<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">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. <div class=""><br class=""></div><div class="">Ed<br class=""><div class="">
<span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div class="">_______________________</div><div class=""><br class="Apple-interchange-newline">Ed Angel<br class=""><br class=""></div><div class="">Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)<br class="">Professor Emeritus of Computer Science, University of New Mexico<br class=""><br class="">1017 Sierra Pinon</div><div class="">Santa Fe, NM 87501<br class="">505-984-0136 (home)<span class="Apple-tab-span" style="white-space: pre; "> </span> <span class="Apple-tab-span" style="white-space: pre; "> </span><a href="mailto:angel@cs.unm.edu" class="">angel@cs.unm.edu</a></div><div class="">505-453-4944 (cell) <span class="Apple-tab-span" style="white-space: pre; "> </span><span class="Apple-tab-span" style="white-space: pre; "> </span><a href="http://www.cs.unm.edu/~angel" class="">http://www.cs.unm.edu/~angel</a><br class=""></div></span></span>
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<div><br class=""><blockquote type="cite" class=""><div class="">On Jul 23, 2020, at 9:03 AM, Frank Wimberly <<a href="mailto:wimberly3@gmail.com" class="">wimberly3@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="auto" class="">p.s. Zeno's Paradox is related to<div dir="auto" class=""><br class=""></div><div dir="auto" class="">1/2 + 1/4 + 1/8 +...</div><div dir="auto" class=""><br class=""></div><div dir="auto" class="">= Sum(1/(2^n)) for n = 1 to infinity</div><div dir="auto" class=""><br class=""></div><div dir="auto" class="">= 1</div><div dir="auto" class=""><br class=""></div><div dir="auto" class="">(Note: Sum(1/(2^n)) for n = 0 to infinity</div><div dir="auto" class=""><br class=""></div><div dir="auto" class="">= 1/(1 - (1/2)) = 2)<br class=""><br class=""><div data-smartmail="gmail_signature" dir="auto" class="">---<br class="">Frank C. Wimberly<br class="">140 Calle Ojo Feliz, <br class="">Santa Fe, NM 87505<br class=""><br class="">505 670-9918<br class="">Santa Fe, NM</div></div></div><br class=""><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Wed, Jul 22, 2020, 8:49 PM Frank Wimberly <<a href="mailto:wimberly3@gmail.com" class="">wimberly3@gmail.com</a>> wrote:<br class=""></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr" class="">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.</div><br class=""><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Wed, Jul 22, 2020 at 7:36 PM Eric Charles <<a href="mailto:eric.phillip.charles@gmail.com" target="_blank" rel="noreferrer" class="">eric.phillip.charles@gmail.com</a>> wrote:<br class=""></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr" class="">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. <div class=""><br class=""></div><div class="">The solution, while clever, doesn't' work if we assert either of the following: </div><div class=""><br class=""></div><div class="">A) When the small-square reaches the limit it stops being a square (as it is just a point). </div><div class=""><br class=""></div><div class="">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. </div><div class=""><br class=""></div><div class="">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: <i class="">Any </i>arbitrary number of squares can be fit inside any other given square. </div><div class=""><br class=""></div><div class=""><br class=""></div><div class=""><div class=""><div dir="ltr" class=""><div dir="ltr" class=""><div class=""><div dir="ltr" class=""><div dir="ltr" class=""><div dir="ltr" class=""><br clear="all" class="">-----------<br class=""><div dir="ltr" class="">Eric P. Charles, Ph.D.<br class="">Department of Justice - Personnel <span class="">Psychologist</span></div><div class="">American University - Adjunct Instructor</div><div class=""></div></div><div dir="ltr" class=""><a href="mailto:echarles@american.edu" target="_blank" rel="noreferrer" class=""></a></div></div></div></div></div></div></div><br class=""></div></div><br class=""><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <<a href="mailto:d00d3rs0n@gmail.com" target="_blank" rel="noreferrer" class="">d00d3rs0n@gmail.com</a>> wrote:<br class=""></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr" class=""><div class="">A kid momentarily convinced me of something that must be wrong today. </div><div class="">We were working on a math problem called Squareland (<a href="https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p" target="_blank" rel="noreferrer" class="">https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p</a>). It basically involved dividing big squares into smaller squares. </div><div class="">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". </div><div class="">I had to admit that I did not know, and that the argument might hold water with more knowledgeable mathematicians. </div><div class=""><br class=""></div><div class="">The calculus trick of taking the limit of something as it gets infinitely small always seemed like magic to me. </div><div class=""><br class=""></div><div class=""><br clear="all" class=""><div class=""><div dir="ltr" class=""><div dir="ltr" class="">Cody Smith</div></div></div></div></div>
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</blockquote></div><br clear="all" class=""><div class=""><br class=""></div>-- <br class=""><div dir="ltr" class="">Frank Wimberly<br class="">140 Calle Ojo Feliz<br class="">Santa Fe, NM 87505<br class="">505 670-9918</div>
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