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Dec 30, 2019, 4:17:02 AM12/30/19

to mathf...@googlegroups.com, Maria Droujkova, wholem...@gmail.com

Hi everybody,

I realized I should write an update on what I've been up to in math.

And I'm curious what others are up to.

Today I was listening to Norman Wildberger's introductory lecture to

Universal Hyperbolic Geometry:

https://www.youtube.com/watch?v=EvP8VtyhzXs

And then the second video lecture on Appolonius and conics:

https://www.youtube.com/watch?v=AjVM5Q-pvjw

And I realized that I should fold up a piece of paper into a cone, and

fix it with a piece of tape, and then try to cut across it and see what

I get. How hard is it to make an ellipse, for example? Well, my

ellipse is not perfect, and I only made one. But I made the cut and then

I set the cone on another piece of paper and drew the ellipse. And then

I set my (slanted) cone on top of it. And I immediately realized that

the tip of the cone seemed to be above one of the foci of the ellipse.

And if I flipped the cone around then the tip would be over the other

foci. Or at least that is what I suspect.

This made me feel (in a good way) very stupid. Why had I never done

this before? And why had I never been taught to do so? And doesn't

this make the foci a completely natural concept, in no way mysterious?

(Of course, it may not be the foci, but it is a starting point for a

very natural investigation.)

I thought I should write to Brad and others who are interested in circle

folding and might also be interested in cone folding.

I also have been wanting to say that I tried out circle folding in a

philosophy class that I taught in the Fall of 2017 at Vilnius Gediminas

Technical University in Lithuania. Rather than teaching the history of

philosophy, I took the initiative to teach my students how to do

philosophy. I had each of them think up their own question and how to

investigate it. Each lesson I taught them different ways of

investigating questions and discussed with them how that might apply to

their particular question. They also read Plato's Republic and I would

relate that to how Plato/Socrates investigated their questions.

In one class, I wanted them to learn about "thinking with their hands".

So I had them do a basic exercise in circle folding. Meanwhile, we had

our discussion. Students liked the circle folding and found it

meditative, contemplative, relaxing. I also realized, and this could be

pursued further, that circle folding occupies a certain part of the

brain, and that leaves discussion to the remaining part of the brain,

with some possible crossover. So I think this is an example where

combining a discussion and an activity can let a different part of the

brain participate in the discussion.

A very practical example of that is taking a pen and paper with me when

I go cycling or jogging. I like to think, but my thinking is relaxed

and scattered and slower paced when I exercise. In the course of an

hour, it is very common that I will get a novel insight from my

unconscious, so I stop and write that down. I'm writing a handbook for

independent thinkers (in the spirit of Lord Baden-Powell's "Scouting for

Boys") so I'm collecting simple activities like that and appreciate any

that we may have.

I am glad to say that I have been making a lot of progress in thinking

about math. I am able to report some very deep things that can be

studied which are at least partly concrete and accessible.

Earlier I had investigated the ways of figuring things out in math:

http://www.ms.lt/sodas/Book/DiscoveryInMathematics

And I need to work on that further. This year I surveyed ways of

figuring things out in physics:

http://www.ms.lt/sodas/Book/PhysicsDiscovery

And that worked out very nicely. The key experiment in physics is

"isolating a system", as with Faraday's pail, where you hang an electric

charge within a pail, and notice how that leads to opposite charges on

the inside and outside of the pail. I also made a poster of the ways of

figuring things out in neuroscience:

http://www.ms.lt/derlius/NeuroscienceDiscovery-Kulikauskas.png

My study of the ways of figuring things out in math led me to look for 4

geometries and 6 transformations between them. Well, there are four

classical Lie algebras/groups and they seem to relate to geometries for

paths, lines, angles, oriented areas. (The latter may be thought of as

"slack", the slack that you need for position and momentum to change

with regard to each other.) I have a mathematical notebook of my

investigations:

http://www.ms.lt/sodas/Book/MathNotebook

One of the main things that I'm trying to understand is why there are 4

classical Lie groups/algebras and so I asked about that at Math Stack

Exchange:

https://math.stackexchange.com/questions/2109581/intuitively-why-are-there-4-classical-lie-groups-algebras

And I'm working on that here:

http://www.ms.lt/sodas/Book/LieAlgebraToGroup

It is very hard for me to understand the literature, and it is taking me

years, but really what I need to know is actually not that hard.

The most important insight is that you can look at the Binomial theorem

in different ways, as I explain here:

http://www.ms.lt/sodas/Book/LieAlgebraToGroup

You can think of 1 3 3 1 as refering to a triangle (and more generally,

a simplex) which has 1 center, 3 vertices, 3 edges, 1 face. Or you can

think of it as a cube on its side, with 1 point leftmost, 3 points left,

3 points right, 1 point rightmost. The mind can think these numbers in

these entirely different ways, especially as it relates to choices. In

the case of the triangle we are choosing (yes or no) for each of three

vertices, and this gives 2^3=8 possibilities. In the case of the cube,

we are choosing (left or right) three times, and this defines the 2^3=8

vertices of the cube. The results are absolutely different as regards

their symmetry or lack of it. In the case of the cube, the choice

between left and right is symmetric (it's just a semantic difference).

But in the case of the triangle, the choice between vertex or no vertex

is not symmetric (no vertices and all vertices are different,

syntactically). Which is to say, the difference between 0 and 1 is just

a matter of names, but the difference between a blank space and a mark

is more than that. Or asking a child, do you want a banana or a hot

dog? is different than asking them, do you want to eat or not?

especially if you want them to eat. In physics, this cognitive

distinction lurks in the math (whether you are modeling "simplexes" or

"coordinate systems") and thus whether you are modeling the observer or

the observed. In nature, there are no gaps, and the electron just

chooses between left and right. But an experimenter can carve up time

as they like and have episodes where "nothing was observed". These are

two extremes which Kirby distinguishes with his Martian tetrahedrons and

Earth cubes (coordinate systems). But there are also two intermediate

cases (cross-polytopes and hypercubes). These four choice frameworks are

the three families of infinite polytopes (simplexes, cross-polytopes,

hypercubes) plus the coordinate systems (the cube on its side).

I have a rather decent understanding of how this relates with Lie

algebras (which are like addition tables for vector systems) but I am

struggling to learn how this gets expressed in Lie groups (which

multiply rotations in various universes). But I can share some

encouraging insights that relate to math which is perhaps accessible.

The goal is to understand intuitively a Dynkin diagram, which in my

case, is just a chain of dots. A single dot is the basic building

block, which in its various forms is sl(2), su(2), SU(2), SL(2). You

could say, metaphorically, it's the hydrogen atom of the periodic

table. Then you can have two dots linked, which is sl(3), su(3), SU(3),

SL(3), which metaphorically would be the helium atom. And then if I

could understand that, then next would be n dots linked in a chain,

which is sl(n), su(n), SU(n), SL(n). That's the plain vanilla

n-dimensional chain. Geometrically, in the Lie algebra, you can imagine

each dimension separated by 120 degrees, and the crucial thing is that

if you go from dimension A by 120 degrees to B, and then by 120 degrees

to C, then if you do it right then you get A and C separated by 90

degrees, which means they are independent, and the chain grows onward,

linking D, E, F, etc.

Now, the plain vanilla chain is modeling "counting", which has a natural

duality, counting forwards and backwards. But if you connect the

forward and backward counting, then you get three ways of linking them

together. Think of the time line in history, the years BC and the years

AD (Anno Domini). The options are:

* you could have a year 0 so that you go ...-3, -2, -1, 0, 1, 2, 3...

(cutting and gluing)

* you could make -1 and 1 be the same year so that you have ...-3, -2,

-1=1, 2, 3... (cutting and fusing)

* you could go straight from -1 to 1 so that you have ...-3, -2, -1, 1,

2, 3... (folding)

This is exactly why there are four classical infinite families of Lie

algebras (if you look at their root systems). Basically they all have

the same chain (counting) and just differ by a doodad at the end (which

lets you turn around the end of the chain and link back). The plain

vanilla case relates to the complex numbers and has their duality. The

cutting cases basically halve the duality to think in terms of real

numbers. And the folding case doubles the duality to have you think in

terms of quaternions.

I don't yet know what that means as regards to Lie groups and the ways

rotations are linked in various dimensions. However, I want to share a

little gold mine that this leads to, which is SL(2), that "hydrogen

atom" that I need to understand.

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation

Basically, SL(2) ("the Mobius transformations") is simply the ways of

composing all the functions that look like f(z) = (az + b)/(cz + d). z

is just a variable, and so the function f is defined by the constants a,

b, c, d, which can be organized as a 2x2 matrix. The major hitch is

that these four constants are all complex numbers. So we are dealing

with 8 real numbers. It is daunting. But if we can intuit this, then

it is like intuiting the hydrogen atom, and all the other atoms aren't

much harder.

One major realization that encourages me is that these Mobius

transformations are exactly what I need to explain six transformations

that came up in my study of moods. Last year I gave a talk on "The

Geometry of Moods" at the World Congress of Philosophy, in Beijing, China:

http://www.ms.lt/sodas/Book/20180815AGeometryOfMoods

The main idea is that our emotions are evoked by our expectations and

how they relate to the boundary of our self and the world. For example,

if your bike is stolen, then I may be surprised, whereas if my bike is

stolen, then I will be sad. I did a study of 39 short, classic Chinese

poems, and showed evidence for 6 geometric transformations (reflection,

shear, rotation, dilation, squeeze, translation) that play with this

boundary. Well, all of these transformations come up in the group

SL(2), see the sections on subgroups and classification:

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Classification

We get subgroups of reflections (circular), rotations (elliptic), shear

(parabolic), dilations (hyperbolic). And then furthermore within an

element, we get squeeze balancing e^t and e^(-t), and translation is

perhaps from the input on the additive, Lie algebra side. Basically, it

seems exactly where I need to look for the answer to the related

question I posed at Math Stack Exchange:

https://math.stackexchange.com/questions/1953312/is-this-set-of-6-transformations-fundamental-to-geometry

Another realization in working through the nitty gritty of the Lie group

SL(2) is that it is variously expressing the imaginary numbers i and -i

in a variety of dualities. Note that -1 has two square roots of equal

value and it's unfortunate that our notation suggests that one (i) is

somehow more basic than the other (-i) when in fact there is no way to

figure which is which until we arbitrarily choose one and call it "i".

Really, complex analysis should be based on the idea that there is not a

single imaginary number but that they are a pair of choices, conjugates,

as with "free will", whereas real numbers don't have that choice and are

"fate". Now there seem to be four ways in which we can express i and

its duality (with respect to the identity). Consider the following (I

write each out plus-or-minus):

(i 0) (-i 0)

(0 i ) (0 -i)

(1 0) (-1 0)

(0 -1) (0 1)

(0 1) (0 -1)

(1 0) (-1 0)

(0 -i) (0 i)

(i 0) (-i 0)

Focusing on the left hand column, the latter three matrices are called

the Pauli matrices. If you multiply two of the four, then you get the

product of the other two. (I have to check that but that sounds right).

Well, in particular, that leads to a three-cycle of matrices, much like

the quaternions, where ij=k, jk=i, ki=j. And those matrices (if you

multiply by i) are the generators for the Lie algebra su(2) where you

then allow for real scalar coefficients. But if you allow for complex

coefficients, then the Lie algebra changes and you no longer have a

three-cycle, but you have a different kind of trinity that looks like this:

(1 0) = H

(0 -1)

(0 1) = X

(0 0)

(0 0) = Y

(1 0)

Here X and Y are "external" opposites, whereas H has opposites built

"internally", inside of itself.

This is very meaningful for me, theologically, because these happen to

model two structures that I discuss in another talk that I gave in

Beijing, "Imagining God's State of Mind as a Question: Is God Necessary?"

http://www.ms.lt/sodas/Book/20180819GodsStateOfMind

The X,Y,H model the following: Imagine God X asking whether God would

exist if God did not exist? and imagine God pulling away to see what

happened. And then imagine a little Godlet Y appearing. How would they

know they are the same? They would if there was a Lens H that said so.

In other words, God X who understands, and Godlet Y who comes to

understand, both understand the same Lens H, and so they are all the

same. But that's how this looks from X's point of view. If you also

consider Y's point of view, and H's point of view, then those three

points of view form a 3-cycle. (And H's point of view is modeled by

those 4 geometries and 6 transformations which have the structure of the

Ten Commandments.) Which is to say, the complex Lie algebra sl(2) models

God with X,Y,H, and the real Lie algebra su(2) models God's

perspectives, the 3-cycle. So that is very promising for my philosophy.

If that's not enough... SL(2) is key to Einstein's theory of

relativity. And also, the Standard Model of particle physics is based

on the product U(1) x SU(2) x SU(3) where U(1) describes

electromagnetism, SU(2) describes the weak force, SU(3) describes the

strong force. In experimental psychology, there is a very potent theory

by Norman H. Anderson that shows how cognition uses Additive,

Multiplicative, and Averaging models, and his student Shu-Hong Zhu in

his thesis demonstrated a sophisticated case of that form (ax + b)/(cx +

d).

The cognitive side of mathematics is not a normal thing to think about.

And there's a vagueness in what I am investigating that is outside

normal mathematical discourse. Yet I want to show that it can be

fruitful and yield deep insights into very concrete mathematical

structures that are of central importance to abstract math but also

model human experience. I think that is the message that comes up in

investigations by Kirby, Brad, Maria and others here.

Andrius

Andrius Kulikauskas

m...@ms.lt

+370 607 27 665

I realized I should write an update on what I've been up to in math.

And I'm curious what others are up to.

Today I was listening to Norman Wildberger's introductory lecture to

Universal Hyperbolic Geometry:

https://www.youtube.com/watch?v=EvP8VtyhzXs

And then the second video lecture on Appolonius and conics:

https://www.youtube.com/watch?v=AjVM5Q-pvjw

And I realized that I should fold up a piece of paper into a cone, and

fix it with a piece of tape, and then try to cut across it and see what

I get. How hard is it to make an ellipse, for example? Well, my

ellipse is not perfect, and I only made one. But I made the cut and then

I set the cone on another piece of paper and drew the ellipse. And then

I set my (slanted) cone on top of it. And I immediately realized that

the tip of the cone seemed to be above one of the foci of the ellipse.

And if I flipped the cone around then the tip would be over the other

foci. Or at least that is what I suspect.

This made me feel (in a good way) very stupid. Why had I never done

this before? And why had I never been taught to do so? And doesn't

this make the foci a completely natural concept, in no way mysterious?

(Of course, it may not be the foci, but it is a starting point for a

very natural investigation.)

I thought I should write to Brad and others who are interested in circle

folding and might also be interested in cone folding.

I also have been wanting to say that I tried out circle folding in a

philosophy class that I taught in the Fall of 2017 at Vilnius Gediminas

Technical University in Lithuania. Rather than teaching the history of

philosophy, I took the initiative to teach my students how to do

philosophy. I had each of them think up their own question and how to

investigate it. Each lesson I taught them different ways of

investigating questions and discussed with them how that might apply to

their particular question. They also read Plato's Republic and I would

relate that to how Plato/Socrates investigated their questions.

In one class, I wanted them to learn about "thinking with their hands".

So I had them do a basic exercise in circle folding. Meanwhile, we had

our discussion. Students liked the circle folding and found it

meditative, contemplative, relaxing. I also realized, and this could be

pursued further, that circle folding occupies a certain part of the

brain, and that leaves discussion to the remaining part of the brain,

with some possible crossover. So I think this is an example where

combining a discussion and an activity can let a different part of the

brain participate in the discussion.

A very practical example of that is taking a pen and paper with me when

I go cycling or jogging. I like to think, but my thinking is relaxed

and scattered and slower paced when I exercise. In the course of an

hour, it is very common that I will get a novel insight from my

unconscious, so I stop and write that down. I'm writing a handbook for

independent thinkers (in the spirit of Lord Baden-Powell's "Scouting for

Boys") so I'm collecting simple activities like that and appreciate any

that we may have.

I am glad to say that I have been making a lot of progress in thinking

about math. I am able to report some very deep things that can be

studied which are at least partly concrete and accessible.

Earlier I had investigated the ways of figuring things out in math:

http://www.ms.lt/sodas/Book/DiscoveryInMathematics

And I need to work on that further. This year I surveyed ways of

figuring things out in physics:

http://www.ms.lt/sodas/Book/PhysicsDiscovery

And that worked out very nicely. The key experiment in physics is

"isolating a system", as with Faraday's pail, where you hang an electric

charge within a pail, and notice how that leads to opposite charges on

the inside and outside of the pail. I also made a poster of the ways of

figuring things out in neuroscience:

http://www.ms.lt/derlius/NeuroscienceDiscovery-Kulikauskas.png

My study of the ways of figuring things out in math led me to look for 4

geometries and 6 transformations between them. Well, there are four

classical Lie algebras/groups and they seem to relate to geometries for

paths, lines, angles, oriented areas. (The latter may be thought of as

"slack", the slack that you need for position and momentum to change

with regard to each other.) I have a mathematical notebook of my

investigations:

http://www.ms.lt/sodas/Book/MathNotebook

One of the main things that I'm trying to understand is why there are 4

classical Lie groups/algebras and so I asked about that at Math Stack

Exchange:

https://math.stackexchange.com/questions/2109581/intuitively-why-are-there-4-classical-lie-groups-algebras

And I'm working on that here:

http://www.ms.lt/sodas/Book/LieAlgebraToGroup

It is very hard for me to understand the literature, and it is taking me

years, but really what I need to know is actually not that hard.

The most important insight is that you can look at the Binomial theorem

in different ways, as I explain here:

http://www.ms.lt/sodas/Book/LieAlgebraToGroup

You can think of 1 3 3 1 as refering to a triangle (and more generally,

a simplex) which has 1 center, 3 vertices, 3 edges, 1 face. Or you can

think of it as a cube on its side, with 1 point leftmost, 3 points left,

3 points right, 1 point rightmost. The mind can think these numbers in

these entirely different ways, especially as it relates to choices. In

the case of the triangle we are choosing (yes or no) for each of three

vertices, and this gives 2^3=8 possibilities. In the case of the cube,

we are choosing (left or right) three times, and this defines the 2^3=8

vertices of the cube. The results are absolutely different as regards

their symmetry or lack of it. In the case of the cube, the choice

between left and right is symmetric (it's just a semantic difference).

But in the case of the triangle, the choice between vertex or no vertex

is not symmetric (no vertices and all vertices are different,

syntactically). Which is to say, the difference between 0 and 1 is just

a matter of names, but the difference between a blank space and a mark

is more than that. Or asking a child, do you want a banana or a hot

dog? is different than asking them, do you want to eat or not?

especially if you want them to eat. In physics, this cognitive

distinction lurks in the math (whether you are modeling "simplexes" or

"coordinate systems") and thus whether you are modeling the observer or

the observed. In nature, there are no gaps, and the electron just

chooses between left and right. But an experimenter can carve up time

as they like and have episodes where "nothing was observed". These are

two extremes which Kirby distinguishes with his Martian tetrahedrons and

Earth cubes (coordinate systems). But there are also two intermediate

cases (cross-polytopes and hypercubes). These four choice frameworks are

the three families of infinite polytopes (simplexes, cross-polytopes,

hypercubes) plus the coordinate systems (the cube on its side).

I have a rather decent understanding of how this relates with Lie

algebras (which are like addition tables for vector systems) but I am

struggling to learn how this gets expressed in Lie groups (which

multiply rotations in various universes). But I can share some

encouraging insights that relate to math which is perhaps accessible.

The goal is to understand intuitively a Dynkin diagram, which in my

case, is just a chain of dots. A single dot is the basic building

block, which in its various forms is sl(2), su(2), SU(2), SL(2). You

could say, metaphorically, it's the hydrogen atom of the periodic

table. Then you can have two dots linked, which is sl(3), su(3), SU(3),

SL(3), which metaphorically would be the helium atom. And then if I

could understand that, then next would be n dots linked in a chain,

which is sl(n), su(n), SU(n), SL(n). That's the plain vanilla

n-dimensional chain. Geometrically, in the Lie algebra, you can imagine

each dimension separated by 120 degrees, and the crucial thing is that

if you go from dimension A by 120 degrees to B, and then by 120 degrees

to C, then if you do it right then you get A and C separated by 90

degrees, which means they are independent, and the chain grows onward,

linking D, E, F, etc.

Now, the plain vanilla chain is modeling "counting", which has a natural

duality, counting forwards and backwards. But if you connect the

forward and backward counting, then you get three ways of linking them

together. Think of the time line in history, the years BC and the years

AD (Anno Domini). The options are:

* you could have a year 0 so that you go ...-3, -2, -1, 0, 1, 2, 3...

(cutting and gluing)

* you could make -1 and 1 be the same year so that you have ...-3, -2,

-1=1, 2, 3... (cutting and fusing)

* you could go straight from -1 to 1 so that you have ...-3, -2, -1, 1,

2, 3... (folding)

This is exactly why there are four classical infinite families of Lie

algebras (if you look at their root systems). Basically they all have

the same chain (counting) and just differ by a doodad at the end (which

lets you turn around the end of the chain and link back). The plain

vanilla case relates to the complex numbers and has their duality. The

cutting cases basically halve the duality to think in terms of real

numbers. And the folding case doubles the duality to have you think in

terms of quaternions.

I don't yet know what that means as regards to Lie groups and the ways

rotations are linked in various dimensions. However, I want to share a

little gold mine that this leads to, which is SL(2), that "hydrogen

atom" that I need to understand.

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation

Basically, SL(2) ("the Mobius transformations") is simply the ways of

composing all the functions that look like f(z) = (az + b)/(cz + d). z

is just a variable, and so the function f is defined by the constants a,

b, c, d, which can be organized as a 2x2 matrix. The major hitch is

that these four constants are all complex numbers. So we are dealing

with 8 real numbers. It is daunting. But if we can intuit this, then

it is like intuiting the hydrogen atom, and all the other atoms aren't

much harder.

One major realization that encourages me is that these Mobius

transformations are exactly what I need to explain six transformations

that came up in my study of moods. Last year I gave a talk on "The

Geometry of Moods" at the World Congress of Philosophy, in Beijing, China:

http://www.ms.lt/sodas/Book/20180815AGeometryOfMoods

The main idea is that our emotions are evoked by our expectations and

how they relate to the boundary of our self and the world. For example,

if your bike is stolen, then I may be surprised, whereas if my bike is

stolen, then I will be sad. I did a study of 39 short, classic Chinese

poems, and showed evidence for 6 geometric transformations (reflection,

shear, rotation, dilation, squeeze, translation) that play with this

boundary. Well, all of these transformations come up in the group

SL(2), see the sections on subgroups and classification:

https://en.wikipedia.org/wiki/M%C3%B6bius_transformation#Classification

We get subgroups of reflections (circular), rotations (elliptic), shear

(parabolic), dilations (hyperbolic). And then furthermore within an

element, we get squeeze balancing e^t and e^(-t), and translation is

perhaps from the input on the additive, Lie algebra side. Basically, it

seems exactly where I need to look for the answer to the related

question I posed at Math Stack Exchange:

https://math.stackexchange.com/questions/1953312/is-this-set-of-6-transformations-fundamental-to-geometry

Another realization in working through the nitty gritty of the Lie group

SL(2) is that it is variously expressing the imaginary numbers i and -i

in a variety of dualities. Note that -1 has two square roots of equal

value and it's unfortunate that our notation suggests that one (i) is

somehow more basic than the other (-i) when in fact there is no way to

figure which is which until we arbitrarily choose one and call it "i".

Really, complex analysis should be based on the idea that there is not a

single imaginary number but that they are a pair of choices, conjugates,

as with "free will", whereas real numbers don't have that choice and are

"fate". Now there seem to be four ways in which we can express i and

its duality (with respect to the identity). Consider the following (I

write each out plus-or-minus):

(i 0) (-i 0)

(0 i ) (0 -i)

(1 0) (-1 0)

(0 -1) (0 1)

(0 1) (0 -1)

(1 0) (-1 0)

(0 -i) (0 i)

(i 0) (-i 0)

Focusing on the left hand column, the latter three matrices are called

the Pauli matrices. If you multiply two of the four, then you get the

product of the other two. (I have to check that but that sounds right).

Well, in particular, that leads to a three-cycle of matrices, much like

the quaternions, where ij=k, jk=i, ki=j. And those matrices (if you

multiply by i) are the generators for the Lie algebra su(2) where you

then allow for real scalar coefficients. But if you allow for complex

coefficients, then the Lie algebra changes and you no longer have a

three-cycle, but you have a different kind of trinity that looks like this:

(1 0) = H

(0 -1)

(0 1) = X

(0 0)

(0 0) = Y

(1 0)

Here X and Y are "external" opposites, whereas H has opposites built

"internally", inside of itself.

This is very meaningful for me, theologically, because these happen to

model two structures that I discuss in another talk that I gave in

Beijing, "Imagining God's State of Mind as a Question: Is God Necessary?"

http://www.ms.lt/sodas/Book/20180819GodsStateOfMind

The X,Y,H model the following: Imagine God X asking whether God would

exist if God did not exist? and imagine God pulling away to see what

happened. And then imagine a little Godlet Y appearing. How would they

know they are the same? They would if there was a Lens H that said so.

In other words, God X who understands, and Godlet Y who comes to

understand, both understand the same Lens H, and so they are all the

same. But that's how this looks from X's point of view. If you also

consider Y's point of view, and H's point of view, then those three

points of view form a 3-cycle. (And H's point of view is modeled by

those 4 geometries and 6 transformations which have the structure of the

Ten Commandments.) Which is to say, the complex Lie algebra sl(2) models

God with X,Y,H, and the real Lie algebra su(2) models God's

perspectives, the 3-cycle. So that is very promising for my philosophy.

If that's not enough... SL(2) is key to Einstein's theory of

relativity. And also, the Standard Model of particle physics is based

on the product U(1) x SU(2) x SU(3) where U(1) describes

electromagnetism, SU(2) describes the weak force, SU(3) describes the

strong force. In experimental psychology, there is a very potent theory

by Norman H. Anderson that shows how cognition uses Additive,

Multiplicative, and Averaging models, and his student Shu-Hong Zhu in

his thesis demonstrated a sophisticated case of that form (ax + b)/(cx +

d).

The cognitive side of mathematics is not a normal thing to think about.

And there's a vagueness in what I am investigating that is outside

normal mathematical discourse. Yet I want to show that it can be

fruitful and yield deep insights into very concrete mathematical

structures that are of central importance to abstract math but also

model human experience. I think that is the message that comes up in

investigations by Kirby, Brad, Maria and others here.

Andrius

Andrius Kulikauskas

m...@ms.lt

+370 607 27 665

Jan 3, 2020, 1:30:02 PM1/3/20

to mathf...@googlegroups.com

Joseph Austin

> On Dec 30, 2019, at 4:17 AM, Andrius Kulikauskas <m...@ms.lt> wrote:

>

> Hi everybody,

> You received this message because you are subscribed to the Google Groups "MathFuture" group.

> To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.

> To view this discussion on the web visit https://groups.google.com/d/msgid/mathfuture/c9763e5f-dbf9-686b-ea9e-baca2b342efb%40ms.lt.

Jan 3, 2020, 3:29:21 PM1/3/20

to mathf...@googlegroups.com

Andrius,

Have you looked at Geometric Algebra [Hestenes, et. al.] particularly in regard to Tensor Product and related concepts? He also has geometric interpretations of Pauli matrices and most of the constructs of modern physics.

I've also been looking for an intuitive intro to Lie Groups, particularly as they seem to be prevalent in Modern Physics. I have the uneasy feeling that the way we teach math makes it much to difficult to understand the basic concepts of modern physics, and perhaps the emerging field of quantum computing.

I'm looking for an approach built on "first principles" [axioms?] instead inventing special matrices of complex numbers because of their "multiplication" properties. Which is to say, I would say if "reality" obeys the axioms of system X and Pauli matrices obey the axioms of system X, I would rather understand Pauli matrices as a consequence of the reality described by X than understand the reality in terms of Pauli matrices, if that makes any sense.

As for groups, I understand finite groups as permutations or, geometrically, rearrangements. So what happens if we add continuity and continuous differentiability or "smooth" reformation?

>

>> On Dec 30, 2019, at 4:17 AM, Andrius Kulikauskas <m...@ms.lt> wrote:

>>

>> Hi everybody,

>>

>> I

Have you looked at Geometric Algebra [Hestenes, et. al.] particularly in regard to Tensor Product and related concepts? He also has geometric interpretations of Pauli matrices and most of the constructs of modern physics.

I've also been looking for an intuitive intro to Lie Groups, particularly as they seem to be prevalent in Modern Physics. I have the uneasy feeling that the way we teach math makes it much to difficult to understand the basic concepts of modern physics, and perhaps the emerging field of quantum computing.

I'm looking for an approach built on "first principles" [axioms?] instead inventing special matrices of complex numbers because of their "multiplication" properties. Which is to say, I would say if "reality" obeys the axioms of system X and Pauli matrices obey the axioms of system X, I would rather understand Pauli matrices as a consequence of the reality described by X than understand the reality in terms of Pauli matrices, if that makes any sense.

As for groups, I understand finite groups as permutations or, geometrically, rearrangements. So what happens if we add continuity and continuous differentiability or "smooth" reformation?

>

>> On Dec 30, 2019, at 4:17 AM, Andrius Kulikauskas <m...@ms.lt> wrote:

>>

>> Hi everybody,

>>

>> I

Jan 3, 2020, 4:32:24 PM1/3/20

to mathf...@googlegroups.com

Dear Joseph,

Thank you for your reply! I also thank Brad for his reply, which I

didn't understand completely, but which I hope he might share here and

we discuss further.

So far the most concrete book that I could find about Lie theory is

"Naive Lie Theory" by John Stillwell:

https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387782141

It still is based on matrices but, truth be told, matrices are much more

concrete than the abstract algebra in most Lie theory books.

Concretely, the entry a_ij of a matrix A can be thought of as the way to

go from i to j in one step. Then matrix multiplication is the way of

generating all paths from i to j. In other words, every finite automaton

can be modeled in terms of matrix multiplication.

A very helpful book I recommend even more is "Visual Complex Analysis"

by Tristan Needham.

https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469

It has a lot about the classification of the Mobius transformations.

Here's a short, beautiful video illustrating the Mobius transformations:

https://www.youtube.com/watch?v=JX3VmDgiFnY

I think those are all real, though, and I need to understand how it

works in complex variables. Although the two real dimensions do combine

into a complex dimension. Now imagine combining two complex inputs!

(But I don't know what I'm talking about...) The video is a nice way to

imagine actions, and combining actions, which is what groups are all about.

I had previously looked a bit at Hestenes and Geometric Algebra, and

Clifford Algebras, and should go back to that. For my philosophy, a

crucial thing to understand is Bott periodicity, which is an eightfold

cycle that is related to clock-shifts in Clifford Algebras.

https://en.wikipedia.org/wiki/Bott_periodicity_theorem

https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras

By coincidence, surely it must relate to the eight-cycle of divisions of

everything that I describe in these papers, which is why I'm interested:

http://www.ms.lt/sodas/Book/20170929TimeSpaceDecisionMaking

http://www.ms.lt/sodas/Book/20171011DisembodyingMind

For me, it's all about the way of adding a perspective (or two or three

new perspectives) to an existing system of perspectives. Our state of

mind is modeled by a self-defining system which has X perspectives,

where 0 <= X <= 7. Being engrossed ("unconsciously stepping in"), being

aware ("consciously stepping out"), and controlling the relationship

between the two modes ("being conscious"), is adding one, two or three

perspectives, respectively. Having 8 perspectives at the same time

makes the system contradictory (as in the logical square saying "all are

good AND all are bad") and the system is empty and collapses to 0

perspectives. Well, in the case of Bott periodicity, it's about adding

a map to a sphere, which I imagine must be related to adding a perspective.

Understanding tensors is a prerequisite for that. There is a three

volume set of books that Vilnius University purchased at my request

which is very good for this. They are translated from Russian, written

by Dubrovin, Fomenko, Novikov. "Modern Geometry - Methods and

Applications". Volume I has a deep discussion of tensors.

http://www.ms.lt/derlius/ModernGeometry-I-DubrovinFomenkoNovikov.pdf

Volume II has sections on Lie theory.

http://www.ms.lt/derlius/ModernGeometry-II-LieGroups-DubrovinFomenkoNovikov.pdf

Volume III has a section on Bott periodicity.

http://www.ms.lt/derlius/ModernGeometryIII-BottPeriodicity.pdf

Groups are systems of actions. So we have to think in terms of verbs

(actions) rather than nouns (states). An example of a group is a

12-hour clock, where +12 is the +0. The actions are additions: +0, +1,

+2, ..., +11. The actions obey the associative law +A + (+B+C) =

+(+A+B) + C. And there is an idenity: +A +0 = +A. And there are

inverses, which is to say, you can undo: +5 +7 = +12 = +0 so +7 = -+5.

And it is closed, which means that combining actions gives you actions.

So this can be continuous, as in the case of rotations. You could have

rotations +X where 0 <= X < 2 pi. And they form a group with identity

+0, with inverses, and so on. This is the simplest nontrivial example

of a Lie group. Except that it is commutative so kind of trivial. In

three real dimensions, rotations are not commutative. But it turns out

that it is the complex dimensions that are most basic for the theory.

In the Lie theory that I'm interested in, especially the "compact" Lie

groups, which don't go off into infinity, but are balled up, it seems

that every action is just a rotation. But those rotations may be quite

bizarre, in multiple dimensions, in the complex numbers, in the

quaternions. If we can understand the abstract theory, then we should

be able to imagine those rotations, and vice versa. The key case here

to be able to think about is SU(2).

A very relevant video by Norman Wildberger, perhaps exactly in the

direction you want, is the following:

https://www.youtube.com/watch?v=f68eYuDCsjw&list=PLC37ED4C488778E7E&index=22&t=0s

I'm making notes about his Universal Hyperbolic Geometry:

http://www.ms.lt/sodas/Book/UniversalHyperbolicGeometry

This is all a very challenging subject. But it seems to be precisely

what my philosophy needs and says.

Thank you for your reply! I also thank Brad for his reply, which I

didn't understand completely, but which I hope he might share here and

we discuss further.

So far the most concrete book that I could find about Lie theory is

"Naive Lie Theory" by John Stillwell:

https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387782141

It still is based on matrices but, truth be told, matrices are much more

concrete than the abstract algebra in most Lie theory books.

Concretely, the entry a_ij of a matrix A can be thought of as the way to

go from i to j in one step. Then matrix multiplication is the way of

generating all paths from i to j. In other words, every finite automaton

can be modeled in terms of matrix multiplication.

A very helpful book I recommend even more is "Visual Complex Analysis"

by Tristan Needham.

https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469

It has a lot about the classification of the Mobius transformations.

Here's a short, beautiful video illustrating the Mobius transformations:

https://www.youtube.com/watch?v=JX3VmDgiFnY

I think those are all real, though, and I need to understand how it

works in complex variables. Although the two real dimensions do combine

into a complex dimension. Now imagine combining two complex inputs!

(But I don't know what I'm talking about...) The video is a nice way to

imagine actions, and combining actions, which is what groups are all about.

I had previously looked a bit at Hestenes and Geometric Algebra, and

Clifford Algebras, and should go back to that. For my philosophy, a

crucial thing to understand is Bott periodicity, which is an eightfold

cycle that is related to clock-shifts in Clifford Algebras.

https://en.wikipedia.org/wiki/Bott_periodicity_theorem

https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras

By coincidence, surely it must relate to the eight-cycle of divisions of

everything that I describe in these papers, which is why I'm interested:

http://www.ms.lt/sodas/Book/20170929TimeSpaceDecisionMaking

http://www.ms.lt/sodas/Book/20171011DisembodyingMind

For me, it's all about the way of adding a perspective (or two or three

new perspectives) to an existing system of perspectives. Our state of

mind is modeled by a self-defining system which has X perspectives,

where 0 <= X <= 7. Being engrossed ("unconsciously stepping in"), being

aware ("consciously stepping out"), and controlling the relationship

between the two modes ("being conscious"), is adding one, two or three

perspectives, respectively. Having 8 perspectives at the same time

makes the system contradictory (as in the logical square saying "all are

good AND all are bad") and the system is empty and collapses to 0

perspectives. Well, in the case of Bott periodicity, it's about adding

a map to a sphere, which I imagine must be related to adding a perspective.

Understanding tensors is a prerequisite for that. There is a three

volume set of books that Vilnius University purchased at my request

which is very good for this. They are translated from Russian, written

by Dubrovin, Fomenko, Novikov. "Modern Geometry - Methods and

Applications". Volume I has a deep discussion of tensors.

http://www.ms.lt/derlius/ModernGeometry-I-DubrovinFomenkoNovikov.pdf

Volume II has sections on Lie theory.

http://www.ms.lt/derlius/ModernGeometry-II-LieGroups-DubrovinFomenkoNovikov.pdf

Volume III has a section on Bott periodicity.

http://www.ms.lt/derlius/ModernGeometryIII-BottPeriodicity.pdf

Groups are systems of actions. So we have to think in terms of verbs

(actions) rather than nouns (states). An example of a group is a

12-hour clock, where +12 is the +0. The actions are additions: +0, +1,

+2, ..., +11. The actions obey the associative law +A + (+B+C) =

+(+A+B) + C. And there is an idenity: +A +0 = +A. And there are

inverses, which is to say, you can undo: +5 +7 = +12 = +0 so +7 = -+5.

And it is closed, which means that combining actions gives you actions.

So this can be continuous, as in the case of rotations. You could have

rotations +X where 0 <= X < 2 pi. And they form a group with identity

+0, with inverses, and so on. This is the simplest nontrivial example

of a Lie group. Except that it is commutative so kind of trivial. In

three real dimensions, rotations are not commutative. But it turns out

that it is the complex dimensions that are most basic for the theory.

In the Lie theory that I'm interested in, especially the "compact" Lie

groups, which don't go off into infinity, but are balled up, it seems

that every action is just a rotation. But those rotations may be quite

bizarre, in multiple dimensions, in the complex numbers, in the

quaternions. If we can understand the abstract theory, then we should

be able to imagine those rotations, and vice versa. The key case here

to be able to think about is SU(2).

A very relevant video by Norman Wildberger, perhaps exactly in the

direction you want, is the following:

https://www.youtube.com/watch?v=f68eYuDCsjw&list=PLC37ED4C488778E7E&index=22&t=0s

I'm making notes about his Universal Hyperbolic Geometry:

http://www.ms.lt/sodas/Book/UniversalHyperbolicGeometry

This is all a very challenging subject. But it seems to be precisely

what my philosophy needs and says.

Jan 3, 2020, 10:16:00 PM1/3/20

to mathf...@googlegroups.com

Andrius and Joseph, see if any of this makes sense to you.

The circle I fold is a compression of the sphere. It is a transformation from one spherical form to another form of the same unity, both 3-D and dynamic.

The first fold of the circle in half generates 4 spherical paths of movement. There are 2 halves, each rotates 360 degrees in both directions making 4 individual and congruent spherical paths around the creased axis. They can be differentiated through intervals of individual rotation. There are 1440 degrees of movement in a single fold we call a 180-degree movement in folding the half-circle.

Because the circle has 2 sides there are then 8 cycles of movement, 2 sets of 2, 2 times. This is a reflection of that single fold where 2 imaginary points touch folding a diameter (2 more points). Those 4 points reveal 6 right-angle triangles unless the touching points are exactly opposite each other, then there are 8 right-angle triangles, thus the square relationship which in circle form is the square grid used in the Mobius video, and is origin for the Cartesian grid.

Using the crease in the first fold and the folded circumference, bring a part of each from opposite ends of the curved edge and creased edge together; straight lines and curved lines can be congruent in space, If you twist one to the other 180 degrees and tape them together you get a Mobius surface form one fold of the circle in half. (2 surfaces become one, and the same is hidden between the 2 halves) This is the same transformation demonstrated by mapping in the Moebius animation based on the perpendicular division using a right-angle function that has become fixed in 2-D imagery. As stated mapping must get off of the flat plane by moving to another dimension making it 3-D. Centuries ago we decided that movement and space were not allowed in the 2-D world, now we are having trouble trying to fit them into our conditioned 2-D and abstracted mindset.

My question is if starting with 3-D would our maps look different, or would they look similar to the 2-D we have developed, but our understanding would be different. As Wildberger stated it is all about the point and line. Folding the circle is about touching points and lines are generated perpendicular to and halfway between them where ever they appear on the circle, which opens the world of symmetry to movement and space and time. As you might have guessed, every fold in the circle is a rotational movement and all we have been able to do is draw straight lines without recognizing the function of rotation in folding.

Joseph, "first principles" are a bit confusing for me since it refers to what comes first, and nobody knows, not being around for it. No, axioms don't do it, they are the limits of somebody's imagination, and dare I say lack of capacity to observe. The sphere is the most comprehensive form I know and as it transforms through compression and is consistent to the first fold, is as far back as I can go with my experience to understand qualities that might be considered principle before everything else.

Brad

To view this discussion on the web visit https://groups.google.com/d/msgid/mathfuture/7cd2586d-c57e-de39-eb93-735c7912d856%40ms.lt.

--

Jan 3, 2020, 10:27:08 PM1/3/20

to mathf...@googlegroups.com

Andrius, thanks for all the leads. I like the perspective of thinking of a group as set of operations, where closure says any two successive (composed) operations can be replaced by another single operation. It helps sort out permutations, were I get tangled up in position, position number, number at position, etc.

How much can I learn about rotations from the rubik's cube? You have a block rotating on it's axis within a larger circuit, all in 90º increments in 3D, 24 discrete states per corner block. Seems like it might be related to Moibus. Or try coiling a hose; there's some crazy compound rotation--interaction between coiling and twisting.

Joseph Austin

> On Jan 3, 2020, at 4:32 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

>

> Dear Joseph,

> To view this discussion on the web visit https://groups.google.com/d/msgid/mathfuture/7cd2586d-c57e-de39-eb93-735c7912d856%40ms.lt.

How much can I learn about rotations from the rubik's cube? You have a block rotating on it's axis within a larger circuit, all in 90º increments in 3D, 24 discrete states per corner block. Seems like it might be related to Moibus. Or try coiling a hose; there's some crazy compound rotation--interaction between coiling and twisting.

Joseph Austin

> On Jan 3, 2020, at 4:32 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

>

> Dear Joseph,

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