Mathematics - 2019-01-22

12:00 AM

Guys, I have a question

when parameterizing a function, we create a mapping and then to eliminate the variable (we compose functions), how can I prove that the composition always yields the function?

Akiva Weinberger

Hm

vimeo.com/20287194

How can we describe these curves mathematically

I'm guessing the computer is solving a differential equation of some sort

ÍgjøgnumMeg

Right, I did a course on PDEs in my undergrad and we plotted the solution to a Bessel equation. The solution looked e x a c t l y how you'd expect a swinging chain to look, which was magical to me

Akiva Weinberger

Or at least, it's modeling the wire as a bunch of line segments, and it's computing some value in terms of the angles between the line segments ("bendiness?") that it's trying to minimize

@ÍgjøgnumMeg Cool

I know nothing about the Bessel equation

though I guess it shows up in physics, the way you're describing it

ÍgjøgnumMeg

yeah that's the only context I've seen it in, as a solution to a particular form of ODE

as in, Bessel functions

Akiva Weinberger

The bent-wire curves almost look like Bézier curves

but they're probably more complicated than that

Leaky Nun

12:06 AM

@AkivaWeinberger reading about curves recently?

Akiva Weinberger

(Bézier curves, by the way, are made from repeated linear interpolation, and they're a really neat idea)

@LeakyNun Well it's a long rabbit hole that started with minimal surfaces

("Soap film surfaces")

Leaky Nun

have you been doing any "proper" maths?

mathsssislife

Guys i'm not considering cases where x=t and y=f(t) where the original equation is y=f(x)

i'm considering equations where this is not the case

LWChris

For a moment I though all the quotes in this chat were from a guy named "jan"... I should quit working for today

Akiva Weinberger

@mathsssislife I'm not sure I understand what you're composing

@LeakyNun There's this really neat minimal surface that repeats in a cubic lattice, called a "gyroid", and it looks really weird

It's used as an infill in 3D printing

(It performs similarly to some other infills, in terms of printing time, weight, and strength, but it beats them all in how cool it looks)

^Statues of the gyroid (which you can buy from Bathsheba Grossman)

ÍgjøgnumMeg

12:10 AM

@Akiva one of my lecturers took a lot of iterations of a construction for a spacefilling curve and them superimposed them on top of each other, connected it all up and 3D printed it

which looks cool

mathsssislife

@AkivaWeinberger Consider the function y=f(x): i'm considering cases for instance the function, x=g(t) and y=h(t) how can I prove h(g(t))=f(x)?

Akiva Weinberger

Animated gyroid structure

►

mathsssislife

never mind

Akiva Weinberger

(When you see a time lapse of it being 3D printing, it looks like that^ but in reverse)

The gyroid partitions space into two pieces. The pieces are called "labyrinths"

Here's a "skeleton" of the labyrinths

vimeo.com/2553639

(so the gyroid weaves between the "skeletons")

Not sure what the green things at the end are. Paths that avoid the gyroid?

I think they go down those spiral paths you see in the skeletons

The spirals are why it's called a "gyroid", by the way

'Cause they "gyrate", I guess? Not 100% clear on that

Nah never mind, the green lines go through the vertices of the skeletons. So I'm not quite sure what they are

Oh, he describes it in the description

"Triply periodic minimal surfaces" is the phrase to Google, by the way

since it repeats in the x, y, and z directions

Triply periodic minimal surface

In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.TPMS are of relevance in natural science. TPMS have been observed as biological membranes, as block copolymers, equipotential surfaces in crystals etc. They have also been of interest in architecture...

People are trying to classify them all but I think they haven't yet succeeded

mathsssislife

Guys what is the formal definition of a parametric equation?

Mike Miller

12:22 AM

what's the informal definition of a parametric equation?

mathsssislife

or a parameter

Rithaniel

I kind of wanted to get the gyroid sculpture from Bathsheba, but the 4" version, is, like, $400.

Is there a term for a space which is "gyroid-like" in geometry, but instead of 2D surfaces bending through 3D space, it's 3D through 4D space?

(Perhaps a bit too inexact of a description)

Akiva Weinberger

If y is some function of x and you represent it parametrically by (h(t), g(t)), then g(t) is that function of h(t) @mathsssislife

Think of it in terms of graphs - for all values of t, you're drawing the point (h(t), g(t)) onto the graph

Compare that to y=f(x), where for all x you draw (x, f(x)) onto the graph

@Rithaniel Quadruply periodic minimal 3-manifolds?

Instead of triply periodic minimal surfaces

Rithaniel

Ah, yes, perfect.

And more specifically, is there an example of such a manifold or family of such manifolds which have similar symmetries to a gyroid?

Akiva Weinberger

@Rithaniel Luckily, the pictures are free

Rithaniel

12:33 AM

Indeed they are.

Akiva Weinberger

@Rithaniel This stuff is hard as it is in 3D - you want this in 4D?!

I mean, if some crazy mathematician has attempted this then I wouldn't be too surprised

or maybe it's easier than I think, who knows

Rithaniel

That might be interesting to conduct some research into that. Perhaps a generalized form of $n$-periodic minimal $m$-manifolds for $m<n$

mathsssislife

How do I prove that eliminating the free variable t in a parametric equation yields a unique function?

Akiva Weinberger

I need to go to bed

Before I do, let me just point out that parametric things don't actually need to satisfy the vertical line test like functions do

mathsssislife

yeah I know, but i'm wondering as to why if they do satisfy the vertical line test then eliminating the variable yields y=f(x)

the function which they trace

i mean

Sir Cumference

1:51 AM

Dumb question, but can every nonempty finite set be well ordered?

Sir Cumference

2:09 AM

@Dair Nice name lol

Dair

2:25 AM

@SirCumference Nice name lol

Daminark

So, every set can be well-ordered in ZFC. If you're using another system maybe something weird can happen. Though regardless of one's opinions on ZFC, finite sets have an honest-to-God well-ordering since they biject to $\{1,\ldots,n\}$, which has its natural well-ordering

Sir Cumference

@Daminark Huh seriously? I may be misunderstanding, but isn't a well order a total order such that every subset contains a minimal element?

How would you define a well order on $\mathbb{R}$?

Wait sorry, for clarification did you mean "every finite set"?

Daminark

Nah, every set, this is called the well-ordering theorem, it's equivalent to the axiom of choice

Do you know Zorn's lemma?

Sir Cumference

Nope, I've only touched the surface of order theory :/

The definition of a poset, total order and well order are about what I know

Daminark

So, Zorn's lemma is the statement that if you have a poset such that every totally ordered subset is bounded above, then there's a maximal element

Sir Cumference

2:39 AM

@Daminark A maximal element of each totally ordered subset?

Daminark

But yeah, so the axiom of choice, Zorn's lemma, and the theorem that every set can be well-ordered are all equivalent

Maximal, it just means there's an element of the set such that no element is strictly larger, not necessarily that it's larger than everyone else

Sir Cumference

I mean we're talking about the subsets having maximal elements right?

Daminark

No, the poset has a maximal element

Sir Cumference

@Daminark Interesting. Do you recommend an introductory textbook on these topics? My uni doesn't have an in-depth course on this stuff

Daminark

I don't really study this stuff myself much, I'm vaguely aware of the absolute basics of set theory and that's about it

Halmos' Naive Set Theory is a book I've heard of

Also Jech has a book that's very hard

Sir Cumference

2:43 AM

@Daminark I maybe being pedantic, but if an element is not comparable to anything else, then nothing is strictly larger than it. Do we consider that to be maximal?

Daminark

I mean yeah that works

Sir Cumference

@Daminark I see, thanks

Daminark

No problem

But yeah so, those books I mentioned are if you specifically care about set theory

There's an amount that every mathematician should know but that amount is very tiny, and if you're only just looking for that amount you can prob just read chapter 1 of some analysis or algebra book floating around

Sir Cumference

I mean yep, set theory and logic and other foundational topics are interesting to me. Shame they are overlooked by our math dept.

@Daminark Yeah, I'm mostly around that minimal amount (at least I think).

Silent

We get done with just 3 matrices, because, in our examples, one entry is 1. But, generally, $\begin{bmatrix}
1 & 0 \\
0 & c \\
\end{bmatrix}
\begin{bmatrix}
b & 0 \\
0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
\frac{d}{c} & 1 \\
\end{bmatrix}
\begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}
=
\begin{bmatrix}
0 & b \\
c & d \\
\end{bmatrix}$

Dair

3:06 AM

@Daminark Clearly, it's all about type theory

type theory is what the real applied metamathematicians use.

but how many schools teach type theory?

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$Mathematics$ Mathematics