Geometric Algebra and Clifford Algebra

[Andrius Kulikauskas]

Dear Steve Lehar,

Thank you for joining us at Math Future! I hereby start a new thread
where we can focus on whatever mosts interests you. Or please start
your own threads.

Maria Droujkova is the founder of the Math Future group of math
educators. For more about her activity, see especially:
http://naturalmath.com and also: http://naturalmath.com/mathfuture/

We learned of your work from Joe Austin:

----------------------------------------

Kirby, Andrius,

I've stumbled on an interesting presentation of Clifford Algebra, by
Steven Lehar.
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
It's special appeal is that it is profusely illustrated with color
diagrams, some animated,
which is only fitting for a *geometric* topic.
So far I have just skimmed it, but I plan to peruse it in more detail.

Lehar identifies himself as"an independent researcher
<http://cns-alumni.bu.edu/%7Eslehar/webstuff/persintro/indep.html>with a
novel theory <http://cns-alumni.bu.edu/%7Eslehar/epist/epist.html> of
mind and brain, inspired by the observed properties
<http://cns-alumni.bu.edu/%7Eslehar/webstuff/persintro/observedprop.html>of
perception."

Joe Austin

--------------------------------------------------------

Steve, I am very grateful to you for your persistence in championing
your mind-brain theory. I'm very interested to see the big picture of
mathematics, its "grand unification". So I look forward to studying
your pages and learning advanced mathematics from you and alongside you.

In particular, I'm intrigued by your writings about characterizing
different kinds of geometry, such as projective geometry:
https://slehar.wordpress.com/2014/06/26/geometric-algebra-projective-geometry/
and conformal geometry
https://slehar.wordpress.com/2014/07/24/geometric-algebra-conformal-geometry/

The latter made me think of the Mandelbrot set in that it does appear to
be an inverse image of a world. That is, it does appear to be connected
together in intricate ways which would make more sense if that universe
was physically "expanding" rather than "shrinking". I imagine the
p-adic integers are exactly like that. That makes me think in my own
work that the point of characteristic p is that it p=0=infinity.

I have noticed about the Mandelbrot set what I think is a great
simplification, that is, that it can be generated directly by plugging
in each complex number z into the generating function for the Catalan
numbers, which happens to count all manner of things that are processed
by context-free grammars (push-down automatas), anything with finitely
many "obligations" that have to be met, as with left hand parentheses
that need to be balanced by right hand parentheses, or walks up tree
branches that need to be balanced by walks back down, thus anything that
makes use of a finite number of memory cells. Here is my letter about
that to this group:
https://groups.google.com/d/msg/mathfuture/ZGkQe_kSkQI/3CtGubQFLwAJ

I am also interested in how you are thinking about infinity.
https://doubleconformal.wordpress.com
I'm starting to realize that the concept of "infinity" is one that I
think could be treated differently, perhaps for what it is, as a
convenient fiction. I'm thinking that it could be just a construct that
means "we have enough vertices" and that concept could be given by a
number p (as in the characteristic for a finite field). That number
could be variable and would, I imagine, be prime simply because
otherwise we could work with a smaller p in whatever we were doing. If
we want to have p be unlimited, then we would set p=0 so that we have a
"field with one element" (a mysterious object that math hasn't figured
out yet what it means) and where 0=p=infinity. I will write about that
separately.

I am very glad that you are succesfully learning by teaching. Over the
last few months I have been trying to understood the tensor product
because I think it is basic to everything. I think that it is "trivial"
in the sense that "linearity" is trivial. So it is the background
assumption in everything, in all of geometry, but I have yet to
understand this triviality. I do have the sense, though, that it is
grounded in a duality of "bottom-up" (building up) and "top-down"
(tearing down) views.

Steve, in my own words, I would say that what you and I and Nunez/Lakoff
are doing in our different ways is a "science of math". That is, we can
look at the enormous output of mathematical activity and say that there
is a way to make sense of the big picture. But to do that we need to
use not only math, but also other tools and approaches, both science and
aesthetics and even some politics to have a relevant conversation.

Math Future is a welcoming place for us because there is an
understanding, I feel, that as educators we need to develop our own
"worlds" and look for how they relate. I think many of these worlds are
very sympathetic to Geometric Algebra. Ted Kosan is leading lessons in
computer algebra software MathPiper http://www.mathpiper.org Kirby
Urner is a modern day Buckminster Fuller who, among so many things, is
interested to learn more about Clifford Algebras, as am I. Joe Austin
champions a teaching approach that is rooted in physical thinking,
including visualizing geometry. Bradford Hansen-Smith is a pioneer of
circle folding, which I'm realizing, is very central. Yesterday was my
niece Ona's birthday so I folded her a sphere (a cuboctachedron) from
four circles which I found instructive (it was surprisingly taut when I
put it together with paperclips). I sent her a link
http://wholemovement.com/how-to-fold-circles and a photo of me with the
sphere and her name on it. Which is to say that we affect each other in
large and small ways.

I want to tell you about three other groups where some day, sooner or
later, you might be successful in starting conversation and
collaboration towards a grand unification of math. They are the
Foundations of Mathematics email list, the nLab wiki and forum, and the
Azimuth Project.

The Foundations of Mathematics email list
http://www.cs.nyu.edu/mailman/listinfo/fom/
is moderated by Martin Davis, one of the solvers of Hilbert's Tenth
Problem. It is dominated by Harvey Friedman, who shares his
thinking-out-loud on the open problems in set theory / foundations of
mathematics which he finds most interesting. The archives are open but
you have to request permission to be a member and have a chance to
write. You should be accepted because you have a Ph.D., as do I. But
my first letter, in which I introduced myself, was rejected because by
the moderator as too long and meandering. I sent the same letter to
Harvey Friedman but he didn't reply. Later they approved my letter on
the Mandelbrot set and the Catalan numbers. Now I'm writing a long
letter to them which I hope they might accept.

The most interesting mathematician, who shares his thinking about all
the math I would like to know, is John Baez.
http://math.ucr.edu/home/baez/
He is famous for his blog
https://johncarlosbaez.wordpress.com/
but especially for his earlier blog "This Week's Finds in Mathematical
Physics", basically 318 essays that he wrote full of mathematical
intuition which I keep bumping up against when I google or read Wikipedia.
http://www.math.ucr.edu/home/baez/TWF.html
He is also one of the founders of the "n-category Lab", which is a loose
affiliation of the initial group blog, the "n-category Cafe"
https://golem.ph.utexas.edu/category/
There is also the "n-Lab" wiki
https://ncatlab.org/nlab/show/HomePage
and the related "n-Forum" https://nforum.ncatlab.org for discussing
changes made to the wiki.

In a sense, that wiki is the closest thing there is to collaborative
work on a "grand unification" of Mathematics. The approach that they
are taking is called "n-category theory". The best source which I have
found for that is the paper by John Baez and Aaron Lauda, "A Pre-History
of n-Categorical Physics"
http://arxiv.org/pdf/0908.2469v1.pdf
See especially page 104 on Baez-Dolan (1995) which discusses this paper:
http://arxiv.org/pdf/q-alg/9503002v2.pdf
Such papers may seem impossible but I am realizing, as I think you as
well, that if I have the fruitful attitude:
"What is the simplest issue of the deepest consequence?"
then I have the machete with which to cut through the thickest weeds.
In my case, it means that nobody knows what the -1 simplex is nor the
"field with one element". In your case, if you can find the right
issue, which relates to what they fail to do in their world, then you
will get the chance to say what you want to say about that and
everything else. So I'm curious not only what your own deepest insights
are (you seem to be able to write about that) but also what particular
math solution you provide might get others interested in our hope for
collaboration on a grand unificiation.

The point of "n-category theory" is that it can formulate mathematical
intuitions in homology, homotopy and other advanced fields which are
completely ignored by the classical "set theory" foundations of
mathematics. N-category theory is related to "homotopy type theory" and
there will be a conference in Munich on "Foundations of Mathematical
Structuralism"
https://golem.ph.utexas.edu/category/2016/05/the_hott_effect.html#more
which I think I'll submit an abstract to, due June 30, 2016. N-category
theory has many layers of abstraction that serve to identify and
describe what it means for mathematical equivalences, transformations,
objects to be "natural". It's just all extremely abstract and
"unnatural" to learn. There are some exchanges by John Baez and Harvey
Friedman where they would like to have a basis for fruitful discussion
but they can't find it and seem to have better things to do. Harvey
Friedman's position is that set theory is what works and that it doesn't
matter which approach is more "natural" but if it can address
mathematicians' problems that the classical foundations can't, then
please speak up. This is why I'm focusing on the field with one element
and the negative-one-dimensional simplex and I think that's proved very
fruitful for me but we'll see what they say. I should mention that I've
learned that the flip-side of the abstractness of "categorification" is
the concreteness of "decategorification" as in algebraic combinatorics,
my own specialty in math. I learned that from Jeff Hicks's e-book
"Categorification":
https://math.berkeley.edu/~jhicks/links/SOTS/jhicks022614.pdf
I'm surprised he's just a grad student. Anyways, in my own work, it
means that a variable q which we use for tracking some feature of
enumerated objects (such as the way a path Pascal's triangle swings left
or right), can be set to equal 1 (in which case we are simply
"counting") or it can be "categorified" in some way to describe the
actual objects, which might be, for example, strings of generators of
the symmetric group. Which is to say, there is a flip-side that is a
concreteness to the abstractness. You might find that in your own work
as well and that might help you communicate it both concretely and
abstractly.

I tried to introduce myself at the nLab wiki
https://ncatlab.org/nlab/show/AndriusKulikauskas
and then I created pages on the "big picture", "beauty", "discovery" but
they were deleted. You can see the discussion at the nForum:
https://nforum.ncatlab.org/discussion/7066/discovery/#Item_0
But I think if you write about the subjects that you know well and link
to your articles then you might be well received. It's worth trying.

I noticed that John Baez is active in the Azimuth Project which he
started for mathematicians and scientists to work together now in
response to climate change. I created a page for myself at the wiki:
http://www.azimuthproject.org/azimuth/show/Andrius+Kulikauskas
And introduced myself at the forum and participated in a couple of threads:
https://forum.azimuthproject.org
I proposed to work together on helping others, and each other, to learn
advanced mathematics, as you are doing, and so is the wiki administrator
David Tanzer. I proposed to work on a graph of all of the areas in
math. I didn't get any response but I wasn't kicked out yet, either. I
should note that John Baez is also known as the author of the "crackpot
index":
http://math.ucr.edu/home/baez/crackpot.html
which is understandable given his active participation in the online world.

Steve, I look forward to your letters!

Andrius

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

--------------------------


Dear Andrius,

Thank you very kindly for alerting me to the existence of the mathfuture
group! And thank you also for promoting my visual introduction to
Clifford Algebra / Geometric Algebra.

You and the group may also be interested in this paper of mine on the
Double Conformal Mapping, an extension to David Hestenes' Conformal
Geometry extension to Geometric Algebra, which relates directly to my
theory that the origins of mathematics lie in the laws of perception.

https://doubleconformal.wordpress.com/

I also have a book in progress (not yet complete) titled The Perceptual
Origins of Mathematics.

https://slehar.wordpress.com/2014/09/12/the-perceptual-origins-of-mathematics/

As with my Visual Introduction to Clifford Algebra, I prefer to explain
math in pictures rather than equations, wherever possible, to clarify
the connection to perception.

Indeed the extraordinary Grand Unification of math accomplished by
Clifford Algebra stems from the discovery that all of algebra is a
branch of geometry, and that most mathematical operations can be
represented as spatial operations on spatial structures. This makes my
writing immediately accessible to the non-professional mathematician.

I intend one day to write a book that explains all the most interesting
aspects of math in simple intuitive terms that most anyone can understand.

Thanks again for making contact with me!

Steve Lehar

--------------------------

On Wed, Jun 15, 2016 at 10:33 PM, Andrius Kulikauskas <m...@ms.lt
<mailto:m...@ms.lt>> wrote:
Dear Joseph Austin,

Thank you for alerting us to Steven Lehar's very helpful page on
Clifford Algebra
https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/
and also his thoughtful independent research which I look forward to
looking over. I take the opportunity to let him know about the
MathFuture google group and share at least the beginning of my letter on
"implicit math" which might interest him and you and Kirby Urner as
well. I am writing the letter to the Foundations of Mathematics group
and so many possibilities are opening up that I simply have to go
through the most basic of them.

Andrius

Andrius Kulikauskas
m...@ms.lt <mailto:m...@ms.lt>
+370 607 27 665 <tel:%2B370%20607%2027%20665>

[Joseph Austin]

Steve, Andrius, Kirby, and others with a philosophical interest--

I'm wondering whether you are familiar with the e-prime [English-prime] movement?

https://en.wikipedia.org/wiki/E-Prime

This is a group of linguists who advocate removing forms of "to be" from the English (and presumably all other) language,

on the ground of ambiguity or over-reach.

I myself once presented a paper relating four uses of "to be" to four "is"s of object-oriented programming,

beginning with "is a " vs "has a":

is a kind of, is a part of (inverse of "has a"), is called (identity/name),  is instantiated (existence).

It certainly raises questions as to the specific meaning of questions such as "what is mathematics", and issues of models, interpretations, etc.

Do mathematical relationships "exist" or are they (merely) attempts to formulate prediction rules for observed patterns in nature?

When we see a supposed pattern in nature, do we say:

(a) that pattern appears to be like this mathematical operation that we already have

or 

(b) let us define a mathematical operation that behaves like the relationship we perceive,

and then integrate it with the ones we already have.

For example, does anyone really believe that the constant e is fundamental to the "rotational" effect that directed physical quantities have on each other,

which we model as rotation of vectors expressed in polar-coordinates in the complex plane, or is "e to an imaginary power" just an artificat of the "algebra" we have formulated to allow us to use our arithmetic methods to calculate quantitative relationships?

Joe Austin

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[kirby urner]

Thanks for introducing me as a next Buckminster Fuller, Andrius.  I'm more the unknown former school teacher who latched on to some of Fuller's mathematical ideas after college, coming from a philosophy background.  Very few were attending to these memes even at his peak of fame as Synergetics is hard to read (for a humanities book yes, but it doesn't have a lot of equations either so it's accessible in that sense -- dense like philosophy is dense, e.g. Heidegger).

At the core of synergetics is a stack of familiar polyhedrons with unusual volume numbers, though with ratios we would recognize. It does not "invalidate" any existing math as it comes with its own way of doing business that's internally consistent, regarding how vectors might be multiplied.  A unit tetrahedron is at the core of volume computations. From there, we get a space-filling rhombic dodecahedron of volume six.

The main philosophical hurdle is seeing how vector multiplication might be other than it is.  But that's really no hurdle at all, given we've had a history of treating things more complicated than single numbers, such as polynomials and complex numbers, to various suites of algorithms, with algebras having group and field properties such as closure and associativity.  Clifford Algebra is somewhat a culmination of research in this direction.

What's useful about applying a "synergetics skin" over some computations is going from a quadrilateral to a triangle, cube to tetrahedron, may be done without changing the numerics -- if one wishes -- because the triangle and tetrahedron have a fixed ratio compared to the geometric figures we usually use.  Applying a constant or simply thinking "in different units" does not have to have many ripple effects, just adds a nudge.  One has more of a Necker Cube effect, a focus of Wittgenstein's philosophy.  New aspects of things or situations simply pop into view.  How does that phenomenon relate to what we call "meaning"?

Turning to Joe's investigation below, I'd suggest that mathematics can be a lot like a pair of glasses in helping sharpen and bring into focus things we might have overlooked.  We can call these things "generalizations".  In the world of software, which we might agree to call "applied math" or "machine executable math" (Mx), we save having to physically build models from metal and wood given we've used mathematics to extract the general ideas, such as scaling, translation, rotation, then adding time and Newton's Laws, more laws as discovered and reduced.  We still need lots of hardware though.  Rotating that letter F as in Peter Farrell's "Hacking Math Class" book, requires the Pi3D library running on a Raspberry Pi.  I may buy one today.

As a project, I'm looking at a nearby mostly unused gigantic former church as a place we could do more learning, teaching and designing.  The Methodists own the property.  I'm wondering if the local science museum would take me on with sponsors donating to my projects there, so that we don't have to create a new nonprofit from scratch first.  We have existing institutions that could pool resources.  I'd have exhibits on some of this Fuller stuff included amidst the bar-code readers and various other "play station" exhibits where hardware and software mix together (robotics etc.).

Kirby

[Joseph Austin]

On Jun 17, 2016, at 11:49 AM, kirby urner <kirby...@gmail.com> wrote:

The main philosophical hurdle is seeing how vector multiplication might be other than it is.

Kirby et al,

The more I think about "multiplication", the more I think the term is being abused.

What we call "scalar" multiplication seems to capture the essence of one understanding: multiplication as repeated addition, the Peano definition.

Physically, this often take the form of a "production" equation:  rate * time = product;  e.g.  60 miles per hour times 4 hours = 240 miles.

(Which, BTW, is the operation of extracting the distance component from a speed, not constructing a composite of distance and time.)

But then there are all sorts of other multiplications.

My question: what do you call "volume conserving" transformations"?  Is there a "branch" of mathematics devoted to their study?

Clearly translations and rotations fall in this category, but only certain kinds of "stretching".

But more generally, consider the shapes assumed by a blob of viscous liquid (of uniform density) tumbling through space or distributing itself over an uneven surface,

My intuition is that "volume" is conserved by nature, even pseudo-volumes composed of multiple types such as kilogram meters  or volt amperes.

And taking a cue from Geometric Algebra, "rotations" are not so much "multiplications" as "additions" of arcs.

As for addition, we see in "vector addition" and more generally in molecular structure, 

that physical "additions" include a component of direction as well as a component of magnitude, so we might call this "construction".

So we have a "constructive" operation of addition, extended to scalar multiplication;

And a "transformative" operation traditionally called "multiplication" but which I would rename as, well, "transformation".

So my "new math" would have operators of "construction" and "transformation", which both can be considered forms of "arrangement",

which probably ends up in group theory!.  

If I were a mathematician, I would probably be able to develop such ideas into a "grand unification of mathematics"!

But for now, I'm still wrestling with the idea that the exponentiation operator has two inverses 

and yields (at least) two new kinds of irrational numbers: radical and complex.  

Complex seems to come from the dimensionality-increasing nature of "multiplication", 

and radical from the notion that "root taking" should be closed and single-valued, i.e. all n factors of x^n equal.

[Eureka!  In my own mind, I had never crystallized the idea into such statements until just now!]

Coupled with the idea that the algebraic sign of the integers is just the 1D version of direction,

we have the concept that "numbers"--or more generally, quantities--have both a magnitude (arithmetic) and a directional  (geometric) component.

So we need to put Euclid and Peano together!

But back to multiplication.

As I have been saying, I think we have been looking at "multiplication" thru the wrong end of the telescope.

I say "nature" starts with multi-dimensional hyper-volumes, and we humans observe and compute the projections.

Which led to my original question: is there a mathematics of "volume conserving" (constant integral) transformations?

Joe Austin

[kirby urner]

Greetings Joe et al --

I enjoy these recent "thinkings out loud" from Andrius, Brad, Joe and myself
because we're each demonstrating Maria's "make math your own" as a
process wherein one gives free rein to intuition. 

We also see a lot of critiquing or setting concepts at a distance where me
might even contest them, re-purpose them. 

I go back to my three Cs:  Comprehend, Critique, Compose (Compute).

Those work better for me than the three Rs.

A way to get at multiplication is to ask oneself "what things multiply?" which

unpacks into the longer "what types of things multiply?" 

A reason to include computer languages in mathematics learning is the
sense of "types" these bring with them.  "What types...?" becomes more
concrete when you have a type( ) function to which any object may be
fed, giving back strings like 'int' or 'float' or 'Fraction'.

In [273]: from fractions import Fraction # template

In [274]: numba = Fraction(3,5)  # give birth to one

In [275]: numba * numba  # self multiplying is "powering"

Out[275]: Fraction(9, 25) 

In [276]: type (numba)

Out[276]: fractions.Fraction

Python didn't always have a Fraction type inside a fractions namespace.

I recall on edu-sig when we were brainstorming a Rational Number type.

Good thing we didn't call them "rat" types. 

I sometimes make a "Q type" (not a Q-tip) from scratch, using the name
Q deliberately to remind of N < Z < Q < R < C where < means
"is contained by".

We could say Real is a sub-type of Complex (C) and Q (rational) is
a subtype of R, meaning complex numbers contain all the reals and
more, whereas rationals are all in reals, but reals contain so much
more than the rationals.

Are vectors any of these types?  They're assembled using these

types.  Computer languages show that process, of assembling

types from component types.  Polynomials:  another assembled

type.  Quaternions, matrices...

The spherical cuboctahedron Andrius made from paper plates is

what Brad spontaneously calls a VE ("vector equilibrium") because

he (like me) has steeped himself in Synergetics more than most,
rolled in that namespace.  As evidence I've rolled around in this
namespace, I keep bringing up AAB = MITE where A = 1/24,
MITE = 1/8 and 8 x MITE = Coupler, a space-filler, oblate.

Actually, more formally, VE represents two ways of acting in balance:

the divergent vectors from the center are pushing out explosively,

while the circumferential vectors are stretched, in tension, a different

disposition of "force".  The pusher vectors could be made of wood,

the puller vectors made of string.  If we make 8 wooden tetrahedrons

pushing out, with those square windows in between, the rods are

seen to pair up and we get 24 of each:  24 on the surface, 8 x 3,

and 24 pushing out, the radial 8 x 3.  24 == 24 hence this name VE

for "vector equilibrium".  Not synonymous with cuboctahedron but

same ballpark.

But then are vectors "allowed" to push and pull in the first place? 
They're conceptual and are designed to communicate "force" we
might say, though in place of "force" feel free to use either "tendency"
or "inclination".  Even "bias" has a role in this language game, as

does "spin".

[ Regarding "spin":  Fuller spent some time at Sperry, the gyroscope

company, because he really wanted to understand the precession

phenomenon.  Synergetics has some long passages on what he

learned. ]

Synergetics leaves timeless Euclidean-Platonic space behind,
much as Einstein space does:  by splashing down through a time
dimension, to our under water or "fallen" world of spatio-temporal
physical objects.  This is Energy World, the world of frequencies,
only hinted at in Platonic forms. 

Once in time (clock ticking, action running "forward" or "backward")
we have tension and compression members, as always co-occurrent
and complementary in Synergetics.  Another kind of conservation

or symmetry constraint.  Push down on a barrel lid and the contained

fluid stains against the hoops that keep the staves wedged together.

The induced compression (pressure) begets tension (straining).

Andrius is exploring the world of N-D polytopes, where N-D means

as many dimensions as you like, all them perpendicular to one

another in a phase space of mutual independence.  None of the

N basis vectors depend on the others, as if they do they may be

expressed as vector sums, not as basic in their own right.  They're

"composed" of more primitive elements.

The quadrays mirror the Cartesian in many ways, in expressing

locations with addresses reflective of basis vector roots.  It's the

same tip-to-tail vector addition operation as in XYZ.  You may

hold a vector (cylindrical, rigid) and stretch or shrink it, but it

resists all attempts to rotate in any way.  Other operators would
be required for that, and the basis vectors do not rotate, period. 

This growing and shrinking is called scalar multiplication and
makes vectors longer or shorter, a dimension we call "magnitude"
and represent thus: |v|.  In the XYZ game we have the magical
scalar -1 where - is the operator of mirroring or reversal. -v
means (-1) * v, the scalar -1 times (multiplication operator)
and v + (-1) * v = the zero vector (additive identity). Quadrays
is like that too.  (1,0,0,0) + (-1) * (1, 0,0,0) = (0,0,0,0).  We

have the same group properties of closure, identity element,

and inverses, addition with associativity.

However mathematicians were historically not content with

only scalar multiplication and wanted to find a way for the
vector type to multiply with other vector types, like complex
numbers were seen to do.  The meaning of  "vector", like

"multiply" is somewhat malleable, so long as we recognize

a "vector field".

The word "product" was employed, which had been tied closely
to "multiply" in that two things multiplied gave a product (progeny)
whereas two things added gave a sum. 

We know of several different ways ways two vectors form to
make a product:   cross product and dot product.  Inner product,
outer product.  Wedge product.  Inner product does not have

closer in that (dot vector vector) -> scalar.  (cross vector vector)

stops being meaningful after three dimensions according to

some sources.

Which operations are defined and make products, and
whether these objects that multiply are still called "vectors"
or not depends on the algebra in question. 

[

http://mathworld.wolfram.com/WedgeProduct.html

(we're now in the neighborhood of Clifford Algebra).

We're also talking about K-forms now, not just vectors:
http://mathworld.wolfram.com/Differentialk-Form.html
https://en.wikipedia.org/wiki/Differential_form

]

N-D 4D is the Coxeter namespace Andrius is exploring.

He's in a Euclidean world.  However I think Joe wants to

have more spatiotemporal content so he can think about

the physics of things and that requires force (tendency,

inclination).  We need action with the advance of time,

mass in motion (mvd or pd).

N-D + Time or Euclid + Time = Newton in a lot of ways.

Newton thought of Universe as a really big room wherein

everything happens on a single timeline or thread.  There

was one clock for all events.  His universe was happy with

a single clock for all the stars and galaxies. His was a one

chip universe.

The consensus of one clock to rule them all started to

break up with Einstein and time became more problematic

in terms of how to add it to the algebra.  The observer

coordinate system had to define itself relative to another

observer coordinate system, a pair at least.  Multiple

observers need not agree on event ordering at first

though perhaps with further communication an ordering

could be imposed by convention / agreement.

Fuller's 4D is Euclidean in the sense that time has yet to
appear (4D is conceptually "pre-frequency") and like XYZ
it has only the three mutual perpendiculars of the tetrahedron
(opposite pairs of six edges). 

There is no fourth perpendicular of N-D Euclidean space,
as we follow Minkowski et al into Time via world lines, with
tension and compression now materializing, allowing vectors
to communicate tendencies, forming relationships we

both observe and participate in (to observe is to participate).

I like Joe's volume-conserving blob, that morphing volume

tumbling through space. I suggest this might represent energy
being conserved (volume) but with the shape itself recording
information in some way, say by looking like a person in the
shape of a bust, a statue of the head and shoulders (we
could use the "Face on Mars" meme, making it Richard Nixon
for comic value). 

This "bust" aspect gives us a measure of increasing entropy: 
as the bust aspect is seen to diminish to where we just get
a faceless lump, not recognizable as a bust, but still with the

same volume.

Kirby

[kirby urner]

On Sat, Jun 18, 2016 at 10:02 AM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>
 

A reason to include computer languages in mathematics learning is the
sense of "types" these bring with them.  "What types...?" becomes more
concrete when you have a type( ) function to which any object may be
fed, giving back strings like 'int' or 'float' or 'Fraction'.

Actually what the type( ) function returns is not a string type object

but something more abstract:

In [277]: type(type(numba))

Out[277]: abc.ABCMeta

 

abc stands for "abstract base class".  There's also some historical

encoding as Python itself "derives" from a language called ABC that

never made it out of its original environment (except as morphed into

Python).

However there's a string representation of these types, controlled

by __repr__ as Peter and I were discussing earlier.

In [273]: from fractions import Fraction # template

In [274]: numba = Fraction(3,5)  # give birth to one

In [275]: numba * numba  # self multiplying is "powering"

Out[275]: Fraction(9, 25) 

In [276]: type (numba)

Out[276]: fractions.Fraction

 

The spherical cuboctahedron Andrius made from paper plates is

what Brad spontaneously calls a VE ("vector equilibrium") because

he (like me) has steeped himself in Synergetics more than most,
rolled in that namespace.  As evidence I've rolled around in this
namespace, I keep bringing up AAB = MITE where A = 1/24,
MITE = 1/8 and 8 x MITE = Coupler, a space-filler, oblate.

"rolled around in" is verbiage that smells a lot like dogs, as dogs

like to roll around in stuff.  Brad and I went to the Lucky Lab, a

brewpub, but that didn't mean kids couldn't come piling in.  I took

this picture while there:

https://flic.kr/p/HVVruB

Here's one with Brad in the foreground, kids in the background

(I imagine them folding paper plates, not drinking beer or overly
sugary drinks, some with genetically modified corn syrup):

https://flic.kr/p/HVVqdt

I lot of marketing has gone into trying to make "Good" be like

"Sweet".  There's also "cute" and "too cute".

In not needing negative numbers at the end of the day, to give

addresses to the whole domain (XYZ's domain), I'm maybe

suggesting an inside-outing through the origin that takes us to

a whole other Universe, where a negative tetrahedron rules,

an "Evil" twin.  I use scare quotes because the more neutral

terms, Left and Right, or Positive and Negative might be

employed.

In full blown Jitterbugging (the core transformation in Synergetics),
there's a collapsing down from the VE (via icosahedron, octahedron,
tetrahedron) to a singularity and a re-expansion into this alternative
IVM (volume or space), a kind of "dumbbell" structure in that we
have full expansion at either end, and a turnaround. 

I tend to go with the dorje or vajra as a representation, given family
history and the investments our household has made in Vajrayana
Buddhism, as seen from the Photostream.

http://buddhism.about.com/od/buddhismglossaryv/g/vajradef.htm
(all math is ethno-math)

Fuller called it "bow-tie Universe" in Synergetics.

Once in time (clock ticking, action running "forward" or "backward")
we have tension and compression members, as always co-occurrent
and complementary in Synergetics. 

When the clock is set to "running too fast" in the sense that we

see phenomena sped up a lot, that's the observer advancing relatively

slowly in time, taking input at a lower rate.  We see this in time lapse

photography, where laws of physics appear "broken" unless we factor

in the increased clock speed and our relative slowness:

http://nofilmschool.com/2016/06/amazing-time-lapse-singapore-took-3-years-shoot

I encourage adding these educational time-lapse films to the GST
library / syllabus where GST = General Systems Theory. 

Human displacement flows, deployments, moving through transit centers,
is sometimes best seen using time lapse.  The twelve million dispersed

from Syria, Libya, other war zones, is a data visualization challenge

and part of climate change.  Supercomputers may need to be enlisted.

Climate = Biosphere.

The quadrays mirror the Cartesian in many ways, in expressing

locations with addresses reflective of basis vector roots.  It's the

same tip-to-tail vector addition operation as in XYZ.  You may

hold a vector (cylindrical, rigid) and stretch or shrink it, but it

resists all attempts to rotate in any way.  Other operators would
be required for that, and the basis vectors do not rotate, period. 

I would not claim that all addresses being positive numbers is an

expression of "all Good". 

Yes, we could have a cartoon in which any negatives in the address
were considered "wrong side of the tracks" such that in XYZ we
have 7/8ths slums and 1/8th all positive place to live, the "right
part of town". 

In that sense, quadrays are less discriminatory in that no address
ends up with a negative sign in the 4-tuple.  That's in the canonical
representation.  We have an equivalence class, like in fractions,
of many ways to say the same thing i.e. (-1, -1, -1, -1) = (0,0,0,0).

In another representation, we may insure that our four coordinates

add to zero, allowing three to 'overshoot" and the last one to

"reel us in" in its negative direction.  I have methods norm0 and

norm in my Qvector class to implement these normalizations.

In [279]: q = Qvector((5, 4, 0, 3))

In [280]: q.xyz()

Out[280]: xyz_vector(x=1.414213562373095, y=-0.7071067811865475, z=2.1213203435596424)

In [281]: q.norm0()

Out[281]: ivm_vector(a=2.0, b=1.0, c=-3.0, d=0.0)

However if people want to get snooty, they can complain about
which slot has the zero.  Maybe a zero in the first slot, called
Quadrant Zero, is the hot place to live and the (0, i, j, k) people
look down on the (i, 0, j, k) people.

We need to just build into our model that people want to be

"insiders" and that means they need "outsiders" to compare

themselves with.  That's what a Castle defines:  a castle keep
(on the inside, where to keep the "crown jewels") and the
expanse beyond the Castle walls (plus a moat).

In XYZ we have eight octants (neighborhoods) whereas with

these quadrays, fanning out from the center of any unit radius

ball, to four voids in between, we get only four neighborhoods.

Maybe a Quadray King or Queen has an easier time as a ruler,

with fewer neighborhoods to govern.

Probably every castle wants to think of itself as an origin, or

(0,0,0,0).  The mappings become relative.  Scene graphs enter

the picture, to maintain a "gods eye" rubric, i.e. all these castle

keeps need to be interconnected with "roads" (vectors of

communication, for the exchange of ideas).  We'll need a Bureau

of Public Roads to make sure no one castle gets too entitled and

tries to reserve all the roads to itself as the privileged user, perhaps
blocking the free exchange of ideas even between other castles.

Sounds like a computer game doesn't it?

The word "product" was employed, which had been tied closely
to "multiply" in that two things multiplied gave a product (progeny)
whereas two things added gave a sum. 

We know of several different ways ways two vectors form to
make a product:   cross product and dot product.  Inner product,
outer product.  Wedge product.  Inner product does not have

closer in that (dot vector vector) -> scalar.  (cross vector vector)

I meant "closure", not "closer". 

The concept of closure goes back to type awareness.  Along the

Lambda Track we'll be sure to keep it CS-friendly and introduce

Group, Ring and Field properties again, to reinforce this "type

awareness".

If  object * object -> object and type(object) always returns the same
type, i.e. we don't "leave the domain" or "kingdom" as a result of
doing our multiplication, then we're well on the way to having a
group, or at least a ring. 

If we're really lucky, we'll get to have a field to rule (govern).

Kirby