Geometry of Moods

[Andrius Kulikauskas]

I'm overviewing my thinking about geometry. I appreciate ideas that you
may have and directions that you might point me in.

I am developing some highly speculative ideas. I will present them in
the form of an art work which I will show and talk about on November 16,
2016 at the Science and Art Festival in Klaipeda, Lithuania. I will be
representing the Klaipeda campus of the Vilnius Art Academy.

My main idea is that our minds naturally (cognitively, metaphysically)
make use of four geometries and six transformations between them. These
geometries are traditionally called affine, projective, conformal and
symplectic. However, it seems to me that these terms are typically
applied in ways that obscure the qualitative distinctions that I would
like to make.

My investigation and my art work are building on my recent talk, "A
Research Program for a Taxonomy of Moods".
http://www.ms.lt/sodas/Book/TaxonomyOfMoods
I showed how geometry is relevant in organizing expectations in a
multitude of ways that evoke corresponding moods. Geometry is inherent
in the distinctions we make between our selves (as when we are happy,
sad, disgusted, hateful or depressed) and the world (as when we are
excited, surprised, frightened, angry or relieved). I am analyzing
Chinese poems from the Tang dynasty to show how slightly more complex
relations between expectations can evoke distinct moods.

Artistically, I plan to build a model of a transparent world which can
serve as a laboratory (or "doll house") for describing combinations of
expectations and then varying them to note changes in the moods they
evoke. The "world" will be a cylindrical transparent wall of 1 m
diameter and 30 cm height, resting on a table. Inside will be
transparent abstract shapes (boxes, cylinders, spheres) representing
houses, beds, villages, etc. and there will also be drinking glasses
representing perspectives, which can be filled with liquids,
representing emotions.

Preliminarily, I note four geometries for describing moods:

1) Sometimes a person's mood can be described simply by the spatial
relations of their expectations. For example, Li Bai's poem "Quiet
Night Thoughts", which I analyzed in my talk, can be interpreted to
evoke a conditional sadness which results upon emphasizing that one's
happy home is in the distant realm of the moon's beauty. I think of
this geometry as "affine" because it can be all described in terms of
outward "vectors".

2) Next, I imagine such a person in dialogue with themselves, as if
looking backwards and forwards along a line, which allows for past and
future. I expect to find such poems. I would call them "projective"
because they are based on two-directional "lines".

3) There may be other people involved with their own expectations and
moods. The people may be "parallel" or "perpendicular" to each other,
or some "angle" in-between. I would think of such a geometry as
"conformal" because it is based on "angles".

4) One person's mood (or a mood attributed to the environment) may be
fixed while another person's mood changes. This reminds me of how an
"area" can be swept out. I would think of such a geometry as
"symplectic" to the extent that it defines a dynamic "area" that can be
thought of as a product of "position" and "momentum". In my talk, I
analyzed the Beatles' song "She Loves You" as a nice example of a
reorientation of mood.

I expect that if I consider several dozen classic poems from the Tang
dynasty, then I will discover many wonderful examples that I can
illustrate with my model world. And surely the examples will push my
thinking forward.

But I also want to show that there are six transformations between these
four geometries. This hypothesis, if correct, would clarify how to
think of these geometries and how they are related. Basically, I
imagine that these transformations as extending a less expressive
geometry so that it matches the sophistication of a more expressive
geometry. The transformations are ways of extending our perspective,
which is, I suppose, the purpose of the subject of geometry. I think
that uncovering such a system of six transformations would clear up
confusion by allowing us to consider what is contributed, qualitatively,
by each geometry and each transformation, that is, by each perspective
and each extension.

For example, I can imagine the transformations below as having the
following purposes:
* Reflection across a (perpendicular) axis interprets an (affine)
one-directional vector within a (projective) two-directional line.
* Shear mapping reshapes an (affine) parallelogram (=two vectors) into a
(conformal) rectangle (=two orthogonal line segments).
* Rotation around origin projects a (projective) line into its
components in a (conformal) coordinate system.
* Dilation from/to the origin adjusts a (conformal) angular shape so
that it has the desired (symplectic) area.
* Squeeze mapping rebalances the contributions of (projective) axes to
the overall (symplectic) area.
* Translation sweeps an (affine) vector across a (symplectic) area.
In each case, the transformation contributes the specific information
needed for the simpler geometry to be interpreted in the more
sophisticated geometry.

I am trying to learn more about affine, projective, conformal and
symplectic geometries. I want to be able to sort out, intuitively and
qualitatively, the concepts, insights and theorems that they
contribute. I appreciate very much Norman Wildberger's videos,
especially on "Universal Hyperbolic Geometry"
https://www.youtube.com/user/njwildberger
and also Stephen Lehar's notes on Clifford Algebras as well as
projective and conformal geometry:
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
and Sylvain Poirer's notes on geometry, from which I drew my list of six
transformations:
http://settheory.net/geometry

However, I am suspecting that they and other mathematicians are
discussing these geometries in a way that overthinks the simpler ones
and muddles their distinctions.
* From my point of view, the admittedly powerful use of (Cartesian)
algebra in geometry presumes a conformal mindset, that is, one that
avails itself of angles and, especially, orthogonal dimensions. I think
that our minds have a sense of projective geometry which, in its "pure"
form, without any transformations, does not, by definition, make use of
any such orthogonal dimensions or algebraic coordinates.
* Norman Wildberger shows the power of adding a single "distinguished"
circle. But I think the point of that circle and the resulting
polarities and perpendicularities is to ground a sense of distance (or
quadrance) and angle (or spread) which does not naturally exist in
projective or affine geometries as our mind applies them.
* Clifford Algebras make use of a geometric product which I think is
symplectic in its very nature.

I have written of a way to distinguish the mathematical structures which
are natural to our mind, namely, those which our minds make use of in
figuring things out in mathematics:
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
I systematized these ways and noticed four ways that we use metalogic to
solve problems. They seem to me to be related to these geometries.

I became interested in these geometries, and geometry, in general, when
I wondered how to overview all of math. I started by trying to organize
the areas in the Mathematics Subject Classification based on how they
depended on each other:
http://www.ms.lt/derlius/MatematikosSakosDidelis.png
and continued, unsuccessfully so far, with a larger "spaghetti diagram":
http://www.ms.lt/derlius/Math/MathWays2/mathways2.html

I realized that geometry must be fundamental. I also saw that Lie
groups/algebras were a key link between algebra and analysis (as is the
Langlands program). I have been trying to intuitively understand why
there are four classical Lie groups/algebras. The most helpful idea in
that regard is that they relate to these four different geometries. In
particular, I am grateful to John Baez for his writings, for example:
http://math.ucr.edu/home/baez/octonions/node13.html
where he discusses isometries of real, complex and Hamiltonian
projective spaces.

In my talk on the ways of figuring things out, I discuss the four
infinite families of polytopes whose symmetry groups are the Weyl groups
of the root systems for the associated Lie algebras. I note the
importance of the "center" and the "totality" of these polytopes and how
the simplexes can be used to define the "vectors" which affine geometry
preserves, the cross polytopes can define the two-directional "lines"
which projective geometry preserves, the cubes can define the "angles"
which conformal geometry preserves, and the demicubes ("coordinate
systems") I suppose could define the "swept out areas" which symplectic
geometry preserves.

I have also been studying the various ways that we use variables:
http://www.ms.lt/derlius/variabletypes.png
These I think should accord with the six transformations. But I need to
write about them separately.

I'm also curious if Grothendieck's six operations might somehow be related.
https://en.wikipedia.org/wiki/Six_operations
I also think of the six natural bases of symmetric functions
(elementary, homogeneous, power, monomial, Schur, forgotten). And I
also consider six visualizations which I have written about
http://www.ms.lt/papers/organizingthoughts.html
by which sequences, hierarchies and networks are used to restructure
each other.

Finally, I return back to moods and the boundary between our selves and
our world. This boundary is apparent neurologically, in that our minds
construct maps of our body (as when our arm gets extended by the hammer
it grasps, or we can physically feel when our car gets too close to
another car) as well as our world.

But the boundary is also there theologically. In my talk on God's dance:
http://www.ms.lt/sodas/Book/GodsDance
I imagine how God investigates, Is God necessary? As in a proof by
contradiction, God steps aside to see if God will arise. Thus there is
the big God who understands (beyond this world) and the little God who
comes to understand (within this world). They are the same God because
they understand the same God, that lens which equalizes the big God and
the little God. So here that lens is very much like that boundary
between our selves (our lives) and our world (which loves us). But this
brings to mind the Field with one element in which 0,1, infinity are all
the same element, as when 0 * infinity = 1. This is what I think about
in considering the larger purpose of geometry.

I am grateful for any thoughts on
* how to understand affine, projective, conformal and symplectic geometries
* why, intuitively, are there four classical Lie groups-algebras
and any ideas this all might inspire.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665