[FRIAM] square land math question

Thu Jul 23 11:40:15 EDT 2020

The point is there is no way to partition a square into two squares.

---
Frank C. Wimberly
140 Calle Ojo Feliz,
Santa Fe, NM 87505

505 670-9918
Santa Fe, NM

On Thu, Jul 23, 2020, 9:17 AM Frank Wimberly <wimberly3 at gmail.com> wrote:

> Right.  When its area reaches zero it's not a square.  That is, there is
> only one square then.
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
> 505 670-9918
> Santa Fe, NM
>
> On Thu, Jul 23, 2020, 9:10 AM Edward Angel <angel at cs.unm.edu> wrote:
>
>> 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
>> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>>
>> 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> 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> 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
>>>> <echarles at american.edu>
>>>>
>>>>
>>>> On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <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).
>>>>> 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
>>> Santa Fe, NM 87505
>>> 505 670-9918
>>>
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