12:02 AM
Hmmmm, I suddenly realize that I personally don't know any vegetarian recipes aside from salads, desserts, beans, rice, and "raw fruit."

Yes, @anakhro. A number. Partly because my sister is vegetarian/vegan.

(For a moment, there, I was wondering why I was the one pinged. :P )

@Eric: I can't argue with you. I do think that your letter-writers might not consider your emotional state and you should take it seriously ... grad school is stressful no matter what.
Because you confuzled me by meddling, @Rithaniel.

I am a master meddler.

It's tough to find vegetarian recipes that are good, I find. Half the time they are crazy vegans who are randomly throwing things into a dish and calling it edible.
Last good one I made was asheh reshteh

12:07 AM
Look at the Green's Cookbook. Fabulous famous restaurant in SF, @anakhro. Also, non-vegetarian cookbooks often have some really good vegetarian recipes (e.g., Julia Child, my idol).

@anakhro But fruit is like literally candy? amirite :P

I've made a few Afghan vegetable dishes that are totally vegetarian.

I just looked at the Asheh Reshteh recipe, best part is that noodles are likely high glycemic so I probably can't even eat that if I wanted. Rip.

I think there are probably plenty of great vegetarian dishes out there, but I don't know if I could enjoy vegan stuff. You lose so much when you cut out milk products.

well, I mean I could, but I would probably just break out.

12:09 AM
Yeah, I'm not looking for vegan ...
Is that true of non-wheat noodles, @Dair? They make noodles out of everything these days. You could even make zucchini "noodles" (thinly sliced so as to resemble noodles) or use spaghetti squash.

@TedShifrin Idk about Zucchini "noodles", but I used to have a glucose monitor on me (long story), rice noodles are definitely a huge huge no-no. But "non-wheat" is a pretty broad thing.

Thanks I will have a look, Ted!

There is a thing called almond flour which is often used in baking cakes or pastries. I wonder if you could use it to make noodles.

i'm allergic to almonds. fml.

Ah darn

12:12 AM
Almond flour probably not great for noodles.

@Dair do you have a condition or something?
I have never heard of acne flairing that bad because of noodles

But they make noodles out of lentils and chickpeas and everything else, @Dair.

@TedShifrin that’s a fair point i didn’t think of

I don't mean to meddle too much, @Eric, but I know you (from afar) pretty well.

@anakhro If you're referring to the glucose monitor: Hypoglycemia.

12:13 AM
well i certainly don’t mind

As for the restrictions and stuff, it's just 'cause of acne lol.

I'm just saying you'll occasionally be stressed, regardless, @Eric, but you need to gauge where you'll have a higher comfort and pleasure level.

for sure

But we need you back doing math soon :P

my family is going to be moving to nearby if i’m in the east coast so that’s another thing :/
ya

12:16 AM
Well, that is a definite factor.

@ÉricoMeloSilva Rip. A moment of silence. :P

@Eric: For me that would have been a negative.
I happily went cross-country to grad school after being 16 miles away for college. :P

i think its a positive for me

@ÉricoMeloSilva Well you would also be wrong. :P

+ my partner is gonna be on the east coast

12:17 AM
Well, then I think I know your decision.
Independent of the factors I think about :P

yeah that’s what i’m leaning to
but i’m still not settled
@Dair u ain’t know me

FYI, I know someone (one of my undergraduate students/stars) who went to Princeton, hated it, and transferred elsewhere after a year or two.

@ÉricoMeloSilva I don't need to know you, the theorem is independent of person. :P

Not clear if you could end up at your other high choice with a transfer, but you're not irreparably stuck.

oooof
what field did they work in

12:19 AM
Number theory.

ah i see

I was admitted to Princeton and happily chose not to go there :P In those days, even with NSF support, they wanted people out in 3 years, and I just didn't want that attitude. I don't regret my choice one iota.

:/ i really don’t know, i did like ppl at stanford quite a bit but i’ll be so isolated from ppl i care about

Yeah, that's not good for you. It's a tough call. I say to give it a shot on the east coast and see how it goes. (You might ask Rafe if they consider transfers if you realize you're miserable ... I honestly don't know the answer.)
@Dair: Eric has no problem making friends. It's significant other + family that's the issue.

yeah^

12:24 AM

@Dair so you have hypoglycemia, or noodles just make your face swell and pus?

@TedShifrin yeah he told me to email him if i’m struggling w my decisions and i’ll probably take him up on it

@anakhro The noodles are bad for the acne. Not the hypoglycemia....

im outta here tho, gotta eat

You definitely should. DEFINITELY.
Bon appétit.

12:26 AM
@Dair what things are bad for the acne?

@anakhro high sugar, high fat.

@Dair: You didn't respond to my alternative pasta and zucchini/spaghetti-squash suggestions. My feelings are hurted.

@Ted Oh sorry, I'm not a big fan of zucchini honestly, but I might try it lol.
i was too preoccupied with the non-wheat portion of that comment.

Try spaghetti squash
It's good.

Yeah, spaghetti squash is great.

12:29 AM
and how rice is actually sugary af.

Just cook it with very little oil
@Dair by sugary, I guess you mean carbs?

never ceases to amaze me how rice can be so bland yet cause spikes in blood sugar so easily...

As for avoiding sugar and fats, try making a curry up to whatever specifications you need/want

I will look at the spaghetti squash.
@anakhro Don't a lot of curries have milk?

No, not really.
Do you run into a problem with potatoes?

12:31 AM
@anakhro You can get away with limited degrees of things depending on how you are able to balance the protein...

Like I mean some curries have milk, but currying is a very generic cooking technique, really.

like protein basically counter acts sugar/carbs.
"currying is a very generic cooking technique, really." Lol, currying, reminds me of currying functions...

Sucks that fats are out.

In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, a function that takes two arguments, one from X and one from Y, and produces outputs in Z, by currying is translated into a function that takes a single argument from X and produces as outputs functions from Y to Z. Currying is related to, but not the same as, partial application. Currying is useful in both practical and theoretical settings. In functional programming languages...

I would highly recommend peanut butter curry.

12:32 AM
@Dair: Some Thai curries have coconut milk. But you can use vegetable broth (or chicken broth if you're not vegetarian).

You can also just use water.

Water is boring, but yeah.
But coconut milk isn't dairy.

I avoid using broths because I never make enough stuff to use the whole container, so I always end up dumping it a week later after it sits in the fridge. :(

It keeps more than a week.

I need to find a good curry chicken recipe.

12:34 AM
And there's stuff in jars.
I think we've turned into CookingStackExchange.

I will try looking for the jarred stuff.

Too vague, @Rithaniel. Thai? Indian? Chinese?

The stuff I get usually smells bad in within a week.

I will stick a thermometer in there and measure that, maybe.

12:36 AM
@anakhro: I used to use Better Than Bouillon. I worry about sodium content now, though.

wait does coconut milk contain actual milk?

No, Dair.

then I can have coconut milk lol.

It's milky from the flesh of the coconut.
That's my point.

I learned how important salt and sugar in spicing was recently, and now I am worried I am going overboard.
In my faux-chinese food I put 2tbsp of it, for about 3-4 people.

12:38 AM
2 tbsp of what?

And it feels really sweet when I drop those two spoons in it. Sugar

That's way too much.
I hardly use any sugar in Chinese cooking.

It tastes so good though D:

Use hoisin sauce, which has a natural sweetness.

Hmmm, let's go with Indian.

12:39 AM
LOL.

I think the key thing in hoisin sauce would be the vinegar.
That would let you get away with less sugar

I think I'm going to leave so the culinary discussion will subside :P

Stevia, you mean
See ya later, alligator

Cya.
I'm sorry I ruined this chat lol.

12:40 AM
No it's good, Dair. I love cooking.

i tend to do that a lot.

I enjoy cooking as well, though most of my passion comes with the cooking of desserts

@Rithaniel re: indian curry chicken, just look up some recipes and sort of meld them together and go by what tastes good to you.

Fair enough, then that is what I will do.

Heavy placement on spices and cooking time, though.
I find high heat kills flavour.
And under spicing leaves you with a horrible dish. But also over spicing is horrible. So test frequently.

12:49 AM
Indeed, I'm pretty competent at finding a good balance of spices, though I hadn't picked up on high heat killing spices (That might explain what happened to my most recent batch of chilli, actually)
I have a slight tendency to over-salt things, though.

Knowing how to deal with that sort of thing is important
Like potatoes suck up salt, you can use something that is acidic, etc.
If you add too much water, you can also suck it up with rice.

Ah, those are some interesting tricks. You can't always put rice or potatoes in everything, but it's useful stuff to keep in mind.

Well with curries you can. ;)

Ah, good point. Too much math. I'm generalizing too much.

3 hours later…
3:37 AM
Is it possible to differentiate $len_i(f(x))$, the length of $f(x)$ under inner product $i$?

2 hours later…
5:11 AM
Can a mobius loop be mapped into $R^4$ in such a way that there are no self intersections when the boundary of that loop is mapped onto a jordan curve?

1 hour later…
6:12 AM
@Isa Since you have mentioned that you were after strike through text and not formula, I have added the tags (formatting) and (markdown) to your post. (The tag (mathjax) seems mor suitable for questions about typing mathematics.)

6:29 AM
Hey everyone!
I have a curve $\gamma\colon[0,1] \rightarrow \mathbb{R}$ that's $C^1$ but not necessarily simple. The length of this curve is $\int_{0}^{1}|\gamma^{'}(t)|dt$. But I want to measure the length of the curve which does not overlap with itself. It seems one can simply define a new curve ignoring the overlaps, which will still be at least piecewise $C^1$ and so we can measure it's length. Is there a name for this? Or a name for measuring the non-overlapping part of a curve?
Ah maybe I can just define $X(t) = 0$ if $\gamma(t) < 0$, $X(t) = 1$ if $\gamma(t) \geq 0$ and then just use $\int_{0}^{1}|\gamma ' (t)|X(t)dt$.
No nevermind, this doesn't work. It only ignores the part of the curve the starts looping back on itself, but not the part of the curve that would start returning after moving back.

6:55 AM
OKay, so I think a simple way of saying what I need would be the length of the image of the curve, but I want it in terms of $\gamma ' (t)$.

@none You can;t determine it purely from $\gamma'$ I think, since it depends not only on how fast it moves but also on where it is located

Why do we need $d(a)\le d(ab)$ in ED definition above?

@Silent Where is that from?
I don't think I have ever seen that referred to as a measure before

Gallian Contemporary Abstract Algebra

7:04 AM
I see

I do not see that condition used in any of the proofs in that section. Maybe they use it in exercises!

It is also not always required

ok

It is not part of the definition in the book I taught from, and all the proofs work fine

wow :)
which book do you use?

7:08 AM
It does seem natural to include, since the motivation is from either the absolute value on the integers or from the degree on polynomials over a field. But since it is not necessary, it can be left out.
Concrete Abstract Algebra by Lauritzen
It is not that great actually, but it was assigned before I took over the course

i see

@TobiasKildetoft I think the reason why I wanted to do this in the first place is the wrong way to go about something, and I was making my life harder than I thought. Thanks for your thought anyway!!
Yes I was looking at the wrong thing before, I can just use the usual length for my purposes. That being said, I'm curious if it's possible to define this non-overlapping curve rigorously. Intuitively you want a recursive definition, and ignore points you visited before. This would be an uncountably recursively define object, I'm not sure if there are things that could go wrong with that.

@TobiasKildetoft. wikipedia says that, that other condition is a consequence!

7:33 AM
@Silent Ahh, that makes sense

7:55 AM
@Silent this font and color reminds me of my school textbooks. it was exactly like this.

8:37 AM
Who was I discussing distances of planets to Earth with?
And how the closest planet to Earth (in terms of average distance) is actually Mercury
@LeakyNun and @Rithaniel I think?
I found this GIF

Was not me, unfortunately, but that is neat.

Oh it was @MarioCarneiro

Hmmmm, makes sense how it commonly ends up being Mercury

The GIF is measuring percent of time that Mercury is closest, rather than comparing average distances, but you get the same answer either way

8:57 AM
@Nick glad to hear that :)
@Rithaniel lol

@AkivaWeinberger can i believe these velocities?
@AkivaWeinberger that's not fair since mercury has more revolutions.

9:15 AM
@Nick That doesn't actually matter in terms of average distance. All that matters is that the two planets don't have the same period, not which one has more
@Nick Doesn't look inaccurate to me

Are the orbital eccentricities and inclinations accurate? /s

Fun fact: if you have two concentric circles of radius $a$ and $b$, and you pick a point from each circle at random, the expected distance between those two points is equal to the $1/2\pi$ times the circumference of an ellipse with semimajor axis $a+b$ and semiminor axis $a-b$

I wonder if it changes if you take the sun into account.

The sun is on average closer to the Earth than Mercury is
(Cont'd) In particular, if $a=b$, then this is the average distance between two points on the same circle, and you end up with $4a/\pi$
(Also: there is no closed form expression for the circumference of an ellipse)
(Don't know if you knew that)

So, on average, the Earth is closer to the sun than it is to any planet. That's kind of wild.

9:27 AM
The average distance from the Earth to the sun is 1AU
(by definition)
All the other average distances are slightly more than that

10:16 AM
Sup there, I gotta find dimension of this linear space:
$V = \{ P \in \mathbb{R}_4[x] : \int_0^2 P(x)dx = P'(-1) = 0 \}$
so I define linear $F: \mathbb{R}_4[x] \to \mathbb{R}_4[x]$ with $F(P) = x\int_0^2 P(x)dx + P'(-1)$ and now $V$ becomes kernel of my $F$
so I use $\dim(\ker (F)) + \dim ( \mathrm{Im} (F)) = \dim( \mathbb{R}_4[x])$

@chandx Why? The condition was they they both needed to be zero, not that their sum should.
or was that an error in the problem statement?

that integral and differential are some plain numbers, so my $F$ is creatin polynomial of degree 1
and that polynomial is equal to zero polynomial only if coefficients are zero
that's what kernel is sayin

@AkivaWeinberger If anyone is good at animation, feel free to try this out: khanacademy.org/computer-programming/solar-system/…
Just a starting point, because I was bored enough :)

so if that's true, I'd conclude that dimension of $V$ is 3

@chandx Ahh, right. I missed the $x$ in front of the integral in the definition of $F$.

10:24 AM
so am I right here?

Right, if you have shown that the image of that map consists of all linear polynomials
(and that the subscript $4$ means all polynomials of degree at most $4$ rather than strictly less than $4$ as it does some places)

yes, means of degree at most 4

@Tobias a question a former classmate asked me and which you probably know the answer to: what are the minimal assumptions on $R$ such that every polynomial in $R[x]$ of degree $n$ has at most $n$ roots? Of course "$R$ is a field" works, but we were wondering whether it can be weakened to maybe UFD

@AlessandroCodenotti integral domain should suffice
(hmm, strike that)

@TobiasKildetoft What's the counterexample? The usual examples of polynomials of degree $n$ with more than $n$ roots are given over rings which are not domains

10:31 AM
So UFD is enough since that implies the polynomial ring is also a UFD and then the usual argument works

I'm not convinced, can't polynomial division be messed up in a UFD?

So the question is if failure of unique factorization in $R$ can be transported to factoring into something like $(x-a)(x-b) = (x-c)(x-d)$

I think maybe $R[x]$ needs to be Euclidean

Roots correspond to linear factors in general (over a domain at least)

Oh, ok, I wasn't sure about the minimal assumptions for that

10:34 AM
the division algorithm always works as long as the divisor is monic
Hmm, so if $a$ and $b$ are roots of $f$, then $b$ is a root of $f/(x-a)$, and we can continue this without needing unique factorization
so indeed being an integral domain suffices
since all we need is that the degree is additive on products

That makes sense, thanks!

Whether it always fails for a non-domain I don't know
Ohh, of course it does, by considering $ax$ for a zero divisor $a$

Here comes Mathei, summoned by the algebra

@AlessandroCodenotti you only need $R$ integral domain
since you can just embed into the quotient field

ahh, an even easier argument
Though I am not sure what people from a country in Wheel of Time have to do with it.

10:44 AM
@MatheinBoulomenos That's a neat argument
(fun fact: domani means tomorrow in Italian)

neat

ah, I learned that vocable some time ago, but forgot it

ma italiano è troppo difficile

That's not bad at all! Just "l'italiano" because we like to put articles everywhere
(and I have the same complaint about algebra!)

10:47 AM
Grazie!

@Silent I have left some stuff about the equivalence of the two definitions of Euclidean domains in the Linear & Abstract algebra chatroom. (But basically I just quoted the relevant part of the Wikipedia article.)

@AlessandroCodenotti non penso che l'italiano è particolarmente difficile, ma imparara una lingua è sempre difficile

che l'italiano sia* (you need a subjunctive in such a sentence but it's a mistake a lot of native speakers make, especially in talking)

ah thanks

I think the conjugation of verbs is a mess (as in every other romance language), but once you get that italian shouldn't be too hard

10:59 AM
I'm really frustrated that I can't roll the R, no matter how much I try

What should I make for dinner? Any thoughts?

but German has weird stuff like three types of adjective declension

three? Isn't it just weak and strong?

no, you also have mixed
strong is e.g. after a definite article, mixed is e.g. after an indefinite article and strong is e.g. when there is no article

I am glad we don't have all of that silly grammar in Danish
(and somehow people still can't seem to get things right, even as native speakers)

11:07 AM
eh, I meant weak is after a definite article

@MartinSleziak Thanks for introducing me to that chatroom!

@AlessandroCodenotti But more importantly: What should I make for dinner. What is a nice and easy to make Italian dish?

Hmm that's a good question, I should figure that out too at some point today

Risotto is one of my favorite Italian dishes

Yeah, risotto is great, but takes a bit more effort to make than I feel like today

11:12 AM
The selection of Rice available in German supermarkets was disappointing, I still have to find good one for a risotto

@AlessandroCodenotti we sometimes have even rice labaled as "risotto rice"

I usually just use parboiled. I could never tell the difference. Neither while cooking it nor while eating it

well I think you want short-grain for risotto

I use whatever is cheap and easy to get a hold of.

The best rices for risotto are vialone nano, carnaroli and arborio (especially the first two in my opinion)

11:17 AM
@Alessandro which supermarket did you check? I just checked and my local Rewe has both vialone nano and carnaroli

So, if I were to make risotto, what sort should I make?

@MatheinBoulomenos Really? I often go to a Rewe, but now I don't remember if I checked the rice there or only at the Edeka... I have to buy groceries anyway later so I'll make sure to check!

@Alessandro I only checked the homepage, so not sure if it's only available if order online or also in the market

11:42 AM
@AlessandroCodenotti also if Aldi has Italian weeks, they seem to sell arborio

@MatheinBoulomenos need some help

I love Aldi.

Suppose we have linear map transformation matrix where all entries are from rational function field then how do we can compute basis for such matrix?
or say polynomial ring from appropriate scaling

what do you mean by "basis for such matrix"?

11:55 AM
Say we have lattice over this field say Q(x) rational field then we get matrix from the basis v1,----vn such B(vi,vj) = (aij) where B is bilinear form symmetric
one can get correspondence from matrix to bilinear form with the help of basis
I am not understanding that what reordering means here?

1 hour later…
1:11 PM
@mathsstudent As far as I can tell that word does not appear anywhere in those images

1:35 PM
Actually @TobiasKildetoft I used reference from book
to rearrange da11,da22,da33 , ... dann we reorder and relabel basis vectors

2:09 PM
Let $X_n(\Bbb{Z})$ be the simplicial complex whose vertex set is $\Bbb{Z}$ and such that the vertices $v_0,...,v_k$ span a $k$-simplex if and only if $|v_i-v_j| \le n$ for every $i,j$. Prove that $X_n(\Bbb{Z})$ is $n$-dimensional
Here's my strategy. I think it suffices to show that $X_n(\Bbb{Z})$ does not "contain" any $(n+1)$-simplex, but I'm not sure how to formulate everything precisely (hence the reason for all the scare quotes). With $k > n$, if $v_0,...v_{n}, v_{n+1},...,v_k$ form a $k$-simplex with, then, in particular, $|v_i-v_j|\le n$ for $i,j=1,...,n+1$. So $v_0,...,v_{n+1}$ "forms" a $(n+1)$-"subsimplex". Thus, if we can show that $X_n(\Bbb{Z})$ doesn't "contain" any $(n+1)$-simplex, then we are finished.
...I'm worried that I haven't stated things precisely/rigorously...I don't even really know what a simplicial complex is....

2:51 PM
In the proof of the Kőnig's lemma here would anyone mind to explain to me how the following is true?
"...There must be one of those adjacent vertices through which infinitely many vertices can be reached without going through $v_1$. If there were not, then the entire graph would be the union of finitely many finite sets, and thus finite, contradicting the assumption that the graph is infinite."

@AlessandroCodenotti @MatheinBoulomenos I was not planning on putting any effort into cooking dinner, but now I decided to make risotto. And it is all your fault.

3:21 PM
I'd say risotto takes more time than effort

3:39 PM
I think wrote I above works, and I can reduce the problem to thinking about $0,...,n$, since translations of simplices are simplices.

@AlessandroCodenotti I was originally planning to find something that I could take straight from the supermarked to the oven and then serve

Ok, risotto takes definitely more effort than that

and yet you two made me want risotto, so now I am making risotto

user389096
x^2-sinx+5=0. Solve this equation.

3:55 PM
No, you solve it

user389096
Don't joke but help me to solve it.

Trasncendental equation do not have analytical solutions

4:11 PM
I NEED HELP!
I do not understand the proof for Jacobi's Sufficient Condition for Weak Extrema in Variational Calculus.
I came here yesterday and got a little help, but I'm truly stumped and can't move on to the next part of my work without understanding this proof.
Can someone help me or point me in the right direction?

4:26 PM
@Secret of course, this assumes said equation has solutions in the first place...
Easy to write down the analytical solutions when there aren’t any
(The equation of interest can be rearranged to x^2+5 = sin(x))

Okay, with the same setup as above, how do I visualize $X_1(\Bbb{Z})$? I am suppose to show that it is contractible, and then inductively show that $X_n(\Bbb{Z})$ is contractible.

4:42 PM
@user193319 What does $X_1(\mathbb{Z})$ mean?
Ohh nvm

5:04 PM
@Perturbative Any thoughts/suggestions on how to visualize $X_1(\Bbb{Z})$ and/or prove that it is contractible?
I don't even know what the topology on $X_1(\Bbb{Z})$ looks like.

5:22 PM
Request to reopen this question as the asker has shown good (and in fact correct) effort, and the question is about the asker's proof, not just a solution of the problem.

Seems like you’d have a bunch of 1-simplices, one for each pair of consecutive integers

a one-dimensional simplicial complex is just a graph

Yeah

I think it's pretty clear that this is contractible, the geometric realization is homeomorphic to $\Bbb R$

I think you can also visualize the case of n=2 without too much effort, and n=3 with a bit more
n=4 and up seems infeasible

5:31 PM
@TedShifrin it was Nowruz yesterday, not the arabic (Islamic) new year!
So it was good luck that I brought up asheh reshteh

(I mean, you can always just add more edges. The graph just doesn’t look particularly nice )

that's $n=2$, drawing only the nondegenerate simplices

Hah, I just sketched the same before looking at this
I guess it’s prettier if you do equilateral triangles but that’s just a visual flourish

I won't try to do n=3 in paint

Newp
pretty much this @MatheinBoulomenos

5:44 PM
yeah

I ran into that while reading up on the 16-cell
Higher dimensional polytopes are weird

Hi @Semiclassical

Why is a 1 dimensional simplicial complex a graph?
What is the topology on a graph?

Because 0-chains are vertices and 1-chains are edges
So if all you’ve got is a 1D complex, then those are your only options

5:49 PM
This is how you get the topology on a [abstract] simplicial complex (technically, you get an associated topological space): en.wikipedia.org/wiki/…

@Semiclassical Do you have a book recommendation for solid state physics?

I should but I really don’t

We use Kittel, but I don't particularly like it

4 hours later…
9:30 PM
I should probably resist the urge to be snarky in response to a main site question
And I will except to say that the exchange is akin to someone asking how to unlock a door by using the key and no other method. “Try turning it upside down.” “I know how to unlock it if I do that, I want to know how to do it without doing so!”

1 hour later…
10:53 PM
@MartinSleziak Oh ok Martin, thanks. It's indeed not a duplicate my question as you have mentioned

Hey chat
Does a two dimensional graph (graph theory) tell you how it's laid out in space

11:16 PM
@Lozansky in my courses we used Kittel and Hungklinger, both books from which I managed to understand exactly nothing. Eventually I needed to get the basics of solid state physics and downloaded some random book from google, it was the one by Gerd Czycholl and here I mananged to understand things
You might also be interested in Alexander Altand's book, which is new, modern, hip etc etc

11:50 PM
@Ultradark all you're given in graph theory are a list of vertices and edges between those vertices
you can ask about what various ways of drawing that graph will look like, but that's not essential
for instance, take 5 vertices and draw edges between all vertices
you can draw that in a whole bunch of different ways, and none of them is "the graph". what defines the graph is that there's 5 points and an edge between each pair of vertices
there are some visual properties, such as the fact that you can't draw said graph on a piece of paper without having at least two edges crossing
but the definition of a graph, in graph theory, doesn't care a whit about that