point-set-topology

[signal venture]

i think i just need to look at examples now to round out my understanding. hartshorne doesn't give many at all. maybe i'll look at that complex log one in detail

[marsh forge]

might help to look up an alternate construction and compare as well

im pretty sure there are at least 2?

[honest narwhal]

Etale spaces are nice

[signal venture]

i guess this is what you're referring to

[noble dock]

Am I correct in thinking that, within the space defined by the 2d surface of a square flat torus in R4, no finite amount of points defines a unique line?

and by “square” I mean that if you unfolded it into a rectangle that rectangle would be a square

For example, imagine you’re given 2 points on a torus that, unfolded into a square, has side length S

like

if you have a square of the 2d plane

and fold it in R4 so it’s like Pac-man in how it loops around

I was under the impression the resulting shape was called a flat torus, or am I wrong?

doesn’t it stretch in R3?

okay then

so that object

say it unfolds into a square with side length S

and you are given two points on it

Define one point as (0,0) and the other as (x,y)

then I would think you could have lines with slopes y/x, or (y+S)/x, or... etc etc, all of which pass through both points

am I correct in thinking that?

for the special case of 2 points?

um

I think it’s the Clifford torus, though I don’t know enough terminology to say for sure

um

I don’t understand the question

The embedding, I think

[gentle osprey]

slimvesus:

[noble dock]

that’s the one that would fold a square so the opposite edges match, right?

oh

whichever one is the one where opposite edges of a square are the same line

and not twisted

oh sorry

quotient of the plane

because for example

[marsh forge]

what the exact question here

[noble dock]

Am I correct in thinking that, within the space defined by the 2d surface of a square flat torus in R4, no finite amount of points defines a unique line?

and he helped me better define what I meant by the torus

[marsh forge]

what is a square flat torus

do you mean a torus?

[noble dock]

if you take a square in the plane, and fold it into a torus

[marsh forge]

sure

thats a torus

no adjectives

[noble dock]

o

sorry

[marsh forge]

now what do you mean by lines?

[dim meadow]

Flat torus is a thing

If you care about curvature

[marsh forge]

ah I see

[noble dock]

^

I don’t fully understand your question what do I mean by a line

[marsh forge]

Yeah okay

I mean more or less what I said unless this term is obvious from a diffgeo perspective

[dim meadow]

And the embedding of the torus in R^3 has a different metric

[marsh forge]

gotcha

[noble dock]

like, imagine you have a square of the plane

[marsh forge]

okay

[noble dock]

and it wraps around like a torus

[marsh forge]

sure

[noble dock]

you start at one point, and you choose a fixed direction to go in until you hit the other point

the line would be your path

well technically the line is infinitely long, not just that segment, but you know what I mean

I’d think any finite number of points allows for infinite lines that cross through them all

[marsh forge]

I think you need more structure here

there are certainly infinitely many maps [0,1]\to T^2

crossing through any finite set of points

[noble dock]

the line is straight sorry

[marsh forge]

but this is all you get if you just take a torus to be a folded plane

If your 'straighness' is induced by R^2 then

[noble dock]

mhm

[marsh forge]

you can take the line to be the unique one in R^2

[noble dock]

but it can wrap

[marsh forge]

Is it allowed to stop?

Basically take your line in the torus

[noble dock]

no, in either direction it doesn’t stop

[marsh forge]

it must lift to the universal cover

this is unique

[noble dock]

so for example

[dim meadow]

Unique up to a basepoint

[noble dock]

let’s say the square is of side length 1

[marsh forge]

we're literally prescribing 2 basepoints liquid lol

[dim meadow]

Okay sure

[noble dock]

and the points on the square are (0,0) and (0,1)

[marsh forge]

actually no

i see your point

i was assuming a fixed basepoint in the [0,1]x[0,1]

[noble dock]

then the straight line from the lower left to the lower right corner would be one

but the straight line from the lower left to upper right corner would be another

[dim meadow]

Yeah there is one lift for every point in the preimage of that basepoint

[noble dock]

is that in response to me or Equivariant?

[dim meadow]

Equivariant

[marsh forge]

no i mean i was fixing

that lift

of the bp

[dim meadow]

Okay

[marsh forge]

anyway

[noble dock]

do you understand my question?

[marsh forge]

im double checking to make sure liquids point isn't clouding my thinking

[dim meadow]

Sorry lol

[marsh forge]

no no i mean

i might be wrong

me too

but in theory we have a choice of lift

OH

but they are all the same

once we pass

yeah

yeah yeah i got there and forgot about it sorry

in a lecture rn

[noble dock]

Okay so like

[old sapphire]

ooo look it’s @marsh forge

N E A T

[noble dock]

Imagine you take the square from (0,0) to (1,1)

and map it onto a torus

Now imagine you take the points (0,0) and (0.5, 1)

that second point is the same as the point “0.5,0”

yes glue the sides

[old sapphire]

Sounds like a question I still have

[marsh forge]

there are infinitely many lifts in the plane that all define the same map in the torus

unless im losing it

[old sapphire]

Is there a space-filling curve that is “toroidal” and resolution independent - i.e. preserves the donut-ness while filling the space and giving the same general point regardless of resolution

[noble dock]

so if you were to look at the equation of the line y = 2x, that line crosses both points

and so would the line y = 0

[marsh forge]

not to give you the same answer again polite

but

a square transforms very nicely into a torus

and you can ofc spacefil a square

[noble dock]

but those two lines cross different sets of points, because the line y = 2x crosses the point (0.25,0.5) and the line y=0 doesn’t

so they’re different lines, right?

[marsh forge]

I will think about tihs when i can actually focus on it

[old sapphire]

I think I will use a fractal five pointed star for my own [0. 1) => (x, y) disk filling curve

T H A N K S @eq

@marsh forge

[marsh forge]

But if you take a square and designate two points, there is a unique line going through both. Likewise any such line in a torus lifts

we don't even need to pass to R^2

square is fine

[noble dock]

Didn’t I just demonstrate that there isn’t a unique line passing through both though?

[marsh forge]

oh wait

duh

sorry

i cant think in coordinates while multitasking

so i ignore that

I actually think you're correct

You can basically abuse the wraparound

[noble dock]

but if you’re on a torus, you can always define one point to be (0,0), right?

and let’s say you have the points (.25,.25) and (.75,.75)

then obviously the line y=x passes through both

that’s not right either why am I failing tbis

sorry that’s not the right line hang on a sec

[marsh forge]

there is a unique line that hits the second point before the boundary of the square

i think

but i dont think its actually unique

oh duh

ofc it is hahaha

thanks for rephrasing it in a much smarter what haha

[noble dock]

y=2.5x - .375

[tired lotus]

An icosahedron (ike) is to a hexacosichoron (ex).
@west brook is that correct for what icosahedron is to?

[marsh forge]

oh youre right

lmao

im all over the place

im literally not capable of thinking about anything not up to homeo

yeah okay

i think we are both correct

but describing different distinct lines

but

i dont think the lines are unique

[noble dock]

I think shortest line is unique, but there are infinite distinct lines

[marsh forge]

yeah

notch i think you're probably correct

[noble dock]

am I right about that?

[marsh forge]

basically you can get inf many by changing how fast you wrap around

[noble dock]

And I think that applies in the more general case of n points for finite n, not just 2 points right?

yeah exactly

[marsh forge]

yeah that worried me

[noble dock]

pi/2 is a number

[marsh forge]

and is why i said probably

[noble dock]

irrational slope wouldn’t be an issue I’d think

[marsh forge]

i dont think so either

certainly if we can wrap around and get to the second point

we can do that twice and three times and four times... as fast

[noble dock]

(0,0) to (1/pi, 1) can have the lines y=0 and y = pi easily

[marsh forge]

and we should be able to wrap around once

oh ive given up on shortest

[noble dock]

you mean like (.999, .001) to (.001,.002) has shortest line wrapping once?

[marsh forge]

please dear god

stop using coordinates

lolll

[noble dock]

sorry

[marsh forge]

just take the origin and (0,.999)

they are almost identified

[west brook]

Yep.
https://polytope.miraheze.org/wiki/Hexacosichoron

[noble dock]

why can I ask?

[marsh forge]

bc it makes me think longer than

"point near the corner"

[noble dock]

well okay lol

fine

so new question

does this always work with finitely many points

actually evidently not

some sets of three points would have zero lines

[marsh forge]

n should be enough

n>>3

[noble dock]

but I’m not sure if it would be limited to 0 and infinity, or if there are cases where there would be just 1 (or even finitely positively many) lines

[marsh forge]

mm

no

wait

you can do the same period trick

in at least some cases

[noble dock]

if the line between the first two points is perpendicular to the line between the second two, I don’t think there’s any lines, right?

[marsh forge]

this gets

increasingly more complicated

and might start depending more on a choice of 'torus'

[noble dock]

like (0.25,0.75), (0.75, 0.75), and (0.75, 0.25)

I don’t think any line can hit all three of those

[marsh forge]

i think there is an interesting theory here but i think it should be nontrivial

[noble dock]

that line doesn’t hit (.75,.75)

[noble dock]

and I think an easy way of looking at it

can you help me phrase it properly on math stackexchange?

and to be clear in a rectangle I think there’s always infinitely many

it’s only in the case of a square that I think the 0-line option is possible

and I don’t know how to phrase “the surface of a torus defined by wrapping a square”

[gentle osprey]

slimvesus:

[noble dock]

Idk if the n>2 is required

why not just n

I think it may be easy to show why the 0-case holds

On the square, there are three cases

-The line travels parallel to the x-axis. In this case, it cannot, by definition, travel along the y-axis at all, so if there are two points of differing y-values, this line cannot hit them both

-the same thing but with x and y swapped

-the line does not travel parallel to either axis. In this case, either it passes through every x-value exactly one time before returning to where it started, or it passes through every y-value exactly one time before returning to where it started. In the case where it passes through every x-value exactly one time, then if two points lie on the same x-value, it cannot pass through them both. Same with y. Therefore, if there exist points such that two points lie on the same x-value, and two points lie on the same y-value, no line can pass through them all.

To be clear I don’t think right angles in general are an issue

just when they’re angled so the lines are parallel to the axes

although... hm

irrational numbers may be an issue

no it isn’t

?

wdym

can you give a counterexample?

before retracing its steps

like

sorry I’m trying to figure out how to phrase it

you don’t count the same point twice

yes

are you sure?

right, that’s why I said irrational numbers do present a problem

but on rational slopes that holds

[marsh forge]

ok ok i think ive figured out a covering approach to the general problem: the set of lines through a finite set of points in the torus is the set of straight lines in R^2 that intersect any of the lifts of these points

thats what was throwing me so off

So the question should be approachable entirely in R^2

[noble dock]

(I’m more curious about rational points anyway)

I’m not actually sure how irrational slopes work in this context

for example, if you have the points (0,0.5) and (0,0.75), does an irrational slope from one ever hit the other?

after how much time?

I don’t think it is finite is the thing

and here’s why-

the distance between the x-components of the points is rational

the distance between the y-components is rational

so by definition, the slope is rational

[marsh forge]

You should be able to tackle this with the fact I posted above. The question is whether an irrational slope can intersect a point that is mod Z^2 equivalent

this should just be whether a certain equation in R^2 can ever hold

[noble dock]

I don’t know what a lift is

so silly question

[marsh forge]

Do you know what R^2/Z^2 means

in this context anyway

[noble dock]

not really

[marsh forge]

ah okay

let me think about this explicitly for a second

[noble dock]

I don’t know the terminology

here’s a question

does this give a method to determine whether e-pi is rational?

are the dense lines all the same line?

how can a dense line miss a point?

yes

well

not in topology

[marsh forge]

Im not sure density helps here

because we care about a specific point

not up to some error term

but anyway we can 'characterize' the possible slopes

[noble dock]

about that

I haven’t come across it in analysis yet either

[marsh forge]

when is there a line with slope alpha

such that

(x_1 + a_1, y_1 + b_1) + t(alpha_1, alpha_2) = (x_2 + a_2, y_2 + b_2) has a solution

where the a,b are integers

[noble dock]

I assume it means that when you draw the line, if you take the any two points on the line with the same x-coordinate, say (x, y1) and (x,y2), then there is a third point on the line (x,y3) such that y1 < y3 < y2, and the same with x and y swapped

[marsh forge]

basically a and b here control our 'lifts'

[noble dock]

dude I don’t know what a lift is either that doesn’t help lol

anyway is my definition a good enough one for now or not?

[marsh forge]

thats why i wrote it

so you dont have to know what a lift is

this is equivalent

[noble dock]

ok then

[marsh forge]

i have no idea what you're saying re: density

i think you should ignore the term

[noble dock]

so here’s a question

[marsh forge]

i think what i posted should lead you to be able to attack this problem number theoretically

at which point i get less interested lol

[noble dock]

do lines with irrational slope only pass through points that are rational multiples of that slope?

bad phrasing

could you use this to test if pi-e was irrational?

[marsh forge]

no

[noble dock]

by working in the square with sides 4-long, and seeing if there was a line that passed through all three of (0,0), (0,e), and (0,pi)?

[marsh forge]

i think you need to know more about this problem

before answering that

[noble dock]

and by “you” do you mean me in particular?

[marsh forge]

well

i also dont know enough

but i am not going to think abt this much harder probably

[noble dock]

can you help me phrase the original question to post on stackexchange?

[marsh forge]

Given a finite set of points in T^2:=R^2/Z^2, when is there a straight line in R^2 whose image under the quotient passes through all of them, and can we count how many? Can we classify the possible slopes of said lines (i.e. rational, irrational, etc)?

[noble dock]

and for the body text, help me phrase “I think there are (0 lines if there are at least two points with the same x-coordinate and two points with the same y-coordinate),(0 lines if there are two points with a difference of x-coordinates of a and two points with a difference of x-coordinates of b such that b and a are not both rational multiples of the same irrational or rational number), (the same but with y), (0 lines if there are two points with an x-coordinate difference of a and two points with a y-coordinate difference of b such that a and b are not rational multiples of the same irrational or rational number), (countably infinite non-dense lines if all sets of two points have rational differences in both their x-coordinates and y-coordinates), and (countably infinite dense lines if all sets of two points have differences in their x-coordinates and y-coordinates that are multiples of the same irrational number)”

[marsh forge]

Don't use the term dense

if you don't know what it means

that goes for any term

[noble dock]

fine

[marsh forge]

I don't think you need any of that

i think what i wrote is fine

[noble dock]

replace non-dense with (lines that go back to their starting point) and dense with (lines that don’t)

and I sort of want to include that in the body text

[marsh forge]

yeah thats just sort of a wall of text that makes me not want to read your question

if anything put it in a separate part of the post

with a subtitle like

"my thinking so far"

[noble dock]

...like the body text?

[marsh forge]

Yes

Its like 40 different conjectures

in like 3 sentences

[noble dock]

dude I already said that’s for the body text

lol

[marsh forge]

im aware

My point is

[noble dock]

the title will just be what you were

wrote

[marsh forge]

the title should be

Can we classify "straight" lines in a torus through different sets of points

or something short like that

the post body should be what i wrote

and then you can put what you want below that

[noble dock]

oh okay

[marsh forge]

Im telling you if I opened a post and saw that mess I'd like

close it instantly lol

If you want to put your hypothesis

make it succinct

and only give one or two

[noble dock]

how do I make that succinct

[marsh forge]

something like

"As far as slopes go, I think the (ir)rationality might matter, depending on whether the points have a rational differnce"

or something like that

[noble dock]

I don’t know how to phrase it and I don’t want to turn people off from answering by my lack of... good phrasing

like formally I mean

[marsh forge]

i mean what i wrote is fine

learning to phrase things formally is the art of revision

you dont need technical language to do it

you just need to be patient and write and rewrite your sentences

until they actually make sense

[noble dock]

what about “lines that hit where they started” vs “lines that don’t”

how can I phrase that formally

[marsh forge]

does a line start anywhere

[noble dock]

like

[marsh forge]

i'd call such a line closed

or a loop

[noble dock]

okay

sorry one last thing

does the way you worded your question earlier necessarily imply a square as opposed to a non-square rectangle?

[marsh forge]

it defines a torus as a canonical quotients of R^2

this ends up being square

[noble dock]

ok thanks

actually

how do I phrase about x-coordinates vs y-coordinates?

@marsh forge

[marsh forge]

?

thats fine

[noble dock]

like how do I phrase “depending on whether the components of each point are all rational multiples of the same rational or irrational number”, or “with an exception made for the case where there are two points with the same x-coordinate and two points with the same y-coordinate”

I can just use the terminology x-coordinate and y-coordinate no problem?

[marsh forge]

yeah

theres no ambiguity

[noble dock]

okay thank you

so final thing

I’m gonna copy my body text for you to tell me if I made a mistake if that’s okay with you?

"I think there are either 0 lines, countably infinite open lines, or countably infinite closed lines, depending on whether the x and y-coordinates of the points are all rational multiples of the same rational or irrational number, with a 0-line exception made for the case when two points share the same x-coordinate and two points share the same y-coordinate."

along with the bit you already posted earlier

a closed line is one that, if you start at (p,q) and begin walking along the line, you'll hit (p,q) again

that's the terminology he used, and it seemed common when looking at other similar-ish questions on stackexchange

is that body text good?

@marsh forge sorry for the ping, this is the last check before I can post

[marsh forge]

dont say open line

personally

[noble dock]

what should I say?

[marsh forge]

i would be way less specific

because all of those conjectures

are basically blind

[noble dock]

I wouldn't say completely, I'd say I have a fairly solid reason to claim all of them honestly

[marsh forge]

but just say "lines that become closed loops in the torus" and "lines that don't"

[noble dock]

thanks

[old sapphire]

This channel seems clear atm?

I need to create a five spoke radially 5-folve symmetric disk space-filling curve

A single input produces an (x, y) output

the disc is a unit circle

I’m stuck at how to make a starfish fractal

😄

I’m very stupid at maths

basically a disc-oriented space filling curve

[marsh forge]

just as a heads up

its not obvious that you can do that

unless you have seen it done

space filling curves are hard to design

[old sapphire]

That is exactly what I’m finding banging against this wall

[marsh forge]

does it need to actually be space filling?

[old sapphire]

I think I need a star fish 5 spoke fractal curve to start with

with a step count parameter

and spoke count parameter

[marsh forge]

thats doable

[old sapphire]

ok

no idear where to go from there

[marsh forge]

but space filling has a very particular mathematical meaning

that might not actually be what you need

[old sapphire]

It’s fine if it’s close enough to space filling

at the limit we don’t really care

we’ll never get there

[marsh forge]

why not just use a spiral then

[old sapphire]

archemedian spirals

I can try an archimedian spiral

that was my first guess

I’ve wanted to avoid code as long as possible

and I think this is just a case of “guess and check” with archimedian spirals...

I’ll try that and see what I Come up with

@marsh forge where would a person who wanted to make a parameterized with spoke count and iteration count start to learn to make a fractal formula?

@marsh forge if you give me a starting point to where this knowledge is

I think I can figure it out from there

I would really prefer a radial fractal over a spiral any day

[marsh forge]

I honestly don't know

[old sapphire]

damn

me neither

[marsh forge]

this is pretty useless to mathematicians

so its not smth people here will likely know much about

[old sapphire]

ok

[marsh forge]

if i were you

[old sapphire]

is matlab a good playground for generating parameterized fractals?

[marsh forge]

i'd make a normal spiral somehow

honestly if i were gonna play just with algos

i'd use processing

[old sapphire]

OK

[marsh forge]

its like ideal for sketching programming art

[old sapphire]

wonderful

I need a fractal

my soul needs one

What is processing?

link?

(also thank you for your help today)

I’m building a physics system

and a disc-filling curve is the center of it

That’s all I can say for now

I’ll google “processing sketching programming art”

that should do it

thanks again

I found it

T H A N K . Y O U !

[marsh forge]

yeah theres a javascript and java version

it's what i used to teach programming back in the day

[uncut geyser]

p5 is more suitable to be a processing substitute than processing.js Id say

[marsh forge]

havent used either since i was 16 lol

[old sapphire]

p5?

I’ll take a look

Joshua, that looks nice.

*Sweet Joshua

*Sweet Baby Joshua

*Sweet 2 lb 5 ounce Baby Joshua

[noble dock]

I'm just gonna ask now

since I feel like I didn't really get a good answer last time (no offense to those who helped me, it was a really enlightening discussion, I just felt like it didn't give the full answer I wanted), if anyone else is willing to help please ping me!

How many straight lines are there on a torus through a finite number of points?

Given a finite set of points in T2:=R2/Z2, when is there a straight line in R2 whose image under the quotient passes through all of them, and can we count how many?

I think there are either 0 lines, countably infinite lines that become closed loops in the torus, or countably infinite lines that don't become closed loops in the torus, depending on whether the x and y-coordinates of the points are all rational multiples of the same rational or irrational number, with a 0-line exception made for the case when two points share the same x-coordinate and two points share the same y-coordinate.

Am I correct in believing these things? If not, what cases have I missed/gotten wrong?

[dim meadow]

Can we characterize when a curve in the torus lifts to a straight line?

[noble dock]

when you unfold the torus into a square on the plane, it's a straight line in that plane

[chrome dew]

imagine the square tiling the whole plane, then where one point lies is at infinitely many points in the plane, so you can draw infinitely many lines

[noble dock]

right

in the n=2 case

but when there's more points, is it always guaranteed you can find a line that does that?

[chrome dew]

what's the difference?

[noble dock]

well

for example

imagine the square is from (0,0) to (1,1)

[chrome dew]

those are the same points

[noble dock]

if you have points at (.25,.75), (.75, .75), and (.75,.25), you'll never be able to draw a line connecting all 3 I believe

I know they're the same point on the torus, just to give a reference

[chrome dew]

oh you specifically want to make 1 line that connects all the points

[noble dock]

yeah

one straight line connecting all points

[chrome dew]

probably depends on if they're rational or irrational

[noble dock]

what I have so far

[chrome dew]

if they're rational, look for the lcm or something

[noble dock]

and I'm not completely confident of this-

-if the distance between the x-components of the points is rational, and the distance between the y-components is rational, you get a countably infinite number of "closed" lines

-if the distances between the x and y components of points are all rational multiples of the same irrational number, you get a countably infinite number of "open" lines

-if the distances between the x and y components are rational numbers of 2 or more irrational numbers, you get 0 lines

-if there are two points with the same X-coordinate and two points with the same y-coordinate, you get 0 lines

I don't know how to check each of those, nor do I know if there are any cases I'm missing even if those are all correct

and a "closed" line is one such that, if you start at point (p,q) on the line and walk in one direction, you'll eventually hit point (p,q) again

an "open" line is one where you won't

oh actually one more case: -if the x or y distances are all rational multiples of 2 or more irrational numbers, but all points are colinear (as in, they would still be colinear if the square didn't tile), then there is exactly one line

[chrome dew]

why are you interested in this question?

[noble dock]

It came up while I was thinking about something else and it seemed interesting

for example, it may theoretically be able to be used to find if numbers such as pi - e are rational, though I kind of doubt it

but ultimately, it just seemed like an interesting problem that might have interesting applications

[dim meadow]

Why would it be able to be used to do that?

Yeah it seems hard to be able to tell a dense line and a loop apart

[noble dock]

More like, if you could have both the distance (pi - e) and rational distances between points, and still be able to construct a line, then (pi-e) must be rational

[dim meadow]

Sorry can you rephrase that?

[noble dock]

like

I'm trying to think of an example, but there may not be one

but if you can have it so points A and B are pi-e apart, and points A and C are a rational distance apart, and points C and B are either pi-e or a rational distance apart, and you can construct a line passing through all points, then you know pi - e must be rational

ah- you have points (e/4, .75), (e/4, .25), (pi/4, 0.5)

obviously the x-distance between the first two is 0, and between either of the first two and the third one is (pi-e)/4, while the y-distances are .5 and .25

and if you can construct a line between those, and all of what I proposed earlier is true, then pi - e is rational

and if you can't construct a line between them and what I proposed earlier is true, then pi - e is irrational

do you agree that if pi - e is rational, then it is possible to construct a "closed" line between all of those points?

sorry actually a better example

the last point is at (pi/4, 1/3)

now do you agree?

wait why did I change that the original was good

imagine constructing a line of slope

let me work out the numbers really quickly

huh

that arrangement with two points sharing a vertical line is really really strange

[noble dock]

I'm too tired I know I'm doing something weird here and I can't figure it out

my equations keep showing that 1/3 is a number that cannot exist, and clearly that's wrong

even if it doesn't work for this, though, the general case of "how many lines are there" is still interesting and still something I'd like to know the results of

[noble dock]

oh I see my mistake

well, at the very least it could test if pi - e is a rational number, such that its simplest form is a/b, where either a is not odd or b is not a multiple of 4

because the line won't work for rational numbers where a is odd and b is a multiple of 4

but it should work for all other rational numbers, showable by the fact that in all other cases there is at least one solution for X and Y in the integers such that .25bx = a(y+.5)

which comes from .25/(a/b) being the slope of the line, and being equal to (y+.5)/x

[golden gust]

does anyone know whether the non-negative reals equipped with the metrics |x-y| and binary xor(x,y) are homeomorphic?

oh I should say bitwise xor

so d(1,2) = 3

we can restrict this to the naturals i think reals is looking difficult

but yeah same question

like um

$\text{xor}(5,7) = \text{xor}(101_2, 111_2) = 010_2 = 2$

[gentle osprey]

xdres:

[golden gust]

_2 is base 2

no worries :)

i mean it's not an urgent question im just genuinely curious

if anyone else has ideas feel free to @ me

[golden gust]

how would you go about finding such a sequence?

[tough imp]

I don’t see how that proves it though

That would only prove they aren’t homeomorphic via the identity map

Maybe I just suck then lol

[golden gust]

surely if they're homeomorphic the bijection is continuous and so a sequence that converges in one has to converge in the other

[tough imp]

Yes, but you said to find a sequence converging in R that doesn’t converge in R_xor

But then aren’t you assuming that if x_n converges in R, that f(x_n) = x_n?

What if the homeomorphism sends that sequence to something else?

[golden gust]

not neccessarily

[tough imp]

The image surely is convergent, but the image might not be the same sequence

[golden gust]

if x_n in R converges then f(x_n) converges, right? and if y_n in R_xor converges then f^-1(y_n) converges?

[tough imp]

Yes

[golden gust]

oh I thought you meant the image of the sequence converges @gritty widget

[tough imp]

Oh then of course but

How do we know what the image is?

[golden gust]

yeah just misunderstanding i guess

[tough imp]

We don’t have a specific map to work with

So unless you wanna range over all continuous maps R-> R_xor

Then I don’t see how you could tackle it that way

Try computing the fundamental group?

Also sorry but I don’t see how the R_xor is even defined

I get it for integers

But what happens for like d(1.479, pi)?

[golden gust]

yeah i restricted to naturals later because of stuff like 0.1111... = 1 in R

so reals seems like a bad idea

[tough imp]

Ahhhh

Wait

You meant 9’s not 1 right?

xdres?

[golden gust]

i meant in binary

[tough imp]

Ah

[golden gust]

but yeah same idea

[tough imp]

Wait but if you restrict to naturals

The subset topology on N is the discrete one right?

(WRT standard topology)

Yeah

Is it true that anything homeomorphic to discrete topology has discrete topology?

Yeah that’s true I think right?

Yeh

So I guess you just have to see if N_xor is discrete

Which turns this into an easy problem

Because if it is you take identity

Well “easy”

We know what we want to look for

Wait

Doesn’t xor lead to distance 0 things that aren’t the same number?

Umm let me think

[golden gust]

no?

[tough imp]

Nvm

Lmao

Thing is 0 iff they’re the same lol

Yeah

I forgot what xor was for a second

And remembered it is 0 for 2 out of the 4 options

And was like “lol wut”

Is there a nice condition for when a metric topology is discrete?

We can show every point is open

I think?

Yeah

So every point is open

@golden gust we solved it

🥳

[golden gust]

:D

thanks guys

[tough imp]

I think once you restricted to N

It became plausible

And possibly even sort of intuitive

[golden gust]

now try dyadic rationals : )

[tough imp]

userformerlyknownasmathemagician has left the room

[golden gust]

yeah the pair (N,xor) forms a metric space and a group

and im investigating if you can get one of those in R too

[tough imp]

Is it a topological group?

As in group operation is continuous?

[golden gust]

idk as in, normal group

xor is associative

[tough imp]

Oh wait it has to be

Lmao

It has discrete topology

So the operation must be continuous

Altho any group is a topological group under discrete topology

So i don’t think there’s anything of note to say there

Oh yeah, what is the identity?

[golden gust]

identity is 0 and every element is its own inverse

no?

[tough imp]

What?

What is 0?

Err not even that

[golden gust]

are we talking about different things

[tough imp]

Take like 101 + 000

= 010

[golden gust]

uhhh yes

hang on

[tough imp]

Also side note nice pfp xdres

Monke gang

[golden gust]

the last airmonke

[tough imp]

I’m the only chmonke

[golden gust]

nice

[tough imp]

Woah

[golden gust]

wait no no 0 is the identity

bitwise xor

if you have a 1 it outputs 1 because they're different

if you have a 0 it outputs 0 because theyre the same

i was really scared for a second

i came across this problem a few weeks ago and ive been trying to learn everything i need to tackle it

and then someone comes along and says 0 isnt the identity

no worries lol i was just worried

yeah R is difficult

could you do Q+ by having a bijection to N, doing the xor there and going back to Q+?

[tough imp]

Wait a second

Oh lmao

I can’t do bitwise xor addition

Was the issue

Oof lmao

I did

The opposite

I think I did nand or whatever

Cuz I said 101 + 000 = 010

Oh sheesh

[golden gust]

theres a really cute bijection i found

[tough imp]

Theoretically you can yeah, but wew

Slim I saw a nice bikection

I looked one up for a set theory class

It wasn’t that bad

Yeah, I don’t think it was a group iso either

Oh also N and Q aren’t iso as groups

Since N is cyclic

And Q isn’t

[golden gust]

let the primes $(2,3,5,...) = (p_1, p_2, p_3,...)$. now, every $n \in \mathbb{N}$ can be written as $n = p_1 ^{a_1} p_2 ^{a_2} \cdots$ where $a_i$ are naturals, and every $q \in \mathbb{Q}$ is $q = p_1 ^{b_1} p_2 ^{b_2} \cdots$ where $b_i$ are integers. so just get a bijection $\mathh{N} \to \mathbb{Z}$

[gentle osprey]

xdres:
Compile Error! Click the reaction for details. (You may edit your message)

[golden gust]

you can get it for dyadic rationals

[tough imp]

Wut is a dyadic rational

[golden gust]

but for maybe rationals and definitely reals, binary expansions arent unique i think

oh it's every fraction n/2^m

[tough imp]

Oh

[golden gust]

so denominators are strictly powers of 2

[tough imp]

Ah

Isn’t this like

N x N /~

[golden gust]

yup

[tough imp]

For some ~ which like puts them to be coprime

#old-network ?

Eh?

CS + Math person

I don’t think a CS person thought about it as a topology haha

Ah sure

TCS = CS + Math change my mind

[golden gust]

TCS?

[tough imp]

Theoretical

So like programming languages

Or like verification

Discrete + doesn’t believe in LEM

For some of them

Since like there’s actual reason not to for that stuff

I remember Shamrock told me about it

Yeah

I feel like it isn’t possible though

Like

An irrational number will always have non terminating binary

Or n-ary expansion

So you have to accept like infinite length strings

So d(x,y) has to be allowed to be infinity

Is my gut instinct

I suppose

Oh yeah

And you can’t round down n-digits down

For the same sort of reason

[golden gust]

fun fact: plotting z = xor(x,y) for naturals x and y gives a cool surface

any guesses before i pull it out?

[tough imp]

Umm

Yeah

Also

I hesitate to call it surface

Because it’s just isolated points

And idk what the “right way” to extend that to a surface is

[golden gust]

sierpinski pyramid (is that the name?)

yeah not surface

[tough imp]

Is it a fractal??

That’s odd yeah

Yeah lmao slim

We got nerd-sniped

[golden gust]

aight thanks guys

[tough imp]

Np

Cheers

[burnt spruce]

@golden gust So it's 'well known' in communication complexity that if you plot the matrix corresponding to the 'disjointedness function' -- disjoint(x,y) = 1 if x and y in {0,1}^n correspond to disjoint subsets, and 0 otherwise -- then you get a Sierpinski triangle. This is a lot like what you are drawing -- the level surfaces of xor(x,y) are basically descriptions of the symmetric differences of the subsets corresponding to those natural numbers (assuming I guessed your definition correctly). So it's not the same thing as disjointedness, but kind of similar in spirit. Anyway, maybe check out Rao's book on communication complexity -- the perspective on looking at 'graphs' of functions like xor in a similar way to what you described is pretty important, and the 'fractally' picture turns into concrete lower bounds on some algorithms.

[golden gust]

oh damn thats pretty cool thank you much, that's a lot to look into

[burnt spruce]

@golden gust See figure I.2 here https://assets.cambridge.org/97811084/97985/excerpt/9781108497985_excerpt.pdf ... you can find the book though I'm sure.

[gritty widget]

hello guys

so, I was looking at the definition of manifold

and it says that it is a locallly euclidean hausdorff space

and it makes me wonder, do you have an example of a locally euclidean space that is not hausdorff?

[gentle osprey]

slimvesus:

[gritty widget]

omg

ty <3

[tough imp]

Bro this is such a sick example

Even in AG lol

[gritty widget]

AG? what does that mean?

[tough imp]

Algebraic Geometry

[gritty widget]

oh, i see, ty <3

[golden gust]

what do you mean by X/~

i haven't come across this yet

[ivory dragon]

modulo the equivalence relation ~

the key term here is "quotient spaces"

essentially its a new space constructed by taking X, but for all a, b in X, a ~ b iff a = b in X / ~

(more formally: X/~ is the set of equivalence classes on X under the relation ~. so, each element of X/~ is some "class" of equivalent objects in X under ~)

[golden gust]

so X/~ becomes a collection of sets?

i might be reading 'each element of X/~ is some "class"' wrong

[tough imp]

you should look into equivalence realtions

and a set theoretic quotient by an equivalence relation

they will come up in many situations

[ivory dragon]

so X/~ becomes a collection of sets?
kind of

what the elements are doesnt matter so much as what they represent

like super-duper-formally X/~ is the space formed by taking X, constructing a bunch of elements [x] for each x in X, and saying for a, b in X, a ~ b iff [a] = [b] (and then equipping it with the appropriate topology)

this isnt really how people think about it though

(this intuition is helpful in more algebraic contexts but not necessarily in topological ones)

a better way to think about it is what slimvesus says

if a ~ b in X, then we say a and b are "the same" in X / ~

(you'll note that i've switched from writing [a] and [b] for elements of X / ~ to just a and b; this is a notational convention)

(and again, represents how we actually think of these spaces)

i feel like a more natural place to learn this stuff is in a group or ring theoretic context

let me give an explicit example though:

consider the set of real numbers R

define an equivalence relation for elements of R by a ~ b iff a - b is an integer

[a \sim b \iff a-b \in \bZ]

[gentle osprey]

Namington:

[ivory dragon]

intuitvely, numbers are equivalent under this relation ~ if their "fractional parts" are equal

so -2 ~ 7, and 3.5 ~ 4.5, and 2.915934 ~ -101.915934

anyway, giving R/~ the appropriate (quotient) topology gives us a space that is actually homeomorphic to the unit circle S^1

if you picture the interval [0, 1)

placing fractional parts on this interval as appropriate

note that this "gluing together" process means that 0 ~ 1

so it makes sense to connect 0 and 1

making the interval [0, 1) a circle

(not a rigorous proof obviously)

(but you can prove that this plays well with how we'd want a circle to behave)

also yeah, that works too

here it's like

we're "rolling" it

we have a ball and the real line gets wrapped around it

such that each full rotation is an interval of size 1

this "glues together" points with the same fractional part

("overlapping" them, if you will)

anyway this is a somewhat contrived example but this is a very powerful concept

it lets us "filter out" what aspects of elements we "care about" and study the structure formed just by those aspects

if we only care about the fractional part of real numbers (for whatever reason), this process lets us assign a structure that represents the properties of just the fractional part

and this structure is very different from the structure of R as a whole

i mean, it's (homeomorphic to) a circle!

(this specific example is also the prototype of the concept of an orbit, but thats better explored in a group theory course)

yeah, its one of the most powerful and versatile ways to construct new spaces

while also giving us an immediate "analogy"

in the sense that, if we already understand the elements of X well, it's often fairly easy to get an intuition for how X/~ should look

whereas if we just construct a random space out of the void by explicitly giving sets and a topology

that intuition is harder to build up

[fair bear]

how would one think about the fundamental group of a torus in R3 (with a point on its surface)? i have a feeling it's closely related to the fundamental group of a circle in R2 with a hole, which is isomorphic to (Z,+), except there is an extra "direction" you can move in (aka through the hole)

[golden gust]

thank you so much namington and slimvesus :D

[velvet finch]

@fair bear did you mean a disk with a hole?

[fair bear]

i think so

[uncut geyser]

is there any topology properties/spaces repository?

[gritty widget]

https://topology.jdabbs.com/ might be like what you're looking for

the book "counterexamples in topology" as well

[uncut geyser]

ah

thanks

[gritty widget]

actually, i think that website gets most of its examples out of the aforementioned book

[uncut geyser]

Ive heard about the book before

[gritty widget]

it's a fun book to look at every once in a while

[marsh forge]

Wikipedia

And topospaces

Are good as well for this

[signal venture]

i'm having trouble taking this identification in. what does im(phi) look like as a subsheaf of G?

like i know that the image presheaf embeds into the image sheaf, and so (im(phi))(U) contains the image of phi(U). but the sheafification process seems to enlarge the image presheaf, so it seems (im(phi))(U) can potentially be bigger than image of phi(U)

maybe i have to look again at my proof of sheafification. this whole thing just seems really messy and i feel like i'm missing something

[tough imp]

That bit is nontrivial

That’s the whole point of the exercise is to show you can

The gist of it is that if you have a “sub presheaf” of a sheaf

In the sense that F(U) < G(U) for all U, G is a sheaf, and F is a presheaf

Then you can construct a subsheaf of G which satisfies the universal property for F^+

I believe is how it goes

And if you remember my description of like “locally representable”

In terms of how like the analytic function z is locally representable by stuff of the form e^f(z) on the punctured plane

That’s actually what it looks like

[signal venture]

Hmm, so I think you're saying in general that if F is a presheaf sitting inside a sheaf G, then with the proper identifications you have inclusions F -> F^+ -> G. Maybe I'll prove that tomorrow

Gonna go to sleep now

Image is just a special case of that then

[tough imp]

Yup

But the F^+ isn’t the same construction as the one in the book

There’s a subsheaf of G which satisfies the same universal property

Namely that maps to sheaves factor through it

By some categorical nonsense this means that the F^+ Hartshorne constructs and this thing are isomorphic uniquely (with respect to commuting with some maps)

[signal venture]

Doesn't the psi in his sheafification proposition let you map F^+ into G

Like he says in that picture im(phi) includes into G, which I'm quite sure is through the map psi

So the F^+ sitting inside of G should be just the (sheaf) image of that psi map

Oh I think I see. The subsheaf you're talking about and the image of F^+ under psi are the same thing, because psi will be injective in this case I think

[tough imp]

Uhh, well image is very nebulous

Precisely because you have to sheafify

But in general with a product “has a unique map to/from such that...”

Is unique up to unique isomorphism

Like the good way to show something is a tensor product is to just show that bilineae maps from M x N factor through it

By providing a map M x N to whatever the object is

And show any bilineae map M x N -> L factors uniquely through this new object

Then this gives you an abstract isomorphism from M (x) N to this other thing

[signal venture]

Hmm. How do you construct the F^+ as a subsheaf of G then?

[tough imp]

Locally representable stuff

It’s stuff in G(U)

Such that for some open cover of U

The element restricted to that part of the cover is equal to something in the subsheaf

Really the issue with not being a sheaf is that you might not be able to glue (or it might not be unique)

Living inside a sheaf tells you that uniqueness is true for the subsheaf

What you don’t have is that elements which agree on intersections actually can glue

But when you view the elements of the subsheaf as elements of the sheaf you actually can glue them

So instead of doing this weird ass way of gluing stuff like artificially (that’s what the sheafificafion in Hartshorne does)

You just use the glued version which exists in G

[signal venture]

Oh that makes a lot of sense

[tough imp]

From this you can show this subsheaf satisfies the universal property for F^+

And then you get this isomorphism from the subsheaf F^+ and the like artificial one

[signal venture]

It seems like Hartshorne intended to arrive there by a different path though? Like he never mentions this construction you're giving. Unless it's implicit somewhere

[tough imp]

He doesn’t but it’s how you solve that problem

He delegates that to an exercise I think so you can figure that out

[signal venture]

Maybe that's what 1.4 was. I was trying to do 1.2 and it had surjective stuff in it so I started pondering this stuff

[tough imp]

Yeah 1.4 is that

[signal venture]

Cool, I'll prove it tomorrow

Using G to get gluing in F makes a lot of sense. That's really cool

[tough imp]

Yeah I only really thought of that now tbh

I just did them before

But after having sat on it for a while what’s being done is way clearer in my head

[signal venture]

That's always a great feeling

[tough imp]

Yeah haha, it’s making sense!

[signal venture]

Going to sleep, it's 2am. Thanks for the help

[tough imp]

Np

[arctic siren]

@hidden arrow so hi, I have a question about hyperbolic geometry

lets say i pick some point in the hyperbolic plane

lets call it A

then I pick a unit vector and go from A in the vector direction in until I reach a distance of lets say N1. Then I put a point B there

[ivory dragon]

weird ping

[arctic siren]

then I pick another unit vector and go in that direction from point B

(I mean so that If the vector were to point at 0,1 in euclid place we would now go straight to the same direction we were already going)

I go into that direction until N2 distance

then I place a point C there

Now, I have 2 questions

1.) How long distance is point C from point

2.) What is the vector that points in the direction of point C from point A

weird ping
Well i just wanna know the answer, i guessed zeno could know something but Im open to answers from everyone

im not sure if my explanation makes sense

if somebody is curious ask me for clarification

[hidden arrow]

you could use the hyperbolic cosine rules for that

@arctic siren https://en.wikipedia.org/wiki/Hyperbolic_law_of_cosines

(probably quite easy to derive the formulas in the hyperboloid model too)

[arctic siren]

Ok

Seems complicated but I am still going to learn it

[sleek thicket]

Can someone check my solution to an exercise about lie groups? Should be pretty simple

I'm supposed to show that [X, Y] = 0 iff exp(tX) exp(sY) = exp(sY) exp(tX) for all s, t in R

Where X, Y are in Lie(G)

My solution is that the flow of X is θ_t(g) = g exp(tX) and the flow of Y is ψ_s(g) = g exp(sY), and [X, Y] = 0 iff the flows of X and Y commute

From which it's immediate

I'm supposed to use this to show if that G is connected, then G is abelian iff Lie(G) is abelian. If G is abelian then exp(tX) exp(sY) = exp(sY) exp(tX) for all X, Y, t, s, so [X, Y] = 0 always. Conversely if Lie(G) is abelian, then exp(X) exp(Y) = exp(Y) exp(X) for all X, Y, and you can restrict exp to a diffeomorphism from some neighborhood U of 0 in Lie(G) to some neighborhood V of e in G, and so there's a neighborhood V of e in G such that gh = hg for all g, h in V. Since G is a connected topological group, it's generated by V, and thus is abelian

[sleek thicket]

The final part of this problem is to give an example where this fails if G isn't connected, i.e. G is nonabelian but Lie(G) is. By the above Lie(G) is abelian iff the connected component of the identity is abelian, so I want a nonabelian lie group G where the component of the identity is abelian

Oh lol I can just take any finite nonabelian group and make it discrete

That's so lame

[uncut surge]

@sleek thicket Mildly less lame: O(2) is non-abelian, but SO(2) is (since it's connected and its Lie algebra is one-dimensional)

[sleek thicket]

Ty

[uncut surge]

saved the day with an optional add-on solution eight hours too late 😎

[sleek thicket]

Haha

I'm just doing problems in Lee for fun, time is far from critical

[chrome dew]

imagine the surface you have defined by F, if you make a slice through it to make a level curve then where it hits the maximum is just a single isolated point, not locally euclidean

similarly for the saddle point, you'll have a crossing there

whatcha mean generically?

oh like outside of 2D

I think so

in terms of the hessian matrix I think

yeah, probably useful here

[pastel linden]

So I'm trying to make sense of a differential 1-form

it's a cotangent field that maps a point p on an open subset to a vector field which assigns it to a vector in the tangent space at p?

[uncut geyser]

that wouldnt be a 1-form tho

[gritty widget]

a 1-form is a section of the cotangent bundle. unwinding the definitions gives you something that assigns to each point p a covector in the tangent space at p (i.e. a linear map T_p M -> R)

that's a more precise definition

i think you were steering in the right direction bacono

[uncut geyser]

isnt it like d: UT_pM -> F st. d|T_pM is linear

[gritty widget]

kind of? but not exactly

a 1 form is a map M -> T*M such that for each p in M, the image is in Tp*M (meaning it is a linear map TpM -> R)

(here, T*M is the union of all the dual spaces to the tangent spaces; aptly named the cotangent bundle)

also someone might ree if you use d to represent a form

cause d usually means the exterior derivative

i.e. a covector field: dual to a vector field giving you a tangent vector at each point, a covector field gives you a tangent covector at each point

[uncut geyser]

wait
where did I say anything wrong
union of tangent spaces to field such that restricted to a tangent space it is a linear functional

but ye d was my bad

personally prefer varphi

(field being field of your vector space)

[gritty widget]

i don't think you said anything wrong, ive just never seen it formulated like that

[pastel linden]

ok yeah I'm a little confused because my text basically said that the differential of a map df can be written as a bilinear form <Xp, f> where Xp is a vector space

then it said that the tangent vector can be seen as a function of the second argument of the bilinear form?

[uncut geyser]

ah

but bacono, what you described before is a vector field

[gritty widget]

yeah what you said seems fine, it just kind of hides the dual-ness to vector fields, and that might help bacono to understand covector fields

[pastel linden]

sorry I meant Xp is a vector field my bad

basically my intuition is that the space of derivations at a point are isomorphic to the tangent space, so that a derivation is essential a sum of coordinates of the tangent vector multiplied by the respective partial derivative

so then wouldn't X_p be the tangent vector, not f?

[uncut geyser]

my bad tterra
that restriction being a l functional gives the dualness

what is X_p
tangent space to X?

[pastel linden]

vector field assigning a point p

[uncut geyser]

ah a vector field

[pastel linden]

I'm just confused because the book never fully explains what it means by an open set we define these concepts on

I've always just assumed there's an implicit dependence on a manifold

[uncut geyser]

but when you say multiplied by the partial derivative, are you taking the usual derivative already?

the derivation is just a vector

that acts upon differentiable functions

[gritty widget]

what book?

[pastel linden]

Tu, introduction to manifolds

[gritty widget]

oh im using that book right now!

[uncut geyser]

what these concepts try to do is take out the dependence on the euclidean coordinates and put it solely on the manifold
wherever it is

ah
theyre taking any derivation to be represented by partial

[pastel linden]

the top part of the page is what has me really not sure

that's what I was talking about

[uncut geyser]

it takes a vector and smooth function and defines an inner product

you can think pf <Xp,.> as a derivation operator that acts on smooth functions (usually taking their directional derivative in the direction of Xp)

[pastel linden]

so is the book just badly worded in saying the tangent vector is a function of f

[uncut geyser]

and you can think of <.,f> as the differential of f which assigned to a derivation gives you the respective derivative associated with a vector in Xp

uhmm

lemme think about what that means

well not really

define <Xp,.> as a function

f -> Xpf

Cinf -> Cinf

wait is the codomain Cinf

should be

Cinf(Rn) to Cinf(R) right?

or am I being dum

[pastel linden]

I believe so

[uncut geyser]

ok
so it would be

$<X_p, \cdot > : C^{\infty}(\mathbb{R}^n) \to C^{\infty}(\mathbb{R}) \ f \mapsto X_p f$\
where $ X_p f : \mathbb{R}^n \to \mathbb{R} \ x \mapsto \sum \frac{\partial f (x)} {\partial x_i} (v_i)_p$

something like this?

jeez
fuk mobile latex reeee

[gentle osprey]

Fractal:

[uncut geyser]

this notation may be better

[pastel linden]

Ok yeah I see what you're getting at

[uncut geyser]

was this where the doubt was?

hadnt seen it like an inner product tbh

[pastel linden]

the differential of f sends f to a derivation along X_p, which is dependent on what vector field we choose in the inner product

I think I understand

[uncut geyser]

well
wym independent

lemme correct somehing

[pastel linden]

$f \longmapsto D_{X_p}(f)$

[gentle osprey]

bacono:

[pastel linden]

and

$D_{X_p}(f) = X_p f = df$

[gentle osprey]

bacono:

[uncut geyser]

not sure if thats df

maybe it is

oh

ah it isnt

[pastel linden]

the book defines

[uncut geyser]

ahhh I gotta recall what the form df does

[pastel linden]

$(df)_p(X_p) = X_p f$

[gentle osprey]

bacono:

[uncut geyser]

ah

then thats better

it didnt have X_p there

df takes the vector Xp to Xpf
its a 1-form