Cone folding, circle folding and update from Andrius

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Andrius Kulikauskas

Dec 30, 2019, 4:17:02 AM12/30/19
to, Maria Droujkova,
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:
And then the second video lecture on Appolonius and conics:

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:
And I need to work on that further.  This year I surveyed ways of
figuring things out in physics:
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:

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

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
And I'm working on that here:
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:
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.

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:
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:
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:

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?"
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 +

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 Kulikauskas
+370 607 27 665

Joseph Austin

Jan 3, 2020, 1:30:02 PM1/3/20

Joseph Austin

> On Dec 30, 2019, at 4:17 AM, Andrius Kulikauskas <> wrote:
> Hi everybody,
> --
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Joseph Austin

Jan 3, 2020, 3:29:21 PM1/3/20
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 <> wrote:
>> Hi everybody,
>> I

Andrius Kulikauskas

Jan 3, 2020, 4:32:24 PM1/3/20
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:
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.
It has a lot about the classification of the Mobius transformations. 
Here's a short, beautiful video illustrating the Mobius transformations:
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.
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:
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.
Volume II has sections on Lie theory.
Volume III has a section on Bott periodicity.

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:
I'm making notes about his Universal Hyperbolic Geometry:

This is all a very challenging subject.  But it seems to be precisely
what my philosophy needs and says.


Andrius Kulikauskas
+370 607 27 665

Bradford Hansen-Smith

Jan 3, 2020, 10:16:00 PM1/3/20
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.

Bradford Hansen-Smith

Joseph Austin

Jan 3, 2020, 10:27:08 PM1/3/20
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 <> wrote:
> Dear Joseph,
> To view this discussion on the web visit
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