for a torus has 14 triangles, 21 edges, and 7 vertices, while the natural delta-complex structure has 2 triangles, 3 edges, and 1 vertex. but once you learn how to compute homology with CW-complexes, all this pretty much won't matter.

forest dove Sep 6, 2018, 6:18 PM

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@sly crane what is it about CW complexes that makes them seemingly preferred to simplicial anyway?

sly crane Sep 6, 2018, 6:28 PM

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do you want to think of the 9-dimensional sphere S^9 as being composed of 11 9-faces, 55 8-faces, 165 7-faces, 330 6-faces, 462 5-faces, 330 4-faces, 165 triangles, 55 edges, and 11 points glued in a certain way?

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or as the 9-dimensional ball with the boundary identified to a point?

forest dove Sep 6, 2018, 6:31 PM

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Ah, I didn't know that simplicial was that bad. And I imagine these delta complexes are still less efficient on the whole?

sly crane Sep 6, 2018, 6:38 PM

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yeah, delta-complexes are more general than simplicial complexes, but cw-complexes are much more general than both. they are all similar in that they are ways of build spaces by inductively gluing n-dimensional cells, but they just differ on how you are allowed to glue things

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they each have their own advantages and disadvantages

forest dove Sep 6, 2018, 6:45 PM

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I see

upper karma Sep 7, 2018, 11:46 PM

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What type of line has an undefined gradient?

zenith ember Sep 7, 2018, 11:57 PM

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Gradients are delta Y/ delta X

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When is that undefined?

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What sort of fractions aren't allowed?

upper karma Sep 8, 2018, 8:08 PM

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ok the drawing is cancer dont judge but the equations should be fine

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idk if its a correct start but i cant seem to solve them up

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https://cdn.discordapp.com/attachments/138016655874719744/488047532170018845/unknown.png

umbral snow Sep 8, 2018, 8:10 PM

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What are we actually solving for?