Proof of Pi = 4 ?

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Draw a circle.
Draw a square around it.  Perimeter = 4
Remove corners.  Perimeter is still 4 !
Remove more corners.  Perimeter is still 4 !
Repeat to infinity
Pi = 4 !   Problem Archimedes ?

The more general question is: If you approximate a curve by a series of straight lines, how do you know when you use smaller and smaller straight lines, in the limit, the sum of lenghs of these small straight lines will be the length of the curve?

I suspect it has something to do with differentiability.

If you approximate a curve by smaller and smaller corners, in the limit, the 1st order differential of the resulting curve is no where continuous. Whereas if you aproximate the curve by the hypotenuse of the small corners, the resulting curve's 1st differential will be continuous when the hypotenuses become smaller and smaller; and the lenghts of the hypotenuses will be able to approximate the length of the curve.

I will leave you to fill in the missing details on why a continuous 1st order defferential will enable the lengths of a series of small straight lines to be able to approximate the length of the curve.

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Fire rainbow

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A rare fire rainbow, by Schristia, from:
http://www.flickr.com/photos/schristia/2516584312/

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How my cats see me

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Tsunami-like fog waves

Tsunami-like fog waves taken at Panama City Beach, Florida on Feb. 5th, 2012, by JR Hott.
http://photoblog.msnbc.msn.com/_news/2012/02/10/10375026-spectacular-cloud-tsunami-rolls-over-florida-high-rise-condos
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Difference between man and woman

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Men are like Bluetooth: He is connected to you when you are nearby, but searches for other devices when you are away.
Women are like WiFi: She sees all available devices but connects to the strongest one.

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Granddad remembers the good old days ...

When I was a boy, my momma would send me down to a corner store with $1, and I'd come back with 5 bags of potatoes, 2 loaves of bread, 3 bottle of milk, a hunk of cheese, a box of tea, and 6 eggs.
You can't do it now.
Too many funking security cameras.
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