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Why is the diagonal of a square not "2"
http://www.lifein19x19.com/viewtopic.php?f=8&t=10330		 Page 1 of 4
Author:  oca [ Fri May 23, 2014 1:12 am ]
Post subject:  Why is the diagonal of a square not "2"
Hi,

It seems that there are a few mathematicans here...

I'm really not strong at all in math, and there is something I cannot understand with the notion of limit...
maybe someone can help me...

Here is my problem : Why is the diagonal of a square 1.4142... and not... 2
Yes I know, I can take a ruler and just check... but...

Let's say I would like to go from "a" to "b" (funny, a goban would be fine to illustate that even if we are not talking about go)

Lets say the side is "1" and that I can only walk on the side...
I will then walk that way and the distance will be 2...
Click Here To Show Diagram Code
[go]$$B
$$ +-----------+
$$ | . . . . b |
$$ | . . . . S |
$$ | . . . . S |
$$ | . . . . S |
$$ | a S S S S |
$$ +-----------+[/go]


Now let say "the grid" on which I can walk is twice finer... I can do that path now...
but the distance is still "2"

Click Here To Show Diagram Code
[go]$$B
$$ +-----------+
$$ | . . . . b |
$$ | . . . . S |
$$ | . . S S S |
$$ | . . S . . |
$$ | a S S . . |
$$ +-----------+[/go]



Now let say the grid size tends to be infinitely small...

Click Here To Show Diagram Code
[go]$$B
$$ +-----------+
$$ | . . . . b |
$$ | . . . S S |
$$ | . . S S . |
$$ | . S S . . |
$$ | a S . . . |
$$ +-----------+[/go]


Ok, the go diagram is not precise engouth here... but you have got the idea I think :D

So I suppose the path would still be "2" even with infinite precision is it ? (I also suppose your answer will be "no")

I understand that "zooming" doesn't change anything and so doesn't help in getting a smaller path... but I'm not sure what that means... The only thing I can think of is that it should be something that cann't be divided anymore at some point... but that seems weird...
Author:  Uberdude [ Fri May 23, 2014 1:54 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
Your fundamental mistake is thinking a diagonal is made up of lots of tiny horizontal and vertical steps, it is not. You can show the diagonal of the unit square is root 2 by geometric arguments from the ancient Greeks, such as http://www.cut-the-knot.org/do_you_know/SqRtOf2.shtml. If you prefer to think about it intuitively: put a piece of string round the 2 edges of a unit square. Then holding the ends at the opposite corners pull the string tight: it will get shorter so the diagonal is less than 2.
Author:  oca [ Fri May 23, 2014 2:27 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
Uberdude wrote:
Your fundamental mistake is thinking a diagonal is made up of lots of tiny horizontal and vertical steps, it is not.


I'm sure you are right, that's just very difficult for me to imagine something that not composed of points..., in my head, any shape in a 2D plan is just made of x,y points... like a computer screen with pixels...

I really like the idea that can be otherwise, but I just cannot "visualize" that...
if it's not tiny points, then... what is this :-? That's stronger than me... space is made of distinct points :D...

That remind me of a book I read that was called "Flatland", a very nice one ! http://en.wikipedia.org/wiki/Flatland
I just feel like that "humble square" not understanding its own world...
Author:  HermanHiddema [ Fri May 23, 2014 2:36 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
oca wrote:
I'm sure you are right, that's just very difficult for me to imagine something that not composed of points..., in my head, any shape in a 2D plan is just made of x,y points... like a computer screen with pixels...


So how long is the diagonal of a square if you rotate it by 45 degrees?
Author:  oca [ Fri May 23, 2014 2:43 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
HermanHiddema wrote:
oca wrote:
I'm sure you are right, that's just very difficult for me to imagine something that not composed of points..., in my head, any shape in a 2D plan is just made of x,y points... like a computer screen with pixels...


So how long is the diagonal of a square if you rotate it by 45 degrees?


hmmm... I would say... "1" :scratch: [edit] at least it should be in my "pixel world" [/edit]
[second edit] so rotating the shape change its length... that should means that rotation is not part of my x,y world... [/second edit]
Author:  Codexus [ Fri May 23, 2014 3:00 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
I think your world made of infinitely small pixels is a kind of fractal geometry.

Did you know the length of the coast of Great Britain is infinite? Well at least in the world of fractals...
Author:  oca [ Fri May 23, 2014 3:08 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
Fractals... so fascinating...
Image
And I like to eat them by the way...
[edit]
Maybe I will try that with illluck in "Non standard 9 or 10 stones handicap placement"... but its 29 stones handicap
and as I'm really not strong using influence, I'm even not sure that will help me much ;-)
Click Here To Show Diagram Code
[go]$$B
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . X X X . . . . . . . |
$$ | . . . . . . . . X . . . X . . . . . . |
$$ | . . . . X X . X . . . . . X . . . . . |
$$ | . . . X . . X . . . . . . X . . . . . |
$$ | . . X X . . . . . , . . X . . , . . . |
$$ | . . . X . . X . . . . . . X . . . . . |
$$ | . . . . X X . X . . . . . X . . . . . |
$$ | . . . . . . . . X . . . X . . . . . . |
$$ | . . . . . . . . . X X X . . . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]

[/edit]
Author:  Uberdude [ Fri May 23, 2014 3:21 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
HermanHiddema wrote:

So how long is the diagonal of a square if you rotate it by 45 degrees?


Stay-at-home Mum discovers amazing new diet technique: rotate through 45 degrees and instantly get slimmer!
Author:  Bill Spight [ Fri May 23, 2014 3:59 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
oca wrote:
Uberdude wrote:
Your fundamental mistake is thinking a diagonal is made up of lots of tiny horizontal and vertical steps, it is not.


I'm sure you are right, that's just very difficult for me to imagine something that not composed of points...,


You just did. You imagined a diagonal composed of line segments. ;)
Author:  leichtloeslich [ Fri May 23, 2014 4:07 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
Ah, somebody found this picture on 4chan:
Attachment:
troll_math.png
troll_math.png [ 47.82 KiB | Viewed 12445 times ]


Interestingly, I was thinking of posting it in that "Paraconsistent logic" thread, but then I decided it was already stupid enough as it is.
Author:  DrStraw [ Fri May 23, 2014 4:38 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
Start at any point on the equator. Travel 6000 miles east and then 6000 miles north. You are at the north pole, 6000 miles from where you started. Therefore the diagonal of a right triangle with two sides of length 6000 is also 6000 thousand. So all right triangles are equilateral?

(Okay, so it is not exactly 6000 - but you get the point.)
Author:  oca [ Fri May 23, 2014 4:46 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
DrStraw wrote:
Start at any point on the equator. Travel 6000 miles east and then 6000 miles north. You are at the north pole, 6000 miles from where you started. Therefore the diagonal of a right triangle with two sides of length 6000 is also 6000 thousand. So all right triangles are equilateral?

(Okay, so it is not exactly 6000 - but you get the point.)


At least I can visualize that one :
Image
Author:  DrStraw [ Fri May 23, 2014 4:57 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
oca wrote:
DrStraw wrote:
Start at any point on the equator. Travel 6000 miles east and then 6000 miles north. You are at the north pole, 6000 miles from where you started. Therefore the diagonal of a right triangle with two sides of length 6000 is also 6000 thousand. So all right triangles are equilateral?

(Okay, so it is not exactly 6000 - but you get the point.)


At least I can visualize that one :

Image


Well, I can visualize the diagonal you are talking about, and it is root 2 in length. But my point was the Euclidean vs. non-Euclidean geometry can create interesting, non-intuitive results.
Author:  oca [ Fri May 23, 2014 5:10 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
Yes... For sure...
BTW I'm quite convinced that our 3 dimensions are actually circular... (but I'm not sure that there are only 3 dimensions... http://topdocumentaryfilms.com/nova-the ... -universe/)

A nice question would also be to imagine what a world with only one dimension of space but two dimensions of time would look like... but that seems more complex then my diagonal's problem :lol:
Author:  jug [ Fri May 23, 2014 5:23 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
oca wrote:
if it's not tiny points, then... what is this :-? That's stronger than me... space is made of distinct points :D...

Only if you try to measure it with discrete points. But for any adjacent 2 points you can construct another point between them. So you can use distinct points to describe space, but you would need an infinite number for it ... and strange things happen with infinity ;-)

With your assumption a circle would have edges, but it does not.

But to come back to your original question "Why is the diagonal of a square not 2 ?" ... The figure describing with only horizontal and vertical movements you would not call a "diagonal" but rather a collection of line-segments.
So rather than calling it or thinking about a diagonal you should think of it as "the shortest distance between points".

Let's apply this scenario to a sphere (like in DrStraw example), for simplicity let's take Earth and we "fly" directly on the surface. Here you don't have a "diagonal", but you want the shortest way from the north-pole to the south-pole (flying on the surface and not digging): You could fly in a straight line from N to S getting the shortest distance. However, if you add in other movements, e.g. taking a side-tour on the equator you increase the distance and you don't get the shortest route.
Similar happens in your "square". Following a grid is not an approximation of the shortest route in geometry.

This is of course different when you only CAN travel on a grid, e.g. you are searching for the shortest route in a city with a square-like street-map. Then your shortest distance might be 2 (some unit). But take a piggy-back ride on Gozilla and let him go in a direct line and you end up with root(2). But it's just another "problem".
Author:  oca [ Fri May 23, 2014 5:47 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
jug wrote:
But take a piggy-back ride on Gozilla and let him go in a direct line and you end up with root(2).

Thanks jug, I can visualize that too ;)

More seriously, thanks everybody for all your explanations, it seems my problem is just to accept that this is not a "pixel world"...
[edit]
but I play guitar not violin so it's not easy ;)
[/edit]
Author:  Uberdude [ Fri May 23, 2014 6:22 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
oca, something for you to ponder: if it were a pixel world, which way are the pixels aligned? Which way is the x direction and y direction? Related to the earth? What about in space? It reminds me of the famous Michelson-Morley experiment which showed that there wasn't this 'aether' as a fixed background reference frame through which the earth moves, rather like your background grid of pixels.

Also when talking about mathematical objects we are not concerned with the physical world so we don't worry about issues like the quantisation of space but instead focus on the 'platonic' world of an ideal square, perfectly straight lines etc.
Author:  Aidoneus [ Fri May 23, 2014 6:43 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
FYI, Taxi Cab Geometry Introduction: http://jwilson.coe.uga.edu/MATH7200/Tax ... xiCab.html
Author:  oca [ Fri May 23, 2014 6:54 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
Uberdude wrote:
if it were a pixel world, which way are the pixels aligned? Which way is the x direction and y direction? Related to the earth? What about in space?


Well, by "pixel world" I mean a world with quantisation in space, state of energy or whatever...
I world where that statement :

jug wrote:
for any adjacent 2 points you can construct another point between them.

would not be true... because you can be in state A or state B but not in between...

Uberdude wrote:
Also when talking about mathematical objects we are not concerned with the physical world so we don't worry about issues like the quantisation of space but instead focus on the 'platonic' world of an ideal square, perfectly straight lines etc


I like abstraction, but sometimes, I'm not sure abstraction reflects our univers... and somehow I think we have got the wrong path at some point... because often when we try to explain something, that leads to complexity rather than simplicity... allways more laws, more theories... answers lead to more questions...

I'm surprised that so many "key numbers" like pi, e, gold number, root(2) are all irational numbers...
I think we are missing something. like if our world is just an imperfect "projection" of something else...

Sorry, my english is a bit limited for that kind of discution...
Author:  Aidoneus [ Fri May 23, 2014 7:16 am ]
Post subject:  Re: Why is the diagonal of a square not "2"
oca wrote:
I like abstraction, but sometimes, I'm not sure abstraction reflects our univers... and somehow I think we have got the wrong path at some point... because often when we try to explain something, that leads to complexity rather than simplicity... allways more laws, more theories... answers lead to more questions...

I'm surprised that so many "key numbers" like pi, e, gold number, root(2) are all irational number...
I think we are missing something. like if our world is just an imperfect "projection" of something else...


Perhaps you will find this old blog of mine concerning discrete (digital) versus continuous concepts interesting: http://www.britannica.com/blogs/2006/10 ... x-answers/
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