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Offline SgtTanner

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Re: three60's Short introduction to... corner
« Reply #15 on: July 25, 2009, 02:03:36 AM »
good read but plz make it more less technical plz in the next one for people who dont know the stuff good.

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #16 on: February 27, 2010, 04:14:00 AM »
Epic bump...

I kind of never got around to the last article I was planning to write, since I felt it was going to be either a very boring read or a possibly inflammatory one.

Having said that, I may actually write another article soon! Is there any particular topic people would like me to write on, otherwise I'll pick my own.
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline xen0n -

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Re: three60's Short introduction to... corner
« Reply #17 on: February 27, 2010, 04:37:29 AM »
Awesome, I remembered this topic from a long time ago and I love to see it come back. I like these kind of scientific facts ;D .

What about something with physics (maybe newton or sombody else), something about the universe would be cool too.

^^ Blast from the past, anyone remember this one? ^^




Offline jim360

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Re: three60's Short introduction to... corner
« Reply #18 on: February 27, 2010, 04:52:50 AM »
Any chance you could be more specific, xen0n?

Having said that, I have a great idea about one article that links with the last one I wrote.
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline xen0n -

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Re: three60's Short introduction to... corner
« Reply #19 on: February 27, 2010, 12:05:04 PM »
maybe something about stars or black holes, I've always found astronomy very interresting.

^^ Blast from the past, anyone remember this one? ^^




Offline jim360

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Re: three60's Short introduction to... corner
« Reply #20 on: February 28, 2010, 05:22:15 PM »
Well here is my next article. It's a bit of a mixed bag, and many of you who read it may have trouble following since - make no mistake - this is hard stuff. I don't follow some of it myself. But hopefully there is some interesting stuff in here.



Space-time, relativity and all that

I expect you’ve all heard of Einstein’s theory of relativity. Some of you may even know that it comes in two parts: the “special” and the “general”. And you must also know E=mc2 too! But what does it all mean, exactly (well, roughly)? What is “space time”? And why do we need this?

In many ways, Einstein’s role in Relativity is a little overstated. It’s been said, at least of the “Special” theory of Relativity, that the whole thing was just waiting to happen and someone somewhere was going to come across it. This is because a lot of the ground work for Relativity Theory had been finished – done and dusted – some 20 years beforehand by mathematicians such as Riemann (who may well get a mention elsewhere) and Lorentz.


I’ll start with the last question I asked – why do we need relativity theory?

The simple fact of the matter is that Newton’s Theories of mechanics (how things move) and Gravity are just not good enough. To be fair, you can work out how the moon moves, how an arrow flies and get a good idea of where the Earth is going to be in a year down to the 3rd decimal place, but the motion of the planet Mercury you can’t get right in Newton’s world. Not only that, but another, at first glance entirely unrelated, field of science called electromagnetism, has some strange stuff too.

You see, electric forces are created when you have charges, and magnetism comes from current, which is just a charge that is moving. So shouldn’t these two essentially be the same? In the 19th century it was known that they were linked – through a set of equations called Maxwell’s equations – and that one of the linking constants was the speed of light, c. Perhaps the most bizarre part of the whole thing was that this speed of light was an absolute constant. That means that it doesn’t matter how you move, the speed light travels at is always the same. Everywhere. You could be moving alongside a beam of light at almost speed and it would still head away from you at the speed of light.


What?? Think about it for a second, that can’t make sense, can it? If you’re running and fire a gun, the bullet travels at the speed you run and then how fast it would travel if you weren’t running put together. But this doesn’t happen when you travel at fast speeds.

Clearly something else, other than Newton’s mechanics, is needed. Enter special relativity, which among other things, says that instead of the normal way of adding speeds (that is, “new speed”=”speed you’re moving at”+”speed the object was moving at before you started running”) you have to do things in a slightly more complicated way. Specifically, you use what’s called the “Lorentz factor” which looks like this:

γ=1/√(1-v2/c2 )


Now the “curly y” bit is just the symbol used for the Lorentz factor (it’s the Greek letter gamma), c is the speed of light, and v is the difference in speed between two different frames. A frame, I should add, is the universe as seen by any particular point in space. So, for example, you could talk about “the frame of the spaceship”, “Earth’s frame” and so on.

What this factor means is:
  • 1. You cannot ever actually travel at the speed of light. If you did, this Lorentz factor becomes 1/(1-1)= 1/0 which works out to be infinity.
  • 2. When v is small the factor works out to be about 1. This means that you can ignore it for small speeds. And in fact Newton’s mechanics only works properly for small speeds.
  • 3.At high speeds the Lorentz factor becomes bigger and bigger. This has the effect, when you work through the maths of the thing, of slowing time down and squashing space. In fact if you were to travel at the speed of light the entire universe would be in the same place and everything would happen instantly. (This idea also pops up elsewhere, in Quantum Mechanics.)
   

Someone called Minkowski said that this means that space and time are linked. This leads to what’s called “space-time”. It also means that the universe is not three-dimensional after all, but in fact has (at least) 4 dimensions: up, forward, left and time.

Even all this exciting stuff still has problems – why does Mercury still not move in the way we want it to?


Let’s step back even further, all the way back to the ancient Greeks, in fact, and Euclid. I mentioned before that all the geometry you’ll have learned, or will learn, at school is Euclidean. That means that the following 5 things are assumed:

1)   It is possible to draw a straight line between any two points.
2)   It is possible to extend any straight line segment indefinitely.
3)   Given any straight line segment a circle can be drawn with the line segment as the radius and one of the endpoints the centre.
4)   Right angles are equal to each other.
5)   Given a line and a point not on the line, only one line through that point is parallel to the first line. [equivalent statement]

These may seem obvious, but I am now going to construct a space where they fail:

To break number (2) is easy. Just draw a straight line on a piece of paper and you’ll find that you have to stop at some point.

To break number one is a little trickier, but here’s one way:
 


The challenge here is to draw a straight line from point A to point B while keeping the line inside the yellow region. You can’t, it’s impossible. So in fact it’s not always possible to draw a straight line between any two points.

Now, in fairness, you aren’t going to walk along and suddenly find this whopping great hole that won’t let you across it because technically it’s not even there. But the point about this is that it’s possible to imagine and create places in which, after all, Euclid’s geometry fails.


The most interesting part of Euclid’s 5 axioms is the fifth one. This says that lines will meet at some point if they aren’t parallel to each other, and if they are parallel then the lines won’t meet, and if you have a line, and any other point in space, there is only one line through that point parallel to the first line:

 

That diagram may help to explain things. As you can see, there’s only one line that is truly parallel.

But now, let’s recall an interesting riddle:

“A man got out of his tent, walked a mile south, tracked a bear eastward for another mile, finally shot it, then took the bear home by walking due north for a mile. What colour was the bear?”

The answer is white, because you were camped at the North Pole. But let’s think about this for a bit: You just walked along the sides of a triangle, taking right-angle turns all the time – that means that the angles in the triangle add up to 270 degrees and not 180.


This means that the geometry on the Earth is not Euclidean either. In particular, all lines joining the North Pole and the South Pole are parallel to each other yet they all meet. So that fifth Axiom of Euclid’s fails on the Earth.

It took about 2000 years for people to realise this. Up until Gauss and Riemann, all geometry was Euclidean. Then those two developed new geometry that wasn’t Euclidean – the two main ones being spherical (where parallel lines can meet) and something called “hyperbolic” (where through any point there are infinitely many parallel lines).  These two geometries, and the whole idea of non-Euclidean geometry, allowed Einstein to describe Gravity.

We come back to Relativity now. You might even have heard of this bit: “every object bends space-time around it”. You can picture this two-dimensionally by taking a net, stretching it out tightly then putting something like an orange on. The net gives way under the weight of the orange. Now look down at the net and you might see something like this:



This helpfully shows that space-time (represented by the netting) has been bent and distorted by the mass of the orange (which is in the middle of the right-hand diagram).

With some horrendously difficult mathematics it’s possible to predict the shapes that this bending will take. And that’s exactly what Einstein did. What General Relativity says, in a nutshell, is that you can view gravity not as a force but as the bending of space around anything with mass. Remarkably, this is just what is needed so solve the problem of why Mercury moves the way it does.

Now xen0n asked about black holes. They fit into this too. The "black hole" is a solution to the equations of general relativity when, essentially, there is a point where you have to divide by zero. That means that instead of space-time (the netting) gently bending under the weight, you have such a heavy weight that the netting - space-time - collapses, breaks and you get a hole. It would look something like this (this image has been grabbed from a website, and also gives the equation for the event horizon of a black hole):



As you can see, there is literally a hole in space-time at the point of the black hole. That's why anyone going too mear a black hole gets sucked in - they fall into the hole.

The great power of General Relativity is that it can help us understand what gravity does near the planet earth much better than simple Newtonian gravity can. That means that space flight, satellites and the Global Positioning system work.



Hope you enjoyed this! Once again, any more ideas, please let me know.
« Last Edit: March 01, 2010, 11:40:16 AM by BFM_three60 »
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline xen0n -

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Re: three60's Short introduction to... corner
« Reply #21 on: March 01, 2010, 10:42:24 AM »
Very nice jim,I realy enjoyed this one ;D I always wanted some explanation on relativity, we're gonna see it soon in school I think (now I'm gonna have a head start :P ). Anyway, keep up the great work, maybe I'll give a new suggestion later ;D


PS: I think you missplaced a diagram :ninja:

^^ Blast from the past, anyone remember this one? ^^




Offline jim360

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Re: three60's Short introduction to... corner
« Reply #22 on: March 01, 2010, 11:12:31 AM »
Glad you liked it.

As for the diagram, good thing too, since I thought about it some more and decided I got it wrong. The corrected image has been inserted.
« Last Edit: March 01, 2010, 11:41:15 AM by BFM_three60 »
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #23 on: March 03, 2010, 04:59:24 PM »
I've written an article on cryptic crossword clues, which will be put up here sometime tomorrow. Those of you who spend your time attempting to solve my BFM-theme clues (more of which to be also put up shortly!) may find this useful!
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #24 on: March 04, 2010, 04:57:52 AM »
How to write a cryptic crossword clue (or at least how I try to)

Recently you may have noticed that in the “riddles game” I’ve been posting cryptic-style clues, taken either from the Times or that I wrote myself. I also found that at least one person was interested in writing them him- or herself. So, here is a rough guide about writing the clues. People who read this may also find useful tips to help them solve the clues I write.

Firstly, what is a cryptic clue? Well, let’s pick a simple example. Here, for example, is a clue taken from the Times March 3rd, 2010:

Hero has sport, turning out a marksman (12)

And here is the same clue as it might appear in a non-cryptic crossword:

Expert marksman (12)

Now, clearly the first thing you see is that both clues share the word “marksman” so this is the important bit. The word that answers both clues is “sharpshooter” which is another word for marksman.

Now, have a look at the letters in “sharpshooter” and rearrange them a bit, and you’ll find that you can make the words “hero has sport”. So what the cryptic clue is trying to say that, “if you were to rearrange the letters of ‘hero has sport’ you would find that you could make a word that means ‘marksman’.” It has done this by the phrase “turning out” that is telling you to rearrange “hero has sport”.

Not all cryptic clues are anagrams – despite the huge number of possible anagrams there are some words which refuse to be rearrange-able (new word ;D) into other words – and even if they did it would be boring solving anagrams all the time. So we might have something different:

One crows, seeing head of organisation in duty list (7)

If you thought of the word “roster” when you saw “duty list” then you might be able to guess at the answer “rooster”. You would be right. The clue is saying, “Put ‘o’ (the first letter of the word ‘organisation’ into a word meaning ‘duty list’ and you make a new word that means ‘something that crows’.”

Hopefully you can see that in general a cryptic clue is trying to describe what happens when you play around with words. At the same time, though, the clue is designed to mean something completely different at face value.

So how would we go about writing such clues? Clearly what we have to do is work backwards – take any word and find some clever way of forming that word, telling the solver what you have done and confusing anyone who doesn’t recognise what’s going on. How to do this is described below.

The first point is that any clue that is written should be solvable. It’s no use having a really clever clue that nobody can possibly solve unless they were sat next to you as you were writing it. So this means that you have to write the clue in such a way that it’s clear what you have done.

Here is an example of a bad clue:

E? (13)

In fact most cryptic crossword enthusiasts would get this instantly since it is a “well-known” example. The answer is “senselessness”. Why? Well, if you write “senselessness” as “’sense’ less ‘ness’”, then you do this “word sum”, you find that you are left with the letter “E”. Hence the clue. But the problem is that, frankly, there’s no way of being able to recognise that based on the clue itself. You would need a leap of faith and a brain wave to solve that clue – it is badly-written. (There are in fact ways to “rescue” this clue but sadly they aren’t really appropriate for these forums.)

So what is allowed? Here is a list of the most common ideas used in cryptic clues I’m used to:

        1. Anagrams. As discussed, if you have a word, say “sharpshooter”, and you can rearrange the letters to something that looks fairly sensible, then an anagram-based clue could be worth a try.

        2. “Buried answer”. What this means is that you have a word, say “happy”, and you put other letters around this word to make new words. In this example, you could add “mis-“ to the front and “-rotechnic” to the end and you get the phrase “mishap pyrotechnic”. Not too promising, but you now add more into the clue to make it “work”. In this example, you could add “despite” to the front: “Despite mishap, pyrotechnic display” – a phrase that makes sense. I’ve also added a comma to break up the answer a bit. Indeed, looking near any punctuation in a clue can be a good trick to discover the answer. We’ll complete this clue later.

        3. Other “buried answers” can have the answer written backwards: as an example you could take the word “tea”, reverse it (“aet”) and write something like “praetor” (a Latin word) that has the letters in the right order.
       
     
   4. Another good trick is “acrostics”. Take a word, say “pain”, and think of words that begin in “p”, “a”, “i” and “n” – say, “Paper and ink needed”.  Again, this can work backwards.

        5. Synonyms and homophony. These work in the same way – you either think of two words that can mean the same thing, such as “pricy” and “darling”, and find a word that means both of these – “dear”. Then the clue could be “pricy darling”. Homophony is when things sound the same – “dear” and “deer”. Then the clue could start “expensive venison” (which would read “dear deer”).

   
    6. Word-building. This is the most complicated type of clue. What you do is you take a word and disassemble it into smaller words and/ or letters. You then find synonyms of these smaller words. This is very useful for compound words. One example might be “shoplifting”, that we can say is “S” plus “hoping” plus “lift”. Then a lift could be a boost, and “hoping” could be “expecting”. The problem now is that we have to be able to connect these words into a sensible phrase – see later.

        7. “Pedantry”. Here you take a word or phrase that is well-known and look at what it might mean if you took it literally. For example: “A rolling stone gathers no moss” could mean something like “Mick Jagger won’t pick Kate [Moss, the model] up”. This clue needs a couple of extra touches but it’s almost complete as it stands.

   
     8. Clues which work backwards. You don’t often see this but what it means is that you look at the answer in a way that could make that the cryptic clue. The clue you write down is then the “answer” to the clue suggested by the answer. This is giving me a headache so let’s find an example:  “officer’s mess”. The word “mess” might mean “make a mess of” so we’ll “make a mess of” the word “officer”. We find that “rice off” works.

        9. The final clue type is what is called “& lit.”. Most clues contain a part that is “wordplay” and the clue itself (anagram plus definition, for example). In some cases it is possible that the clue is contained inside the wordplay. Take “hairdresser”. By some amazing coincidence, “hairdresser” is an anagram of “shears drier”. We can then write “I manipulate shears and drier” as our clue. The clue is to be read both literally (the “& lit.”) and cryptically.


There aren’t many more techniques used than the ones listed above – and the key one is number 6, which you see  all the time.

Now that we have some idea how clues work, we can set about trying to write some. Let’s start by completing the example clues I have given so far, starting with happy:

     - When we moved on I had got as far as “Despite mishap, pyrotechnic display”. This is good, but to complete the clue we now need to put something in explaining to the solver what we have done. So we look for words that mean things like “contain”, “inside” or something similar. I think that “in part” might be a good fit here. That gives is “Despite mishap, pyrotechnic display ...in part...”. Still not done yet, we now add onto the end of this a word that means “happy” – one example might be “pleasant”. Even now we are not quite done because we want the clue to make sense. So we finish the clue as follows: “Despite mishap, pyrotechnic display partially pleasant”. That’s it!

    - The next example was “tea” which I had put backwards in “praetor”. To finish this clue we need words that mean “hold backwards”, and a word that means “tea” – say, “drink”. One way of doing this is to write, “Praetor holds back drink”. That’s all you need to do to finish that one off!

     - The next example was “pain” which we had started to clue by writing “Paper and ink needed”. Now we need a word that tells the solver to “look at the first letters of...”, say “at first”, and a word that means “pain”, perhaps “torture”. A clue might then be “Paper and ink needed at first to give torture”.  Another finished clue!

     - The synonym clues tend to be very short – for our first clue we can just say “Pricy Darling” and leave it at that. For the second one we need one more word to make it clear that a homophone is involved: “Venison is very expensive, reportedly”. The “reportedly” is the indicator that makes this clue read as “word that means ‘venison’ and sounds like a word meaning ‘expensive’.” The important thing is that if instead we write, “Venison, say, is very expensive” then the answer would be “dear” not “deer” (the clue now reads “a word, that sounds like a word meaning ‘venison’, meaning ‘expensive’”).

   
  - Now we come to the clue for “shoplifting”. We disassembled this to “S”, “hoping”, “lift”, and said that “expecting” means “hoping” and “boost” could mean lift. The “hoplifting” part of the word has the word “lift” inside the word “hoping”, as you can see. So we put this down as “[hoping] receives [lift]”, or “wishing to receive boost”. Now what about the “S”? We can get round this problem by, say, choosing a word that begins with ”S” and saying “take the first letter of this word”. Now let’s also remember that “shoplifting” is a type of Crime, so our clue might look like “Crime S____ at first expecting to receive boost”. Why not pick “Squad”? This creates the final clue as: “Crime squad at first expecting to receive boost”.  This is, in fact, the clue as it appeared in the Times a few months ago.
 
     - To complete the clue for “A rolling stone gathers no moss” based around “Mick Jagger won’t pick Kate up”, we add a question mark at the end, and insert the word “perhaps”. The clue in its final version reads, “Mick Jagger, perhaps, won’t pick Kate up?” Why the “?”? This clarifies that the clue should not be read in a “normal” cryptic way – words like “up” and “pick” could be read cryptically as, say, “read from bottom to top” and “put one word inside another”. The “perhaps” says that we think of groups of which Mick Jagger is a member. This completes the clue.

 
    - Our final example was “officer’s mess”. We saw that “rice off” was an anagram of “officer”, so let’s write the clue as “rice off, here?” The “?” again indicates that the clue works differently from normal cryptic clues, and the “here” is the “clue word”. I think anyone getting this clue would be a cryptic genius. I certainly wouldn’t.


Those are the basic forms of clues. Now let’s give a few more examples. Here I have picked 3 words that I don’t think I have seen clued before, and tried to write clues for them. The thought process is included throughout.

      1. Thoroughfare. (This means “main road or highway”). “Thorough” sounds like “furrow, “fare” like “fair”; “trench” means “furrow” and “even” means “fair”, so we can say “Trench even, we hear, in highway”. “we hear” is the part that indicates the homophone.

      2. Dragonrider. I have already used this clue in the Riddle thread. How I got to it was as follows. I noticed that “roaring red D” was an anagram of “dragonrider”. This didn’t seem too promising, but dragons can be red and do tend to roar, so I saw potential in this. The clue was completed because “D” is the first letter of “Dstroyr”. So I had “Dstroyr’s first ____ roaring red _______”. The words “to sit” tells you that “D” is the first letter of the answer. So I put that in. Then I used “strange” to indicate the anagram: “Dstroyr’s first to sit on strange roaring red thing”. To complete the clue to Times standards I would have to add one other word to make it clear that the answer is indeed a BFM admin. Hence the final clue should read something more like “BFM Dstroyr’s first to sit on strange roaring red thing”. The word “thing” is surplus and means that this clue, perhaps, is not quite perfect. But I enjoyed it and that’s what matters.

      3. Gordon Banks (England Keeper, 2nd best in the 20th century according to one poll). Gordon Brown being the current UK Prime Minister, and “banks” being a verb, we might try something that means “Gordon banks”. Something simple that works could be “Brown bets on a great saver!” The exclamation point? – let’s face it, Banks deserves one. Also exclamation marks tend to indicate double meanings.

And that’s it! To write your own clue, find a word, look at it in another way, write a short sentence that contains all the information the solver will need to solve the clue, and try to make the sentence make sense in another way.

It may be hard, but it’s great fun!!

[readers may be interested to know that there were exactly 2337 words in this post until I added this comment.]

Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #25 on: March 05, 2010, 06:24:38 AM »
I hope people enjoy both of the last two articles. Has anyone any suggestion for the next one I could write?
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #26 on: March 08, 2010, 07:19:40 AM »
In lieu of anyone having other ideas, over the next few weeks (probably around Easter time when I have nothing better to do) I'm hoping to write a series of articles leading to a very famous equation. Each one will focus on mathematical topics that were covered when I was at high-school, so in theory at least they should be accessible to a lot of people. The final article will link the first articles in a single equation.

Edison will probably be able to guess what equation that is. :P The rest of you who may not know of it yet...I hope you look forward to it!!
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #27 on: April 09, 2010, 11:15:04 AM »
Update for those interested: the first of the three or four articles in this series, on trigonometry, is due out tomorrow or Sunday, with the next one to follow within the coming week.
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #28 on: April 12, 2010, 01:07:44 PM »
This article is the first in a series of four articles with the hope that at the end I will be able to introduce to you the “most beautiful equation in mathematics”. How I will go about doing this is by taking the three subjects in mathematics that lead to this equation in turn, all of which seem to stand fairly separately, and finally tie them together. These topics are: trigonometry, calculus, and complex numbers. The equation at the end is known as “Euler’s Identity”.

Let’s start at the end. Euler’s Identity states that:

ei*(pi)+1=0

All very well, but... what is e? What is i? And what, for that matter, does having i and pi to the top right of the e mean? Why does this have anything to do with triangles?

All these questions and a few more I hope to be able to answer for you in the course of these four articles. So let’s begin!



Trigonometry

When you get around to starting your own civilisation, sooner or later you’ll find that you need to be able to measure something. Not only that, but a bit of experiment shows that in fact one of the “strongest” shapes there is is the triangle. So, the early civilisations concluded, why not see if we can figure out everything there is to know about triangles?

This is the reason why trigonometry is so important. Even for the layman measuring is useful, in DIY, etc.. In fact as we shall see later on trigonometry has far more uses than just measurement.

Triangles come in several varieties, though we’ll start with (and indeed stick to) right-angled triangles. It’s a good place to start because for every rectangle there is a right-angled triangle that goes with it, so we could also find out stuff about rectangles into the bargain.

First, a few definitions:



The symbol next to the marked angle is the Greek letter theta, which is the usual symbol used to mean an angle.

In any triangle there are seven measurements – three sides, three angles and the area. Cool thing we shall find out is that so long as we know at least 3 of these measurements (of which at least one is a side) we can find out all the others. The answer to the question how is trigonometry.

The easiest way to start trigonometry off is by noticing that when you measure the sides of a triangle and find them to be, say, 3cm, 4cm and 5cm, then drew a bigger triangle and measured its sides to be 3m, 4m and 5m, then both triangles look exactly the same.



This suggests that what matters isn’t so much the length of any side, but its length compared with the other sides. In fact, the angles in both these triangles are the same and it’s not too hard to see that the sizes of the angles depend on the lengths of the sides.

Because of this what we do is define the following three ratios:



In days of old, before calculators, there were also several other ratios used, with glorious names such as the arc cotangent and the half versed sine, that now aren’t so useful because we have calculators.

These ratios link the angles to the sides and we could now work out what they are by drawing triangles with different angles, measuring the sides and dividing these answers – and that’s how it was done, in fact, for thousands of years. It’s more sensible, though, to fish out a calculator. The ratios are usually abbreviated to sin, cos and tan.

What can we do with these? That’s what High School maths spends some weeks doing. The basic gist, though, is that now you can work out how high buildings are, how far away stars are, and so on. To work out the angles from a given ratio you use the "inverse ratio" which appears on calculators as "sin-1", for example.

Since the world isn’t made up of right-angled triangles you might wonder how we deal with other ones. Actually it’s not too hard. What you can do is split up the triangle into smaller, right-angled triangles and work from there. This simple trick means that any and every triangle can be “solved” in almost no time at all – so long as you know the length of a side, of course.

But what if you don’t? It’s not the end of the world, because what you can still do is find out the values of the ratios of the sides.

There are one or two other useful facts we can find out about triangles, in fact, by ignoring how long the sides actually are and choosing the length of the longest side to be equal to 1. Now if you remember that in a right-angled triangle with sides a, b and c (c the longest side), a squared plus b squared equals c squared, then we get the following:



This neatly links all the ratios together. In High School again, you’ll see that these two facts allow you to prove all sorts of other interesting, if generally useless, relationships.

So far there’s not been anything that gripping, and I’m sorry about that because the details are a bit boring. However the idea of making the hypotenuse have length 1 is quite useful as I shall now show. If you increase one of the angles all the way from 0 degrees to 90 degrees, and look at the path the opposite corner follows, you get this shape:



Yes, it’s circular! This quirky-looking coincidence says something far deeper – that in fact triangles and circles are very deeply related. We’ll come back to this in a couple of articles’ time, but for now keep this fact in mind.

The most immediate use of this is that we can now extend our definitions of sine, cosine and tangent to angles of any size, though only if you allow the confusing idea of "negative lengths". What you do is keep draggin that far corner round and measure the lengths of the sides, while saying that all sides to the left of and below the central point are "negative" in length. This leads to the “sine curves” and “tan waves” that will be or have been introduced at some point in High School.  There are several uses of this, the most appealing being the cool fact that if you turned a sine wave into a sound it would be – it is – the purest form of musical note you can have.

You’ll find that, far from being restricted to triangles and measurement, these ratios have a use in anything involving waves and motion! The shape of a wave on the sea is, roughly, a sine wave, while the motion of a pendulum in a clock is also described in terms of sine waves - in fact you need sine waves to tell the time!

There is one last thing I will introduce into this article and that is the idea of a slightly nicer way of measuring angles. Degrees, you see, are all very useful in their way but are very artificial.

Instead, a far more useful method of measuring angles is as follows:



A well-known fact about circles is that the circumference is twice the radius times pi, so there are 2*pi radians in 360 degrees.


That concludes the article on trigonometry. The main conclusions I would like you to draw from it are:

 - That despite the apparent fact that trigonometry is useful but not interesting, it turns out that even this subject has opened up all sorts of new possibilities.
 - That rectangles and circles are somehow linked through trigonometry.


The second fact is the most profound. Mathematics is a lot about linking areas that seem unconnected, and the more links the better!



That concludes our brief look at trigonometry. There is of course a lot more to this that you will explore or have explored at High School.

Next article we will be taking a brief look at another area of high-school mathematics, calculus.

« Last Edit: April 12, 2010, 01:29:51 PM by BFM_three60 »
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

Offline jim360

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Re: three60's Short introduction to... corner
« Reply #29 on: November 02, 2010, 12:57:26 PM »
Things have come up over the last few months that have kept me from posting here with the second, third and fourth articles in this sequence. Sorry about that, but can't be helped. Hopefully over the next week or two I'll dig up the files for the next post. Until then, stay tuned...
Check out my Short introduction... corner and my "Historical figures who should perhaps be better-known" thread!!

Exciting videos: 1.1 / 1.2 / 2 / 3 / 4 / 5 / 6



              

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