The Geniuses' Thread

Insect Overlord

Pygmalion

Insect Overlord wrote: »

sock puppet

Insect Overlord wrote: »

Is it wrong because it's not a regular polygon so the limit isn't actually a circle? Or something else?

Pygmalion

sock puppet wrote: »

Is it wrong because it's not a regular polygon so the limit isn't actually a circle? Or something else?

Well the problem is that you can't really just say "at infinite this is a circle", because while it will look exactly like a circle (as long as you don't zoom in too close) it won't ever actually gain the properties of a circle (there will always be points further from the center than others, they just get less and less noticable).

I saw a proper disproof of this once, but all I remember of it was that there was some differentiation and a lot of hand-waving, the word "clearly" was used a lot where the results weren't actually that clear.

I liked the π = ∞ response a bit better though >.>

Davidius

I think the mistake is assuming the shape it converges to has an arc length equal to the perimeter.

It can converge to a circle in that as you go to infinity the shape gets closer to that shape which has all points equidistant from the centre. Any nth term in the sequence isn't a circle, but it does approach it (as far as I can tell).

An intuitive disproof is:
A circle doesn't have corners. Even if you make the length of each side of the polygon, it will still have corners - that's part of the definition of a polygon.
Therefore, the polygon will never be a circle.

As far as I can tell, the area of the polygon will approach that enclosed by the circle, but the perimeter is greater, as we can see in both of the strips.

Richard Cranium

My first thought was that it had something to do with pi being transcendental. That is, it can never be the solution to a polynomial. And as far as I know the equation of any polygon is a polynomial.

Maybe?

*waves hands and says "clearly..." some more*

By the way, why are latex tags not working for me?!:(

Richard Cranium

Richard Cranium wrote: »

My first thought was that it had something to do with pi being transcendental. That is, it can never be the solution to a polynomial. And as far as I know the equation of any polygon is a polynomial.

Maybe?

*waves hands and says "clearly..." some more*

By the way, why are latex tags not working for me?!:(

Actually, good point. Since there are other ways to disprove this, perhaps the disproof of this could serve as a proof that pi is transcendental?
That would be awesome, as one of our lecturers told us that the algebraic proof of this is something we'll be lucky to see as undergrads.

Having checked up elsewhere online, it's easy to prove that the n-polygon's area converges to that of the circle's as n->infinity.
To do this, just look at the difference between the area of the n-polygon and an n-polygon with vertices on the circle as in the case of the other polygon, but with every other vertex inside the circle.
Clearly, as n->infinity, the difference tends to 0, so the two n-polygons have a common limit, the circle.

However, the perimeter of the n-polygon does not tend to the perimeter of the circle.
Intuitively, it's because an n-polygon has 'corners' and a circle doesn't. More rigorously, but still a short argument, is the fact that each of the lines in the n-polygon has curvature 0 and a circle has curvature 1/radius.

KnifeWRENCH

Currently downloading LaTeX onto my laptop. AAARGH WHY IS IT TAKING FOREVER? :mad:

And once it's on, I have to figure out how to actually use it. By tomorrow at 3.30pm. I can't see this going well at all...

Hopefully my supervisor will only expect me to have the basics.

I've used a program that's quite good for learning it, as it drops in code similar to how the boards.ie editor inserts bb code.

If it appeals to you, it's here.

KnifeWRENCH

Cool, thanks for that. I'll give it a look anyway; I know nothing about computers or computer programmes, so anything that makes it easier will be hugely appreciated.

(And now I remember why I nominated you for nerdiest poster! :cool:)

**Timbuk2**

I really like LaTeX, but I only know the basics and I'm not that quick at writing in it, but that will improve with practice I'm sure!

In case anyone doesn't know, there are latex tags on boards, namely [noparse][latex]...your latex code here...[/latex][/noparse] - which is very useful for inserting Maths equations into posts!

EDIT: KnifeWRENCH, you might find the LaTeX Forum helpful (seriously what doesn't boards have :O)

But I like statistics even better than LaTeX, and on that note:

BBC4 are showing a program called "The Joy of Stats" tonight at 9pm. Here is a taster, which looks extremely interesting!

https://www.youtube.com/watch?v=jbkSRLYSojo

**Timbuk2**

**Timbuk2** wrote: »

I really like LaTeX, but I only know the basics and I'm not that quick at writing in it, but that will improve with practice I'm sure!

In case anyone doesn't know, there are latex tags on boards, namely [noparse][latex]...your latex code here...[/latex][/noparse] - which is very useful for inserting Maths equations into posts!

EDIT: KnifeWRENCH, you might find the LaTeX Forum helpful (seriously what doesn't boards have :O)

But I like statistics even better than LaTeX, and on that note:

BBC4 are showing a program called "The Joy of Stats" tonight at 9pm. Here is a taster, which looks extremely interesting!

https://www.youtube.com/watch?v=jbkSRLYSojo

Thanks. I've been wondering how to get the latex tags for ages. (In fairness, I should have guessed or asked in the maths forum).

KnifeWRENCH

**Timbuk2** wrote: »

EDIT: KnifeWRENCH, you might find the LaTeX Forum helpful (seriously what doesn't boards have :O)

There's a LaTeX forum? :eek: Wow! Cheers for that.

On a semi-related note, I was reading the Wiki article on Group Theory (have an exam in it on Thursday) and apparently one of the leading experts in the field currently is called Jacques Tits.

Getting cheap laughs from people's names; is there anything maths can't do? :pac:

**Timbuk2**

No problem! I found this in one of the stickies in the LaTeX forum a few minutes ago, but it's amazing

http://detexify.kirelabs.org/classify.html

You draw a symbol, and it gives you the closest approximations in LaTeX. For example, you want to know what [latex]\sum[/latex] is in latex, draw it in the box and it will show you that it is \sum (ok an easy one to guess, but still an incredibly handy site!).

Also a handy hint if you are posting Latex to boards (this probably is useful elsewhere, I'm not sure). To make things bigger, start your code with \displaystyle. For example

[latex]\frac{a^2-b^2}{c}[/latex] looks better as [latex]\displaystyle \frac{a^2-b^2}{c}[/latex]. I think in the real world, you can rewrite \frac as \dfrac to get the same effect (for fractions!!) but it doesn't seem to work on boards.

Pygmalion

Deleted User wrote: »

Facebook's stupid OAuth system is putting me off, but I'm working on it. May as well not do it the hacky way, and learn some GTK and WebKit while I'm at it.

But using Webkit for it defeats the point of using the Graph API :O :O :O

Also GTK sucks balls, use Qt.

Pygmalion

Pygmalion wrote: »

But using Webkit for it defeats the point of using the Graph API :O :O :O

Also GTK sucks balls, use Qt.

I still need to use the Graph API.

The idea is to use Webkit to let the user log in. Then, my program will use the Graph API to retrieve the relevant data, and then, since I'm already using the web interface, I may as well design a web GUI to use in my creeping program.

Maybe I'll use Qt.

Deleted User

Deleted User wrote: »

I still need to use the Graph API.

The idea is to use Webkit to let the user log in. Then, my program will use the Graph API to retrieve the relevant data, and then, since I'm already using the web interface, I may as well design a web GUI to use in my creeping program.

Maybe I'll use Qt.

Ok, I'm definitely using Qt. This is the first time I've ever checked it out and it's ridiculously simple.

Also, since there's lots of negative-marking MCQ related anger in the Pet Hates thread, I thought I'd show that they're fair, but I'll do it here where it's more relevant and people are less likely to get annoyed at me. :pac:

Take a 4-option MCQ with 3 marks for every right answer and -1 for every wrong one.
If you look at somebody who knows nothing, and guess random answers, you would expect them to get the question wrong with probability 3/4 and right with probability 1/4.
Therefore, the average mark they'll get for each question is 3(1/4) - 1(3/4) which is 0.
Thus they get an overall mark of 0, even if they guess some right.

A person who knows 100% of the answers will get 100%, similarly.

When it comes to people in the middle, well, they'll get full marks for every answer they know, and an average of 0 on every one they don't and just guess.
In fact, since their guesses are likely to be educated, you would expect their marks to be slightly higher than if they chose at random, so even for questions they don't know, they should pick up some marks.

Therefore, we can conclude that the negative-marking MCQ grade is a slightly biased estimator for a students' knowledge of the content, biased towards higher grades.

sock puppet

Deleted User wrote: »

Also, since there's lots of negative-marking MCQ related anger in the Pet Hates thread, I thought I'd show that they're fair, but I'll do it here where it's more relevant and people are less likely to get annoyed at me. :pac:

Take a 4-option MCQ with 3 marks for every right answer and -1 for every wrong one.
If you look at somebody who knows nothing, and guess random answers, you would expect them to get the question wrong with probability 3/4 and right with probability 1/4.
Therefore, the average mark they'll get for each question is 3(1/4) - 1(3/4) which is 0.
Thus they get an overall mark of 0, even if they guess some right.

A person who knows 100% of the answers will get 100%, similarly.

When it comes to people in the middle, well, they'll get full marks for every answer they know, and an average of 0 on every one they don't and just guess.
In fact, since their guesses are likely to be educated, you would expect their marks to be slightly higher than if they chose at random, so even for questions they don't know, they should pick up some marks.

Therefore, we can conclude that the negative-marking MCQ grade is a slightly biased estimator for a students' knowledge of the content, biased towards higher grades.

Ah but if you chose randomly in a 20 question test theres still ~40% chance of you coming away with a negative mark. I'd also suggest that it's really the expected utility of each answer rather than the expected value that you should be concerned with.

sock puppet

sock puppet wrote: »

Ah but if you chose randomly in a 20 question test theres still ~40% chance of you coming away with a negative mark. I'd also suggest that it's really the expected utility of each answer rather than the expected value that you should be concerned with.

That's a good point, but the chance of coming away with a ~40% grade is only indicative of what a bad idea it is to go into an exam with no knowledge.

Examining the expected utility would be the correct approach in trying to determine a particular students' score, but from the point of view of trying to capture a student's knowledge fairly, examining the expected value is the correct approach, I would say.

First Version of my Mandelbrot Set Viewer. There aren't many features yet because I had to figure out threading.

**Timbuk2**

^ Looks like a snowman gone horribly wrong

This question probably fits here:

If the 'cold snap' is over (for the moment anyway) and the temperatures are above 0 at the moment, then why is there still patches of ice around, such as along footpaths? Is this where the sun doesn't get to? Even still - it should melt. Is it because the inside of the ice is < 0*C, or some other reason?

marglin

im no expert but id imagine yeah that the core temp in the patches is still less than 0, you usually see those patches in places that dont get much sun, they're kinda like icebergs in that the surface is melting slowly, those patches you see around are melting but we'd need at least a few days of above zero temps day and night for all of them to go away.

also white ice patches reflect sunlight slowing down melting more, of course if we had just 1 really nice day they'd probably all melt(like mid teens or something)

doesnt look likely i heard someone say we're getting snow again for christmas yay!

amacachi

Well for one thing the temperatures used in weather forecasting and recording are from the shade anyway, so the sun isn't very relevant. But anyway, some frost usually happens even at 2 and 3 degrees, so water can freeze at this temperature.

Also the cold spell is coming back now, reckon there could be a 10 degree temperature drop in the space of an hour or two tomorrow as a double cold front move south over the country. Pity it wasn't a day later as that would suit me perfectly finishing college but I've plenty of supplies to get through it. ^_^ There's the potential for the next week to have the biggest snow event in more than half a century.

**Timbuk2**

Oh no not cold weather again

I was up at 6am this morning, and went outside to throw our rubbish bags in the bin. It was ridiculously cold this morning. Was fine though later on in the day, even was a good bit better at around 8.30am or so when I was walking to my exam!!

**Timbuk2**

**Timbuk2** wrote: »

Oh no not cold weather again

I was up at 6am this morning, and went outside to throw our rubbish bags in the bin. It was ridiculously cold this morning. Was fine though later on in the day, even was a good bit better at around 8.30am or so when I was walking to my exam!!

It was bitterly cold when walking back from my exam there.

amacachi

LOL, Christmas exams. :pac:

Chuchoter

Can someone explain to me what f(x) means in the scheme of HL maths? I can't do any of the part c's that involve it but I can manage the ones that are like show q-p=r type questions. Help!!

Screaminmidget

crayolastereo wrote: »

Can someone explain to me what f(x) means in the scheme of HL maths? I can't do any of the part c's that involve it but I can manage the ones that are like show q-p=r type questions. Help!!

basically, f(x) means 'y'. Conorstuff would be a better man to help you though

sock puppet

crayolastereo wrote: »

Can someone explain to me what f(x) means in the scheme of HL maths? I can't do any of the part c's that involve it but I can manage the ones that are like show q-p=r type questions. Help!!

It's just a function. I presume you're used to using y then? You won't lose any marks for substituting it with y in your answer. If you don't do that then the correct notation would be f'(x) if you're differentiating and f''(x) for second derivative and so on.