Internet & Space

Merlin

So you try to confuse me with fiction he, Watch and wonder as the Confusee confuses the confuser using merely the facts.(and maybe some brackets.)

Let's Start at the Start, (for as I say (and say Well) I am using nought but Facts and Brackets, (Starting at the end would be unnecessery(and unfair))) Think of 2 boxes, each with lots of apples in them, your job is to check that both boxes have the same number of apples in them, you have 2 empty boxes to aid you. What's the best way of checking. Easy, you take one apple from each box and put it into one of the other boxes, making sure all the apples from one box go into one empty box and all the apples in the other go into the other. You keep doing this one pair of apples at a time until one box runs out of apples, if there are the same number of apples in each box then both boxes will be empty, if there are not the same number in each box then one of the boxes will have some left over. and vice versa.

What did we actually do here, we proved that if we have2 sets of objects and we can match each object in set A with a unique object in set B with none left over Then Set A and Set B have the Same number of Things in them, or more usefully.

2 Sets A and B have the same order(number of elements) iff each member of Set A can be mached with a unique element of set B and each element in set B an be matched with a unique element in set A Hope you're with me so far, this is the easy bit.

Now Let's imagine 2 Sets with an infinite number of elements, let's say the Set of Natrual Numbers(1,2,3,4,....) (call this N) And The Set Of Even Numbers,(2,4,6,8,....) (Call This E) Which has More elements? E you Cry E, But Let's Put this to Our Test,

Every Element In N Can Be Matched With A Unique Element in E, and Every Element in E can be Matched with a unique Element in N. (1,2) (2,4) (3,6), .... (N,2N).



What About Fractions, Surely There Are More Fractions than Whole Numbers,

Every whole Number Definatly Matches With a Fraction, a whole number N Matches With N/1.

But What about Matching the Fractions With whole Numbers

Let's write Some Code

Code:

---

Public Sub ListFractions()

Dim i As Integer
Dim j As Integer
Dim n As Integer

i = 2
n = 2

MsgBox "1/1 Matches With 1"

Do While True

    For j = 1 To i - 1
   
        MsgBox j & "/" & i & " Matches With " & n
   
        n = n + 1
   
        MsgBox i & "/" & j & " Matches With " & n
   
        n = n + 1
   
    Next j
   
    MsgBox i & "/" & i & " Matches With " & n
   
    n = n + 1
    i = i + 1

Loop

End Sub

---

If We Pretend There's not Going to be an Overflow Error when VB runs out of integers, then Ther is No Fraction We Won't Get Eventually Matched up to a unique whole Number.

So there are the Same Amount of Whole Numbers as there are Fractions.


There aren't very many numbers left, How about Irrational Numbers, These are numbers that Can't be expressed as Fractions, They are infinite Strings of Decimals that go on forever without Repeating, like pi or sqr(2). Are there more of These than there are Whole Numbers?

[you - No, I'm Sensing a pattern here, you're going to find a clever way of listing them all]
[Me - Wrong , Ha Ha Ha Fooled You, There's no way of listing them all]
[you - Proove it]
[me - I'm glad you asked]

Imagine I found a way of listing all the irrational numbers between 0 and 1.

Code:

---

1  -  0.45789274894209347207937394720......
2  -  0.89024652934520349572034572394......
3  -  0.03462945239475620873456023456......
4  -  0.34987234785203456238452346529......
5  -  0.34753745629834562934759232373......
etc

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OK, now I'm going to think of one that's not on the list, take the first digit of the first number(4) and add 1(5) now take the second digit of the Second number (9) and add 1, in this case it's 10 so we'll make it 0, then the 3rd digit of the 3rd number and keep going for all the numbers on the list (you won't reach the end of the list but there's an infinite number of digits so we can match each digit with a number on our list.

so we get the number 0.50594........

it's not the first number because the first digit is different, it's not the second number because the second digit is different, it's not the 3rd number because the third digit is different, and so on all down the list.

So whatever way you find of listing the irrational numbers I can show you a number you've missed out. So there are more irrational numbers than whole numbers.


So What about the Numbers between 0 and 1 and all Numbers(Rational and Irrational (including -ve ones) Are there more irrational Numbers in total than there are between 0 and 1


Of Course, take any number between 0 and 1, subtract 0.5
take the inverse and then if it's +ve subtract one, if it's -ve add one, there we go, every real number has a unique partner between 0 and 1. and Vice Versa.

Now, Any Point on a line Can be Identified by a Number the Distance From it's end, Or the Distace Form a particular point if it's infintely long. So the number of points on a line Is always the Same, No matter How Long it is.

A Line is a one Dimension al Space. I Can't remember the proof but there are more points on a plane than there are in a line, In Fact.

There are more points in an x dimensional Space than there are in a y dimensional space iff x>y

It can also be shown that

any 2 x Dimensional Spaces Contain the same number of points and for any 2 spaces containing the same number of points if one is x dimensional so is the other


Now we've got that sorted (there'll be proofs somewhere on the web if you're desperate) we can Go on to what a Dimension is.

Take an X Dimensional Object, transform it so that you can split it up into n Equal sized pieces all the same shape as the original (for example a square can be split up into four smaller squares.

Now, Measure the length of one of the Sides of these objects (If it's a square take one of the sides, a cube one of the edges etc) find the ratio of the same length on the original shape / the new length

Eg if you split a square into 4 small squares the original sides will be twice the length

so for the squares the ratio of the sides is 2 and the shape was divided into 4 pieces

. take natrual logs of both sides and divide the 2 to get the number of Dimensions

Code:

---

NumberOfDimensions = Log(NumberOfPieces) / Log(RatioOfLengths)

---

This Is How We Define Dimensions
OK, That's Enough BackGround, Let's Look at Fractals and Why They're nothing like the universe.


Let's look at a very simple fractal, A Spvenski Gasket (I don't know If I spelt it right)

Start a new project, Add a little Command button to your form, and a Text Box, set the AutoRedraw Property to true and text1.Text to 5

now add this

Code:

---

Option Explicit



Private Sub DrawGasket(x1 As Single, y1 As Single, theta As Single, Length As Single, depth As Single)
Dim x2 As Single
Dim y2 As Single

x2 = x1 + (Length * Cos(theta))
y2 = y1 - (Length * Sin(theta))
Form1.Refresh
DoEvents
If depth = 0 Then

    Form1.Line (x1, y1)-(x2, y2)
   
Else

    Call DrawGasket(x1, y1, theta, Length / 3, depth - 1)
    Call DrawGasket(x1, y1, theta + 1.0471975511966, Length / 3, depth - 1)
    Call DrawGasket(x1, y1, theta - 1.0471975511966, Length / 3, depth - 1)
    Call DrawGasket(x1, y1, theta, Length / 3, depth - 1)
   
End If

x1 = x2
y1 = y2
End Sub

Private Sub Command1_Click()
Form1.Cls
Form1.ForeColor = vbBlack
Call DrawGasket(0, Form1.Height / 2, 0, Form1.Width, Text1.Text)
'Form1.ForeColor = vbRed
'Call DrawGasket(0, Form1.Height / 2, 0, Form1.Width, 1)
End Sub

---

Hit the CommandButton, and You Get a Gasket, If you wanna make the form big you can increce the number in the Text box for more detail.

Now Imagine it in infinite detail, the Closer you zoom in on the Line the More detail you get, following that set pattern.

Now You Can Divide this shape into Four Seperate Bits of the Same Shape (UnComment the Commented lines and you'll get some red lines to help.)

So, If We Divide the Shape up into 4 pieces each one 1/3 of the Length of the Original from end to end.

this gives us Log(4)/Log(3) = 1.2618595 Dimensions?

As you can See, There's not as many points as in a plane, But More than in a line If you tried to map all the Points in it onto a line, you couldn't. In Order to beable to get all these points into the 2D Space you Have to Leave a lot of Space but it must be to Complex to be a line, to accomidate this exta part we need it to be infinitly Complex, The Universe isn't infinitley Complex, It's Very Complex But not infinatley, which makes it Exactly 3 Dimensions (Or 4, 5 or 10 depending on what book you read I use 3 because it's easier to visualize)
This is why the Universe is nothing like a fractal And You know what A Dimension Is And What A fractal is. (if you Look at the About Creation and AI and Evolution For those who Care threads between me and Gen-x you'll se why I think it's neccesery to get the big guns out first on the Dimension issue)


Zaphod, When We Say the Universe is expanding it means that all the Stuff in it Is Flying appart, The Space itself isn't expanding, and the Universe Isn't infinate as such, but it is flying appart at the Speed of light at the edges so we can never get to nor see the end, so for all intents and purposes it is infinite, but there's only a finite amount of stuff in it.

I am the master of clarity, your head is too full of fog to know when you are confused or not. I have told the wind to blow through your ears to dispell the mists of mistery

I just had a thought:

If the edges of the universe are travelling away from the origin of the big bang at the speed of light, and photons are massless, then how would the energy contained within them ever return for another big bang (as per the closed universe theory/theorem (whichever it is))? Although they can be affected by the EM force, the guage bosons of the EM force are photons, so they would not catch up with them. If I'm wrong and there is a restrictive force acting upon them, then what about neutrinos, whose rest mass is believed to be zero, and are only affected by the weak force which has limited range? (some GUTs suggest the neutrino may have non-zero mass)

If this was the case, then wouldn't that mean that the universe could not have been in existence for infinite time, since all the energy would have been dispersed?

hi, Sam

i shall conteract your wind with littening to strike confusion to the very depths of your heart and soul


Merlin ?

Harry

It's A Good Point, I Don't Fully Undersand The Physics Behind Must of these Theories, but I Think it's something to Do With The Idea That When A Gravitational Field Changes You Don't Get Instant Effect Everywhere, so You Can Imagine The Change Spreading out Along the Universe, This Is a Little Bit like a Wave, and as we all know waves and particles are the Same thing so they manifest themselves as Gravitons. This would just mean that the Photons at the Edge of the Universe are acting under the Gravitational field of the Big Bang itself, eg the whole universe as a point mass, so they would be pulled in.

Merlin

Ha I've got My anti Lightnig Boots On, I've magiced up some books on "Life Made Less Confusing" and a Man with a big stick to force you to read them.

If photons are massless, though, then I don't see how they could be affected by the gravitational force, especially since photons are the guage bosons of the electromagnetic force.

Another thing I was thinking, though, is that the speed of light is said (by Einstein I believe) to be constant relative to the observer. Could a photon be considered to be an observer?

I don't fully understand the way something this complex works, for instance is it actually possible to decelerate a photon? The speed of light is constant after all. But then light cannot escape a black hole, so it must be either decelerated or bent by the gravitational field. But that means that the gravitational force is acting on a massless particle (or wave, whichever way you want to think about it). I think this must be something to do with bent space, something Gen-X mentioned earlier, in which light always travels in a straight line, but the way space is bent means that that straight line is itself bent and therefore perceived to be curved.

Clearly, this is difficult to explain, but do you see what I mean? Is it possible that this bent space could alter the path of photons at the very edge of the universe?

HI, sam

does this man have a long grey beard, and a little crystal on top of his stick (no rudeness intended)

Merlin ?

Harry

The Trouble With Questions like thyese is the more that get answered the more you can think of, I'll give these ones a go.

Quote:

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If photons are massless, though, then I don't see how they could be affected by the gravitational force, especially since photons are the guage bosons of the electromagnetic force.

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Under Classical(normal) Physics Light isn't affected by Gravity, It was never observed to do anything else until Einstein predicted that it should and he set Up his Faous Experiment. (I can't remember what it was called but he pointed out that a certain star would be hidden by the sun at one point and 2 images of the star were observed because the light bent round it) Essentially in Relitivity the Concept of mass is meaningless, as you say later photons go in a straight line and space itself is bent, I don't claim to be able to undersand it fully, If you just imagine rolling a snooker ball round a rubber sheet with large objects on and seeing what path it takes.

Quote:

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Another thing I was thinking, though, is that the speed of light is said (by Einstein I believe) to be constant relative to the observer. Could a photon be considered to be an observer?

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Yes, Is the short answer, If you do all the maths taking a photon as an observer then you see the photon at rest and everything else going at light speed in the opposite direction. (everything, including other photons, unless they are going parallel to the observer.

The Maths isn't that hard for this, there's 2 speeds for every object, a speed u which is the speed you would expect something to be doing if it weren't for reletivity, and an observed speed v which is the speed you observe it to be going. there is a formula u = F(v) which get's the observed speed from the expected speed

Code:

---

u = Sqr(((c * c) * (v * v)) / ((c * c) + (v * v)))

---

and to get the other way round use

Code:

---

v = u / Sqr(1 - ((v * v) / (c * c)))

---

if you plug an observed speed of c into the equation you find u to be infinite, this means that because the expected speed is infinite you can never reach it, and if you have something going at that speed and you try to decellarate it you get infinity - x = Infinity, plug that back in you still get the speed of light.

Trying to imagine relitivity getting out of a black hole you have to imagine it as a wave, a wave has a certain amount of energy, calculated from it's wavelength, the less energy it has the longer its wavelength, so as it get' sout of a black hole it's wavelength gets longer and longer until eventually it becomes infinate then it flips round and befomes -ve, this means it is travelling in the opposite direction(imagine the graph y = 1/x, going form 1 to -1 the number gets bigger and bigger then becomes infinite and goes -ve, every point has a value except 0, this is what's happening to the light)

That's the very very basics, I'm sure this will spring more questions and I probably won't be able to answer them, If you get a book about special Relitivity you should be able to understand the maths (If you can sole a set of 4 Simultanious equations you can do Special Relitivity, it's just gravity that makes it hard)


Merlin No, He's very big and very hard and has "I hit people with sticks unless they read the books I give them" Tattooed on his forehead.

So under relativity, photons are affected by gravity. I see. I've done some particle physics so I know a bit about basic relativity, but I'm not sure if I can solve that kind of equation you mentioned. Could you give an example please?


Btw Sam, do you understand general relativity?

hi Sam

Oh i am scared now, i bet that he has a tatoo saying "i love Mum" in a heart shape on his arm, i have a Dragon that eat men like yours for breakfast, and also if you annoy him he will blow red fire at you and eat you as toast.

Merlin ?

I do Special Relitivity, I've not studied General Relitivity though, I think I get the General Principal, The Idea is that you assume the laws of special Relitivity and then assume that there'a no difference between going in a straight line at a constant speed and falling freely under Gravity, Then You do some hideous maths and you find that every point in spacetime has a gradient, this Gradient has no units (ie it's a length divided by a length which is a ratio) and you have to introduce another Dimension along which space is curved.

Do you mean you want an example of a set of 4 simultanious equations? basicly imagine a set of equations like this

Code:

---


 X + 2Y - 3Z + 7T = 6

3X + 4Y + 4Z -  T = 18

7x +  Y +  Z +  T = 12

 X +  Y +  Z +  T = 6

---

the solution being X=1,Y=2,Z=2,T=1
you should be ablt to write a program t solve any set of equations like that.

If you can do that you won't have any problems with Special Rel, If not, try a set of equations like this

X +2Y = 3

3X - 4Y = 4

the answer is X=2, Y=0.5

If you can do that, look up a Topic Called Linear Algbra, That'll get you to set's of 4 quite quickly.

Oh right, that doesn't look too hard then, just do some substitution and stuff. I've done A-Level maths so I can do that I think.

Quote:

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Originally posted by Sam Finch
Harry

Yes, Is the short answer, If you do all the maths taking a photon as an observer then you see the photon at rest and everything else going at light speed in the opposite direction. (everything, including other photons, unless they are going parallel to the observer......

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So, Is that..

Einstein's Prediction that nothing having a mass can reach the velocity of light is not valid when you consider a Photon as an observer???

Active has been doing some archeology off late :D

Mass is a bit of a funny concept in Relitivity and Quantum Mechanics, it doesn't mean the same thing as in classical physics.

There are actually 2 sorts of mass.

Gravatational Mass (or rest mass.)

This is the mass that dictates gravitational pull, it is constant no matter the velocity of the object.

Inertial Mass
This is the objects momentum divided by its speed, which dictates how much force is needed to accelarate the object. As momentum is a vector and speed is a scalar the objects inertial mass is also a vector.


A photon has a rest mass of 0 (but they can never be at rest, or any other speed other than c) so they can go at the speed of light. As they have no gravitatial mass and no inertial mass perpendicular to their velocity they can be affected by gravity, but only in one plane, using the laws of clasical physics to determine their accelaration under gravity we get the indeterminate form 0/0. And so we must use General Relitivity to predict their movements.

This is one of those beard-strokey concepts.

This also way over my head :( But I'm almost catching up :)

Well thanks for that.. but i have a real basic question
de-stabilizing the foundation of my understanding on relativity.

Is there any mathematical or a theoretical proof for
light should have a constant velocity relative to an
observer.

I don't think Michelson-Morley experiments and the practical experiments that followed are the
only supporting factors for this principle of relativity.
So where can i find a Mathematical treatment for the
same..any links you recommend ??

I read Stephen Hawking's Brief History of time...but
he just says in one line that...

"All observers in relativity must agree on the same
speed of light "

I could understand the Relative Space concept...
But to understand that time is relative...I have to
clear this doubt on the Constancy of Speed of light
relative to an observer independent of the velocity of
the source of the light.

Hope you could help me ! :)

Quote:

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Originally posted by Active
Well thanks for that.. but i have a real basic question
de-stabilizing the foundation of my understanding on relativity.

I don't think Michelson-Morley experiments and the practical experiments that followed are the
only supporting factors for this principle of relativity.
So where can i find a Mathematical treatment for the
same..any links you recommend ??

Hope you could help me ! :)

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I think what you are looking for are Maxwell' equations - try http://www.britannica.com/eb/article...61&tocid=69308 for an introduction.

As you say these theoretical constructs are backed up empirically and robustly up by the Michelson-Morley Experiment - if you can find a hypothesis that explains the absence of the Aether Wind better than the equivalence of all inertial frames I'd be keen to hear more -
http://galileoandeinstein.physics.vi...michelson.html [updated link I put in earlier was an old broken one]

Anyone who wants to catch up of why active asked this question start with
http://www.phys.virginia.edu/classes.../spedlite.html before reading the above

Quote:

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Originally posted by Matthew Gates
In the 1st, 2nd, and 3rd dimensions, we have up and down, left and right, and three-dimensional. What is the 4th dimension?

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About a hundred and twenty years ago a British math teacher Edwin A. Abbott wrote a book to help schoolkids (and anyone else really) to understand and manipulate concepts in higher dimensions. Its written as a sort of tongue-in cheek adventure story and is very easy to read and follow. The quaint little tales are designed to make the intelligent high-school age (and above) reader think through the issues involved in different dimensions

It is called Flatland and you can find a copy on http://sailor.gutenberg.org/by-title/xx548.html

Wow..Kzin the Links are really Wonderful...

Just 5 pages more and I am Sure I 'll come back with
a better Understanding !