so i was thinking on my walk to work yesterday
there's gotta be a zeno-like paradox for a manhattan distance of increasing sharpness not eventually leading to a straight line.
ie
go down 1 right 1, total distance travelled 1+1 = 2
go down 1/2 right 1/2 down 1/2 right 1/2 = 1/2*4= 2
go down 1/4 right 1/4 ... = 1/4*8= 2
go down 1/n right 1/n ... = 1/n*(2n)= 2
lim n-> infty (1/n)(2n) = ....sqrt(2)?
that has never sat right in my head
there's gotta be a zeno-like paradox for a manhattan distance of increasing sharpness not eventually leading to a straight line.
ie
go down 1 right 1, total distance travelled 1+1 = 2
go down 1/2 right 1/2 down 1/2 right 1/2 = 1/2*4= 2
go down 1/4 right 1/4 ... = 1/4*8= 2
go down 1/n right 1/n ... = 1/n*(2n)= 2
lim n-> infty (1/n)(2n) = ....sqrt(2)?
that has never sat right in my head
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This already exists.
What you are calling "Manhattan distance" is more formally called "Taxicab geometry" and the paradox you are describing is called the "Staircase paradox". It is also sometimes called the "Diagonal paradox"
@freemo Cannot GET /CoastlineParadox.html link seems broken
@jeffcliff Huh, the link I sent wasnt to the coastline paradox (though it does link to that paradox as well)
@freemo sorry both are broken for me that's a copypaste error on my part but https://mathworld.wolfram.com/DiagonalParadox.html is also broken
@jeffcliff That is rather strange, its a well known site and doesnt go down often. It is up for me. Might be a problem on your end.
@freemo weird
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Error</title>
</head>
<body>
<pre>Cannot GET /DiagonalParadox.html</pre>
</body>
</html>
is pretty definitely a 'their end' sorta thing
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>Error</title>
</head>
<body>
<pre>Cannot GET /DiagonalParadox.html</pre>
</body>
</html>
is pretty definitely a 'their end' sorta thing
@jeffcliff very strange indeed. Still loading fine for me...
