<div dir="auto">Right. When its area reaches zero it's not a square. That is, there is only one square then.<br><br><div data-smartmail="gmail_signature">---<br>Frank C. Wimberly<br>140 Calle Ojo Feliz, <br>Santa Fe, NM 87505<br><br>505 670-9918<br>Santa Fe, NM</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Thu, Jul 23, 2020, 9:10 AM Edward Angel <<a href="mailto:angel@cs.unm.edu">angel@cs.unm.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div style="word-wrap:break-word;line-break:after-white-space">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><br></div><div>Ed<br><div>
<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;text-align:-webkit-auto;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><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;text-align:-webkit-auto;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px"><div>_______________________</div><div><br>Ed Angel<br><br></div><div>Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)<br>Professor Emeritus of Computer Science, University of New Mexico<br><br>1017 Sierra Pinon</div><div>Santa Fe, NM 87501<br>505-984-0136 (home)<span style="white-space:pre-wrap"> </span> <span style="white-space:pre-wrap"> </span><a href="mailto:angel@cs.unm.edu" target="_blank" rel="noreferrer">angel@cs.unm.edu</a></div><div>505-453-4944 (cell) <span style="white-space:pre-wrap"> </span><span style="white-space:pre-wrap"> </span><a href="http://www.cs.unm.edu/~angel" target="_blank" rel="noreferrer">http://www.cs.unm.edu/~angel</a><br></div></span></span>
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<div><br><blockquote type="cite"><div>On Jul 23, 2020, at 9:03 AM, Frank Wimberly <<a href="mailto:wimberly3@gmail.com" target="_blank" rel="noreferrer">wimberly3@gmail.com</a>> wrote:</div><br><div><div dir="auto">p.s. Zeno's Paradox is related to<div dir="auto"><br></div><div dir="auto">1/2 + 1/4 + 1/8 +...</div><div dir="auto"><br></div><div dir="auto">= Sum(1/(2^n)) for n = 1 to infinity</div><div dir="auto"><br></div><div dir="auto">= 1</div><div dir="auto"><br></div><div dir="auto">(Note: Sum(1/(2^n)) for n = 0 to infinity</div><div dir="auto"><br></div><div dir="auto">= 1/(1 - (1/2)) = 2)<br><br><div data-smartmail="gmail_signature" dir="auto">---<br>Frank C. Wimberly<br>140 Calle Ojo Feliz, <br>Santa Fe, NM 87505<br><br>505 670-9918<br>Santa Fe, NM</div></div></div><br><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" target="_blank" rel="noreferrer">wimberly3@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">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><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" rel="noreferrer noreferrer" target="_blank">eric.phillip.charles@gmail.com</a>> wrote:<br></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">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><br></div><div>The solution, while clever, doesn't' work if we assert either of the following: </div><div><br></div><div>A) When the small-square reaches the limit it stops being a square (as it is just a point). </div><div><br></div><div>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><br></div><div>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>Any </i>arbitrary number of squares can be fit inside any other given square. </div><div><br></div><div><br></div><div><div><div dir="ltr"><div dir="ltr"><div><div dir="ltr"><div dir="ltr"><div dir="ltr"><br clear="all">-----------<br><div dir="ltr">Eric P. Charles, Ph.D.<br>Department of Justice - Personnel <span>Psychologist</span></div><div>American University - Adjunct Instructor</div><div></div></div><div dir="ltr"><a href="mailto:echarles@american.edu" rel="noreferrer noreferrer" target="_blank"></a></div></div></div></div></div></div></div><br></div></div><br><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" rel="noreferrer noreferrer" target="_blank">d00d3rs0n@gmail.com</a>> wrote:<br></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"><div>A kid momentarily convinced me of something that must be wrong today. </div><div>We were working on a math problem called Squareland (<a href="https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p" rel="noreferrer noreferrer" target="_blank">https://docs.google.com/presentation/d/1q3qr65tzau8lLGWKxWssXimrSdqwCQnovt0vgHhw7ro/edit#slide=id.p</a>). It basically involved dividing big squares into smaller squares. </div><div>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>I had to admit that I did not know, and that the argument might hold water with more knowledgeable mathematicians. </div><div><br></div><div>The calculus trick of taking the limit of something as it gets infinitely small always seemed like magic to me. </div><div><br></div><div><br clear="all"><div><div dir="ltr"><div dir="ltr">Cody Smith</div></div></div></div></div>
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