Presumably this is the length of some discretisation of the curve, perhaps one of millimeter-long line segments. (The exact length is uncomputable!) Really cool though!
Well, it doesn't in theory, but it does in practice. Natural phenomena are only fractal over a certain range of scales. Outside that range they follow normal rules.
Yeah, one could draw a tight spiral in any of the open spaces of unbounded length. But "discretization" is probably a good angle to interpret it, since such a counterexample would be discretized to a point.
I can imagine the practical appeal of a device like this when applied to something like a map. Especially over large journeys, this could likely cut down on your error considerably.
Wow it really is great. Plan to share with everyone I know who would find this interesting. Already watched a few of them and they are all just as enjoyable. Hope he gets a bunch more subs and keeps making videos.
He groans with disappointment in the video that the longimeter and the map measurer don't give the same result, but we don't find out which was right. When he measures the curve he drew, the longimeter gives 28.9 mm but the map measurer (a.k.a. opisometer, a hand-held instrument for measuring curves on paper) says 34.5 mm; that's a 16% difference, so pretty big.
To do a real test, you could print out a curve with a mathematically calculated length, say a sine curve, then measure with both methods (and other methods like placing a string on top) to find out what works best in practice.
Hm. That 16% difference reminds me of the 60% point Miles Mathis makes in his infamous 'pi'='4' [single quotes INTENTIONAL.] paper (which ISN'T what people think. Numerous people tried to debunk it and I would absolutely LOVE to see a proper debunking of it, but each 'debunker' actually only skimmed his writing, or only read one of his papers on it instead of all of them, and constructed bad counter arguments as a consequence.)
The short version has a simple but fundamental error in the use of limits, and so as long as the author claims that this contains the essence of his argument, there seems little point in wading through the full work, especially as it is written in an extremely discursive manner, to put it charitably.
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[ 3.5 ms ] story [ 78.5 ms ] thread[citation needed]
https://books.google.it/books?id=sFzCAgAAQBAJ&pg=PA121#v=one...
https://www.youtube.com/watch?v=jMvEOmpy8Kw
http://www.rolatape.com/us/en/home/measuring-wheels.html
To do a real test, you could print out a curve with a mathematically calculated length, say a sine curve, then measure with both methods (and other methods like placing a string on top) to find out what works best in practice.
http://milesmathis.com/pi.html http://milesmathis.com/pi2.html http://milesmathis.com/pi3.html https://sagacityssentinel.wordpress.com/ (the debunk reply there is from 2011 and doesn't factor in the fact that Mathis indeed didn't realized the rather obvious link to the Hilbert/Taxicab metric until 2012, see the pi2.html page there.)
Also see page 41 of Differential Geometry of Curves and Surfaces by Manfredo Do Carmo