Adventures in Circle Folding
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Andrius Kulikauskas
Jun 17, 2016, 1:14:04 PM6/17/16
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Dear Bradford and all,
Every so often I would like to do some circle folding and share my
findings. Thank you, Bradford, for opening up this very new way of
thinking for me.
As I wrote, I sent my niece Ona a circle folding link for her birthday:
http://wholemovement.com/how-to-fold-circles
I made a "sphere" with her name on it and I sent her this photo:
http://www.ms.lt/derlius/HappyCirclesOna800.jpg
As I mentioned, the surprise for me was that the sphere was so taut
after I assembled the four circles with paper clips. Another surprise
was that I would need to use paper clips. And then I was curious what I
had actually built. I counted 14 holes. I realized that 8 of the holes
had 3 sides and 6 of the holes had 4 sides. And they were laid out like
the 6 faces and 8 corners of a cube. (Equivalently, the 8 faces and 6
corners of an octahedron). So my "sphere" is basically a cuboctahedron.
Each circle (actually, it's a disk) contributed 2 corners
(tetrahedrons). And then, apparently, it also contributed 1 1/2 corners
of the "octahedron". They aren't cubical corners in the sense that if I
try to mount them on a book, I see that the angles are too acute, they
aren't "right angles". (I wonder if Kirby's Martians are left-handed
and they call the tetrahedron angle the "left angle".) The sphere can
be thought of as two half-spheres and the plane in between is divided
into 6 angles, which I see need to be equal, so these "left angles" are
360/6 = 60 degrees. They need to be equal because the side of each
"square hole" matches the side of a "triangular hole" and the triangle
at the top of the sphere is clearly equilateral.
Putting together the 4 circles creates the 6 "square holes". Basically,
each circle contributes 3 "quarter-squares". So 6 x 3/4 = 4. So we see
that the circles pull together two different worlds, the "intra-world"
of the holes in the circles and the "inter-world" created by the spaces
between the circles. This distinction may be very meaningful in the
field of "homology" which is a very abstract branch of mathematics that
is basically the study of "holes". And it's a very difficult, abstract
subject perhaps because the holes "are not there" from the point of view
of set theory, so it's hard for set theory to talk about them.
It's also very helpful to be able to build models of the
three-dimensional polytopes because I've been thinking a lot about
polytopes as they turn out to be central to the distinctions I am
looking for, many of which Kirby discusses.
Theoretically, I can think of two ways in which circle folding makes
tangible fundamental ideas in math.
Folding the circle at once creates "implicit opposites" very much like i
and j, the square roots of negative one. One point that I would make to
Steve which I think isn't apparent in his exposition of Clifford
Algebras is that there is, a priori, no distinction to be made between
the two square roots of negative one as they are both indistinguishable
in every way. The distinction i and -i is artificial and misleading if
they make us think that one rotation is more preferable or natural than
another. Whereas the distinction between +1 and -1 is completely valid
because -1 x -1 = 1 whereas 1 x 1 = 1 and so there is a real distinction
to be made. So circle folding gives us practice with the implications
of such implicit opposites. It models for us the complex plane. Then
things like "complex conjugation" become natural. Or looking at the
"upper half plane" of the complexes likewise, I imagine. Also, complex
multiplication may make more sense. Perhaps we could try to draw a
"unit circle" on the circle to consider how that multiplication works.
Or perhaps, better yet, we could consider one side of the circle as the
In-Side (centered on 0) and the flip side of the paper circle as the
Out-Side (centered on infinity). Now we have a representation of the
complex plane where 0 and infinity are naturally identifiable and motion
away from the unit circle and towards the unit circle is, I expect,
equivalent. The two operations involved in complex multiplication -
moving towards the center or the edge - and rotating along the edge -
then seem completely natural in a way that is unnatural in the "square"
Euclidean plane.
Circle folding also comes up, I think, in terms of a more general
"sphere-folding" where the sphere can be of any dimension N. In the
letter that I am writing I am describing about a Center which generates
four families of polytopes An, Bn, Cn, Dn. The Center is the -1
simplex, which is to say, it is the unexplained unique -1 simplex which
should be accompanying as an "empty set" every k-simplex made up of
1-dimensional vertices, 2-dimensional edges, 3-dimensional faces and
such in k-dimensions.
https://en.wikipedia.org/wiki/Simplex
Look for the table of simplexes, see how it relates to Pascal's
triangle, and note the unexplained -1 dimensional simplex. I note that
the -1 simplex can be conceived as the unexpressed unique center which
we can associate with each simplex. For example, it is the center of a
point, the center of an edge, the center of a triangle, the center of a
tetrahedron, and so on. We can also imagine the Totality of the
simplex, which for a 2-dimensional triangle would be the area, and for a
3-dimensional tetrahedron would be the volume, whatever is defined by
all of the vertices. So we can imagine that first the Center is all by
itself, and then it generates vertices along with the Totality of these
vertices, so that the Center expresses itself but is never expressed,
always moving on as the new Center of the vertices it has created. The
Center is never expressed because each vertex it creates is distinct
from the rest and so the Center is ever new.
Well, it turns out that this is just one of four "theological" models of
how the Center can behave. We can think of the Center as modeling God
and the Totality as modeling Everything. And this particular process
generates the Simplexes, which are known as polytope family An because
the symmetries of these simplexes are given by the Symmetric group.
Which is to say, every vertex is linked to every other vertex, and the
only symmetry is that the vertices could be relabelled or renumbered in
any possible way. The lack of any additional symmetry also means that
each vertex is uniquely "positive" as Kirby keeps saying. So this is a
model that "God is good" in every direction.
But there are four families of such polytopes:
An: n-dimensional simplexes
Bn: n-dimensional cubes
Cn: n-dimensional cross polytopes (such as the octahedron)
Dn: n-dimensional demicubes (demihypercubes) which I find are best
thought of as half-cubes (hemicubes) with double edges.
The symmetries of these four families are captured by their symmetry
groups (Coxeter groups of reflections), which are the Weyl groups of the
root systems, which are vector bases for the Lie algebras, whose vector
"addition" is the analogue (by way of the exponential function) of the
matrix "multiplication" in the Lie groups, which are the continuous
groups, which are restricted by the possible, allowable "short-cuts" for
inverting (undoing) a continuous action, so that there could be a
continuous geometry in which nothing gets ripped apart. In other words,
I think that the symmetries of the four families of polytopes dictate
four geometries, such as Steve distinguishes. And it is interesting
what geometries are yielded by the 5 exceptional Lie groups.
There is a paper by Victor Kac
http://arxiv.org/pdf/math/9912235.pdf
"Classification of infinite-dimensional simple groups of supersymmetries
and quantum field theory" where in his final section (page 20) on
"Speculations and Visions" he notes that "Each of the four types W, S,
H, K of simple primitive Lie algebras (L, L0) correspond to the four
most important types of geometries of manifolds: all manifolds, oriented
manifolds, symplectic and contact manifolds." I found this quote
through a link by John Baez
https://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_3.html
about a video lecture given by his co-teacher James Dolan as part of
their "Geometric Representation Theory Seminar"
http://math.ucr.edu/home/baez/qg-fall2007/
which discusses the relevance of the polytopes to logic and geometry.
John Baez has a series of essays about these geometries which I suppose
starts here:
http://math.ucr.edu/home/baez/week181.html
Note, however, that he swaps the letters Cn and Dn from the way I and
many others use them.
An yields projective geometry
Bn yields conformal geometry
Cn yields conformal geometry
Dn yields symplectic conformal geometry. Basically, symplectic is
relevant when two quantities "position" and "momentum" are related, as
by time, energy. Symplectic is, I think, what happens when possiblities
tracked in the "complex" quantum reality get manifested (through a
natural or human "measurement") as a "real" actuality. Dn is where I
imagine circle-folding is relevant, again.
The Center and the Totality distinguish An, Bn, Cn, Dn in the following
way. The An simplexes have both a Center and a Totality. For example, a
tetrahedron has 1 Center (the center), 4 vertices, 6 edges, 4 faces and
1 Totality (the volume). Well, that's a row in Pascal's triangle. And
we see that in each row there is always 1 Center and always 1 Totality
at the opposite ends of Pascal's triangle.
Let's consider next the "pascal triangle" which counts the pieces of the
cross-polytopes (the orthoplexes, the octahedrons). Here in each row
there is a power of two that keeps increasing as follows. An octahedron
has 6 vertices, 12 edges and 8 faces. So the pascal triangle is:
1 x 2^0 3 x 2^1 3 x 2^2 1 x 2^3
In other words, it has 1 center, 3 x 2 vertices, 3 x 4 edges and 1 x 8
faces. And it has no Totality, no volume!
This makes sense if we consider what the Center is doing here. It is
creating two vertices at a time. We can think of them as "implicit
opposites" because there is no way of telling them apart. However, later
on, we will see how Dn labels them as "positive" and "negative".
Anyways, each new dimension yields 2 new vertices that are connected to
all of the previous vertices to keep them all distinct. The Center
yields first two unconnected points, then a square, then the octahedron,
then the tetracross (the 16-cell). So each vertex is directly connected
to every vertex except for its opposite. If you were to connect all of
the opposites, then you would have a simplex (but the dimension would be
twice as large because the simplex is generated one vertex at at time).
Anyways, when all the vertices are related to each other, (except for
their opposites!), then we have the 8 faces of the octahedron. That's
why the octahedron has, by definition, as we can see from Pascal's
triangle, no concept of totality, no volume. It can't because the
opposites aren't allowed to be related. They don't need to be related
because they are, by definition, opposite.
So that's a rather typical example of where physical reality can be
misleading or not the most helpful model. I first came to that
conclusion when I tried to understand electricity in terms of "running
water". Finally, a teacher explained to me that it can't be thought of
that way because the relationship between voltage, amperage, resistance,
etc. is simply different and the analogy fails spectacularly.
Similarly, the spin of an electron doesn't relate to our physical
models. And that seems to be the case with most of modern physics. In
general, the Nunez/Lakoff view that mathematical thinking arose to match
the experience of the human body seems to me very foreign. Instead, it
seems natural to me that mathematical structures reflect our internal,
spiritual modeling. Steve's focus on vision seems fruitful and then
there arises the question as to whether our visual evolution was driven
by our spiritual inner life or our practical outer life. And I'm
curious what mathematical thinking is like for the blind and for those
conscious in the womb. But I think the latter could be "playing" with
the type of philosophical Center and Totality issues that I'm
discussing. It's enough to abstractly think two or three abstract
"things" and the various ways they may relate.
Now the cubes Bn are just like the cross-polytopes but with the rows of
the "pascal triangle" reversed. A cube has 8 points, 12 edges, 6
faces. This means, by analogy above, that is has a Totality but it has
no Center! So I will explain what seems to be happening.
Let us start with a Totality and imagine it as an unfolding mirror. The
Totality is itself the one initial mirror. We can think of it as
defining two opposite directions. Now imagine that mirror dividing
itself into two and moving out into those two opposite directions.
Imagine that it is thereby opening up a mirror in the perpendicular
direction. Now we have a second dimension which also defines opposite
directions. Now imagine the totality of all the mirrors dividing to
reveal a new dimension with a new mirror. As this process continues, we
get 2^N quadrants, that is, N pairs of "implicit opposites". I see that
we really don't get vertices, edges, faces. Indeed, the quadrants keeps
getting refined at each step. We end up drawing them as "vertices" but
it's not like the simplex or octahedron where we add new vertices at
each step. Rather, we multiply each quadrant by 2 at each step. Also,
in counting the "pieces", we have to be mindful of the different "paths"
or "sequences" by which, for example, the faces of a cube arose. We
have a Totality, but there isn't any Center because, frankly, there
aren't any vertices, either. It's just pairs of mirrors sliding out (to
infinity?) and opening up more pairs of mirrors.
Now, theologically, we can imagine that we have the issues that Kirby
raises which I suppose come up amongst his Martian and Earthling
theologians. Namely, the Martians suppose that in every direction, "God
is good". There is no concept of "bad", of negative. But Earthlings
have polar valuations. At this point there isn't any moral
distinction. It's like "up" and "down", "right" and "left", "forwards"
and "backwards", without any moral preference (ignoring connotations!)
In the case of the octahedrons (cross-polytopes), each dimension has a
single, independent polarity and the Center keeps adding new
dimensions. In the case of the cubes, the polarities build on each
other, so that each "quadrant" participates in all of the polarities.
We could flip our thinking around in the last couple of sentences
because these are dual structures, that is, we could say that each
triangular face in an octahedron involves a choice from the three
polarities (dimensions) and thus defines a "quadrant". In every way
these two families are dual to each other. The difference is that the
Center is building bottom-up whereas the Totality is building top-down.
This distinction appears in the Simplex where the Center creates
vertices whereas the Totality could be thought of as getting rid of
vertices, in the opposite direction.
Finally, we can explain the Dn polytopes. I have found that these are
actually poorly or inconsistently defined. How could that be for such
an important object? Well, partly, it seems that this math seems
"peripheral", too concrete and simple yet messy for those mathematicians
and physicists trying to make big discoveries without having to imagine
a "big picture". But also, there is a great book by Imre Latakos,
"Proofs and refutations", which shows how in his dialogue about Euler's
characteristic formula how mathematicians make up their definitions as
they go along, and then remake them as they feel they need, quite
arbitrarily, in fact. So I've found inconsistent descriptions of these
polytopes, especially as to whether they have single edges or double
edges. It's just a philosophical distinction because the symmetry group
is the same in each case, which is what matters to them. So I offer a
philosophical answer.
Typically, the Dn polytopes are defined as the demicubes
(demihypercubes) which are gotten by starting with a cube and taking
half of its vertices. In three dimensions, this means that a cube's
vertices can be split into the vertices of two tetrahedrons, for
example, if we color alternating vertices white and black. This can be
done for an N-dimensional cube as well, in which case we take half of
the vertices, say, the black ones, and relink them by adding new edges
between "second nearest neighbors". The object we get will not be a
regular polytope, that is, not all the faces will be the same, but there
will be at least two different kinds, as with the cuboctahedron.
However, every vertex will look the same, nevertheless, and so it will
be a uniform polytope. And the symmetry group will be almost the same
as for the cube. The symmetry group for the N-cube includes the
symmetric group SN which relabels all of the "vertices" but also the
group of reflections Z2^N because you can reflect the cube (or
octahedron) in each dimension (they are built of opposites!) which is
not true for the simplex (no sense of opposites). Well, for the Dn
polytopes we only allow even reflection sequences of reflections. If
you reflect once, then you have to reflect in some direction yet again,
because we've trashed every other vertex, or so to say, we've prohibited
odd sequences of reflections.
Let's instead think of folding the cube! (or sphere!) In order to do
this, we have to distinguish a "vertex" (a quadrant) in our cube. And
we will identify it with the vertex which is opposite to it in every
way, in every dimension. Now they are one and the same, and likewise,
we identify in this direction each pair of vertices. So we get a spiky
construct that looks like, and is, a coordinate system. Each edge is
actually a double edge. Indeed, from the "pascal triangle" for the Dn
polytopes, it is most elegant to have double edges instead of single
edges. Then this pascal triangle is actually the sum of two pascal
triangles, one for bottom-up simplexes and the other for top-down
cubes. These spiky constructs are similar to the "half cubes"
(hemicubes). However, we have to add a double edge between the tips of
each "vector" in the coordinate system. This makes each coordinate
system into a simplex (all vertices are related!) with double edges and
a distinguished "origin". The angles look 90-45-45, though, because the
origin is special. What are the the pieces of our coordinate system?
There are two kinds because we have a double perspective:
* We have 2^(N-1) vertices because we fused opposite vertices. For each
of these vertices we have our "coordinate system" which we made into a
symplex and is understood to have all of the usual pieces of a simplex.
* We also keep in mind the original cube, the original Totality, and its
pieces (except for vertices, edges and two-dimensional faces). So for
the three-dimensional D3-polytope we have just the "volume cell", for
the four-dimensional D4-polytope we have one 4-D cell and 4 3-d faces,
all as we would for the cube as a whole.
Instead of imagining this as a "coordinate system" with the vector tips
related (which reminds me of Kirby's "closing the lid" operator), we can
also imagine these as cubes with double edges and also "reinforcements"
on every 2-dimensional face which connects the opposite corners, making
an X of double edges on each face. That seems to yield the same construct.
Even as I described it, it's messier than the others. I found an open
access paper on it by R.M.Green, "Homology representations arising from
the half cube"
http://www.sciencedirect.com/science/article/pii/S0001870809001017
and there is also a paper by Daniel Pellicer on a very much related
construct, "The Higher Dimensional Hemicubeoctahedron":
https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false
The latter construct is very similar: Imagine the vertices of an
n-dimensional cube and make squares out of every pair of opposite edges
(carving up the middle). Look at the "cube" as if it were an
octahedron and place, alternating, half of the simplexes that you would
need to cover it all. So there is a "cubic" inside and a "simplex"
outside.
Philosophically and even theologically, I think that Dn is the whole
point of An, Bn, Cn. We started with An having both a Center and a
Totality. We can think of it has having arisen from an initial tension
in the original "implicit opposites", Center and Totality, before they
found expression. Then we saw that instead, if we created in terms of
such "opposites", we could generate a series Cn with a Center but no
Totality, as with the octahedrons. Or we could generate a series Bn
with a Totality but no Center. Now, understandably, Dn is the series
with no Center and no Totality. That is, it has an Anti-Center and an
Anti-Totality. Each pair of opposite vertices can be fused together to
create a Coordinate System, an Origin which is an Anti-Center. With
regard to that coordinate system, suddenly every polarity becomes
explicit. One direction is with the vectors of the coordinate system,
and one direct is against. So we have "good" and "bad" made plain in
every direction. But there is one direction which is above it all,
namely the two fused vertices who formed the Origin. Apparently, they
chose one corner, and from that corner everything emmanates as "good"
towards the other corner, but "bad" in the opposite direction. Or as we
say in math, "positive" and "negative". And, apparently, you can't try
to get out of choosing, you can't just meet in the center, because you
would then lose a dimension. The Totality insists on the space being
full, one way or the other, and all of the N-cube's cells likewise must
assert which way is the "good" way. Of course, which direction is
"good" is completely arbitrary. But the Center is above it all. I
expect that the point of this all is to model what I think is the big
truth, which is that, "God doesn't have to be good. Life doesn't have to
be fair."
A related way to think about this all is in terms of Christopher
Alexander's books "The Nature of Order". Here is a picture of a
sequence of "wholeness preserving transformations" which very much bring
to my mind what I've described for An, Bn, Cn...
http://www.ms.lt/derlius/MatematikosGrozis/06.jpg
Well, suppose we already live in a built environment. Then the question
is, how can we restructure it? And that is what I think Dn is all about.
So the Dn is a kind of "sphere folding". But I suppose what's important
is also "sphere unfolding"? That is, abandoning our particular
Anti-Center, our particular key dimension for orienting ourselves as to
what is "good" and what is "bad". I think that is what I mean when I
say: God doesn't have to be "good".
I started writing this letter to share my experiences about circle
folding. I hope I've at least suggested that I have reason to believe
it can be informative about the most central issues in math and life.
I've expressed that in ideas and language that I personally am more
familiar with. I've ended up writing here about my own explorations.
For my own sake, I'm encouraged that the kinds of models that I think
are most basic for life seem to be at the very heart of mathematics and
help us sense how and why it all gets generated.
I wish to offer some pictures at some point.
Thank you to all of us for giving us our various worlds.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
Every so often I would like to do some circle folding and share my
findings. Thank you, Bradford, for opening up this very new way of
thinking for me.
As I wrote, I sent my niece Ona a circle folding link for her birthday:
http://wholemovement.com/how-to-fold-circles
I made a "sphere" with her name on it and I sent her this photo:
http://www.ms.lt/derlius/HappyCirclesOna800.jpg
As I mentioned, the surprise for me was that the sphere was so taut
after I assembled the four circles with paper clips. Another surprise
was that I would need to use paper clips. And then I was curious what I
had actually built. I counted 14 holes. I realized that 8 of the holes
had 3 sides and 6 of the holes had 4 sides. And they were laid out like
the 6 faces and 8 corners of a cube. (Equivalently, the 8 faces and 6
corners of an octahedron). So my "sphere" is basically a cuboctahedron.
Each circle (actually, it's a disk) contributed 2 corners
(tetrahedrons). And then, apparently, it also contributed 1 1/2 corners
of the "octahedron". They aren't cubical corners in the sense that if I
try to mount them on a book, I see that the angles are too acute, they
aren't "right angles". (I wonder if Kirby's Martians are left-handed
and they call the tetrahedron angle the "left angle".) The sphere can
be thought of as two half-spheres and the plane in between is divided
into 6 angles, which I see need to be equal, so these "left angles" are
360/6 = 60 degrees. They need to be equal because the side of each
"square hole" matches the side of a "triangular hole" and the triangle
at the top of the sphere is clearly equilateral.
Putting together the 4 circles creates the 6 "square holes". Basically,
each circle contributes 3 "quarter-squares". So 6 x 3/4 = 4. So we see
that the circles pull together two different worlds, the "intra-world"
of the holes in the circles and the "inter-world" created by the spaces
between the circles. This distinction may be very meaningful in the
field of "homology" which is a very abstract branch of mathematics that
is basically the study of "holes". And it's a very difficult, abstract
subject perhaps because the holes "are not there" from the point of view
of set theory, so it's hard for set theory to talk about them.
It's also very helpful to be able to build models of the
three-dimensional polytopes because I've been thinking a lot about
polytopes as they turn out to be central to the distinctions I am
looking for, many of which Kirby discusses.
Theoretically, I can think of two ways in which circle folding makes
tangible fundamental ideas in math.
Folding the circle at once creates "implicit opposites" very much like i
and j, the square roots of negative one. One point that I would make to
Steve which I think isn't apparent in his exposition of Clifford
Algebras is that there is, a priori, no distinction to be made between
the two square roots of negative one as they are both indistinguishable
in every way. The distinction i and -i is artificial and misleading if
they make us think that one rotation is more preferable or natural than
another. Whereas the distinction between +1 and -1 is completely valid
because -1 x -1 = 1 whereas 1 x 1 = 1 and so there is a real distinction
to be made. So circle folding gives us practice with the implications
of such implicit opposites. It models for us the complex plane. Then
things like "complex conjugation" become natural. Or looking at the
"upper half plane" of the complexes likewise, I imagine. Also, complex
multiplication may make more sense. Perhaps we could try to draw a
"unit circle" on the circle to consider how that multiplication works.
Or perhaps, better yet, we could consider one side of the circle as the
In-Side (centered on 0) and the flip side of the paper circle as the
Out-Side (centered on infinity). Now we have a representation of the
complex plane where 0 and infinity are naturally identifiable and motion
away from the unit circle and towards the unit circle is, I expect,
equivalent. The two operations involved in complex multiplication -
moving towards the center or the edge - and rotating along the edge -
then seem completely natural in a way that is unnatural in the "square"
Euclidean plane.
Circle folding also comes up, I think, in terms of a more general
"sphere-folding" where the sphere can be of any dimension N. In the
letter that I am writing I am describing about a Center which generates
four families of polytopes An, Bn, Cn, Dn. The Center is the -1
simplex, which is to say, it is the unexplained unique -1 simplex which
should be accompanying as an "empty set" every k-simplex made up of
1-dimensional vertices, 2-dimensional edges, 3-dimensional faces and
such in k-dimensions.
https://en.wikipedia.org/wiki/Simplex
Look for the table of simplexes, see how it relates to Pascal's
triangle, and note the unexplained -1 dimensional simplex. I note that
the -1 simplex can be conceived as the unexpressed unique center which
we can associate with each simplex. For example, it is the center of a
point, the center of an edge, the center of a triangle, the center of a
tetrahedron, and so on. We can also imagine the Totality of the
simplex, which for a 2-dimensional triangle would be the area, and for a
3-dimensional tetrahedron would be the volume, whatever is defined by
all of the vertices. So we can imagine that first the Center is all by
itself, and then it generates vertices along with the Totality of these
vertices, so that the Center expresses itself but is never expressed,
always moving on as the new Center of the vertices it has created. The
Center is never expressed because each vertex it creates is distinct
from the rest and so the Center is ever new.
Well, it turns out that this is just one of four "theological" models of
how the Center can behave. We can think of the Center as modeling God
and the Totality as modeling Everything. And this particular process
generates the Simplexes, which are known as polytope family An because
the symmetries of these simplexes are given by the Symmetric group.
Which is to say, every vertex is linked to every other vertex, and the
only symmetry is that the vertices could be relabelled or renumbered in
any possible way. The lack of any additional symmetry also means that
each vertex is uniquely "positive" as Kirby keeps saying. So this is a
model that "God is good" in every direction.
But there are four families of such polytopes:
An: n-dimensional simplexes
Bn: n-dimensional cubes
Cn: n-dimensional cross polytopes (such as the octahedron)
Dn: n-dimensional demicubes (demihypercubes) which I find are best
thought of as half-cubes (hemicubes) with double edges.
The symmetries of these four families are captured by their symmetry
groups (Coxeter groups of reflections), which are the Weyl groups of the
root systems, which are vector bases for the Lie algebras, whose vector
"addition" is the analogue (by way of the exponential function) of the
matrix "multiplication" in the Lie groups, which are the continuous
groups, which are restricted by the possible, allowable "short-cuts" for
inverting (undoing) a continuous action, so that there could be a
continuous geometry in which nothing gets ripped apart. In other words,
I think that the symmetries of the four families of polytopes dictate
four geometries, such as Steve distinguishes. And it is interesting
what geometries are yielded by the 5 exceptional Lie groups.
There is a paper by Victor Kac
http://arxiv.org/pdf/math/9912235.pdf
"Classification of infinite-dimensional simple groups of supersymmetries
and quantum field theory" where in his final section (page 20) on
"Speculations and Visions" he notes that "Each of the four types W, S,
H, K of simple primitive Lie algebras (L, L0) correspond to the four
most important types of geometries of manifolds: all manifolds, oriented
manifolds, symplectic and contact manifolds." I found this quote
through a link by John Baez
https://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_3.html
about a video lecture given by his co-teacher James Dolan as part of
their "Geometric Representation Theory Seminar"
http://math.ucr.edu/home/baez/qg-fall2007/
which discusses the relevance of the polytopes to logic and geometry.
John Baez has a series of essays about these geometries which I suppose
starts here:
http://math.ucr.edu/home/baez/week181.html
Note, however, that he swaps the letters Cn and Dn from the way I and
many others use them.
An yields projective geometry
Bn yields conformal geometry
Cn yields conformal geometry
Dn yields symplectic conformal geometry. Basically, symplectic is
relevant when two quantities "position" and "momentum" are related, as
by time, energy. Symplectic is, I think, what happens when possiblities
tracked in the "complex" quantum reality get manifested (through a
natural or human "measurement") as a "real" actuality. Dn is where I
imagine circle-folding is relevant, again.
The Center and the Totality distinguish An, Bn, Cn, Dn in the following
way. The An simplexes have both a Center and a Totality. For example, a
tetrahedron has 1 Center (the center), 4 vertices, 6 edges, 4 faces and
1 Totality (the volume). Well, that's a row in Pascal's triangle. And
we see that in each row there is always 1 Center and always 1 Totality
at the opposite ends of Pascal's triangle.
Let's consider next the "pascal triangle" which counts the pieces of the
cross-polytopes (the orthoplexes, the octahedrons). Here in each row
there is a power of two that keeps increasing as follows. An octahedron
has 6 vertices, 12 edges and 8 faces. So the pascal triangle is:
1 x 2^0 3 x 2^1 3 x 2^2 1 x 2^3
In other words, it has 1 center, 3 x 2 vertices, 3 x 4 edges and 1 x 8
faces. And it has no Totality, no volume!
This makes sense if we consider what the Center is doing here. It is
creating two vertices at a time. We can think of them as "implicit
opposites" because there is no way of telling them apart. However, later
on, we will see how Dn labels them as "positive" and "negative".
Anyways, each new dimension yields 2 new vertices that are connected to
all of the previous vertices to keep them all distinct. The Center
yields first two unconnected points, then a square, then the octahedron,
then the tetracross (the 16-cell). So each vertex is directly connected
to every vertex except for its opposite. If you were to connect all of
the opposites, then you would have a simplex (but the dimension would be
twice as large because the simplex is generated one vertex at at time).
Anyways, when all the vertices are related to each other, (except for
their opposites!), then we have the 8 faces of the octahedron. That's
why the octahedron has, by definition, as we can see from Pascal's
triangle, no concept of totality, no volume. It can't because the
opposites aren't allowed to be related. They don't need to be related
because they are, by definition, opposite.
So that's a rather typical example of where physical reality can be
misleading or not the most helpful model. I first came to that
conclusion when I tried to understand electricity in terms of "running
water". Finally, a teacher explained to me that it can't be thought of
that way because the relationship between voltage, amperage, resistance,
etc. is simply different and the analogy fails spectacularly.
Similarly, the spin of an electron doesn't relate to our physical
models. And that seems to be the case with most of modern physics. In
general, the Nunez/Lakoff view that mathematical thinking arose to match
the experience of the human body seems to me very foreign. Instead, it
seems natural to me that mathematical structures reflect our internal,
spiritual modeling. Steve's focus on vision seems fruitful and then
there arises the question as to whether our visual evolution was driven
by our spiritual inner life or our practical outer life. And I'm
curious what mathematical thinking is like for the blind and for those
conscious in the womb. But I think the latter could be "playing" with
the type of philosophical Center and Totality issues that I'm
discussing. It's enough to abstractly think two or three abstract
"things" and the various ways they may relate.
Now the cubes Bn are just like the cross-polytopes but with the rows of
the "pascal triangle" reversed. A cube has 8 points, 12 edges, 6
faces. This means, by analogy above, that is has a Totality but it has
no Center! So I will explain what seems to be happening.
Let us start with a Totality and imagine it as an unfolding mirror. The
Totality is itself the one initial mirror. We can think of it as
defining two opposite directions. Now imagine that mirror dividing
itself into two and moving out into those two opposite directions.
Imagine that it is thereby opening up a mirror in the perpendicular
direction. Now we have a second dimension which also defines opposite
directions. Now imagine the totality of all the mirrors dividing to
reveal a new dimension with a new mirror. As this process continues, we
get 2^N quadrants, that is, N pairs of "implicit opposites". I see that
we really don't get vertices, edges, faces. Indeed, the quadrants keeps
getting refined at each step. We end up drawing them as "vertices" but
it's not like the simplex or octahedron where we add new vertices at
each step. Rather, we multiply each quadrant by 2 at each step. Also,
in counting the "pieces", we have to be mindful of the different "paths"
or "sequences" by which, for example, the faces of a cube arose. We
have a Totality, but there isn't any Center because, frankly, there
aren't any vertices, either. It's just pairs of mirrors sliding out (to
infinity?) and opening up more pairs of mirrors.
Now, theologically, we can imagine that we have the issues that Kirby
raises which I suppose come up amongst his Martian and Earthling
theologians. Namely, the Martians suppose that in every direction, "God
is good". There is no concept of "bad", of negative. But Earthlings
have polar valuations. At this point there isn't any moral
distinction. It's like "up" and "down", "right" and "left", "forwards"
and "backwards", without any moral preference (ignoring connotations!)
In the case of the octahedrons (cross-polytopes), each dimension has a
single, independent polarity and the Center keeps adding new
dimensions. In the case of the cubes, the polarities build on each
other, so that each "quadrant" participates in all of the polarities.
We could flip our thinking around in the last couple of sentences
because these are dual structures, that is, we could say that each
triangular face in an octahedron involves a choice from the three
polarities (dimensions) and thus defines a "quadrant". In every way
these two families are dual to each other. The difference is that the
Center is building bottom-up whereas the Totality is building top-down.
This distinction appears in the Simplex where the Center creates
vertices whereas the Totality could be thought of as getting rid of
vertices, in the opposite direction.
Finally, we can explain the Dn polytopes. I have found that these are
actually poorly or inconsistently defined. How could that be for such
an important object? Well, partly, it seems that this math seems
"peripheral", too concrete and simple yet messy for those mathematicians
and physicists trying to make big discoveries without having to imagine
a "big picture". But also, there is a great book by Imre Latakos,
"Proofs and refutations", which shows how in his dialogue about Euler's
characteristic formula how mathematicians make up their definitions as
they go along, and then remake them as they feel they need, quite
arbitrarily, in fact. So I've found inconsistent descriptions of these
polytopes, especially as to whether they have single edges or double
edges. It's just a philosophical distinction because the symmetry group
is the same in each case, which is what matters to them. So I offer a
philosophical answer.
Typically, the Dn polytopes are defined as the demicubes
(demihypercubes) which are gotten by starting with a cube and taking
half of its vertices. In three dimensions, this means that a cube's
vertices can be split into the vertices of two tetrahedrons, for
example, if we color alternating vertices white and black. This can be
done for an N-dimensional cube as well, in which case we take half of
the vertices, say, the black ones, and relink them by adding new edges
between "second nearest neighbors". The object we get will not be a
regular polytope, that is, not all the faces will be the same, but there
will be at least two different kinds, as with the cuboctahedron.
However, every vertex will look the same, nevertheless, and so it will
be a uniform polytope. And the symmetry group will be almost the same
as for the cube. The symmetry group for the N-cube includes the
symmetric group SN which relabels all of the "vertices" but also the
group of reflections Z2^N because you can reflect the cube (or
octahedron) in each dimension (they are built of opposites!) which is
not true for the simplex (no sense of opposites). Well, for the Dn
polytopes we only allow even reflection sequences of reflections. If
you reflect once, then you have to reflect in some direction yet again,
because we've trashed every other vertex, or so to say, we've prohibited
odd sequences of reflections.
Let's instead think of folding the cube! (or sphere!) In order to do
this, we have to distinguish a "vertex" (a quadrant) in our cube. And
we will identify it with the vertex which is opposite to it in every
way, in every dimension. Now they are one and the same, and likewise,
we identify in this direction each pair of vertices. So we get a spiky
construct that looks like, and is, a coordinate system. Each edge is
actually a double edge. Indeed, from the "pascal triangle" for the Dn
polytopes, it is most elegant to have double edges instead of single
edges. Then this pascal triangle is actually the sum of two pascal
triangles, one for bottom-up simplexes and the other for top-down
cubes. These spiky constructs are similar to the "half cubes"
(hemicubes). However, we have to add a double edge between the tips of
each "vector" in the coordinate system. This makes each coordinate
system into a simplex (all vertices are related!) with double edges and
a distinguished "origin". The angles look 90-45-45, though, because the
origin is special. What are the the pieces of our coordinate system?
There are two kinds because we have a double perspective:
* We have 2^(N-1) vertices because we fused opposite vertices. For each
of these vertices we have our "coordinate system" which we made into a
symplex and is understood to have all of the usual pieces of a simplex.
* We also keep in mind the original cube, the original Totality, and its
pieces (except for vertices, edges and two-dimensional faces). So for
the three-dimensional D3-polytope we have just the "volume cell", for
the four-dimensional D4-polytope we have one 4-D cell and 4 3-d faces,
all as we would for the cube as a whole.
Instead of imagining this as a "coordinate system" with the vector tips
related (which reminds me of Kirby's "closing the lid" operator), we can
also imagine these as cubes with double edges and also "reinforcements"
on every 2-dimensional face which connects the opposite corners, making
an X of double edges on each face. That seems to yield the same construct.
Even as I described it, it's messier than the others. I found an open
access paper on it by R.M.Green, "Homology representations arising from
the half cube"
http://www.sciencedirect.com/science/article/pii/S0001870809001017
and there is also a paper by Daniel Pellicer on a very much related
construct, "The Higher Dimensional Hemicubeoctahedron":
https://books.google.lt/books?id=HarWCwAAQBAJ&printsec=frontcover#v=onepage&q&f=false
The latter construct is very similar: Imagine the vertices of an
n-dimensional cube and make squares out of every pair of opposite edges
(carving up the middle). Look at the "cube" as if it were an
octahedron and place, alternating, half of the simplexes that you would
need to cover it all. So there is a "cubic" inside and a "simplex"
outside.
Philosophically and even theologically, I think that Dn is the whole
point of An, Bn, Cn. We started with An having both a Center and a
Totality. We can think of it has having arisen from an initial tension
in the original "implicit opposites", Center and Totality, before they
found expression. Then we saw that instead, if we created in terms of
such "opposites", we could generate a series Cn with a Center but no
Totality, as with the octahedrons. Or we could generate a series Bn
with a Totality but no Center. Now, understandably, Dn is the series
with no Center and no Totality. That is, it has an Anti-Center and an
Anti-Totality. Each pair of opposite vertices can be fused together to
create a Coordinate System, an Origin which is an Anti-Center. With
regard to that coordinate system, suddenly every polarity becomes
explicit. One direction is with the vectors of the coordinate system,
and one direct is against. So we have "good" and "bad" made plain in
every direction. But there is one direction which is above it all,
namely the two fused vertices who formed the Origin. Apparently, they
chose one corner, and from that corner everything emmanates as "good"
towards the other corner, but "bad" in the opposite direction. Or as we
say in math, "positive" and "negative". And, apparently, you can't try
to get out of choosing, you can't just meet in the center, because you
would then lose a dimension. The Totality insists on the space being
full, one way or the other, and all of the N-cube's cells likewise must
assert which way is the "good" way. Of course, which direction is
"good" is completely arbitrary. But the Center is above it all. I
expect that the point of this all is to model what I think is the big
truth, which is that, "God doesn't have to be good. Life doesn't have to
be fair."
A related way to think about this all is in terms of Christopher
Alexander's books "The Nature of Order". Here is a picture of a
sequence of "wholeness preserving transformations" which very much bring
to my mind what I've described for An, Bn, Cn...
http://www.ms.lt/derlius/MatematikosGrozis/06.jpg
Well, suppose we already live in a built environment. Then the question
is, how can we restructure it? And that is what I think Dn is all about.
So the Dn is a kind of "sphere folding". But I suppose what's important
is also "sphere unfolding"? That is, abandoning our particular
Anti-Center, our particular key dimension for orienting ourselves as to
what is "good" and what is "bad". I think that is what I mean when I
say: God doesn't have to be "good".
I started writing this letter to share my experiences about circle
folding. I hope I've at least suggested that I have reason to believe
it can be informative about the most central issues in math and life.
I've expressed that in ideas and language that I personally am more
familiar with. I've ended up writing here about my own explorations.
For my own sake, I'm encouraged that the kinds of models that I think
are most basic for life seem to be at the very heart of mathematics and
help us sense how and why it all gets generated.
I wish to offer some pictures at some point.
Thank you to all of us for giving us our various worlds.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
Bradford Hansen-Smith
Jun 18, 2016, 8:51:32 AM6/18/16
to mathf...@googlegroups.com
Andrius,
Thank you for sharing your experience. I rarely get feed back, I think most people will not fold and/or do not observe what has been generated.
I see the “holes” in the spherical VE as open plane triangles and squares that can then treated as positive sets, like “closing the lid” I suppose. This creates definable volume. Nothing is empty, not even a hole. The choice is then a flat plane or a curved plane that corresponds to spherical volume.
http://wholemovement.com/blog/item/735-the-other-circle
--“Perhaps we could try to draw a "unit circle" on the circle to consider how that multiplication works. Or perhaps, better yet, we could consider one side of the circle as the In-Side (centered on 0) and the flip side of the paper circle as the Out-Side (centered on infinity). Now we have a representation of the complex plane where 0 and infinity are naturally identifiable and motion away from the unit circle and towards the unit circle is, I expect, equivalent.”
Use two-sided colored paper, be consistent in folding. The circle arrangement of squares and triangles reveals implicit opposites that may cause confusion with the inside/outside idea of a complex plane because the folded circle is not just a concept. While complex plane 2-D means one thing, the circle is a complex object in space.
http://wholemovement.com/blog/item/731-paradigm-shift-from-part-to-whole
There are two different ways to pin the 4 circles together, one allows you to remove four pins and it will spiral down collapsing to a pile of circle, the other does not. They both look exactly the same, it is a matter of orientation of pinning together. There are many levels of subtle differences that are not visually apparent.
The process of making the VE sphere shows complex multiplication as a function of division. Not being very conversant in math I do see the concepts being talked about in the circle, which is why I find your observations about folding the circle valuable. As far as I can tell every abstraction is grounded in relationships inherent to the circle.
--“Circle folding also comes up, I think, in terms of a more general "sphere-folding" where the sphere can be of any dimension N. In the letter that I am writing I am describing about a Center which generates four families of polytopes An, Bn, Cn, Dn”
Can you explain "sphere-folding"? I don’t know what you mean by the sphere having any dimension N unless you are talking about infinite lines of symmetry, any number of which can be separated out and used to substantiate a variety of concepts. Maybe you are referring to compressing the sphere to a circle that can then be folded? Since compression changes nothing but the form you might call folding the circle “sphere-folding.”
What is “the center all by itself”? How can you have an isolated center without a context to give location? It is an abstraction that has no potential until located, then it is not all by itself, it but must reside somewhere among other. Forgive my naivety but not being a math person I get to ask these kinds of questions.
This center that is in no place, “…generates four families of polytopes An, Bn, Cn, Dn.” What are they,where are they, where do they come from, and why? The first fold in the circle generates four points. Two randomly placed points (local centers?) on the circumference generated two more that are at 90 degrees from given distance D between the first two. The circumference is a center given the concentric nature of the circle into and out from itself. The 2-D information in the fold is very different than the 3-D information, yet both of the same context and movement. I know we are talking about different systems but I come to rely on numbers to reveal patterns connections that are not visually obvious and cross formal boundaries. Can you explain more about four families of polytopes?
--“Instead, it seems natural to me that mathematical structures reflect our internal, spiritual modeling.”
I would agree, not just mathematical structures but everything that math abstractly represents is a result of spiritual modeling to a priori patterns of organization where the 3-D circle itself is the center of spherical alignment.
--“"God is good." There is no concept of "bad" of negative. But Earthlings have polar valuations. At this point there isn't any moral distinction.”
God is good with one o. There is nothing redundant about God or creation, as we understand; everything is uniquely different which gives value to the idea of unity. You are right “there isn't any moral distinction” and there needs to be, otherwise we will continue the separation and fragmentation in abstracting mathematics to be used for any purpose, good /bad, positive and negative. The moral nature of man is inherent in the tools we make, not just how we use them. The circle is not a man-made tool. Folding circles works or it does not, demonstrating spiritual modeling both in the mechanical ordering and with individual relationships sustainable as parts to whole that determines the interrelationships between parts. Without structural ethics and moral intent inherent in the context what we separate out and fragment is destine to fail. You are asking big questions about math I do not often hear in the same way from others, Kirby has his own way. Maria says make math your own.
Brad
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Andrius Kulikauskas
Jun 18, 2016, 11:34:45 AM6/18/16
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Bradford,
Thank you! I appreciate your long and thoughtful response.
I have been looking for places to have such conversations. I realized
today, though, that I need to develop part of my own wiki in English
where I can "make math my own", as Maria teaches. Then I can write my
essays there and also add pictures.
I will create some pictures, organize my first essays, fold some more
circles, and then reply to you and share new thoughts.
Thank you!
> polytopes An, Bn, Cn, Dn.” What are they,where are they,where do they
> m...@ms.lt <mailto:m...@ms.lt>
> +370 607 27 665 <tel:%2B370%20607%2027%20665>
Thank you! I appreciate your long and thoughtful response.
I have been looking for places to have such conversations. I realized
today, though, that I need to develop part of my own wiki in English
where I can "make math my own", as Maria teaches. Then I can write my
essays there and also add pictures.
I will create some pictures, organize my first essays, fold some more
circles, and then reply to you and share new thoughts.
Thank you!
> +370 607 27 665 <tel:%2B370%20607%2027%20665>
>
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> Bradford Hansen-Smith
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