[FRIAM] square land math question

Thu Jul 23 11:10:40 EDT 2020

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)		 	angel at cs.unm.edu <mailto:angel at cs.unm.edu>
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> On Jul 23, 2020, at 9:03 AM, Frank Wimberly <wimberly3 at 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 <wimberly3 at gmail.com <mailto:wimberly3 at 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.charles at gmail.com <mailto:eric.phillip.charles at 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:echarles at american.edu>
> 
> On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <d00d3rs0n at gmail.com <mailto:d00d3rs0n at 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
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> -- 
> Frank Wimberly
> 140 Calle Ojo Feliz
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> 505 670-9918
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