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So that's why they call 'em fractals!
March 17, 2012 7:39 AM   Subscribe


 
That's interesting. I had only thought of fractals as being self-similar (and defined by recursive functions), I didn't know about the fractional dimension aspect.

Now I'm looking at Wikipedia and found out about Multifractals, where a single dimension isn't enough to describe them, and you have a whole spectrum of exponents over a continuum, and also strange attractors which has a very cool picture associated with the article. (Okay the article is about attractors in general, but the picture is of a strange attractor, which have fractional dimensions)
posted by delmoi at 8:01 AM on March 17, 2012 [3 favorites]


Nice and clear and not at all obscure! Thanks.
posted by flabdablet at 8:03 AM on March 17, 2012


Very interesting; thanks for posting it.
posted by languagehat at 8:04 AM on March 17, 2012


Is this different from the fractal dimension?
posted by scose at 8:26 AM on March 17, 2012


I like all the pretty colours he used and his British accent.
posted by Fizz at 8:33 AM on March 17, 2012


Is this different from the fractal dimension?

I don't think so, sounds like the same thing from the video.
posted by delmoi at 8:38 AM on March 17, 2012


Very cool. I knew what fractals were, but it would've saved me a lot of headaches if someone had explained it this clearly to begin with.

My favorite quote: "That's a little math equation! I can play with that math equation, and all is delightful! Here goes... so exciting I'll change color pen."
posted by Riki tiki at 8:40 AM on March 17, 2012 [1 favorite]


This is one of those things where, the idea is very simple, but mathematicans tend to describe it in a complex way. The first sentence of the Wikipedia article says:
A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension[1] and may fall between the integers.
That coincides with what's in the video, but a term like "topological dimension" might not be obvious to people. If you click the link for "Topological Dimension" you get redirected to the page on Lebesgue covering dimension, which are describe thusly:
Lebesgue covering dimension or topological dimension is one of several inequivalent notions of assigning a topological invariant dimension to a given topological space.
Under examples they say:
The n-dimensional Euclidean space En has covering dimension n.
Which is the same that the normal dimensions we think of (which are in Euclidean space) are examples of Lebesgue covering dimensions/topological dimensions

Now it's probably the case that "Fractals" cover all kinds of stuff that happens outside of euclidean space, so a defining them without mentioning all that would be incomplete. But still, it gives a false impression that you can't understand the basic idea.
posted by delmoi at 8:47 AM on March 17, 2012 [4 favorites]


The dimensions occurring in fractals are Hausdorff dimensions, not Lebesgue covering dimensions: the thing that's kind of mindblowing about Hausdorff dimension, and fractals, is that the dimensions don't have to be integers! So you end up with, say, 1.58-dimensional spaces (cf the Wikipedia article).
posted by Frobenius Twist at 8:54 AM on March 17, 2012 [1 favorite]


That was really well done, and didn't require skipping ahead by Wadsworth's constant either!
posted by spacewrench at 8:55 AM on March 17, 2012 [1 favorite]


Do MiFi comments have a fractal dimension?
posted by sammyo at 9:13 AM on March 17, 2012


Well, the reason for the 'many inequivalent' definitions of topological dimension is that in our day-to-day world there are a lot of different properties that correspond to dimension. When we start thinking about not-so-day-to-day things, we have a lot of choices for which of these properties to talk about when we speak of 'dimension.'

For example, the scaling idea in the video gives one notion of dimension: what happens to the 'volume' of a thing when you scale it up by a factor of (say) 2? This depends on a notion of volume, which depends on having a notion of distance, which we don't have in a general topological space. So we have to go for something else.

Another notion of dimension is this: We have a notion of an n-dimensional ball, which we should probably agree really is n-dimensional. Then given a point in some space that we're interested in, we can try to draw a little n-dimensional ball around that point, with n as big as possible. If we can draw in an n-dimensional ball for every point, then we say that the space is n-dimensional. But unfortunately, this only helps us with spaces called 'manifolds,' which are pretty close to our usual intuition....

Then you start looking at more obscure properties of dimension which might hold in places that aren't manifolds; the Lesbegue covering dimension is based on one such property, which is well-illustrated in the wikipedia article, regardless of the text!

And yeah, then there's Hausdorff dimension, which is getting at this notion of volumes, and does require a notion of distance. The tension is between defining dimension in general topological spaces, without any reference to length, and the extra properties that come into play when distance is allowed into the mix.

The surface of a tree, topologically, is very two dimensional. But the tree is filling up a volume the best it can. If you allow arbitrary stretching of a tree (as happens when there's no notion of distance), there's no way we can really say that the tree is filling up that volume, because we can distort it in all kinds of funny ways. But once that notion of distance is allowed, the tree can do its best to fill up the available space, and thereby maximize its access to sunlight...
posted by kaibutsu at 9:15 AM on March 17, 2012 [1 favorite]


This is one of those things where, the idea is very simple, but mathematicans tend to describe it in a complex way.

No, it's one of those things where, the idea is very simple, and mathematicians tend to describe it in a simple way, using some words that not everyone knows, because those words are what's necessary to describe the actual idea and not some approximation to it.

Not to take anything away from this video, which is great.
posted by escabeche at 9:24 AM on March 17, 2012 [4 favorites]


one of several inequivalent notions of assigning a topological invariant dimension to a given topological space.

This.

This makes me crazy about wikipedia math topics, they seem authoritative, well written and probably quite accurate, but if you need to take a post graduate class to understand an, ahem, encyclopedia article, then it's NOT the right level.

Then to demoli's excellent point that many rigorously defined concepts are very approachable but just need a bit of common folk speak to get rough feel of the essence of the idea. So why couldn't there be a wikipeida section that could just say that a 3-Manafold is just a frigg'n bowling ball?
posted by sammyo at 9:27 AM on March 17, 2012


Because it's _not_ just a bowling ball.......... A donut (minus the surface) is also a 3-manifold.
posted by kaibutsu at 9:29 AM on March 17, 2012 [1 favorite]


This makes me crazy about wikipedia math topics, they seem authoritative, well written and probably quite accurate, but if you need to take a post graduate class to understand an, ahem, encyclopedia article, then it's NOT the right level.

The right level for what? Thanks to Wikipedia, math students and researchers all over the world have access to a constantly updated and amazingly accurate and exhaustive compendium of basic definitions and facts about mathematical objects. It would be great if the Wikipedia article about notions of dimension linked to this video or an exposition like it. It would be terrible if the Wikipedia article about notions of dimension were this video or an exposition like it, because this video doesn't actually tell you what fractional dimension is.
posted by escabeche at 9:37 AM on March 17, 2012 [1 favorite]


A donut (minus the surface) is also a 3-manifold

Someone please register this name on meta.
posted by Fizz at 9:37 AM on March 17, 2012 [1 favorite]


No, it's one of those things where, the idea is very simple, and mathematicians tend to describe it in a simple way, using some words that not everyone knows, because those words are what's necessary to describe the actual idea and not some approximation to it.
Well, that's what I said:
Now it's probably the case that "Fractals" cover all kinds of stuff that happens outside of euclidean space, so a defining them without mentioning all that would be incomplete
So I don't see how you're disagreeing with me here. But Wikipedia is not a website written only for mathematicians. There would be nothing wrong with structuring an article in a way that first gives an approximation, then gets into a more rigorous definition. Especially since 'approximation' here would just be the restriction to euclidean space anyway, it wouldn't even be imprecise, just incomplete (Unless the video was wrong)
posted by delmoi at 9:39 AM on March 17, 2012 [1 favorite]


Yes, certainly, but 3-4 mundane examples in a wikipedia article is not going to ruin the field of algebraic topology.

No, really, is it that hard to give a general gut feeling example of, oh say, cohomology?
posted by sammyo at 9:40 AM on March 17, 2012


Then to demoli's excellent point that many rigorously defined concepts are very approachable but just need a bit of common folk speak to get rough feel of the essence of the idea. So why couldn't there be a wikipeida section that could just say that a 3-Manafold is just a frigg'n bowling ball?

Drrrrrrr
posted by alex_skazat at 9:50 AM on March 17, 2012 [1 favorite]


No, really, is it that hard to give a general gut feeling example of, oh say, cohomology?

I'll be here, all day :)


Actually, that may be a good example of a gut feeling made by eating chili too late in the evening after a large amount of drinking. If only we could edit the article, and make it clearer...
posted by alex_skazat at 9:53 AM on March 17, 2012


But Wikipedia is not a website written only for mathematicians. There would be nothing wrong with structuring an article in a way that first gives an approximation, then gets into a more rigorous definition.

I'm sympathetic to this, but at the same time I think Wikipedia should not do that. Mathematicians are -- with good reason! -- highly wary of approximations, because approximations are pretty much always wrong, mathematically speaking; and intuitive approximations have historically caused serious problems in mathematics (cf the italian school of algebraic geometry).

Also, the Wikipedia article on topological dimension that you linked to is not just for mathematicians: it's written at a level that an undergrad student in mathematics who has taken an introductory topology course could understand, so I think it's at a reasonable level.
posted by Frobenius Twist at 9:54 AM on March 17, 2012 [1 favorite]


No, really, is it that hard to give a general gut feeling example of, oh say, cohomology?


I think so, yes.
posted by escabeche at 9:55 AM on March 17, 2012 [1 favorite]


Topology can be fun. I once used it to prove that a pair of underwear that you're wearing can be removed, without taking off the pants worn over top, in two different ways, as long as you're flexible and the underwear is sufficiently stretchy. Both methods give you a hell of a wedgie though, and one risks strangulation if you're not careful.
posted by radwolf76 at 10:03 AM on March 17, 2012 [1 favorite]


Homology I can give a brief description of, but cohomology's a different beast....

Two curves on a surface a 'homologous' if they co-bound a surface, ie, they trace out a nice surface between them. So, for example, two lines of latitude on a sphere have a kind of belt, so they're homologous. But you're also allowed to stretch and move around the curves freely, which makes it a bit more obscure. So two lines of longitude, which intersect, can be pushed around until they don't intersect, and then they co-bound a belt on the sphere, just like the lines of latitude did. In fact, on the sphere, all curves are homologous. Homology is interested in how many different 'kinds' of curves you can put on a surface, where two curves are of the same kind if they're homologous.

On the surface of a donut, there are three different kinds of curves: Curves which you can shrink down to a point, curves which run all the way around the donut 'horizontally,' and curves which run around the donut 'vertically.' (See the red lines in this picture.)

So that's curves on surfaces. For higher-dimensional things, you look at k-dimensional things which cobound (k+1)-dimensional things. This gives the k-homology of a space. So what I've described here is the 1-homology of a surface. The 0-homology is about points cobounding curves on the surface: This happens whenever you can draw a line on the surface between two points. As a result, the 0-homology counts the number of connected pieces of the surface. (So a single surface always has 0-homology with just one thing in it...)

The thing that makes homology really interesting is that it has some extra algebraic structure. By doing a kind of obscure algebraic trick, one turns homology into co-homology. And the co-homology has a product structure on it, which the homology doesn't: you can take two different cohomology classes, multiply them together, and get another cohomology class back again. And that extra structure lets us do quite a lot. In the case of manifolds, there's a nice connection between the intersection of curves and the product structure: notice that the two red curves on the torus intersect in a point; this intersection can never be smoothed away, and corresponds to the mutliplication of the cohomology classes associated to those curves. But to do this in more generality, it's really the algebra that takes over.

A prof of mine was very active when algebraic topology was first taking off, and big results started coming down the pipe... Apparently the old guard was very suspicious of the algebraic approach, which was driven more by algebraic ideas than geometric intuition, but ultimately the algebraic approach produced really amazing results. To the point where there's a joke that runs, 'Those who do not understand homology are doomed to re-invent it.'
posted by kaibutsu at 10:08 AM on March 17, 2012 [4 favorites]


No, really, is it that hard to give a general gut feeling example of, oh say, cohomology?

I think so, yes.

Agreed. The beauty of math is its abstraction. But that's also exactly the reason that it's hard -- if not impossible -- to give a reasonable, common-sense explanation of mathematical gadgets like cohomology. To give an appropriate background to cohomology requires topology, group theory, homological algebra, and ideally category theory, and although you could explain what cohomology is by using those tools, it's the why that's so much harder. Why do mathematicians want to create these groups/vector spaces associated to manifolds or algebraic varieties (whatever those are)? What is the purpose? There is a very good reason, but unfortunately it's not at all easy to give a coherent explanation.

Also, oh my god, the simple wikipedia article on cohomology that alex_skazat links to is just horribly depressing.
posted by Frobenius Twist at 10:09 AM on March 17, 2012


What are you talking about? I make cohomologies of items with my machine all the time, even sometimes when I understand the item just fine. Sometimes I try to put multiple items in my machine just because they look nice together, with their different cohomologies. (j/k)
posted by kaibutsu at 10:11 AM on March 17, 2012 [2 favorites]


(this is me working really hard on that paper which is apparently not writing itself right now.)
posted by kaibutsu at 10:14 AM on March 17, 2012


If I hear one more person say something like "I've never understood this, but now I watched this video, and the guy explained it so clearly, why don't mathematicians do that more frequently", I might just kill myself. I explain shit in simple terms all the time. No one ever fucking listens. They'd rather sit there and furiously copy down notes while refusing to pay attention to any of the words that I'm speaking.

Maybe it's just the curse of sitting in a classroom. No one wants to learn while sitting in those things.

This video was OK. He too quickly is labeling parts of his picture of the house, and very quickly talking about area and volume, and reuses variables he's already used and so forth. If I did something like that, there would be 900 complaints (and rightly so, it's irresponsible).

There's something to be said for being able to speak spontaneously about a topic that you truly understand, but this video was painfully unrehearsed.
posted by King Bee at 10:19 AM on March 17, 2012


To give an appropriate background to cohomology requires topology, group theory, homological algebra, and ideally category theory

Here I disagree -- I think cohomology is appropriate background for category theory, not the other way around; while the basic definitions of category theory are in some sense completely independent of any context, I don't think they're very easy to understand without some such context in mind, and cohomology (or at least homology) seems to me the most natural choice.
posted by escabeche at 10:44 AM on March 17, 2012


King Bee: No one ever fucking listens.

You got that right.
posted by sneebler at 11:06 AM on March 17, 2012


Also, oh my god, the simple wikipedia article on cohomology that alex_skazat links to is just horribly depressing.


No kidding. This is the entirety of the article:
Cohomology is something in higher math which is used to solve math problems. When there is an item which a mathematician doesn't know enough about, he or she can make a cohomology out of it using a machine. The mathematician then takes objects he or she knows more about, puts them into the same machine, and looks at the objects which have a cohomology like the first item.
I may have to go cry.
posted by leahwrenn at 11:19 AM on March 17, 2012


You can give a good sense of what cohomology is by the end of a good undergraduate multivariable calculus class. Integrating dθ around the unit circle to get 2π, and understanding Green's Theorem, is about all you need to get the essence of it. Also, in lots of settings, cohomology classes are explainable by intersection theory, so you can "do cohomology" with nothing more than counting and +/- signs. It is good to remember that all that algebra built up around truly simple ideas.
posted by Wolfdog at 11:20 AM on March 17, 2012 [1 favorite]


I once used it to prove that a pair of underwear that you're wearing can be removed, without taking off the pants worn over top, in two different ways, as long as you're flexible and the underwear is sufficiently stretchy.

obligatory
posted by flabdablet at 11:34 AM on March 17, 2012 [2 favorites]


There are some things that you cannot and will not understand if you are not a mathematician and sometimes even then.

There are worlds, and not just mathematical ones, which take a lifetime to even partially comprehend and translating these worlds into something understandable by lay persons, even highly educated lay persons, loses a lot of necessary detail.

This is not always the fault of the translator. Sometimes, it goes with the territory.

While you might understand the brilliant analogies to discuss protein folding, Relativity Theory, or Quantum Mechanics, you do not fully understand those domains unless you actually do the math, as they say.

There is nothing wrong with the fact that there are things which you will forever be ignorant of and to which you will forever be uninitiated, despite your and others most strenuous efforts. Nothing wrong, that is, except your unreasonable self-expectation.
posted by mistersquid at 11:53 AM on March 17, 2012 [3 favorites]


So I liked the video and feel I have an understanding of something I didn't before. Based on the conversation above, exactly how stupid does this make me? Use whatever dimensions you like.
posted by maxwelton at 12:34 PM on March 17, 2012 [1 favorite]


Metafilter: There is nothing wrong with the fact that there are things which you will forever be ignorant of and to which you will forever be uninitiated, despite your and others most strenuous efforts. Nothing wrong, that is, except your unreasonable self-expectation.

hahahahahahahahaha! I had to post that!
posted by Multicellular Exothermic at 12:48 PM on March 17, 2012


Fizz: I like all the pretty colours he used and his British accent

His accent wasn't easy to discern, but seeing how "length" sounded more like "lingth" to me, I'd say the accent was more Kiwi than British.
posted by salmacis at 1:26 PM on March 17, 2012


So I liked the video and feel I have an understanding of something I didn't before. Based on the conversation above, exactly how stupid does this make me?

That's not what I was talking about. I was talking about people who watch these videos, get the feeling you got, then feel the need to bash everyone who has tried to explain a complex mathematical topic to them in the past.
posted by King Bee at 1:34 PM on March 17, 2012


There are worlds, and not just mathematical ones, which take a lifetime to even partially comprehend and translating these worlds into something understandable by lay persons, even highly educated lay persons, loses a lot of necessary detail.

This is not always the fault of the translator. Sometimes, it goes with the territory.

While you might understand the brilliant analogies to discuss protein folding, Relativity Theory, or Quantum Mechanics, you do not fully understand those domains unless you actually do the math, as they say.

There is nothing wrong with the fact that there are things which you will forever be ignorant of and to which you will forever be uninitiated, despite your and others most strenuous efforts. Nothing wrong, that is, except your unreasonable self-expectation.
Wow, could you try making that more arrogant and obnoxious? The reason I want to see math described for people in a way they can understand is so they can appreciate it. The stuff presented in the video only requires high-school algebra to understand. Obviously, at the same time fractals can exist in other dimensions with spaces with other topologies.

But the problem is that by not making it clear that the concept, when applied to everyday euclidean space is pretty easy to grasp. (The only drawback of the video is that the narrator doesn't make it clear that there are different 'types' of dimension)

That may be the case with cohomology, but it clearly isn't the case when it comes to fractional dimensions, as explained in the video. Since fractals are something lots of people are interested in, even if they're not mathematicians, it would be a good example of a page that would benefit from an explanation for 'ordinary' people.
Also, the Wikipedia article on topological dimension that you linked to is not just for mathematicians: it's written at a level that an undergrad student in mathematics who has taken an introductory topology course could understand, so I think it's at a reasonable level.
Right, while that may be good for someone specifically looking up topological dimension, it's also a link that's in the basic definition of fractal. So if you're reading about fractals, you click that link, and get that page. That happens all the time if you just click around on science topics. You'll start out on one page at one level, then you end up on another that's like half done and what's there is at a PhD level.

The other thing is that you don't need to 'approximate' if you if you simply restrict the definition to 2-d or 3-d surfaces, then it's easy to explain, as was done in the video. If you are trying to give a complete definition that applies to all spaces to which the concept applies, then obviously you can't do that. But I think it would be good if they could try to have a 'simple explanation' section along with the more rigorous definitions. And actually the article on fractals does have a good "Introduction" section, but that's 'below the fold'.
That's not what I was talking about. I was talking about people who watch these videos, get the feeling you got, then feel the need to bash everyone who has tried to explain a complex mathematical topic to them in the past.
At the same time, though, you do have to recognize that there plenty of bad explanations out there too. When I was trying to learn abstract algebra, I was working my way through some free e-book, and I was having trouble with it. When I switched to a different book the same concepts seemed much, much easier to understand.
Drrrrrrr
posted by alex_skazat at 11:50 AM on March 17 [+] [!]
No, really, is it that hard to give a general gut feeling example of, oh say, cohomology?
I'll be here, all day :)


Actually, that may be a good example of a gut feeling made by eating chili too late in the evening after a large amount of drinking. If only we could edit the article, and make it clearer...
posted by alex_skazat at 11:53 AM on March 17 [+] [!]
Uh what? As people have pointed out, the articles on the simple english Wikipedia don't actually even tell you anything. The article on Chomology says:
Cohomology is something in higher math which is used to solve math problems. When there is an item which a mathematician doesn't know enough about, he or she can make a cohomology out of it using a machine. The mathematician then takes objects he or she knows more about, puts them into the same machine, and looks at the objects which have a cohomology like the first item.
Which as far as I can tell, doesn't even tell you anything. On the other hand, Wikipedia:
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory.

From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra. The terminology tends to mask the fact that in many applications cohomology, a contravariant theory, is more natural than homology. At a basic level this has to do with functions and pullbacks in geometric situations: given spaces X and Y, and some kind of function F on Y, for any mapping ƒ: X → Y composition with ƒ gives rise to a function F o ƒ on X. Cohomology groups often also have a natural product, the cup product, which gives them a ring structure. Because of this feature, cohomology is a stronger invariant than homology, as it can differentiate between certain algebraic objects that homology cannot.
The two articles don't seem to have anything to do with eachother.
posted by delmoi at 1:47 PM on March 17, 2012 [1 favorite]


I think Wikipedia's math articles could benefit from more 'real world' applications of the concepts in the articles. I can't tell you how many times I've watched lectures where the professor immediately introduces a math concept, explains the general gist of it, and then uses it to solve a problem or finish a proof. I watch that, and in 15 minutes, I get a decent understanding of the concept and know at least one example of how it's useful. Then I'll go to the Wikipedia article for more information, and it's just a mess of jargon and formulas and links to other articles which are also a mess of jargon and formulas.

Susskinds physics lectures are great for that. Try understanding Christoffel symbols from the Wikipedia articles. Then watch susskind explain them in his general relativity courses, which assume you know nothing more than calculus. (it's probably best if you start from the first lecture in the series, but even so, you can see that he explains it in plain English with a minimum of jargon.

Granted, it's more verbose, but words don't cost anything. Who cares if it takes 1000 more words to explain it in plain English? There should be room for that and for the precisely defined mathematical definition both.
posted by empath at 2:17 PM on March 17, 2012


I think one of the problems at play in the whole morass of explaining (difficult) mathematical concepts is the fact that lay people require 'plain English' explanations, and mathematicians are used to rigor. In fact, part of the process of doing math is learning how to be excruciatingly rigorous about proving things. Which basically means mathematicians are trained to be bad at giving plain English explanations, because the kind of hand-waviness required to do that is one of the things that makes for a poor proof (which is really no proof at all).

All of which is not to say that it's impossible (as this video I think is rather good despite it being a little unrehearsed), just that it's not surprising that the guy who really knows algebraic topology or whatever is probably going to suck at giving an elevator-pitch explanation of cohomology.
posted by axiom at 2:32 PM on March 17, 2012


Wait, there's a Simple Wikipedia? Separate from the regular Wikipedia? I had no idea. Sadly, there is no simple.wikipedia entry for Christoffel symbols.
posted by exphysicist345 at 2:54 PM on March 17, 2012


Wow, could you try making that more arrogant and obnoxious?
I didn't think I was being arrogant and obnoxious. I consider myself fortunate that your accusation doesn't make me feel I was being so. What I wonder is what is driving you to take so oppositional a stance to so many different people in this thread? Honestly, I don't think it's me.

Fractals are a fairly simple concept to understand. The video helped you understand them. Awesome. FWIW, James Gleick's 1980s Chaos has a great explanation for lay persons, too.

Still, I'm guessing the complexities of topology and set theory are probably beyond your reach. They certainly are beyond mine. I just don't accuse mathematicians of using unnecessarily complex words because I don't have the background to fully understand those domains.
posted by mistersquid at 2:57 PM on March 17, 2012


At the same time, though, you do have to recognize that there plenty of bad explanations out there too. When I was trying to learn abstract algebra, I was working my way through some free e-book, and I was having trouble with it. When I switched to a different book the same concepts seemed much, much easier to understand.

I do. There are. There are also more great explanations out there than people seem to realize. I know this, because I will have students in my classroom who don't understand a thing, and then those same students will come to my office later saying they are lost. I'll explain the topic again, using exactly the same words, and they will say "wow, it's that simple?"

I think, yes, all you have to do is listen.
posted by King Bee at 3:04 PM on March 17, 2012 [1 favorite]


This is delightful! Thanks for posting.
posted by odinsdream at 3:10 PM on March 17, 2012


Wait, there's a Simple Wikipedia? Separate from the regular Wikipedia? I had no idea. Sadly, there is no simple.wikipedia entry for Christoffel symbols.
It's the "Simple English" Wikipedia, written with a minimal vocabulary, I think the idea is for it to be useful to people for whom English is a second language. There's no restriction on the complexity of the topics, but it just doesn't get nearly the attention as the main English Wikipedia, for obvious reasons.
I didn't think I was being arrogant and obnoxious.
Seriously? Let's review:
There is nothing wrong with the fact that there are things which you will forever be ignorant of and to which you will forever be uninitiated, despite your and others most strenuous efforts. Nothing wrong, that is, except your unreasonable self-expectation.
Which reads a lot like "You are stupid. You are so stupid that your belief that you might understand these things is totally unreasonable." Which seems arrogant and obnoxious to me. How could you not expect that calling someone irredeemably ignorant might come across as at least a little insulting insulting?

It's true that there will be some gaps in everyone's knowledge, but for a smart person, each individual gap could be closed over time, but given a finite lifetime, you have to pick and choose. If something like that is all you meant, it was phrased poorly.
I consider myself fortunate that your accusation doesn't make me feel I was being so. What I wonder is what is driving you to take so oppositional a stance to so many different people in this thread? Honestly, I don't think it's me.
I went back and re-read my comments, I don't really see where I'm being "oppositional" to anyone else, unless you just mean I disagree with them.
posted by delmoi at 4:04 PM on March 17, 2012


Mathematicians are up themselves. There are only so many numbers, and once you learn all the numbers you basically know maths. So why do they make things so hard to understand?
posted by Joe in Australia at 4:26 PM on March 17, 2012 [1 favorite]


A lot of people believe ignorance is a badge of shame. I am always often when I'm having a conversation and I profess ignorance and someone says, "No, you're not ignorant" when in fact I know I am on whatever topic is being discussed.

I don't agree that very intelligent people might close all gaps in knowledge over time because information is potentially infinite and intelligence in certain domains is to some extent innate. For example, some people have a higher capacity than others to learn human languages. Some dancers move through the world in ways that other people simply cannot, not for any physical reason but because of timing, cadence, and poise. I know scientists who cannot figure out the first thing about poetry and poets who cannot write a line of code even with step-by-step instructions.

I do equate serial disagreement as "being oppositional" and from my perspective your accusing me of being arrogant for pointing out the unavoidability and ineluctability of ignorance contributes to that sense of "being oppositional". But, totally, I often find myself simply disagreeing with much of what's been said; I just tend not to speak out everywhere I disagree. To my sensibilities serially pointing out your disagreement with others seems a bit "fighty".

To get back on topic, I think instructional aids that bring lay persons closer to an understanding of the beauty of mathematics are fantastic. I enjoy watching them myself. However, having studied mathematics as an undergraduate, I have no illusion that much of what is being presented is the mathematical material with which mathematicians work. Often not even close.

One really good example are the popularized explanations of Quantum Theory as regards Thomas Young's famous Double-slit experiment. The idiomatic name of the experiment and some popular representations of the experiment are quite understandable. The separation between physicists and lay persons comes when one explains that the original experiment has nothing to do with slits and Young’s original experiment verified the wave aspect of unquantized light. However the "double-slit" experiment and its variations are used to explain the particle-wave duality of light as well as complementarity.

However modern verifications of the implications of Young's "Double-slit" experiment do not involve (visible) light or slits at all. My lay person's understanding is that verifications of quantum complementarity use x-rays, crystals, highly advanced mathematics, and non-photographic means of detection. At this point, however, I'm WAY beyond my depth and I'm sure I've made a number of lay person misstatements, which is my point.

Reducing highly scientific phenomena to concepts understandable by lay person's almost always reduces the scientific usability of those concepts. There is no substitute for the math and blaming mathematicians for not explaining things clearly misses the point that there is no way to translate scientific principles into common language and maintain those scientific principles qua scientific principles.

You may very well understand the analogy and the analogy may bear superficial resemblance to the subject at hand, but it's not the subject as such.

Given you, delmoi, are a programmer, it's sort of like the difference between code and pseudocode. Except, that's not quite it because it's an analogy a programmer might understand.
posted by mistersquid at 4:49 PM on March 17, 2012


I was ready to be skeptical of this video, but found myself pleasantly surprised when the narrator did some actual calculation. Mathematics benefits more from showing than telling.
posted by grog at 5:09 PM on March 17, 2012


The other thing is that you don't need to 'approximate' if you if you simply restrict the definition to 2-d or 3-d surfaces, then it's easy to explain, as was done in the video.

Well, so here's the thing, delmoi -- that's not right. There are lots of subspaces of 2-dimensional or 3-dimensional (or for that matter 1-dimensional!) Euclidean space which have fractional dimension and which are nothing like the ones described in the movie. I support the movie. The movie is good. The movie gives you a very nice example of a case in which a property you might want a "dimension" to have requires you to assign non-integral dimensions to certain subsets of the plane. I like all this! But I wouldn't hold out much hope that anyone, no matter how much native intelligence and mathematical intuition they had, would understand what a fractal was based on watching this movie. I think they would understand one important thing about fractals -- which is great! But it's not, for instance, what an encyclopedia entry is supposed to do.
posted by escabeche at 6:41 PM on March 17, 2012 [1 favorite]


Two curves on a surface a 'homologous' if they co-bound a surface, ie, they trace out a nice surface between them. […] On the surface of a donut, there are three different kinds of curves

Wouldn't a curve that wraps around the torus twice the short way and once the long way be a distinct kind of curve as well? (And if so, presumably there'd also be a distinct kind for every number of windings?)
posted by hattifattener at 10:13 PM on March 17, 2012


Reducing highly scientific phenomena to concepts understandable by lay person's almost always reduces the scientific usability of those concepts

Like when I explained to my nonmusical friends how I could play the piano, and my piano then became less useful for making music.
posted by scrowdid at 10:28 PM on March 17, 2012 [1 favorite]


There seems to be some serious confusion here. It's as if some people in this thread want to argue about the value of "analogies" or "approximations". At no point did I say that Wikipedia articles should have "analogies" or "approximations". What I said was that I thought where possible they should examples of things in Euclidian 2-space or 3-space so that people can understand the basic idea, before jumping into a jargon-laden technical that applies to every type of topological space or metric space or other generalization of 'space' to which the concept applies. Apparently this outraged some people.

The article on Eigenvectors does this. There's a diagram showing a red and blue vector superimposed two images superimposed on the Mona Lisa, the second had an affined transform applied to it. The red vector changes direction, while the blue vector does not, because the blue vector is an eigenvector of the transform, and the red vector isn't.

And, that's correct. It's a concrete and correct example of an eigenvector. It's not an "approximation" and it certainly isn't an "analogy".

I agree that approximations and analogies don't give you "true" understanding of what something is. But at the same time, concrete examples can help you understand what's happening as you're learning and coming to an understanding. No one was born understanding eigenvectors, you have to learn it at some point.

I mentioned earlier when I was learning abstract algebra, I tried one free e-book that mostly talked about symbolic manipulation without many concrete examples. I was getting stuck in a few places and switched to a different book.

In the first chapter of this other book they started out talking about the different ways you could rotate a 2d square in 3d space so that the four corners lined up. You could do it in 8 different ways (rotating 0,90,180 or 270 degrees in it's plane, rotating along x=0, y=0 and the two diagonals). And the thing is, those eight different rotations formed actually an algebraic group. If you 'multiplied' the rotations to each other, you ended up with the equivalent. Of one rotation. It's also something you can easily illustrate visually.

And that's the thing: It's not an approximation to an algebraic group, it's not and "analogy" to an algebraic group It's an actual example of an (abelian group) algebraic group. It's one that's simple and intuitive for people, and can use it to explain the basic principles of algebraic groups. By the time I got to the place where I was getting stuck in the first book (the concept of an ideal in a ring) it seemed like the most intuitive, natural idea ever.

It's the same thing with this video: They take a couple of simple examples of fractals in 2D space, and use them to illustrate the concept of the fractal dimension. It's easy to understand and (as far as I know) isn't just an approximation. The only flaw might be in not telling people that there are other possible types of fractals in other types of topological spaces, or not being to clear with.


(And by the way, I didn't even intend for it to be that big of critique, I just thought it was kind of funny that, if you clicked the link for 'topological dimension' you had to wade through a lot of jargon before you found out that ordinary euclidean dimensions were an example - That's something that happens on Wikipedia a lot, where you start out reading an article at one level, then you click a link that you need to click in order to understand what you're looking at, and you end up at an article that is far more in depth then what you need at that moment)
I wouldn't hold out much hope that anyone, no matter how much native intelligence and mathematical intuition they had, would understand what a fractal was based on watching this movie. I think they would understand one important thing about fractals -- which is great! But it's not, for instance, what an encyclopedia entry is supposed to do.
Well, before Wikipedia, an encyclopedia would really just give a brief summary of something, a paragraph or two or just a couple sentences. There might be a specialist encyclopedia for mathematicians out there, but you'd have the expectation that anyone reading it would already have a detailed understanding of mathematics, and everything would probably written at about the same level.

So Wikipedia is really the first ever thing that has massively detailed articles on mathematical topics as well as catering to everyone as a kind of general encyclopedia. So I don't really know if it makes sense to talk about what an encyclopedia is supposed to do when there's never really been anything with the scope and depth of Wikipedia before.

Secondly, sure, of course people won't understand everything there is to know about fractals, nor will someone who sees the eigenvector illustration understand everything there is to know about eigenvectors, or someone who understands the tile example will understand everything there is to know about algebraic groups. But it can help them get a sense of what it means, without sacrificing accuracy, and it's helpful if they actually want to learn more.

--
You may very well understand the analogy and the analogy may bear superficial resemblance to the subject at hand, but it's not the subject as such.

Given you, delmoi, are a programmer, it's sort of like the difference between code and pseudocode. Except, that's not quite it because it's an analogy a programmer might understand.
Let me see if I'm understanding you properly here: You think that, if someone is a programmer, they can't possibly understand the concept you're talking? Therefore they need an analogy that they "might understand?" but which wouldn't ever capture the full depth of your awesome understanding of the concept?

Except in this case the "concept" you're trying to describe is simply the relationship between analogies to mathematical results and the results themselves, which both isn't complicated and isn't even rigorously defined to begin with.

But extending the pseudocode analogy: pseudocode in a textbook or academic paper is actually supposed to completely capture and communicate the algorithm. It's supposed to be like a mathematical proof, containing enough information for someone to precisely reconstruct the algorithm. It's not supposed to be an approximation or an analogy to the concept of the algorithm itself, which is actually mathematical/conceptual object in the first place. Maybe somewhere people use "pseudocode" to describe half-baked concepts or something, but I've never seen it outside of textbooks or academic papers. But of course there's no rigorous quantitative definition of what is and is not pseudocode.
A lot of people believe ignorance is a badge of shame. I am always often when I'm having a conversation and I profess ignorance and someone says, "No, you're not ignorant" when in fact I know I am on whatever topic is being discussed.
This is completely ridiculous. First of all you didn't just say 'ignorant' you said 'forever ignorant, which is actually kind of a misuse of the word "ignorant". Generally just means a 'lack of knowledge', usually with the implication that the person could, in theory gain that knowledge.

Saying someone is ignorant and will always be ignorant is basically just saying they are unintelligent, or at least not intelligent enough to understand whatever it is you're talking about.

And of course, calling someone unintelligent is insulting. Duh. If you don't see that you must be pretty "ignorant" when it comes to communication (see how that works?). You can't just insult people and then try to redefine terms such that they are no longer insulting.

Furthermore, in your followup comment you clarified that you really did think that some people were clearly just too unintelligent to understand the concepts, period:
I don't agree that very intelligent people might close all gaps in knowledge over time because information is potentially infinite and intelligence in certain domains is to some extent innate. For example, some people have a higher capacity than others...
I'm certainly not arguing that everyone is really equally intelligent, but there are probably lots of concept that people could understand (correctly) if they were explained to them at the level they were already at. Obviously if they don't understand hyperbolic 4 dimensional spaces, they wouldn't understand fractals in hyperbolic 4 dimensional spaces. But that doesn't mean they can't understand fractals restricted to topological spaces they do understand.

The other kind of obnoxious thing about what you're writing is that you seem to think that I personally had trouble understanding the Wikipedia article or something. That's not the case at all, I was merely pointing out that it would be intimidating for people who weren't familiar with mathematical terminology and give them the false impression that the topic was too complicated for them to understand the basic concept presented in the video and in the 'introduction' section on wikipedia. They may not ever understand multifractals in hyperbolic n dimensional spaces, but they should hopefully be able to understand the basic concept of what fractional dimension means in the context of ordinary Euclidian space.

Anyway, rather then reading what I wrote you seem to really want to rail against anyone who wants approximations or analogies to mathematical concepts and call all of them stupid. Not only did I not say I wanted approximations and analogies, I specifically said that wasn't what I was talking about at all. So none of what you've been saying even applies what I clearly stated in the first place.

Secondly, it's obviously not at all helpful when trying to get ordinary people interested in math. Not every mathematical concept is equally complex or difficult to understand. I think it's unfortunate that more people don't ever get a chance to appreciate mathematics. Simply telling people that they are too stupid to ever understand some concept is clearly not at all helpful, especially when it's not even correct.
posted by delmoi at 6:51 AM on March 18, 2012


I like the idea of understanding it eventually . Thanks !
posted by nicolin at 7:11 AM on March 18, 2012


By the way, you don't always need to know the 'complete' definition of a mathematical concept in order to use it in practice. For a hypothetical example, imagine an architect working with a builder. The architecture decides he wants a ceiling with a fractal design, and you need to figure out how much paint it's going to take in order to cover it. Maybe the architect is saying it scales up 2-dimensionally, and won't be expensive, while you think it scales up 3-dimensionally and will be expensive. If you can figure out the fractional dimension of the surface, you can figure out how much paint you are going to need to buy.

It's kind of a contrived example, since paint is not very expensive. (Maybe if they were using gold leaf instead). There are probably lots of hypothetical examples where engineers might want to use fractional scaling in 2 or 3 euclidean dimensions.

People do use advanced mathematical concepts all the time for various practical purposes, even if they don't understand the 'full breadth' of the concept.
posted by delmoi at 8:16 AM on March 18, 2012


Bleh, I noticed a few errors in my earlier post:
an actual example of an (abelian group) algebraic group.
I actually meant non-abelian. Natural numbers are a great example of an abelian group, but they also have a lot more algebraic structure, so if you use them as an example you're not illustrating what groups are apart from more complicated structures like fields or integral domains (like only illustrating the concept of 'vehicle' by only showing cars, and not bicycles)
The only flaw might be in not telling people that there are other possible types of fractals in other types of topological spaces, or not being to clear with.
In that case I meant not being clear with the specific terms for the various dimensions (topological vs. hausdorff dimensions)

clearly I need to spend time proofing longer comments, but at the time I was kind of like "okay, I've already spent way too much time writing this, I just need to post it and stop wasting time"... I'd also just gotten out of bed :P
posted by delmoi at 8:29 AM on March 18, 2012


You keep using words like "obnoxious" and "arrogant." I think you are projecting your feelings onto phrases like "forever ignorant" (which are facts, not judgements) rather than simply reading the words for what they mean. Clearly you are deeply offended by ignorance.

Also: "fighty".
posted by mistersquid at 9:35 AM on March 18, 2012


Wouldn't a curve that wraps around the torus twice the short way and once the long way be a distinct kind of curve as well? (And if so, presumably there'd also be a distinct kind for every number of windings?)

Good point! It turns out that wrapping around k times in one of the two directions gives different classes of curves, but wrapping around in multiple directions allows you to unwind some of the wrapping. (try it!)
posted by kaibutsu at 12:14 PM on March 18, 2012


I think you are projecting your feelings onto phrases like "forever ignorant" (which are facts, not judgements)
Huh? You actually think you can run around saying things like that and not expect people to take offense? That's kind of hard to believe, and I think most socially functional people understand that telling someone they will be forever ignorant is not going to go over well.
posted by delmoi at 12:26 PM on March 18, 2012


Quite the schism I triggered with a little math derail. mistersquid, I do think it would be worth your while to review your comments at some point, I needed to step away to avoid making an embarrassing pejorative comment. At least for me, kaibutsu's comment gave me a wonderful insight into the derail topic, highly appreciated. On another thread Blasdelb made an incredible succinct, illuminating, and I thought very understandable discussion of quite a technical issue. Not to mention the original post!

It has been my experience that no matter how esoteric, a person with a full and true, comprehensive knowledge of a field is able to relay to an interested party.

While on one level I have to agree with mistersquid's thesis, if only in that none of us has the time in a single lifetime to delve deeply into every single topic of interest, I highly contest that with determination any specific idea/theorem/field of thought is not approachable by a motivated amateur. Kinda, sorta the principle behind MiFi, no?
posted by sammyo at 6:35 PM on March 18, 2012


I highly contest that with determination any specific idea/theorem/field of thought is not approachable by a motivated amateur.

It just takes time. The question is really about pre requisites. I think that a lot of wiki articles on science and math are kind of circular and rely on knowledge that you're assumed to know, and which isn't actually available on Wikipedia. I think to build this base of knowledge up, you have to start with a laymans explanation of something, and then build on it, until you've got a solid map of the domain of knowledge in your head. But Wikipedia really lacks articles that can be used as entry points into a field.

I think it might be interesting, as a project, to systematically go through some higher level math articles and enumerate all the concepts and definitions you need to understand it, beyond, say, basic calculus. Not just all the links and references from the page, but all the concepts you need to understand those, and so on.

I think you'll find there are going to be some pretty significant gaps.

I think creating a landing page for different fields of math and science with a college freshman introduction would be really useful, and maybe a glossary, and some links to college lectures and standard textbooks.
posted by empath at 7:59 PM on March 18, 2012 [1 favorite]


systematically go through some higher level math articles and enumerate all the concepts and definitions you need to understand it, beyond, say, basic calculus

This just isn't possible. Different people use different examples to refine their understanding of concepts. There is no one true mathematical cannon. No single source can be everything to everybody, and it is folly to try. Finding the course of explanations that optimizes for your personal understanding is non-trivial, and you shouldn't expect a single source to do it all for everyone. Thus Wikipedia must strike a balance between simplicity and explicity in mathematics, which ironically maps the least common denominator to a relatively high level of understanding.
posted by grog at 2:29 PM on March 20, 2012


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