Why would you call the limit of the increasing smaller squares a “square”? Would you still say it has a dimension of 2? It has no area and no perimeter. In fractal geometry we can create objects with only slightly different constructions that in the limit have a zero area and an infinite perimeter.
Ed _______________________ Ed Angel Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab) Professor Emeritus of Computer Science, University of New Mexico 1017 Sierra Pinon Santa Fe, NM 87501 505-984-0136 (home) an...@cs.unm.edu <mailto:an...@cs.unm.edu> 505-453-4944 (cell) http://www.cs.unm.edu/~angel <http://www.cs.unm.edu/~angel> > On Jul 23, 2020, at 9:03 AM, Frank Wimberly <wimber...@gmail.com> wrote: > > p.s. Zeno's Paradox is related to > > 1/2 + 1/4 + 1/8 +... > > = Sum(1/(2^n)) for n = 1 to infinity > > = 1 > > (Note: Sum(1/(2^n)) for n = 0 to infinity > > = 1/(1 - (1/2)) = 2) > > --- > Frank C. Wimberly > 140 Calle Ojo Feliz, > Santa Fe, NM 87505 > > 505 670-9918 > Santa Fe, NM > > On Wed, Jul 22, 2020, 8:49 PM Frank Wimberly <wimber...@gmail.com > <mailto:wimber...@gmail.com>> wrote: > Incidentally, people are used to seeing limits that aren't reached such a > limit as x goes to infinity of 1/x = 0. But there are limits such as limit > as x goes to 3 of x/3 = 1. The question of the squares is the latter type. > There is no reason the area of the small square doesn't reach 0. > > On Wed, Jul 22, 2020 at 7:36 PM Eric Charles <eric.phillip.char...@gmail.com > <mailto:eric.phillip.char...@gmail.com>> wrote: > This is a Zeno's Paradox styled challenge, right? I sometimes describe > calculus as a solution to Zeno's paradoxes, based on the assumption that > paradoxes are false. > > The solution, while clever, doesn't' work if we assert either of the > following: > > A) When the small-square reaches the limit it stops being a square (as it is > just a point). > > B) You can never actually reach the limit, therefore the small square always > removes a square-sized corner of the large square, rendering the large bit > no-longer-square. > > The solution works only if we allow the infinitely small square to still be a > square, while removing nothing from the larger square. But if we are allowing > infinitely small still-square objects, so small that they don't stop an > object they are in from also being a square, then there's no Squareland > problem at all: Any arbitrary number of squares can be fit inside any other > given square. > > > > ----------- > Eric P. Charles, Ph.D. > Department of Justice - Personnel Psychologist > American University - Adjunct Instructor > <mailto:echar...@american.edu> > > On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <d00d3r...@gmail.com > <mailto:d00d3r...@gmail.com>> wrote: > A kid momentarily convinced me of something that must be wrong today. > We were working on a math problem called Squareland > (https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p > > <https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p>). > It basically involved dividing big squares into smaller squares. > I volunteered to tell the kids the rules of the problem. I made a fairly > strong argument for why a square can not be divided into 2 smaller squares, > when a kid stumped me with a calculus argument. She drew a tiny square in the > corner of a bigger one and said that "as the tiny square area approaches > zero, the big outer square would become increasingly square-like and the > smaller one would still be a square". > I had to admit that I did not know, and that the argument might hold water > with more knowledgeable mathematicians. > > The calculus trick of taking the limit of something as it gets infinitely > small always seemed like magic to me. > > > Cody Smith > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > <http://bit.ly/virtualfriam> > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > <http://redfish.com/mailman/listinfo/friam_redfish.com> > archives: http://friam.471366.n2.nabble.com/ > <http://friam.471366.n2.nabble.com/> > FRIAM-COMIC http://friam-comic.blogspot.com/ > <http://friam-comic.blogspot.com/> > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > <http://bit.ly/virtualfriam> > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > <http://redfish.com/mailman/listinfo/friam_redfish.com> > archives: http://friam.471366.n2.nabble.com/ > <http://friam.471366.n2.nabble.com/> > FRIAM-COMIC http://friam-comic.blogspot.com/ > <http://friam-comic.blogspot.com/> > > > -- > Frank Wimberly > 140 Calle Ojo Feliz > Santa Fe, NM 87505 > 505 670-9918 > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC http://friam-comic.blogspot.com/
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