How to get the perimeter of a circle from 8r to 2 $pi$ r [duplicate]The staircase paradox, or why $pine4$How many n square can fit into a square of side NWhat is the largest square that can be inscribed in an annulus?Constructing a circle from a squareExistence of a Triangle with a MultiplicityWhat shape results from this deformation of a circle?What is the area percentage of a ⌈square in a circle? What is the area percentage of a ⌈circle in a square?Difficulty with circle geometryGetting a wrong value of $pi$How to work out the diameter of a circle from a rectangleSimple proof that the first digit of pi is 3
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How to get the perimeter of a circle from 8r to 2 $pi$ r [duplicate]
The staircase paradox, or why $pine4$How many n square can fit into a square of side NWhat is the largest square that can be inscribed in an annulus?Constructing a circle from a squareExistence of a Triangle with a MultiplicityWhat shape results from this deformation of a circle?What is the area percentage of a ⌈square in a circle? What is the area percentage of a ⌈circle in a square?Difficulty with circle geometryGetting a wrong value of $pi$How to work out the diameter of a circle from a rectangleSimple proof that the first digit of pi is 3
$begingroup$
This question already has an answer here:
The staircase paradox, or why $pine4$
20 answers
A friend showed me a circle inside a square such that the diameter of the circle is equal to the side length of the square. The perimeter of the square is 8r. He then proceeded to take small squares of the corners of the original square, maintaining perimeter as 8r. As this continues the original square begins to appear like a circle, somehow making it seem like the perimeter of the circle is 8r, which I know isn't true.
What is the flaw in the method above to show that it the perimeter is not 8r?
geometry
asked Mar 21 at 12:46
CipherBotCipherBot
1012
$endgroup$
marked as duplicate by Ethan Bolker, Arthur, Wouter, Robert Soupe, José Carlos Santos Mar 21 at 15:46
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
The staircase paradox, or why $pine4$
20 answers
A friend showed me a circle inside a square such that the diameter of the circle is equal to the side length of the square. The perimeter of the square is 8r. He then proceeded to take small squares of the corners of the original square, maintaining perimeter as 8r. As this continues the original square begins to appear like a circle, somehow making it seem like the perimeter of the circle is 8r, which I know isn't true.
What is the flaw in the method above to show that it the perimeter is not 8r?
geometry
asked Mar 21 at 12:46
CipherBotCipherBot
1012
$endgroup$
marked as duplicate by Ethan Bolker, Arthur, Wouter, Robert Soupe, José Carlos Santos Mar 21 at 15:46
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
This is a well known paradox with several explanations on this site that I can't find right now. Someone will. The issue is essentially that approximating a slanted line by a staircase with small steps does not approximate the length of the line.
$endgroup$
– Ethan Bolker
Mar 21 at 12:50
$begingroup$
The same faulty argument demonstrates that the length of the square’s diagonal is equal to double its side length. Can you work out the argument’s flaw in that case?
$endgroup$
– amd
Mar 21 at 18:37
add a comment |
$begingroup$
This question already has an answer here:
The staircase paradox, or why $pine4$
20 answers
A friend showed me a circle inside a square such that the diameter of the circle is equal to the side length of the square. The perimeter of the square is 8r. He then proceeded to take small squares of the corners of the original square, maintaining perimeter as 8r. As this continues the original square begins to appear like a circle, somehow making it seem like the perimeter of the circle is 8r, which I know isn't true.
What is the flaw in the method above to show that it the perimeter is not 8r?
geometry
asked Mar 21 at 12:46
CipherBotCipherBot
1012
$endgroup$
This question already has an answer here:
The staircase paradox, or why $pine4$
20 answers
A friend showed me a circle inside a square such that the diameter of the circle is equal to the side length of the square. The perimeter of the square is 8r. He then proceeded to take small squares of the corners of the original square, maintaining perimeter as 8r. As this continues the original square begins to appear like a circle, somehow making it seem like the perimeter of the circle is 8r, which I know isn't true.
What is the flaw in the method above to show that it the perimeter is not 8r?
This question already has an answer here:
The staircase paradox, or why $pine4$
20 answers
geometry
geometry
asked Mar 21 at 12:46
CipherBotCipherBot
1012
asked Mar 21 at 12:46
CipherBotCipherBot
1012
asked Mar 21 at 12:46
CipherBotCipherBot
1012
asked Mar 21 at 12:46
CipherBotCipherBot
1012
asked Mar 21 at 12:46
CipherBotCipherBot
1012
1012
marked as duplicate by Ethan Bolker, Arthur, Wouter, Robert Soupe, José Carlos Santos Mar 21 at 15:46
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Ethan Bolker, Arthur, Wouter, Robert Soupe, José Carlos Santos Mar 21 at 15:46
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
This is a well known paradox with several explanations on this site that I can't find right now. Someone will. The issue is essentially that approximating a slanted line by a staircase with small steps does not approximate the length of the line.
$endgroup$
– Ethan Bolker
Mar 21 at 12:50
$begingroup$
The same faulty argument demonstrates that the length of the square’s diagonal is equal to double its side length. Can you work out the argument’s flaw in that case?
$endgroup$
– amd
Mar 21 at 18:37
add a comment |
$begingroup$
This is a well known paradox with several explanations on this site that I can't find right now. Someone will. The issue is essentially that approximating a slanted line by a staircase with small steps does not approximate the length of the line.
$endgroup$
– Ethan Bolker
Mar 21 at 12:50
$begingroup$
The same faulty argument demonstrates that the length of the square’s diagonal is equal to double its side length. Can you work out the argument’s flaw in that case?
$endgroup$
– amd
Mar 21 at 18:37
$begingroup$
This is a well known paradox with several explanations on this site that I can't find right now. Someone will. The issue is essentially that approximating a slanted line by a staircase with small steps does not approximate the length of the line.
$endgroup$
– Ethan Bolker
Mar 21 at 12:50
$begingroup$
This is a well known paradox with several explanations on this site that I can't find right now. Someone will. The issue is essentially that approximating a slanted line by a staircase with small steps does not approximate the length of the line.
$endgroup$
– Ethan Bolker
Mar 21 at 12:50
$begingroup$
The same faulty argument demonstrates that the length of the square’s diagonal is equal to double its side length. Can you work out the argument’s flaw in that case?
$endgroup$
– amd
Mar 21 at 18:37
$begingroup$
The same faulty argument demonstrates that the length of the square’s diagonal is equal to double its side length. Can you work out the argument’s flaw in that case?
$endgroup$
– amd
Mar 21 at 18:37
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Arclength isn't continuous with respect to uniform convergence.
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
$endgroup$
$begingroup$
“Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx.
$endgroup$
– steven gregory
Mar 21 at 12:55
$begingroup$
Hooray for Captain Spaulding!
$endgroup$
– Umberto P.
Mar 21 at 12:58
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Arclength isn't continuous with respect to uniform convergence.
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
$endgroup$
$begingroup$
“Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx.
$endgroup$
– steven gregory
Mar 21 at 12:55
$begingroup$
Hooray for Captain Spaulding!
$endgroup$
– Umberto P.
Mar 21 at 12:58
add a comment |
$begingroup$
Arclength isn't continuous with respect to uniform convergence.
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
$endgroup$
$begingroup$
“Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx.
$endgroup$
– steven gregory
Mar 21 at 12:55
$begingroup$
Hooray for Captain Spaulding!
$endgroup$
– Umberto P.
Mar 21 at 12:58
add a comment |
$begingroup$
Arclength isn't continuous with respect to uniform convergence.
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
$endgroup$
Arclength isn't continuous with respect to uniform convergence.
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
answered Mar 21 at 12:48
Umberto P.Umberto P.
40.3k13370
40.3k13370
$begingroup$
“Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx.
$endgroup$
– steven gregory
Mar 21 at 12:55
$begingroup$
Hooray for Captain Spaulding!
$endgroup$
– Umberto P.
Mar 21 at 12:58
add a comment |
$begingroup$
“Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx.
$endgroup$
– steven gregory
Mar 21 at 12:55
$begingroup$
Hooray for Captain Spaulding!
$endgroup$
– Umberto P.
Mar 21 at 12:58
$begingroup$
“Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx.
$endgroup$
– steven gregory
Mar 21 at 12:55
$begingroup$
“Well, art is art, isn't it? Still, on the other hand, water is water! And east is east and west is west and if you take cranberries and stew them like applesauce they taste much more like prunes than rhubarb does." - Groucho Marx.
$endgroup$
– steven gregory
Mar 21 at 12:55
$begingroup$
Hooray for Captain Spaulding!
$endgroup$
– Umberto P.
Mar 21 at 12:58
$begingroup$
Hooray for Captain Spaulding!
$endgroup$
– Umberto P.
Mar 21 at 12:58
add a comment |
$begingroup$
This is a well known paradox with several explanations on this site that I can't find right now. Someone will. The issue is essentially that approximating a slanted line by a staircase with small steps does not approximate the length of the line.
$endgroup$
– Ethan Bolker
Mar 21 at 12:50
$begingroup$
The same faulty argument demonstrates that the length of the square’s diagonal is equal to double its side length. Can you work out the argument’s flaw in that case?
$endgroup$
– amd
Mar 21 at 18:37