Why is the diagonal of a square not "2"

oca · Post by **oca** » Fri May 23, 2014 1:12 am

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

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...

Uberdude · Post by **Uberdude** » Fri May 23, 2014 1:54 am

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.

oca · Post by **oca** » Fri May 23, 2014 2:27 am

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

...

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...

HermanHiddema · Post by **HermanHiddema** » Fri May 23, 2014 2:36 am

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?

oca · Post by **oca** » Fri May 23, 2014 2:43 am

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]

Codexus · Post by **Codexus** » Fri May 23, 2014 3:00 am

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...

oca · Post by **oca** » Fri May 23, 2014 3:08 am

Fractals... so fascinating...

[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 ;-)

[/edit]

Uberdude · Post by **Uberdude** » Fri May 23, 2014 3:21 am

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!

Bill Spight · Post by **Bill Spight** » Fri May 23, 2014 3:59 am

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.

leichtloeslich · Post by **leichtloeslich** » Fri May 23, 2014 4:07 am

Ah, somebody found this picture on 4chan:

: troll_math.png (47.82 KiB) Viewed 13167 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.

DrStraw · Post by **DrStraw** » Fri May 23, 2014 4:38 am

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.)

oca · Post by **oca** » Fri May 23, 2014 4:46 am

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 :

DrStraw · Post by **DrStraw** » Fri May 23, 2014 4:57 am

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 :

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.

oca · Post by **oca** » Fri May 23, 2014 5:10 am

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:

jug · Post by **jug** » Fri May 23, 2014 5:23 am

oca wrote:if it's not tiny points, then... what is this That's stronger than me... space is made of distinct points ...

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".

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