Serious math question: Why is the diagonal in a rectangle shorter?

reply to post by Wide-Eyes


That's what I imagine I will end up looking like if I spend much more time stalking on the internet.

reply to post by primus2012


lol


reply to post by VekTorVik


There you go, perfect example of the one line rule being broken.

Mods, feel free to pull my pants down and spank me!

If you never actually take the diagonal line & instead walk from one corner to the other by correcting your course only @ 90 degree angles you are effectively traveling the distance of both sides no matter now small the increment. The difference is distance vs. the straight line is the wasted distance traveled by deviating from the most direct path (the straight line from one corner to the other)
That deviation will always add up to the sides because you are inside a limited space that was determined by the distances of it's side and by traveling parallel (taking 90 degree turns at each direction correction) to those sides you MUST travel the distance that the space has been determined by...unless it's a straight line, then there is no deviation and the wasted distances go away & you start talking about Pythagoras

The answer to your question is rather a simple one. You see in your hypothesis you are asking why is to shorter to go linear in a straight line as compared to going 90 degrees vertical and the 90 degrees horizontal. Well if you travel 10cm 90 degrees south on a vertical plane and then 10cm 90 degrees west on a horizontal plane you have traveled 20 cm altogether. Compared to 14.14 cm in a linear line.

the equation can be summed in a^2 + b^2 = c^2

reply to post by Domo1

When I was 9 my Dad told me it was impossible to get from point A to point B because there was no way you could ever cross half the distance as there was no way to ever measure every half. Or something. Same thing?

Yes, this is a re-formulated version of Zeno's paradox, which is the official name for what your Dad told you. Actually, what he told you is that you can keep on halving distances infinitely, but you will never reach a half-distance that measures zero.

The solution of Zeno's paradox lies in the blunt fact that there is no such thing at an infinitesimal point in nature – at some point one reaches a 'half-distance' that's shorter than a footprint, or a molecule, or an atom, or a Planck length, or whatever. At that point you have to stop halving distances – you've arrived!

The same solution works for the OP's riddle. At some point, his stepwise zigzag smoothes out into a straight line.

As Phage implies, this is the way the calculus works.

Next time I walk across a square I'm going to walk it like this:



Cos that makes infinitely more sense than walking across it diagonally in a straight line.

Suddenly I want to play snake, where's my old Nokia 3200?

reply to post by Wide-Eyes


LOL. I think this OP is trying to find out why in his mind why the equation states the fastest way to two points is a straight line. Well its not really the fastest way in a 1 dimensional world. In a 2 or 3 dimensional world it is. In a one dimensional world you would fold the points on top of each other and then you would go through the zero dimension to get to the second point without traveling a cm.

That will bake his brain for a while.

reply to post by Astyanax

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise

Ahh. Thank you.

“ That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ” —Aristotle, Physics VI:9, 239b10 Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

Link

If you only look at the horizontal vectors, they will always add up to 10. Then add 10 for vertical.
A diagonal line is a singular vector though.
Think about this now;
If you unraveled your dna it would stretch to the sun and back 4 times over.

Originally posted by Astyanax
The solution of Zeno's paradox lies in the blunt fact that there is no such thing at an infinitesimal point in nature – at some point one reaches a 'half-distance' that's shorter than a footprint, or a molecule, or an atom, or a Planck length, or whatever. At that point you have to stop halving distances – you've arrived!

The same solution works for the OP's riddle. At some point, his stepwise zigzag smoothes out into a straight line.

Ecaxtly this is my problem. If there is no infinitesimal point in nature and everything is made out of something (no matter how small this part is) than how can I move in a straight line? How can I move in a absolutely straight line if I have to step along those atoms or obey the Planck length. This would result in a zigzag line (although it would look like a straight line as we can't see these absolutely small parts much smaller than an atom) and thus be 20cm long.

I know the formulas, Pythagoras and from my everyday life that the straight line is indeed shorter but wouldn't this require a world without a smallest part. If something in this universe is forced to move along some smallest parts (no matter if you call it strings, atoms or whatever) then it wouldn't be possible to take the straight line.

So it's more a physical problem/question than a math one i quess.

reply to post by UnixFE

I look at it this way: you have a 10cm x 10cm square and you lay a 1cm x 1cm grid over your square.
Take a piece of string 20cm long, pin one end of the string to the top left hand corner of the square, lay the string 1cm down, pin it in place, move 1cm to the right and pin the string again, and so on until you reach the bottom, the string will reach the bottom right corner of the square exactly. The distance is 20 cm.

if you take all the pins out except the top left hand corner and pull the sting straight, the point where the string passes the bottom right corner will be as you say, 14.14cm (in a straight line).

Now if you lay a grid of .5cm x .5cm you can stil lay the string from top left to bottom right because the distance has not changed. you simply create a grid with a greater frequency of 'steps'.

it does not matter what the frequency is (how small the steps can get is infinite), the total distance (length) is fixed (finite) by the measure of the square.

If you travel along the string making more frequent (smaller) steps, this may increase the TIME it takes to travel the lenght of the string due to the time needed to make the turns, but the distance travelled will be the same. so you will never need a longer piece of string, just more time.

reply to post by UnixFE


is this even a thing?

This thread, I mean... Really?

Seriously, Op... you never asked why the diagonal from A to Z is the same distance as going around the perimeter....

You asked why Going from A to B to C to D to E to F to G to H to I to J to K to L to M to N to O to P to Q to R to S to T to U to V to W to X to Y to Z..... is the same distance as walking around the perimeter...

It's just a nonsensical question.



reply to post by Domo1

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise.

Zeno's Paradox
In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"
The mathematician said: "Never."
The physicist said: "In an infinite amount of time."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."

reply to post by UnixFE

Your problem is that you assume for small a, b the side length a + b = c contrary to pythagorean a^2 + b^2 = c^2 for all a, b.

reply to post by Phage


This is it in a nutshell.

OP, you, in essence, asking why the hypotenuse on an isosceles triangle is longer than its shortest side. Study the Pythagorean Theorum for your answers.

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It changes, if you look at the destination you go staight to it. In you diagram you are aiming for it blind. It's to do with direction. Up, left, up, left, or you could go upish leftish in a straight line. The diference is direction and changes when you only look in the middle.

Originally posted by UnixFE

Originally posted by Astyanax
The solution of Zeno's paradox lies in the blunt fact that there is no such thing at an infinitesimal point in nature – at some point one reaches a 'half-distance' that's shorter than a footprint, or a molecule, or an atom, or a Planck length, or whatever. At that point you have to stop halving distances – you've arrived!

The same solution works for the OP's riddle. At some point, his stepwise zigzag smoothes out into a straight line.

Ecaxtly this is my problem. If there is no infinitesimal point in nature and everything is made out of something (no matter how small this part is) than how can I move in a straight line? How can I move in a absolutely straight line if I have to step along those atoms or obey the Planck length. This would result in a zigzag line (although it would look like a straight line as we can't see these absolutely small parts much smaller than an atom) and thus be 20cm long.

I know the formulas, Pythagoras and from my everyday life that the straight line is indeed shorter but wouldn't this require a world without a smallest part. If something in this universe is forced to move along some smallest parts (no matter if you call it strings, atoms or whatever) then it wouldn't be possible to take the straight line.

So it's more a physical problem/question than a math one i quess.

maybe thats why things in space travel in waves?? waves are like a cool smooth zig zag

The diameter of a circle is shorter than the circumference. Can the zigzag theory work with that shape.

I think I get the OP question.
He question is, if you make those steps smaller and smaller, eventually at the size of your screen pixels and then smaller, where is the point where the zigzag line of 20 cm became a straight line of 14.4, and where the the remaining 5.6 cm do go. Did I get it right?

Now the answer will be this: the diagonal is the straight line between two corners of the rectangle. No matter how many steps you create inside the rectangle, to find out the diagonal you will have to measure their diagonal, not their perimeter. What you did is to run along their sizes, which will give you, obviously, the perimeter. What you should have done is to draw their diagonal, and measure that. The diagonal of the big rectangle will be made of the diagonals of your inside steps, and together they will sum up the same distance. So no matter how minuscule your steps will became, it has no importance at all, since you will always measure the diagonal between them. Does it make sense?

And if they are so small that cannot be drawn anymore, you can still mathematically assume their value, and then deduce the value of their diagonal, ad infinitum. In theory, they will never became the straight line that is their diagonal.

Hope you will get your sleep back [

By taking "steps", no matter how small, you're traveling in one direction at a time. By taking a straight line, you're traveling in two directions at a time.

"steps" is a course, "line" is a direct route.

A line is an infinite series of points. By making an infinite series of "steps" (lines), you're essentially traversing the entire plane, or in this case, the length + width.

EDIT: By traveling 2 directions at once, you're traversing the length + width simultaneously.