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# Visually stunning math concepts...

Didn't Sean Bean die before he got his hands on any Rings? I mean I...

I'll... just show myself out. Back into 3-space, you know....

posted by GenjiandProust at 3:31 PM on April 7 [3 favorites]

Well, the question didn't ask for easy explanations, just that they exist. You have to be careful of the distinction when talking to mathematicians.

I actually think the Seifert surface one is something you could almost explain to elementary/middle school kids. There is already a common demonstration where you take strips of paper and form loops with different numbers of twists, then cut them down the middle. A strip with no twist produces two separate loops. A strip with one half twist produces a single loop. A strip with two half twists produces two loops, but now they are linked. Three half twists produces a single loop with a knot in it.

The idea is that the strips are various surfaces, and cutting down the middle is a rough approximation of taking the boundary of the surface. There are a couple of issues, though. First is that the Moebius strips, with an odd number of half twists, are not technically Seifert surfaces. Seifert surfaces must be orientable, i.e. two-sided, and the Moebius strips have only one side. There always is a two-sided Seifert surface for any knotted loop or set of linked loops, but that gets to the second problem with the demonstration: those surfaces are not easy to construct from paper. Even if you could, finding the boundary would not be as simple as cutting down the center of a loop. Still, it seems like someone could find a way do it.

posted by eruonna at 4:19 PM on April 7 [6 favorites]

Actually, I just tried it for the trefoil knot, and it worked rather well. You won't get anything like the picture for the Borromean rings, but definitely something you could do with middle schoolers.

posted by eruonna at 4:57 PM on April 7

I shall dispute this in the scholarly method in which I am most thoroughly versed which is by hollering NUH HUH IT DID TOO

but really it did; "easy to understand and explain" was a direct quote from the question

posted by ook at 7:12 PM on April 7

Those numbers (1,2,3,4) are diameters of the circle. The circumference of the circle (one turn) is pi, or 3.14 diameters.

posted by weapons-grade pandemonium at 7:15 PM on April 7

Math stack exchange is really not for the general reader.

posted by empath at 1:42 AM on April 8 [1 favorite]

Post

# Visually stunning math concepts...

April 7, 2014 2:36 PM Subscribe

Math trigger warning..big time.

posted by jnnla at 3:01 PM on April 7 [2 favorites]

posted by jnnla at 3:01 PM on April 7 [2 favorites]

I found it interesting how many of the posters in that thread have completely lost track of what "easy to understand and explain" means. My favorite is the topologist whose explanation for his impenetrable diagram is "Every link (or knot) is the boundary of a smooth orientable surface in 3-space. This fact is attributed to Herbert Seifert, since he was the first to give an algorithm for constructing them. The surface we are looking at is bounded by Borromean rings" OH RIGHT BORROMEAN RINGS OF COURSE I TOTALLY GET IT NOW

posted by ook at 3:03 PM on April 7 [19 favorites]

posted by ook at 3:03 PM on April 7 [19 favorites]

Once again I am reminded that pretty much any alternate proof of the Pythagorean Theorem is easier to follow that the stupid glue three squares to a triangle one they teach in school.

posted by ckape at 3:03 PM on April 7 [1 favorite]

posted by ckape at 3:03 PM on April 7 [1 favorite]

I like the ones that don't move, because I can stare at them until the analogy finally clicks. That, and comment #36's stories about the math teacher that made her students derive their own formulas by measuring things.

Incidentally, this same teacher introduced us to the concept of pi by asking us to find something circular in our house (“like a plate or a coffee can”), measuring the circumference and the diameter, and dividing the one number by the other. I can still see her studying the data on the chalkboard the next day – all 20 or so numbers just a smidgeon over 3 – marveling how, even though we all probably measured differently-sized circles, the answers were coming out remarkably similar, “as if maybe that ratio is some kind of constant or something...”posted by Kevin Street at 3:10 PM on April 7 [10 favorites]

*OH RIGHT BORROMEAN RINGS OF COURSE I TOTALLY GET IT NOW*

Didn't Sean Bean die before he got his hands on any Rings? I mean I...

*Borromean*....

I'll... just show myself out. Back into 3-space, you know....

posted by GenjiandProust at 3:31 PM on April 7 [3 favorites]

I really like the beauty of the multiplication table modulo m. Modular multiplication is simple enough that kids can understand it, but the pictures it generates can be really beautiful.

posted by wittgenstein at 4:14 PM on April 7

posted by wittgenstein at 4:14 PM on April 7

the one demonstrating a line integral blew my mind when i first saw it. wish i had seen it while i was actually taking multivariable calculus.

posted by triceryclops at 4:16 PM on April 7

posted by triceryclops at 4:16 PM on April 7

*I found it interesting how many of the posters in that thread have completely lost track of what "easy to understand and explain" means.*

Well, the question didn't ask for easy explanations, just that they exist. You have to be careful of the distinction when talking to mathematicians.

I actually think the Seifert surface one is something you could almost explain to elementary/middle school kids. There is already a common demonstration where you take strips of paper and form loops with different numbers of twists, then cut them down the middle. A strip with no twist produces two separate loops. A strip with one half twist produces a single loop. A strip with two half twists produces two loops, but now they are linked. Three half twists produces a single loop with a knot in it.

The idea is that the strips are various surfaces, and cutting down the middle is a rough approximation of taking the boundary of the surface. There are a couple of issues, though. First is that the Moebius strips, with an odd number of half twists, are not technically Seifert surfaces. Seifert surfaces must be orientable, i.e. two-sided, and the Moebius strips have only one side. There always is a two-sided Seifert surface for any knotted loop or set of linked loops, but that gets to the second problem with the demonstration: those surfaces are not easy to construct from paper. Even if you could, finding the boundary would not be as simple as cutting down the center of a loop. Still, it seems like someone could find a way do it.

posted by eruonna at 4:19 PM on April 7 [6 favorites]

That's the kind of animation that can be gimmicked up to show that pi is 3.

posted by Obscure Reference at 4:55 PM on April 7 [3 favorites]

posted by Obscure Reference at 4:55 PM on April 7 [3 favorites]

No one suggested Newton's method! I hate just about everything about Newton's method, except the fact it is the one thing where a) the picture makes sense to me and b) I truly never bothered learning the formula because it's derivable from the picture. Never fear, Wikipedia has it.

posted by hoyland at 4:56 PM on April 7 [1 favorite]

posted by hoyland at 4:56 PM on April 7 [1 favorite]

*[T]hose surfaces are not easy to construct from paper.*

Actually, I just tried it for the trefoil knot, and it worked rather well. You won't get anything like the picture for the Borromean rings, but definitely something you could do with middle schoolers.

posted by eruonna at 4:57 PM on April 7

Yeah, I don't get how that rolling circle has to land where it does. Am I missing something?

posted by benito.strauss at 5:51 PM on April 7

posted by benito.strauss at 5:51 PM on April 7

Wooooooooooooooo! Nice pix!

posted by zscore at 6:21 PM on April 7 [1 favorite]

posted by zscore at 6:21 PM on April 7 [1 favorite]

The animated trig example was mind blowing. So, so easy to get your head around once you see the relationships in motion. The Fourier transform animations were likewise, "crap... I get it."

posted by Slap*Happy at 6:40 PM on April 7

posted by Slap*Happy at 6:40 PM on April 7

*well, the question didn't ask for easy explanations, just that they exist*

I shall dispute this in the scholarly method in which I am most thoroughly versed which is by hollering NUH HUH IT DID TOO

but really it did; "easy to understand and explain" was a direct quote from the question

posted by ook at 7:12 PM on April 7

*Yeah, I don't get how that rolling circle has to land where it does. Am I missing something?*

Those numbers (1,2,3,4) are diameters of the circle. The circumference of the circle (one turn) is pi, or 3.14 diameters.

posted by weapons-grade pandemonium at 7:15 PM on April 7

Right, it asked for easy to understand and explain. It just didn't ask for the easy explanations.

posted by eruonna at 7:18 PM on April 7 [1 favorite]

posted by eruonna at 7:18 PM on April 7 [1 favorite]

I guess I just don't find that rolling circle as compelling as the others. For the others, if they put any other values they would just look wrong, but not for this. Like one of the comments says "That the circle's perimeter is π

posted by benito.strauss at 8:32 PM on April 7

*d*is the definition of π, so I wouldn't say this is an explanation of the fact; rather it's an illustration of what the definition means".posted by benito.strauss at 8:32 PM on April 7

ook: "

Yep, thanks for this. You're spot on.

posted by Conrad Cornelius o'Donald o'Dell at 8:51 PM on April 7 [1 favorite]

*I found it interesting how many of the posters in that thread have completely lost track of what "easy to understand and explain" means.*"Yep, thanks for this. You're spot on.

posted by Conrad Cornelius o'Donald o'Dell at 8:51 PM on April 7 [1 favorite]

As far as losing track of 'easy' goes, I think A visual display that 0

Cool picture though. But Fourier transform of the light intensity due to a diffraction pattern caused by light going through 8 pinholes... is flat-out gorgeous, if even less edifying, if that's even possible.

Can anyone here explain those 'explanations' at all?

posted by hap_hazard at 12:14 AM on April 8

^{0}=1 is the most wtf for me. I'm not even sure what it is he's trying to explain, never mind whether he did or not. The comments on the post, and the general... demeanor... of his website are not helping me sort that out, either.Cool picture though. But Fourier transform of the light intensity due to a diffraction pattern caused by light going through 8 pinholes... is flat-out gorgeous, if even less edifying, if that's even possible.

Can anyone here explain those 'explanations' at all?

posted by hap_hazard at 12:14 AM on April 8

*I found it interesting how many of the posters in that thread have completely lost track of what "easy to understand and explain" means.*

Math stack exchange is really not for the general reader.

posted by empath at 1:42 AM on April 8 [1 favorite]

I like doing estimates. The square root of 10 is 3.16(227766...). Remember that number (conveniently close to PI) to help with estimations of the sides of square areas.

sqrt(10) = 3.16

sqrt(100) = 10

sqrt(1,000) = sqrt(100)*sqrt(10) = 10* 3.16 = 31.6

sqrt(10,000) = 100

sqrt(100,000) = sqrt (10,000)*sqrt(10) = 100*3.16 = 316

etc.etc.

So when you hear "the search area covers 400,000 square miles" you can estimate sqrt(4*10*10,000).

2*316 ≈ 632 miles on a side.

posted by Twang at 6:11 AM on April 8 [4 favorites]

sqrt(10) = 3.16

sqrt(100) = 10

sqrt(1,000) = sqrt(100)*sqrt(10) = 10* 3.16 = 31.6

sqrt(10,000) = 100

sqrt(100,000) = sqrt (10,000)*sqrt(10) = 100*3.16 = 316

etc.etc.

So when you hear "the search area covers 400,000 square miles" you can estimate sqrt(4*10*10,000).

2*316 ≈ 632 miles on a side.

posted by Twang at 6:11 AM on April 8 [4 favorites]

Borromean rings are cool -- they're 3 loops that are interconnected such that breaking any one unravels the whole thing. For the Seifert surface, think about dipping the rings it into a soap solution -- the theorem says that you can get a single sheet of soap film anchored to all the rings. The picture shows you that soap film.

Cool thing about the fourier transform of the square wave is that it shows the Gibbs phenomenon -- as the number of sines increases, the overshoot/undershoot at the jump gets narrower but stays (more or less) at the same height.

0

posted by phliar at 12:54 PM on April 8

Cool thing about the fourier transform of the square wave is that it shows the Gibbs phenomenon -- as the number of sines increases, the overshoot/undershoot at the jump gets narrower but stays (more or less) at the same height.

0

^{0}is only defined in some areas of math...posted by phliar at 12:54 PM on April 8

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I think if you look at this animation and think about it long enough, you'll understandI beg to differ.

As in, I see how aesthetically pleasing they are, especially the first one, and it's a cool post. But the maths part of my brain has tumbleweeds blowing through it and much as I stared all I saw were pretty pictures. I am, as always, in awe of people who speak the language of maths.

posted by billiebee at 2:57 PM on April 7 [3 favorites]