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square land math question

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cody dooderson

square land math question

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-2

Re: square land math question

Off the top off my head. As long as the small square isn't of zero area the larger square isn't a square. When the smaller square reaches area zero there is only one square.

What do you think?

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

505 670-9918
Santa Fe, NM

On Tue, Jul 21, 2020, 5:59 PM cody dooderson <[hidden email]> 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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

Eric Charles-2

Re: square land math question

In reply to this post by cody dooderson

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

[hidden email]

On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <[hidden email]> 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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

Frank Wimberly-2

Re: square land math question

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 <[hidden email]> 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
[hidden email]

On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <[hidden email]> 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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918

Frank Wimberly-2

Re: square land math question

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 <[hidden email]> 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 <[hidden email]> 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
[hidden email]

On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <[hidden email]> 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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

--
Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918

Edward Angel

Re: square land math question

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 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) [hidden email]

505-453-4944 (cell) http://www.cs.unm.edu/~angel

On Jul 23, 2020, at 9:03 AM, Frank Wimberly <[hidden email]> 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 <[hidden email]> 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 <[hidden email]> 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
[hidden email]

On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <[hidden email]> 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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

--
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/

Frank Wimberly-2

Re: square land math question

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 <[hidden email]> 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) [hidden email]
505-453-4944 (cell) http://www.cs.unm.edu/~angel

On Jul 23, 2020, at 9:03 AM, Frank Wimberly <[hidden email]> 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 <[hidden email]> 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 <[hidden email]> 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
[hidden email]

On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <[hidden email]> 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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

--
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/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

Frank Wimberly-2

Re: square land math question

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 <[hidden email]> 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 <[hidden email]> 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) [hidden email]
505-453-4944 (cell) http://www.cs.unm.edu/~angel

On Jul 23, 2020, at 9:03 AM, Frank Wimberly <[hidden email]> 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 <[hidden email]> 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 <[hidden email]> 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
[hidden email]

On Tue, Jul 21, 2020 at 7:59 PM cody dooderson <[hidden email]> 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
- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
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/

--
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/

- .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
FRIAM Applied Complexity Group listserv
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gepr

Re: square land math question

I'm surprised EricC didn't say "it all depends on the definition of 'square'". I regard a point as a degenerate square (also a degenerate sphere, cube, etc.). It's the same sort of object as the empty set or an identity like 0 (for +) or 1 (for *).

If all we need for a square is an object with 4 sides of the same length, then a point is clearly a square and the kid is correct. But Cody's also correct. You can't divide a finite square into TWO finite squares. But you can divide it into an infinity of infinitesimal squares. And it's similarly degenerately trivial to divide a point into 2 points.

That's the beauty of math, all you need for the object is for it to satisfy its definition. All that excess meaning y'all are piling onto "square" and its vernacular referent is irrelevant. If you stopped using the word "square" and called it XYZ, then you'd be freer to see its membership.

On 7/23/20 8:40 AM, Frank Wimberly wrote:

> The point is there is no way to partition a square into two squares.
>
>
> On Thu, Jul 23, 2020, 9:17 AM Frank Wimberly <[hidden email] <mailto:[hidden email]>> wrote:
>
> Right. When its area reaches zero it's not a square. That is, there is only one square then.
>
>
> On Thu, Jul 23, 2020, 9:10 AM Edward Angel <[hidden email] <mailto:[hidden email]>> 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.
>

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gepr

Re: square land math question

But a *relevant* question for me is whether or not you can divide an infinitesimal point into an infinity of points? My *guess* is that a point divided an infinite number of times is like a power set and is a greater infinity than the point, itself. But I still haven't read a book I bought awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read many of the English intros and such and a few of the proofs ... but Whew! It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up and just cherry-pick sections that I only kindasorta understand by analogy at this point. At least with math papers I don't feel like such a failure when I give up on reading it ... another way papers are better than books!

On 7/23/20 8:48 AM, uǝlƃ ↙↙↙ wrote:
> And it's similarly degenerately trivial to divide a point into 2 points.

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Frank Wimberly-2

Re: square land math question

points are indivisible. Pardon the tone of authority.

On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙ <[hidden email]> wrote:

But a *relevant* question for me is whether or not you can divide an infinitesimal point into an infinity of points? My *guess* is that a point divided an infinite number of times is like a power set and is a greater infinity than the point, itself. But I still haven't read a book I bought awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read many of the English intros and such and a few of the proofs ... but Whew! It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up and just cherry-pick sections that I only kindasorta understand by analogy at this point. At least with math papers I don't feel like such a failure when I give up on reading it ... another way papers are better than books!

On 7/23/20 8:48 AM, uǝlƃ ↙↙↙ wrote:
> And it's similarly degenerately trivial to divide a point into 2 points.

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thompnickson2

Re: square land math question

So, we’ve finally come to the essential question:

How many points can dance on the head of a point?

Nick

Nicholas Thompson

Emeritus Professor of Ethology and Psychology

Clark University

[hidden email]

https://wordpress.clarku.edu/nthompson/

From: Friam <[hidden email]> On Behalf Of Frank Wimberly
Sent: Thursday, July 23, 2020 10:28 AM
To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
Subject: Re: [FRIAM] square land math question

points are indivisible. Pardon the tone of authority.

On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙ <[hidden email]> wrote:

But a *relevant* question for me is whether or not you can divide an infinitesimal point into an infinity of points? My *guess* is that a point divided an infinite number of times is like a power set and is a greater infinity than the point, itself. But I still haven't read a book I bought awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read many of the English intros and such and a few of the proofs ... but Whew! It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up and just cherry-pick sections that I only kindasorta understand by analogy at this point. At least with math papers I don't feel like such a failure when I give up on reading it ... another way papers are better than books!

On 7/23/20 8:48 AM, uǝlƃ ↙↙↙ wrote:
> And it's similarly degenerately trivial to divide a point into 2 points.

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Santa Fe, NM 87505
505 670-9918

gepr

Re: square land math question

In reply to this post by Frank Wimberly-2

Ha! I can't pardon the tone because the authority is simply wrong. Besides, asserting such things with no justification is not merely a tone.

On 7/23/20 9:28 AM, Frank Wimberly wrote:
> points are indivisible. Pardon the tone of authority.
>
>
> On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙ <[hidden email] <mailto:[hidden email]>> wrote:
>
> But a *relevant* question for me is whether or not you can divide an infinitesimal point into an infinity of points? My *guess* is that a point divided an infinite number of times is like a power set and is a greater infinity than the point, itself. But I still haven't read a book I bought awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read many of the English intros and such and a few of the proofs ... but Whew! It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up and just cherry-pick sections that I only kindasorta understand by analogy at this point. At least with math papers I don't feel like such a failure when I give up on reading it ... another way papers are better than books!
>

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Steve Smith

Re: square land math question

In reply to this post by thompnickson2

So, we’ve finally come to the essential question:

How many points can dance on the head of a point?

We've come full circle again...

https://friam-comic.blogspot.com/2017/10/truthiness-games.html

Frank Wimberly-2

Re: square land math question

In reply to this post by gepr

In R2 a point is an ordered pair. How can (1,1) be decomposed into other points.

I am correct, goshdarnit. When I was about 9 I said that word in the presence of my Southern Baptist grandfather. He said, "Say Goddamit. It means the same thing and it sounds better."

On Thu, Jul 23, 2020 at 10:34 AM uǝlƃ ↙↙↙ <[hidden email]> wrote:

Ha! I can't pardon the tone because the authority is simply wrong. Besides, asserting such things with no justification is not merely a tone.

On 7/23/20 9:28 AM, Frank Wimberly wrote:
> points are indivisible. Pardon the tone of authority.
>
>
> On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙ <[hidden email] <mailto:[hidden email]>> wrote:
>
> But a *relevant* question for me is whether or not you can divide an infinitesimal point into an infinity of points? My *guess* is that a point divided an infinite number of times is like a power set and is a greater infinity than the point, itself. But I still haven't read a book I bought awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read many of the English intros and such and a few of the proofs ... but Whew! It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up and just cherry-pick sections that I only kindasorta understand by analogy at this point. At least with math papers I don't feel like such a failure when I give up on reading it ... another way papers are better than books!
>

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505 670-9918

Prof David West

Re: square land math question

In reply to this post by gepr

maybe of interest:

In the 1630s, when the Roman Catholic Church was confronting Galileo over the Copernican system, the Revisors General of the Jesuit order condemned the doctrine that the continuum is composed of indivisibles. What we now call Cavalieri’s Principle was thought to be dangerous to religion.

Why did the Church get involved in evaluating the “new math” of indivisibles, infinitesimals, and the infinite? The doctrine of indivisibles was on the side of Galileo. Besides opposing the Church about whether the earth went around the sun, Galileo treated matter as made of atoms, which are physical indivisibles. Bonaventura Cavalieri, who pioneered indivisible methods in geometry, was among Galileo’s followers. Furthermore, Catholic theology owes much to Aristotle’s philosophy, and Aristotle, arguing for the potentially infinite divisibility of the continuum, had explicitly ruled out both indivisibles and the actual infinite. So it is no wonder that Jesuit intellectuals opposed using indivisibles in geometry.

davew

On Thu, Jul 23, 2020, at 10:34 AM, uǝlƃ ↙↙↙ wrote:

> Ha! I can't pardon the tone because the authority is simply wrong.
> Besides, asserting such things with no justification is not merely a
> tone.
>
> On 7/23/20 9:28 AM, Frank Wimberly wrote:
> > points are indivisible. Pardon the tone of authority.
> >
> >
> > On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙ <[hidden email] <mailto:[hidden email]>> wrote:
> >
> > But a *relevant* question for me is whether or not you can divide an infinitesimal point into an infinity of points? My *guess* is that a point divided an infinite number of times is like a power set and is a greater infinity than the point, itself. But I still haven't read a book I bought awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read many of the English intros and such and a few of the proofs ... but Whew! It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up and just cherry-pick sections that I only kindasorta understand by analogy at this point. At least with math papers I don't feel like such a failure when I give up on reading it ... another way papers are better than books!
> >
>
> --
> ↙↙↙ uǝlƃ
>
> - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. .
> FRIAM Applied Complexity Group listserv
> Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam
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>

Steve Smith

Re: square land math question

In reply to this post by gepr

Glen -

Ha! I can't pardon the tone because the authority is simply wrong. Besides, asserting such things with no justification is not merely a tone.

Can you unpack that in the light of Euclid's definition of a point, to whose authority I presume Frank was deferring/invoking.

I'm curious if this is a matter of dismissing/rejecting Euclid and his definitions in this matter, or an alternative interpretation of his text?

αʹ. Σημεῖόν ἐστιν, οὗ μέρος οὐθέν. 1. A point is that of which there is no part

I'm always interested in creative alternative interpretations of intention and meaning, but I'm not getting traction on this one (yet?)

- Steve

gepr

Re: square land math question

In reply to this post by Frank Wimberly-2

Well, at least in this post, you *try* to define things such that you'd be right. Although normally considered a rhetorical fallacy, programming into the premises the conclusion you seek is a perfectly reasonable thing to do in math. As long as you actually *do* it ... make the definitions, then your assumed conclusions will be just fine.

On 7/23/20 9:47 AM, Frank Wimberly wrote:

> In R2 a point is an ordered pair. How can (1,1) be decomposed into other points.
>
> I am correct, goshdarnit. When I was about 9 I said that word in the presence of my Southern Baptist grandfather. He said, "Say Goddamit. It means the same thing and it sounds better."
>
> On Thu, Jul 23, 2020 at 10:34 AM uǝlƃ ↙↙↙ <[hidden email] <mailto:[hidden email]>> wrote:
>
> Ha! I can't pardon the tone because the authority is simply wrong. Besides, asserting such things with no justification is not merely a tone.
>
> On 7/23/20 9:28 AM, Frank Wimberly wrote:
> > points are indivisible. Pardon the tone of authority.
> >
> >
> > On Thu, Jul 23, 2020 at 10:12 AM uǝlƃ ↙↙↙ <[hidden email] <mailto:[hidden email]> <mailto:[hidden email] <mailto:[hidden email]>>> wrote:
> >
> > But a *relevant* question for me is whether or not you can divide an infinitesimal point into an infinity of points? My *guess* is that a point divided an infinite number of times is like a power set and is a greater infinity than the point, itself. But I still haven't read a book I bought awhile ago: "Applied Nonstandard Analysis". It's a bit dense. 8^D I've read many of the English intros and such and a few of the proofs ... but Whew! It's almost exactly like Alexandrov's "Combinatorial Topology". I've given up and just cherry-pick sections that I only kindasorta understand by analogy at this point. At least with math papers I don't feel like such a failure when I give up on reading it ... another way papers are better than books!

uǝʃƃ ⊥ glen

Steve Smith

Re: square land math question

In reply to this post by Frank Wimberly-2

On 7/23/20 10:47 AM, Frank Wimberly wrote:
> In R2 a point is an ordered pair. How can (1,1) be decomposed into
> other points.
>
> I am correct, goshdarnit. When I was about 9 I said that word in the
> presence of my Southern Baptist grandfather. He said, "Say Goddamit.
> It means the same thing and it sounds better."

My grandfather taught me "Gauldarnitt!" at age 5 while teaching me how
to lose at checkers.   His daughter (my mother) was not impressed. I
had no clue what the utterance meant, and it probably took me many years
to associate it with the profane analog...   but I *did* know it had
some kind of arcane power.

... ShuckeyDarn

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gepr

Re: square land math question

In reply to this post by Prof David West

Yes! That is of interest. I've been trying to understand a claim I've heard that *actual* infinities are required for full 2nd order math. I.e. potential infinities (which I suppose are necessary for intuitionism and/or program-as-proof) limit the 2nd order operators you can use.

I shouldn't be surprised that the Church got involved. Thanks.

On 7/23/20 9:47 AM, Prof David West wrote:
> maybe of interest:
>
> In the 1630s, when the Roman Catholic Church was confronting Galileo over the Copernican system, the Revisors General of the Jesuit order condemned the doctrine that the continuum is composed of indivisibles. What we now call Cavalieri’s Principle was thought to be dangerous to religion.
>
> Why did the Church get involved in evaluating the “new math” of indivisibles, infinitesimals, and the infinite? The doctrine of indivisibles was on the side of Galileo. Besides opposing the Church about whether the earth went around the sun, Galileo treated matter as made of atoms, which are physical indivisibles. Bonaventura Cavalieri, who pioneered indivisible methods in geometry, was among Galileo’s followers. Furthermore, Catholic theology owes much to Aristotle’s philosophy, and Aristotle, arguing for the potentially infinite divisibility of the continuum, had explicitly ruled out both indivisibles and the actual infinite. So it is no wonder that Jesuit intellectuals opposed using indivisibles in geometry.

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