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Old 19th April 2014
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Question How is Pythagoras so wrong

Pythagoras says that the distance from one corner of a square, to the one furthest away is 1.414... times the distance along the side.

The Drill Sargent will not allow anyone to walk diagonally across his parade ground, saying it's the same distance to walk around the two sides.

He will however let you walk North, turn 90 degrees, walk East, turn 90 degrees and repeat this at whatever stride and number of steps you choose.

So if the parade ground is 100 yards by 100 yards, you can cross from one corner to the furthest by walking 200 yds around the edge,

OR

50yds North, 50yds East, 50yds North & 50yds East = 200yds

20yds North, 20yds East, 20yds North, 20yds East, etc, etc, etc = 200yds

10yds North, 10yds East, 10 times over = 200yds

1 yard North, 1 yard East, 100 times over = 200yds

You can now see a pattern emerging in which as the size of the steps gets ever smaller, while we are moving only in the directions North & East it would appear that we are moving diagonally.

In much the same way as a line of square pixels form a line, but lets not digress into image quality

SO:

As we approach infinitely small steps the distance is still 200 yards.

Therefore had Pythagoras been using a digital approach to his calculations, we may never have known that the shortest distance between two points is a straight line...
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Old 19th April 2014
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Re: How is Pythagoras so wrong

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Originally Posted by Graham_of_Rainham View Post
You can now see a pattern emerging in which as the size of the steps gets ever smaller, while we are moving only in the directions North & East it would appear that we are moving diagonally.
Is the key to the apparent paradox not in the word 'appear'? We are not 'actually' moving diagonally.
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Old 19th April 2014
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Re: How is Pythagoras so wrong

You need a huge AA filter to make it work. And then you need to know how to sharpen as the parade ground will be soft.
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Old 19th April 2014
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Re: How is Pythagoras so wrong

You aren't travelling diagonally. You are still travelling along the sides of infinitesimally small squares which when totalled add up to the length of the two original sides.
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Old 19th April 2014
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Re: How is Pythagoras so wrong

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You aren't travelling diagonally. You are still travelling along the sides of infinitesimally small squares which when totalled add up to the length of the two original sides.
That's what I thought, but didn't comment because I thought I hadn't 'got' it!

'Thompson's Lamp' is a nice one:

Switch a lamp on for one second, then off for half a second, on for a quarter - and so on. At two seconds the state of the lamp cannot be known!

Jim
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Old 20th April 2014
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Re: How is Pythagoras so wrong

The Drill Sargent is wrong. But try telling him that
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Old 20th April 2014
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Re: How is Pythagoras so wrong

It's another statement of Zeno's Paradox. I remember struggling with this in undergraduate engineering maths and still can't give concise solution but there is an old joke

A mathematician, a physicist and an engineer were asked to answer the following question. A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance d apart at time 0, they are d/2 at t=10, d/4 at t=20, d/8 at t=30, and so on.) When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes.
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Old 20th April 2014
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Re: How is Pythagoras so wrong

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Originally Posted by Jim Ford View Post

Switch a lamp on for one second, then off for half a second, on for a quarter - and so on. At two seconds the state of the lamp cannot be known!

Jim
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Old 20th April 2014
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Re: How is Pythagoras so wrong

All this reminds me of the way we were taught to remember Pythagoras' Theorem at school (long long ago ).

There were three Indian squaws. One sat on a Bison hide, one sat on a Cow hide and one sat on Hippopotamus hide.

The squaws who sat on the Bison and Cow hides each had one son whilst the one who sat on a Hippopotamus hide had two sons.

Therefore:

"The sons of the squaw on the Hippopotamus are equal to the sum of the sons of the squaws on the other two hides."

Worked for me.
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Old 20th April 2014
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Re: How is Pythagoras so wrong

What a great thread...I feel justified in taking a career in engineering rather than science now......this reminds me that an engineers glass is never half full or half empty....it's just the wrong size.
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Old 21st April 2014
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Re: How is Pythagoras so wrong

If you really want to get your brain spinning, google 'monty hall problem'. It's a simple idea, but has had PhD mathematicians arguing in the past! It goes:

You're in a game show. The host shows you three doors. Behind one door is a car and behind the other two are goats. You can chose one door. Before the door is opened to reveal what's behind, the host opens one of the other doors, revealing a goat. You are now given the opportunity of changing your original choice. What should you do?

The answer is surprising. By changing your choice you increase your chance of winning the car!

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Old 21st April 2014
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Re: How is Pythagoras so wrong

It's a bit of an unsatisfactory problem - the result is different depending on whether or not the host knows which box the car is in. Since you are not told this in the statement of the problem, it makes a bit of a nonsense of it.

John the pedantic mathematician
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Old 21st April 2014
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Wink Re: How is Pythagoras so wrong

Quote:
Originally Posted by Bikie John View Post
... the result is different depending on whether or not the host knows which box the car is in....
John the pedantic mathematician
The "host" always knows

If they were to use cats; then by opening the door and making the observation, you may have unwittingly altered its state...

or is that "a whole nother story"
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Old 21st April 2014
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Re: How is Pythagoras so wrong

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Originally Posted by Graham_of_Rainham View Post
The "host" always knows
Yes, I should have mentioned it - but then it's pretty obvious anyway!

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Old 21st April 2014
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Re: How is Pythagoras so wrong

Lovely, statistical strangenesses. The Monty Hall problem is an absolute belter, it's not an easy one to investigate. This one's easy. How many people have to gather together so that there's a better than evens chance two of them having the same birthday?
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