more about 4D coordinates in ordinary volume

[kirby urner]

I've appended some verbiage shared with another dimension theorist (not Karl Menger who has passed on), giving more of the flavor of 4D Q-rays or "quadray coordinates".

In pushing this topic to the art schools, I'm reconciling myself to the fact that pure mathematicians don't want to touch anything redolent of Synergetics, which J.H. Conway dismissed out of hand as a work in the humanities (which it is, but not without mathematical content). 

Notice the semantic network Andrius came up with, purporting to cover all of contemporary math, has no room for Synergetics.

H.S.M. Coxeter worked closely with Fuller in a couple chapters, including at University of Toronto, as is documented in The King of Infinite Space.  Coxeter's first impression of Fuller was negative because Coxeter's own son was trying to break into the radome business (cite DEW line) and came up against Fuller's patents. 

"You can't patent naturally occurring patterns" was Donald's attitude, anticipating open source.

When they met in person though, Donald's attitude softened, and he later gave permission to Fuller to dedicate Synergetics to him.  He even contacted Fuller about building a geodesic dome home, when his own home burned down late in the game.

From my standpoint, a rational whole number volumed concentric hierarchy of polyhedrons is a very useful tool to have in inventory, with space-filling rhombic dodecahedrons of volume 6 and so on. 

To review:

Tetrahedron (1)  edge = V

Cube (3); face diagonal = V

Octahedron (4); edge = V

Rhombic Dodecahedron (6); long diagonal = V

Icosahedron (18.51...) <--- incommesurable / "irrational"); edge = V

Cuboctahedron (20); edge = V

What could be easier to teach in 3rd grade?  The volume formulae now are so complicated, why not streamline and nest them?  One may always convert to XYZ volumes in another chapter. 

Learn there's more than one way to do it, early, before hardening of the mental arteries sets in.

Getting to this volumes table, however, requires a unit-volume tetrahedron and that means understanding a different model of powering i.e. 2 x 2 may be represented as a triangle of edges 2, and 2 x 2 x 2 may be shown as a tetrahedron. 

http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html

None of the arithmetic changes, but we no longer say "2 squared" and "2 cubed" as a matter of un-reflective reflex, as is done in mathematics.

When it comes to reaching 3rd graders, this "quadpod" is useful:

http://controlroom.blogspot.com/2009/03/quadpod.html

Based on the power sum from the thrusters, this pod is free to fly around in any fly-like pattern.  Four fixed thrusters are mathematically sufficient to power motion in any direction.  Three would not be, unless we introduce rotation in addition to vector sum based translation.  

In the Cartesian system of six thrusters we have both x and -x directions i.e. the six thrusters may point at 180 degrees from one another in order to move in any direction.  This is not as economical, meaning six thrusters are needed instead of just four.  A third grader could grok that I think.

My own approach, given mathematics has been unable to digest a unit-volume tetrahedron, has been to do an end run through the philosophy department, which proved a winning strategy.

http://controlroom.blogspot.com/2015/07/ramping-up.html

From philosophy, we move to other areas in the humanities, art history and sculpture especially.

Kirby

To another webmaster:

...  showing how the quadray apparatus squishes to flat and linear is useful.
 

On a line, we could say left is like (1, 0) and to the right is (0, 1).  If you want to go far to the right, go (0, 10.1). 

We see how the need for negatives is replaced by two rays pointing oppositely, each with its own "slot" in the (Q0, Q1) "tuple". 

A "tuple" is just an ordered list of numbers in this case e.g. (1,2,3) or (0,1,2,3).  Every basis vector gets a slot, so in Q-rays we use four for volume, three for flat, two for line.  All of these sculptures occur in the same observer-observed "tent" -- we're free to alter our angle of observation.

On a plane, three rays fan out from (0,0,0) and each sector is bordered by two rays.  One may move only in the direction a basis ray points (like on a chess board but here three choices), but for any distance. 

(45, 19, 0) is a point in a sector that has no border with the 3rd basis vector (so it's unneeded and set to 0).

In volume, four rays fan out from (0,0,0,0) at about 109.47 degrees to one another (caltrop).  Every quadrant with a buzzing fly (P) is like XYZ but with the basis vectors splayed wider apart.  Otherwise the game, of moving only in alignment with basis vectors, is the same. 

In XYZ, we get to any point by moving in the direction of each basis vector as far as we need.  Same in Q-rays.

(1, 1, 0, 0) is on the plane betwixt the plane formed by (1,0,0,0) and (0,1,0,0).

(1, 1, 1, 0) is out in a quadrant, floating. 

We could actually write it as -(0,0,0,1) but that's not the canonical form, is more applying - as a unary operator to flip it 180 degrees.  Once flipped, an all-positives canonical address is always possible.

[Joseph Austin]

On May 21, 2016, at 11:33 AM, kirby urner <kirby...@gmail.com> wrote:

Getting to this volumes table, however, requires a unit-volume tetrahedron and that means understanding a different model of powering i.e. 2 x 2 may be represented as a triangle of edges 2, and 2 x 2 x 2 may be shown as a tetrahedron.  

None of the arithmetic changes, but we no longer say "2 squared" and "2 cubed" as a matter of un-reflective reflex, as is done in mathematics.

And let's not forget that Euclidean space itself is a mathematical abstraction.

So far as we  know, there is no "Euclidean space" in actual physical space--everything is curved.

Even the Nile Delta, the supposed realization of Euclid's geometry, is now "known" to be a portion of a curved surface of a gigantic sphere.

And space is not even isotropic: physical behavior in the radial direction is significantly different than in circumferential directions, due to "gravity".

And what if "gravity" is essentially "inertial resistance" to an accelerating expansion or a spinning of the universe?

(We know we can create "artificial gravity" by spinning a container.)

I've noted elsewhere that the "product" of two lengths is not a length, but an area. So who decides that "area" should be measured in units of "squares" instead of units of "triangles"? (or discs, or whatever?)  "Area" as a concept is related to the amount of "surface" occupied and can remain invariant among a variety of shapes of it's boundary, including irregular shapes ("area under a curve"), even many non-contiguous regions.  We could even go further, and consider surfaces that are not flat, or of non-uniform curvature and even non-uniform density--what does even "area" mean then?

Joe Austin

[kirby urner]

On Sat, May 21, 2016 at 1:07 PM, Joseph Austin <drtec...@gmail.com> wrote:

On May 21, 2016, at 11:33 AM, kirby urner <kirby...@gmail.com> wrote:

Getting to this volumes table, however, requires a unit-volume tetrahedron and that means understanding a different model of powering i.e. 2 x 2 may be represented as a triangle of edges 2, and 2 x 2 x 2 may be shown as a tetrahedron.  

None of the arithmetic changes, but we no longer say "2 squared" and "2 cubed" as a matter of un-reflective reflex, as is done in mathematics.

And let's not forget that Euclidean space itself is a mathematical abstraction.

So far as we  know, there is no "Euclidean space" in actual physical space--everything is curved.

Even the Nile Delta, the supposed realization of Euclid's geometry, is now "known" to be a portion of a curved surface of a gigantic sphere.

And space is not even isotropic: physical behavior in the radial direction is significantly different than in circumferential directions, due to "gravity".

And what if "gravity" is essentially "inertial resistance" to an accelerating expansion or a spinning of the universe?

(We know we can create "artificial gravity" by spinning a container.)

Yes, good point. 

Einstein-Minkowski space introduces the "time" variable such that we have "time-space" and a four-coordinate view of the universe in which (x, y, z, t) pertains to each event, relative to some "inertial" coordinate system (in general relativity that "inertial system" goes away more than in special relativity). 

We develop the concept of "world lines" wherein in spatial existence stretches out behind us in the time domain.  We might use a stick-figure ballerina to demonstrate these "world lines":

https://youtu.be/38iz0-dopSg
https://youtu.be/3WehC6LxZe8

(both from my First Person Physics portfolio, part of a CDROM at an AAPT conference some years back, AAPT being American Association of Physics Teachers).

A lot of lay-folk get confused when confronted with two 4D memes:  Einstein's, as above, and "extended Euclidean" as in Coxeter.  That latter is not especially concerned with time, as Euclidean geometry is essentially timeless.

As Coxeter himself pointed out, in 'Regular Polytopes':

Little, if anything, is gained by representing the fourth Euclidean dimension as time. In fact, this idea, so attractively developed by H.G. Wells in The Time Machine, has led such authors as J. W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity. Minkowski's geometry of space-time is not Euclidean, and consequently has no connection with the present investigation.

H.S.M. Coxeter. Regular Polytopes. Dover Publications, 1973. pg. 119

 

I've noted elsewhere that the "product" of two lengths is not a length, but an area. So who decides that "area" should be measured in units of "squares" instead of units of "triangles"? (or discs, or whatever?)  

Exactly.

Lets choose triangles, why not?  They're the minimum polygon after all.

Here's a Youtube I made, on the model of Khan Academy, showcasing a triangular model of A x B:

https://youtu.be/2B1XXV2Eoh8

I notice your comment that it's easier to read off the area when the angle is 90 degrees.  But if you're used to a 60 degree canonical apex at the origin, and just count up the intervals along each edge (or fractions thereof), it's about the same "mental process" (I hesitate to introduce one, a "mental process" that is -- I'm Wittgenstein trained and he discouraged over-imagining thought as "processes").

Another spin on the 4D meme came after the Einstein-Coxeter fork, if we peg it formally to the publication of Synergetics in the late 1970s (two volumes, index in the 2nd volume, now on the web [1]).  That's the geometry wherein volume reduces to the four planes of the minimum enclosure i.e. the topologically minimum "stick figure", the tetrahedron.

We're left with three meanings of 4D by the end of the 20th Century:

4D:  relativity theory, includes a time variable

4D:  higher dimensional polytope geometry ala Coxeter and n-dimensional sphere packing

4D:  Fuller's emphasis on the tetrahedron as conceptually primitive

Using namespace notation, one might write:

Einstein.4D   (time is a variable, in addition to 3D space)

Coxeter.4D   (extended Euclidean, timeless, all axes spatial)

RBF.4D        (tetrahedron as primal and timeless, frequency adds time/size)

thereby anchoring the 4D meme to a contextualizing namespace in each case (we choose a principal for brevity and shared recognition, but of course many contributed to each namespace). RBF = R. Buckminster Fuller (a familiar-enough acronym in the literature).

 

"Area" as a concept is related to the amount of "surface" occupied and can remain invariant among a variety of shapes of it's boundary, including irregular shapes ("area under a curve"), even many non-contiguous regions.  We could even go further, and consider surfaces that are not flat, or of non-uniform curvature and even non-uniform density--what does even "area" mean then?

Joe Austin

All good questions. 

Coming form a background in physics, we know better than to consider "perfect solid" a reasonable concept.  What are we really talking about here, in terms of energetic phenomena?  Atoms?  Quanta?

If the phenomenon is non-energetic i.e. "imaginary only" then need we concern ourselves with it to the same degree?  Logical positivists might want to cut their losses and say "no, if it's not modeled anywhere energetically, it's but a figment of the imagination and we have better things to consider than mere fairy tale figments." 

I'm not joining this chorus, as I put a lot of value in "mere figments" but I do understand why things that don't exist might be of less interest to empirical science.  Energetic phenomena have an edge, in being real enough to be realized. 

Yet information content and entropy are real as well in their own way, suggesting another arrow besides the physical-energetic one. 

The same energy may contain more or less information, one might say.  All novels or papers of exactly N words are not equally "sensical" although we might imagine an AI-bot that generates equally "non-sensical" (yet grammatically correct) papers. [2]

Kirby
 
[1]   http://www.rwgrayprojects.com/synergetics/synergetics.html 
(check semantic web for how it integrates with math maps like Andrius is working on)

[2]  http://www.elsewhere.org/journal/pomo/ 
(hit reload to generate a new paper in this namespace)

[Andrius Kulikauskas]

Kirby,

I think your quadpod is a magnificent concept for illustrating your
points. It's very vivid and fun, too.

I am impressed by your geometry
http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html
which is intriguing and persuasive.

However, if you line up the corners of the squares and also of the
cubes, then you get a progression which is very helpful for teaching
calculus, namely, if you consider a square x and grow it by one more bit
h so you have a square of sides x+h, then:
(x + h)**2 = (x + h)(x + h) = x2 + 2hx + h2 which all make
geometric sense, and then you can see why you can ignore the h2 and upon
subtracting x2 you are left with 2hx which, when compared with h, gives
you the derivative 2x.

Similarly,
(x+h)**3 = (x+h)(x+h)(x+h) = x3 + 3x2h + 3xh2 + h3 and discarding
the small stuff and substracting x3 you are left with 3x2h and dividing
by h gives the derivative 3x2.

This for me is a very powerful way to illustrate differentiation in a
very real sense. And also these binomial expansions are very worthwhile
to spend time with and very meaningful for problems in probability,
heads and tails: (h+t)**3 or recessive and dominate genes, blue eyes
b and brown eyes B (b+B)(b+B) for example.

So I'm curious if your triangular thinking has a nice way to talk about
this all, perhaps?

P.S. I forgot to say in my other letter that I'll be suggesting a talk
at a philosophy (aesthetics) conference here in Vilnius, Lithuania, most
likely to be accepted, about mathematical beauty. I appreciate thoughts
on mathematical beauty (I suppose through a new thread). My main
thought so far is that mathematicians (at least me) would typically not
consider the Mandelbrot set as beautiful because you can only see it,
you can't imagine it. Whereas Galois theory is beautiful because it
empowers the imagination. So I want to explain what it takes for the
Mandelbrot set to become beautiful for a mathematician. Also, I want to
link in with architect Christopher Alexander's 15 principles of life.

Andrius

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

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[Andrius Kulikauskas]

Hi Kirby, Joseph and all,

Kirby, thank you for mentioning Synergetics. I will look into that and
add it to my map of math areas.
http://www.azimuthproject.org/azimuth/show/Andrius+Kulikauskas
I hope to work on that tomorrow. Also, I want to highlight some areas
that I think relate to my philosophy. I just learned of the "field of
one element":
https://en.wikipedia.org/wiki/Field_with_one_element
It's a nonexistent mathematical concept (fields are supposed to contain
at least two distinct elements, 0 and 1) which apparently has spurred
quite a bit of research. It suggests itself in different situations as
a limiting initial case. I hope to learn more about it and report. But
my impression is that it relates to my concept of a God who goes beyond
himself into himself, who asks, Is God necessary? Would I be even if I
wasn't?

Another key concept for me is the idea of an "unmarked opposite" vs. a
"marked opposite".
* We can have what is beyond a system be identified with an opposite
within the system. For example, a blank sheet of paper can be noted by
the empty set. The empty set is opposite to nonempty sets. And there is
a sense in which the empty set is preferred, is central. Or we can have
the identity element of a group which expresses no action at all.
Similarly, good can be distinguished from bad by claiming that God is
good, where God is what is beyond the system. So here in this sense I
say that good is the marked opposite, the one identified with what is
beyond the system.
* Also, in a different sense, in a system, a marked opposite is when you
have two opposites (choices) that are clearly distinguished and one is
the default (thus preferred) because it is unmarked, whereas the other
one is marked to distinguish it and thus secondary. For example, 1 and
-1. 1 is unmarked and -1 is marked (and it actually has an extra
mark). And they are clearly distinct: -1 x -1 = 1 whereas 1 x 1 = 1.
* Finally, we can have "unmarked opposites" where two choices are
distinct but otherwise not distinguishable. They have yet to be
marked. For example, the two square roots of -1. One will imagined
clockwise, the other counterclockwise, perhaps. But which is which
doesn't matter. Only when we name them using + and -, only when we
attribute such a prejudice to them, do we lose touch with their original
indistinguishability, thus ending up with i and -i, forgetting that -i
is no less basic than +i.

So I'm very interested where such dualities and opposites come up in math.

Andrius

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


2016.05.21 18:33, kirby urner rašė:
>

[Bradford Hansen-Smith]

And let's not forget that Euclidean space itself is a mathematical abstraction.

So far as we  know, there is no "Euclidean space" in actual physical space--everything is curved.

---As far as we know all mathematics is an abstraction, and does not exist at all, except as ideas in the mind---and what ever it is when we experience "actual physical space."

Triangles, squares, 90 degree angles, etc, all ideas about what people have decided for us about our experience. Kirby and Andrius are at least open to deciding for themselves some of these things that otherwise go unquestioned.

The triangle and all polygons are truncated circles, but it has been decided for us that circles are semiotic symbols of nothing, useful in thinking math and for cutting into 2-D  pieces to move around geometrically to prove any number of give ideas. 

The traditional circle, primary to all geometric figures, is a picture of a spatial object. Life is said to be about folding and unfolding in space, DNA, protein, and so on.  I don't know if that is true or not, but the first fold and crease in the paper circle object " pancake" I do know to be a tetrahedron pattern of dual nature, meaning there are two; both with two open triangle planes and two closed triangle planes, positive and negative to distinguish between them, though both are positive to the front and back of the circle. They are not separate but discernible only one at a time. This gives presidents to the triangle over the square. The folded crease is a perpendicular movement between any two points on any flat plane to the distance between the two points. Triangulation in the form of dual tetrahedra where 90 degrees is about movement of relationship explains the nonstructural nature of the square and cube unless triangulated which put them into the tetrahedron camp that is grounded in the circle revealed through movement.

Andrius, I suggest you do some serious folding of circles so you will know where to put that on your semiotic chart of math classifications, along with where to place Synergetics. Until we collectively put time into folding circles, at least as much as we have spent drawing pictures of them, now having computers do that for us, we will continue to have no idea about the extraordinary amount of math information, reformations, and transformations that we have missed while developing math from 2-D images.

Brad

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Bradford Hansen-Smith
www.wholemovement.com

[kirby urner]

On Sat, May 21, 2016 at 3:22 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

Kirby,

I think your quadpod is a magnificent concept for illustrating your points.  It's very vivid and fun, too.

Re: 
http://controlroom.blogspot.com/2009/03/quadpod.html

Yes, thank you.

I was fortunate this artist, known as Hop, was following
the thread and imagining the quadpod as I did, plus he
had the skills to render it so tastefully as an animated GIF.

The quadpod helps show why just three fixed direction
thrusters (i.e. basis vectors) are insufficient to "fly as a
fly flies" i.e. omni-directionally in whatever curlicue pattern. 
We would have to allow thruster rotation (change in direction),
which in this language game I'm disallowing.

With just three positive thrusters and no rotation, you
can move however you want on a plane (plain), by
keeping the three thrusters co-planar, ditto with two
thrusters on a line. 

Turning the three thrusters into a "corner" i.e. making
the hub an "apex" adding concavity / convexity to the
picture does not solve the challenge of being able to
create any vector sum i.e. a thrust in any direction. 

We have what I sometimes call "amputated XYZ",
helpless to map any but the (+ + +) octant.

Splaying four thrusters to the corners of a tetrahedron
is the minimum one needs to have exactly the deltas
necessary to sum to *any* vector in *any* desired direction. 

That's another way to feed intuitions about the inherent
4Dness of space.

In XYZ, we maybe "over-compensate" for not having that
fourth thruster by introducing three more, but then claiming
only three basis vectors are needed because multiplication
by -1 has the power to "reverse thrust" (?) effectively adding
another basis vector (thruster) -- actually three more, all

secondary to the three originals (the positives).

Your earlier point, that we sometimes mark a negative or
inverse to show it is secondary somehow, applies when
we say "three basis vectors are sufficient to map 3D space". 
We really mean three basis vectors that also act as "reverse
thrusters" (whatever that means), thanks to scalar multiplication

including -1.

But just one more thruster would have done the job,
e.g. -(i + j + k) would have worked the same way a quadpod
does.  If we re-arrange these four vectors for greater symmetry,
the regular tetrahedron conformation used in Q-rays is born.

Note that I'm not saying XYZ is "wrong" in any way, just when

attempting to get across the 4D meme as Fuller used it, the

nomenclature comes in for deconstruction a little more than

usual.  But in the end, we leave it be (Martian Math is not
about "dismantling" Earthling Math so much as investigating

its foundational underpinnings -- all philosophy majors should

be clued).

 

I am impressed by your geometry
http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html
which is intriguing and persuasive.

Welcome to Synergetics and its alternative model of 2nd and
3rd powering.
 

However, if you line up the corners of the squares and also of the cubes, then you get a progression which is very helpful for teaching calculus, namely, if you consider a square x and grow it by one more bit h so you have a square of sides x+h, then:
(x + h)**2 = (x + h)(x + h)  =  x2  + 2hx  + h2  which all make geometric sense, and then you can see why you can ignore the h2 and upon subtracting x2 you are left with 2hx which, when compared with h, gives you the derivative 2x.

That's an excellent question and I drew this diagram to show
how the identity (x + h)(x + h) = x**2 + 2xh + h (where ** is "to
the power of, borrowed from Python notation).

https://flic.kr/p/Ho7pyX  (showing this identity with a triangular model)

 

Similarly,
(x+h)**3  =  (x+h)(x+h)(x+h) = x3 + 3x2h + 3xh2 + h3   and discarding the small stuff and substracting x3 you are left with 3x2h and dividing by h gives the derivative 3x2.

I have not diagrammed this in terms of the planes of a tetrahedron
yet but the algebra has to be the same i.e. there will be no conflict. 

Swapping in a tetrahedron for a cube, triangle for a square, when
modeling 3rd and 2nd powering results in no contradictions, but one
does need to keep track of which convention is being used, if there's
any ambiguity.

That's why I have the storyboard about the Earthlings and Martians
collaborating on building a dam across a canyon, and for the purposes
of pouring concrete (among other things) agreeing on what to call
a "unit sphere" with radius R and diameter D such that D = 2R. 

The agreement is not on the volume of the sphere, but on the linear
distance R.

https://flic.kr/p/8thDHW  (shows the canyon and our "two cultures")

The Martians then construct a tetrahedron with edges D and call
that D**3 or unit volume (this has always been their practice). 

In doubling all edges, its volume goes up 8 fold.  Triple the edges
and the volume is 27 and so on. 

Here's a picture (note scale; their tetrahedron is pretty big!):

https://flic.kr/p/8teDJ4

The Earthlings, for their part, build a cube with edges R and
call that R**3 or unit volume for them.  The same rules apply
in that doubling all edges makes the cube (2R)**3 = 8R**3
i.e. 8 times the original volume. 

Martians and Earthlings agree that volume increases / decreases
as a 3rd power of the change in the controlling linear dimension
(as measured in "intervals" or "frequency" -- the distances D or R

could be units).

So then the question is, what's the conversion constant betwixt
a cube of edges R, known as Unit on Earth, and a tetrahedron
of edges D known as Unit on Mars? 

It turns out the D-edged tetrahedron has slightly smaller volume,
sqrt(8/9) the R-edged cube's to be precise.

Another figure to use is the "one page book with triangular book
covers".  The front and back covers lay flat on the table, and a
single equi-angular triangular page is free to turn such that its
tip traces the 180 arc of a circle, thereby defining two
complementary equi-volumed tetrahedrons. 

At 90 degrees, two equi-sized right tetrahedra are defined.  If all
edges are D, this right tetrahedron has volume equal to the Earthling
cube of edges R.  That's what I show in this picture:

https://flic.kr/p/fwdt8t  (triangular book covers)

The Martians don't use the mostly D-edged right tetrahedron's
volume as their Unit however (same as R**3 cube's).  They use
the all-D-edged tetrahedron as their Unit (the regular tetrahedron,
not the right one).

When adopting the Martian tetrahedron as unit, thanks to this
"currency conversion ratio" of sqrt(9/8) (known as S3), the cube
R**3, and likewise the right tetrahedron of five edges D, have
a volume of sqrt(9/8).

https://youtu.be/9f3bpFrZkhQ  (note my laptop wallpaper might
get scare the passenger next me huh [1] )

https://youtu.be/FaKpOmKJ6Yw  (continues to ramble on about
these "triangular book covers")

The Martians have a canonical cube of course, which is the one
their unit of volume inscribes in as face diagonals.  Its volume is 3. 

They use this volumes table (also leaned by Earthlings working
on this collaborative project, the better to understand the ETs

they're working with):

Tetrahedron (1)  edge = V

Cube (3); face diagonal = V

Octahedron (4); edge = V

Rhombic Dodecahedron (6); long diagonal = V

Icosahedron (18.51...) <--- incommesurable / "irrational"); edge = V

Cuboctahedron (20); edge = V

 

This for me is a very powerful way to illustrate differentiation in a very real sense.  And also these binomial expansions are very worthwhile to spend time with and very meaningful for problems in probability, heads and tails:  (h+t)**3   or recessive and dominate genes, blue eyes b and brown eyes B    (b+B)(b+B) for example.

So I'm curious if your triangular thinking has a nice way to talk about this all, perhaps?

Yes, but I still need to show the tetrahedron dissection for (x + h)**3 ...
or someone else might have done that already?

 

P.S. I forgot to say in my other letter that I'll be suggesting a talk at a philosophy (aesthetics) conference here in Vilnius, Lithuania, most likely to be accepted, about mathematical beauty. I appreciate thoughts on mathematical beauty (I suppose through a new thread).  My main thought so far is that mathematicians (at least me) would typically not consider the Mandelbrot set as beautiful because you can only see it, you can't imagine it. Whereas Galois theory is beautiful because it empowers the imagination.  So I want to explain what it takes for the Mandelbrot set to become beautiful for a mathematician.  Also, I want to link in with architect Christopher Alexander's 15 principles of life.

Andrius

Don't forget the Mandelbulb, a way to generate a "3D fractal"
that many Youtubes fly around in.

I use the Mandelbrot set to introduce the complex plane.  It's
the complement of a corresponding Julia Set.

For Julia sets, the function is o = o**2 + z, where z is some fixed value, and o ranges through a rectangle of values (some part of the complex plane).

For the Mandelbrot set, o starts at a fixed point (typically (0,0), and z ranges through a rectangle of values.

http://www.4dsolutions.net/ocn/fractals.html

Kirby

[1]  http://www.foxnews.com/us/2016/05/08/flight-delayed-when-math-mistaken-for-terrorism-by-passenger.htm.html

[kirby urner]

On Sat, May 21, 2016 at 9:14 PM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>
 

At 90 degrees, two equi-sized right tetrahedra are defined.  If all
edges are D, this right tetrahedron has volume equal to the Earthling
cube of edges R.  That's what I show in this picture:

https://flic.kr/p/fwdt8t  (triangular book covers)

Just to clarify, "all edges" refers to the page edges (D) and the

book covers (D), however, the page tip to book cover tip is not

D when the dihedral angle twixt page and cover is 90 degrees.

The altitude of the base triangle (= area of page or either book
cover) is then h = sqrt(3)/2 * D, for an area A, using bh/2 i.e. 
(base D * height)/2 giving (D * (sqrt(3)/2) * D)(1/2) = (sqrt(3)/4)*D**2.

1/3 Ah for the tetrahedron is then 1/3 (sqrt(3)/4) * (sqrt(3)/2) D**3

== (1/8)D**3 i.e. R**3 since R**3 is 1/8th of D**3 using the

cube model.

 

The Martians don't use the mostly D-edged right tetrahedron's
volume as their Unit however (same as R**3 cube's).  They use
the all-D-edged tetrahedron as their Unit (the regular tetrahedron,
not the right one).

<< SNIP >>
 

Tetrahedron (1)  edge = V

Cube (3); face diagonal = V

Octahedron (4); edge = V

Rhombic Dodecahedron (6); long diagonal = V

Icosahedron (18.51...) <--- incommesurable / "irrational"); edge = V

Cuboctahedron (20); edge = V

See Wikipedia / Synergetics.
 

Kirby
 

[kirby urner]

On Sat, May 21, 2016 at 9:14 PM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>
 

That's an excellent question and I drew this diagram to show
how the identity (x + h)(x + h) = x**2 + 2xh + h (where ** is "to
the power of, borrowed from Python notation).

https://flic.kr/p/Ho7pyX  (showing this identity with a triangular model)

 

Sorry:   (x + h)(x + h) = x**2 + 2xh + h**2

It's right in the diagram.

Kirby

[Joseph Austin]

On May 21, 2016, at 7:08 PM, kirby urner <kirby...@gmail.com> wrote:

We're left with three meanings of 4D by the end of the 20th Century:

4D:  relativity theory, includes a time variable

4D:  higher dimensional polytope geometry ala Coxeter and n-dimensional sphere packing

4D:  Fuller's emphasis on the tetrahedron as conceptually primitive

Using namespace notation, one might write:

Einstein.4D   (time is a variable, in addition to 3D space)

Coxeter.4D   (extended Euclidean, timeless, all axes spatial)

RBF.4D        (tetrahedron as primal and timeless, frequency adds time/size)

Kirby,

There's a lot to consider in your reply.

Let me focus first on "dimension".

In physics we sometimes understand dimension as "degrees of freedom".

Doesn't the tetrahedral "4D" still have only 3 degrees of freedom?

You can position a point along only 3 axes at a time, as the planes divide the space into 4 distinct regions with 3 axes each.

Or have we introduced an additional dimension, similar to homogenous coordinates, as it were embedding 3D "cubic" geometry in a 4D "tetrahedral" space?

BTW, speaking of dimensions, 

one interpretation of "multiplication" is a dimension-increasing [see note below] operation:

e.g. length x length -> area.

Given this interpretation, what is the meaning of a polynomial, being a sum of terms of different "dimensions"?

Or are we to understand the coefficients are always of such units or dimensions to render the polynomial homogeneous?

[note] Actually, I prefer to suppose that the "dimension" starts as maximal, 

and what we call "multiplication" is really just "factorization".

This may correspond to the idea of "lumpy" space--the only "primitive" is the lump; 

all lower-dimensions are mere projections. 

Joe

[Joseph Austin]

On May 21, 2016, at 7:08 PM, kirby urner <kirby...@gmail.com> wrote:

<snip>

Kirby,

There's a lot to consider here. 

Let me focus first on "dimension".

In physics we sometimes understand dimension as "degree of freedom".

Doesn't the tetrahedral "4D" still have only 3 degrees of freedom?

You can position a point along only 3 axes at a time, as the planes divide the space into 4 distinct regions with 3 axes each.

Or have we introduced an additonal dimension, as in homogenous coordinates, as it were embedding 3D "cubic" geometry in a 4D "tetrahedral" space?

[kirby urner]

You ask good questions Joe, thanks for diving in.

Given my humanities / art school audience, I'm more doing

anthropology than physics.  Meaning I'm less interested in

saying what's really so, than in making sense out of what

different tribes say is really so.

Consider our degrees of freedom limited to a plane (plain).

How many degrees of freedom is that?

The City tribe is used to two way streets that meet at city

intersections forming almost 90 degree or exactly 90

degree angles.  Two streets cross, and we may grid the

city in terms of two directions, understanding each

direction is itself two ways.

The Mountain tribe thinks of water up-welling from a

spring and symmetrically dividing the surrounding plain

into three regions.  Three rivers leave the source and

all surrounding territories may be calibrated in terms

of the two rivers that bound it.  The rivers flow only one

way ("down").

Here is a picture, looking down at the two tribal icons,

City and Mountain:

https://flic.kr/p/H2yFcu  (ethno-math)

Both tribes notice that, because we're looking down

and seeing motion constrained to a plane, we must

have another dimension separating the observer from

the observed.  Lets call that altitude.  We want to

acknowledge we're in a giant room or container of

some kind and want the freedoms to steer inside

that.

The City tribe says "add another street" and calls it
Z Street (the other two being X Street and Y Street). 

The Mountain (or Martian) tribe says "add another

river" making a fourth.  To preserve symmetry around

the source, the angles are re-organized such that

the four rivers spring forth at about 109.47 degrees

to one another.  We can make them light beams or

lasers instead, all shining from a common origin. 
Zero-gravity.

The City tribe now knows that volume is 3D because

of the X, Y and Z streets it uses.  The Martian (or

Mountain) people are content with their Four Rivers

analysis, with each Region of space bounded by

three of them, analogous the the situation in the

Plain, where two rivers so served.

As an anthropologist going back and forth between

these two tribes, I'm more interested in making them

intelligible to one another.  I'm more the diplomat.

To the City people I say "do you see why the Mountain

people say space is 4D?"  And to the Martians I say:

the Earthlings say volume is 3D, why argue?  Lets

just get on with building this hydro-dam.  Nobody

said we all need to agree on everything in order to

get work accomplished.

I sort of mixed Martian and Mountain in that story,

not too confusing I hope.

Kirby

[Bradford Hansen-Smith]

4-D
up/down
side/side

front/back

movement

Without movement no number of dimensions has any meaning.

3-D + movement = 4-D

Why not? Then you can add any number of hypothetical dimensions to that with a means of getting there.

There are seven rotational axis for the tetrahedron. Doesn't that mean 7 degrees of freedom or 14 vector directions is kind of fundamental?

Bradford Hansen-Smith
www.wholemovement.com

[kirby urner]

On Mon, May 23, 2016 at 6:47 PM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

4-D
up/down
side/side

front/back

movement

Without movement no number of dimensions has any meaning.

3-D + movement = 4-D

That sounds like 3D + Time, no?  Change == Time.

Einstein.4D

 

Why not? Then you can add any number of hypothetical dimensions to that with a means of getting there.

Or why not 4D = the Four Rivers?  We have a published lineage using 4D that way:  Fuller.4D

Then add time/size in one move as the moment you have a specific size you need specific time to traverse it.

4D + Frequency = Scenario (these inter-twine)

That's a language game that makes sense to readers of synergetic geometry.
 

There are seven rotational axis for the tetrahedron. Doesn't that mean 7 degrees of freedom or 14 vector directions is kind of fundamental?

I'm all for spinning our polyhedra around their axes and seeing what geodesic networks we get.

Kirby

[Andrius Kulikauskas]

Joe,

Thank you very much for your discussion of dimensions. I'm grappling
with this very much. But basically I think that you are describing the
distinction between the "top-down" view of a space (in which we start
with the whole space and break it down) and the "bottom-up" view of a
space (in which we start with an empty space and build it up). That's
what at's the heart of tensors. They combine the two points of view.
They break up an n dimensional space into p bottom-up (contravariant =
vector) and q bottom-up (covariant = covector = hyperplane = reflection)
points of view.

I appreciate your thinking on this difference.

I look forward to sharing some great thoughts about this as regards the
binomial expansion of the simplexes
https://en.wikipedia.org/wiki/Simplex
I will explain the missing diagonal "-1". It's basically the imagined
center of the simplex, from which we keep creating a new point. We can
expand (unlabelled + labelled)**N and when they are all unlabelled, we
get the center. When we treat the center as a new point, then we get
the next simplex.

I'm thinking that will relate somehow to the "field with one element",
an imaginary object in mathematics:
https://en.wikipedia.org/wiki/Field_with_one_element

Also, the simplexes seem to be very much related to the unitary groups
An, the most important of the Lie groups:
https://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagram

All of this to confirm the importance of the "tetrahedral" point of view.

Andrius

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

[Bradford Hansen-Smith]

On Mon, May 23, 2016 at 6:47 PM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

4-D
up/down
side/side

front/back

movement

Without movement no number of dimensions has any meaning.

3-D + movement = 4-D

That sounds like 3D + Time, no?  Change == Time.

Einstein.4D

Kirby, no. Time is a concept used to measure movement through space regardless of scale, rate of change, or change at all. Measuring is a calculation about movement, it is not movement. To move through space from one location to another is also moving space we carry with us.

If all we go by is "published lineage using 4D that way:  Fuller.4D,"  then we are tied to the past in ways that may prevent us from our own experience and understanding, neither of which anyone can give to us. As to Fuller reference; "No More Secondhand God"

I'm all for spinning our polyhedra around their axes and seeing what geodesic networks we get.

Me too, that is motion without change, some call symmetry. The primary non-centered system in the closes packing of spheres is four spheres in that arrangement that show seven lines of division, or projecting a line over the distance between points of connection to other points. Spinning an isolated polygon separated from context is much like spinning the facts to change one's understanding about the facts that has not changed the facts at all. We do this by taking things out of context and putting them in a context where they do not belong. Even rotating on all seven axes in a moment in time will not change the tetrahedron, only our perceptual imagination about it.

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www.wholemovement.com

[Bradford Hansen-Smith]

Andrius, the problem with top down (in which we start with the whole space and break it down) is that one can not break down what is already whole. The problem with bottom up (in which we start with an empty space and build it up) is that there is never enough to build with to make a whole. One has not the experience of the other. The whole has not the experience of parts anymore than parts have the experience of the whole. there will never be an equalization or reconciling, at best each to the needs of the other for fulfillment.

Nothing of the other is everything. There is no equality or middle ground, only the experience of one towards the other. I suspect confusion in generalizing and simplifying in math creates the need for disambiguation in attempting to understand what is not possible for one to experience of the other. In this light I am not sure what you mean by "whole space" and "empty space."

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[kirby urner]

On Tue, May 24, 2016 at 5:11 AM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

On Mon, May 23, 2016 at 6:47 PM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

4-D
up/down
side/side

front/back

movement

Without movement no number of dimensions has any meaning.

3-D + movement = 4-D

That sounds like 3D + Time, no?  Change == Time.

Einstein.4D

Kirby, no. Time is a concept used to measure movement through space regardless of scale, rate of change, or change at all. Measuring is a calculation about movement, it is not movement. To move through space from one location to another is also moving space we carry with us.

Without change we have no clocks and no time.  Time requires change.

Fuller associated shape-only, not changing, with angles.  Time/size with frequency.  The shape-only was 4D because the most primitive concept of a container is tetrahedron shape.  Add frequency and you have the basis for energy flows i.e. change i.e. time.

I diagram it like this:

4D  (timeless)
===========

Frequency (time/size added)  <-- 4D+ we could say

That's what's communicated in Synergetics.  It's not just an ink blot.  I know what I'm talking about.

 

If all we go by is "published lineage using 4D that way:  Fuller.4D,"  then we are tied to the past in ways that may prevent us from our own experience and understanding, neither of which anyone can give to us. As to Fuller reference; "No More Secondhand God"

I'm familiar with that book. 
 

I'm all for spinning our polyhedra around their axes and seeing what geodesic networks we get.

Me too, that is motion without change, some call symmetry. The primary non-centered system in the closes packing of spheres is four spheres in that arrangement that show seven lines of division, or projecting a line over the distance between points of connection to other points. Spinning an isolated polygon separated from context is much like spinning the facts to change one's understanding about the facts that has not changed the facts at all. We do this by taking things out of context and putting them in a context where they do not belong. Even rotating on all seven axes in a moment in time will not change the tetrahedron, only our perceptual imagination about it.

I'm all for spinning the facts to change one's understanding of the facts without changing the facts.

We do this by providing a different context that may highlight something we were missing.

Here's spinning a tetrahedron:

http://www.rwgrayprojects.com/synergetics/s09/figs/f87230.html

Kirby

[Bradford Hansen-Smith]

That's what's communicated in Synergetics.  It's not just an ink blot.  I know what I'm talking about.

I'm sorry Kirby, I did not mean to imply your do not know what you are talking about. It is clear you do.

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[kirby urner]

On Mon, May 23, 2016 at 7:25 PM, kirby urner <kirby...@gmail.com> wrote:

On Mon, May 23, 2016 at 6:47 PM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

4-D
up/down
side/side

front/back

movement

Without movement no number of dimensions has any meaning.

3-D + movement = 4-D

That sounds like 3D + Time, no?  Change == Time.

Einstein.4D

 

Why not? Then you can add any number of hypothetical dimensions to that with a means of getting there.

Or why not 4D = the Four Rivers?  We have a published lineage using 4D that way:  Fuller.4D

4D + Time = 5D at minimum (one could say, in the Synergetics namespace), time being required to have experience in the first place.  From experience, we intuit an unchanging shape:  the 4D tetrahedron.

 

Then add time/size in one move as the moment you have a specific size you need specific time to traverse it.

4D + Frequency = Scenario (these inter-twine)

Quoting more Synergetics:

529.01 Time is experience. Time can be expressed only in relative magnitude ratios of relevant experiences. Time can be defined only in terms of the relative frequency of reoccurrence of relative angular changes of the observer's environment, the relative frequency-of-occurrence rate being referenced to any constantly recycling behavior of any chosen subsystem of Universe.3 All experiential realizations are conceptually definable in degrees of angulation change and in relative frequency-of-occurrence rates in respect to the observer's optionally chosen axis of conceptuality and to his specifically identified time-recycling rate. (Footnote 3: E.g., a clock.)

529.02 Distance is measured in time. Time increments are calculated in respect to a variety of cyclic regularities manifest in our environmental experiences. Experimentally demonstrable cyclic regularities, such as the frequencies of the reoccurrence of radiation emissions of various atomic isotopes, become the fundamental time-increment references of relative size measurement of elemental phenomena.
 

That's a language game that makes sense to readers of synergetic geometry.
 

In contrast to Einstein.4D and Fuller.4D we have the namespace Coxeter.4D. 

After the 1900s shakeout, these were the three pillars left standing, with Fuller.4D the most abstruse (because hardly anyone has time for Synergetics, no one paid to study it in any department).

We also get many laypeople commonly confusing 3D + Time with hypercube-type polytopes (because led on by science fiction, which needs to conflate these two as a plot-driving device (see Regular Polytopes, page 119)).

The Youtube below is clearly in the Coxeter.4D namespace (not the same as Fuller.4D's but partially overlapping). 

Andrius, can you make any sense of this one?  It mentions Lie Algebras:

https://youtu.be/mVVcRDv414M

Kirby

[Joseph Austin]

Interesting, but beyond simpler formulas for area and volume,

what is the advantage?

I suppose you can still do vector addition,

but how do you compute components?

Do you have a kind of vector math?

Maybe I missed something.

I'm possibly chasing a different drummer--

trying to figure out how to put the units used in  physics into the math used in math,

and whether that makes a different "math".

My first clue is that length x length is not length, but something else, so "physical multiplication" in not "closed".

That should be scary to a math theory based on groups and fields and "closed" operations.

Joe

[kirby urner]

On Tue, May 24, 2016 at 4:48 PM, Joseph Austin <drtec...@gmail.com> wrote:

Interesting, but beyond simpler formulas for area and volume,

what is the advantage?

We have this idea of "mental arithmetic" where it's easy to do some computations

given practice.  What about "mental geometry" that's simple enough to both imagine

and see frequently reinforced.  A space-filling shape of volume 6 that corresponds

the the CCP / FCC lattice:  that by itself buys a place in the sun for the 4D IVM geometry.

The logical gap that needed bridging was how to get a tetrahedron to be unit volume

in the first place, where's the beef?  Once the beef was provided, the gap bridged,

Fuller.4D went on to a bright future in the 21st century.  That'd be the narrative.

I suppose you can still do vector addition,

but how do you compute components?

Do you have a kind of vector math?

Vector addition is exactly the same in Q-rays:  tip-to-tail.  Scale a basis
vector however you like (no need to rotate / reverse direction) and add
it to others so scaled, and provide every point in space with its unique
(a, b, c, d) address. 

Every point in space is reached with a mix of four directions, with at least
one arrow not involved per any point.  Only three border any of the four
quadrants, just as in XYZ where three rays from the origin border each
octant. 

Permutations of {2, 1, 1, 0} -- all 12 of them -- are at the corners of a
cuboctahedron, calibrated at 20 volume vis-a-vis the space-filling rhombic

dodecahedron of 6.  A wrinkle is the canonical basis vectors are not unit

length (or one could say are unit in a measurement system we throw

away):  it's the edges of the home base tetrahedron that we set at 2R

or 1D (R = unit sphere radius, D = 2R).

Again, the FCC matrix is the focus with balls layering around a nuclear
ball in sequence 1, 12, 42, 92, 162....  (that's the same as the CCP). 

Q-rays allow every CCP ball to have only positive integer coordinates,

as combinations of the 12 permutations above.  That's because the

center of every CCP ball spokes out to 12 neighbors and so on (every

ball is the same, every edge is the same length -- why some of us call

it -- the skeleton of rods -- the isotropic vector matrix or IVM, Bradford

knows).

The tetrahedrons and octahedrons so formed in said matrix have

volumes 1 and 4 respectively.

You can see I've got this mental geometry down.  I'm eager to share it
with more Earthlings.  Where are the cartoons from Sesame Studios or
whatever?  What's wrong with sharing STEM with more kids?

I'm not the grinch here to away anyone's XYZ (sub t for time).  We
have room for more than one namespace (i.e. language game) in this
world, just as we have room for more than one crypto-currency (I'm
watching more videos about bitcoin etc. this afternoon, eager to dig
deeper into Stellar...

https://www.stellar.org/blog/stellar-consensus-protocol-proof-code/

).

 

Maybe I missed something.

Again, the context could be history and art school.

I would contend that knowing the history of ideas means tracing the
New England Transcendentalist movement forward, from Margaret Fuller
to her grand nephew Bucky Fuller and the Cold War integration of the
latter's patented geometry into the US defensive infrastructure (DEW line
radomes especially -- enter Coxeter's son, stage left). 

This same Bucky guy, said to be a "utopian" (a label he rejected) later
comes out with these two thick volumes (Macmillan) about 4D this and
tetrahedron that, and your average 20th Century readers, already plenty
confused by the 4D of Einstein vs. the 4D of Coxeter mix-up, don't grok it. 
Given my training in philosophy at Princeton I was one of several who
took it on to help them grok it, and now many more of them do.  Q-rays
(not my invention, I contributed to their open source emergence) helped
with that project.  HP4E was useful too (another media campaign). 

 

I'm possibly chasing a different drummer--

trying to figure out how to put the units used in  physics into the math used in math,

and whether that makes a different "math".

There's an aspect of my own thinking which I think meets up with yours.

Thinking in Newtonian units, I notice mvd is action and think how we say

"lights, camera, action!" to shoot film.  Each frame captures a bunch of

deltas, or lets say change, and that's "per time" i.e. action / time or mvd/t

which works out to units of E i.e. energy.  A frame of film is like a bucket of

energy, some action per a frame of time.  1/t = f = frequency.  E = hf.

You can play a film too fast and people see right away the laws of physics

are being broken.  Things don't really move like that.  How fast the buckets

go by is E per t i.e. energy-frames per time E/t, or units of power, like watts.

In First Person Physics, we're interested in the human body as a burning

bulb, more or less bright depending on energy expenditure.  Any given human

body is akin to a 150-300 watt bulb, depending on resting or exerting.  That's

burning energy through time.  One may burn the same energy more slowly,

less wattage in that case.

All of this energy stuff occurs in the synergetics namespace when we

bathe our 4D eternal Platonic stuff in energy i.e. add E (frequency or time)

to the picture.  Still frame static stuff like an icosahedron or cuboctahedron,

not moving at all, is pure 4D (convex / concave, not much else, a container).

Adding time is creating special case instances, episodes, scenarios, films.
 

My first clue is that length x length is not length, but something else, so "physical multiplication" in not "closed".

That should be scary to a math theory based on groups and fields and "closed" operations.

Joe

I think a place to begin is with any shape, no matter how complicated,

say a sewing machine. 

Now double all its linear dimensions so it's effectively twice as tall, twice
as high.  It's self similar in shape, just a lot bigger.  It's been "scaled" (that's
the verb we use). 

The area will have increased as a 2nd power of this linear increase
(picturing parabola) and the volume will have increased as a 3rd power
(picturing the steeper 3rd power curve).  Increase linear dimensions by

phi (1.618...) and volume goes up by a factor of phi to the 3rd power.

Phi-scaled modules of just a few tetrahedral shapes assemble a large
vocabulary of familiar polyhedrons.

This relationship always obtains, between linear, areal and volumetric
aspects.  We could call it the "Power Rule" 1:2:3 and include it with common
core, right in there with Euler's V + F = E + 2 for ordinary omni-symmetrical
wire-frame polyhedrons, such as the cube, octahedron, tetrahedron and so
on. 

And lets not forget Descartes' angular deficit of 720 degrees (constant,
same as in a tetrahedron).  Add the angles around every vertex (of a cube

say, or octahedron), compute each difference from 360 at each angle, add
all those differences, and get 720 every time.

These are all elementary school topics in Mountain / Martian math, at

the basis of any serious STEM education.  We need a semantic network

that connects them.  In the art world, we have one.  We're willing to share

it with the math world.

Kirby

[kirby urner]

"to take away..." (fixing typo)...

 

We
have room for more than one namespace (i.e. language game) in this
world, just as we have room for more than one crypto-currency (I'm
watching more videos about bitcoin etc. this afternoon, eager to dig
deeper into Stellar...

https://www.stellar.org/blog/stellar-consensus-protocol-proof-code/

).

 

Maybe I missed something.

Again, the context could be history and art school.

I would contend that knowing the history of ideas means tracing the
New England Transcendentalist movement forward, from Margaret Fuller
to her grand nephew Bucky Fuller and the Cold War integration of the
latter's patented geometry into the US defensive infrastructure (DEW line
radomes especially -- enter Coxeter's son, stage left). 

As recounted in The King of Infinite Space by Siobhan Roberts about
Donald Coxeter, he was incensed that some clown in the Imperial-minded
USA (he was Canadian) would try to colonize Euclidean geometry with
patents, and that the Pentagon would actually honor them. 

But that's intellectual property for ya, what the Pentagon is charged with
protecting and defending (the freedom to privately own property, including
metaphysical property). 

Bucky effectively patented the CCP itself in the form of an architectural
system, as well as the geodesic sphere and dome (his better known patents,

the ones winning his companies the radome contracts).

By his own accounts he was trying to advertise the effectiveness of 4D

geometry i.e. more tetrahedron-based, and sensitive to tension vs. compression.

He patented as much as he could afford, not an easy process, as a proof

on concept.  For a philosopher, he was going to great lengths to achieve
credibility (Heidegger never tried anything so bold), so it made more sense
to treat him as an architect or engineer.

http://www.amazon.com/King-Infinite-Space-Coxeter-Geometry/dp/0802714994

Anyway, the domes were certainly part of the US tool chest in more

ways than one.  Aside from the radomes, just having a geodesic dome

in Moscow sounded a hopeful note that Khrushchev picked up on, ordering

one made.  The dome reappear in the World's Fair of Kabul at a later date,

again advertising that the US had Martian allies (or something along those

lines).

The US later pulled out of the World's Fair business, no longer wanting

to compete with its former self.  How to top the Montreal 67 Pavilion?

Even the Disney EPCOT Spaceship Earth couldn't quite do that, in terms
of sheer size.

https://en.wikipedia.org/wiki/Spaceship_Earth_%28Epcot%29

Kirby

[Joseph Austin]

So many roads diverging in the yellow wood!

Complex sphere.
Tetrahedral coordinates.
Homogeneous coordinates.
Minkowski coordinates.
Geometric Algebra.
Tensors.

Re RBF:

I grew up in St. Louis--at one time RBF lived nearby.
They built one of this Geodesic Domes as a "Climatron" (greenhouse garden).
I also knew I guy who lived in a geodesic home.
In my younger days, I always imagined I would live in one myself one day,
but practicality and economics overruled.

Re physics:

It occurred to me that the density of the surface of a sphere obeys the inverse-square law.
Central "forces" obey the inverse square law in classical physics. Coincidence?
Einstein says gravity is geometry.
Quantum theory says everything is groups.
Geometry has symmetry groups; so does quantum physics.
Are the "fundamental particles" just different geometric shapes after all?!
Is the only thing "moving" US--drifting through a multidimensional space, sampling various "viewpoints" like strolling through a sculpture gallery?

So, where does one "start" with your math program? Do you have a book, or at least a syllabus with suggested readings?
(My daughter is an artist. But I had become convinced that she is missing the "math" gene.
She'd be a college graduate but for not passing the required math course.
In spite of having earned a living as a CAD draftsperson.)

Joe

> On May 24, 2016, at 9:12 PM, kirby urner <kirby...@gmail.com> wrote:
>

[kirby urner]

On Tue, May 24, 2016 at 6:57 PM, Joseph Austin <drtec...@gmail.com> wrote:

 

I grew up in St. Louis--at one time RBF lived nearby.

I just flow back to Portland from there on May 10. 

I really like St. Louis (having only seen it in April-May).
 

They built one of this Geodesic Domes as a "Climatron" (greenhouse garden).

Yes, I paid a visit to said Climatron on a previous trip a year ago:

http://worldgame.blogspot.com/2015/04/garden-of-eden.html
 

I also knew I guy who lived in a geodesic home.
In my younger days, I always imagined I would live in one myself one day,
but practicality and economics overruled.

Have you seen the Dymaxion House at the Henry Ford Museum in Dearborn MI?

That wasn't a dome, more a hexagon suspended from a utility mast.

You might enjoy this excerpt from I talk I gave on the dome homestead:

https://youtu.be/QV4m76Om7bk  (it does have audio)

It also shows up in Martian Math at Reed College (another curriculum access point):

http://www.4dsolutions.net/satacad/martianmath/mm4.html

Kirby

[kirby urner]

On Tue, May 24, 2016 at 6:47 PM, kirby urner <kirby...@gmail.com> wrote:

Here's TIME putting another spin on it.  After assuring us they're uneconomical,
we read there's some hope for a World's Fair or Expo in North America down
the road a ways:

http://time.com/79600/the-fall-of-the-fair/

This documentary explores the issue in more detail:

http://www.imdb.com/title/tt2415482/

Reviewed:

http://controlroom.blogspot.com/2014/12/wheres-fair-movie-review.html

So, where does one "start" with your math program?  Do you have a book,
or at least a syllabus with suggested readings?

Yes to all of the above.  You're in the thick of my curriculum already.  Well along.

My daughter is an artist.  But I had become convinced that she is missing the "math" gene.

I think learning by treasure hunt or geocaching is fun.  All my links lead to other links. 

Suggest she Google on Grunch maybe?  Or Hexapent in Google Images? 

Some of those are mine.

Kirby

[kirby urner]

Kirby again, post Pycon.

Inspired by what everyone was doing around Portland Pycon 2016,

I dug out an old Quadrays implementation code in Python from

Python 2.x days (2.x is retiring) and modifying it to run in Python 3.x.

Here's a fragment of dialog, chatting with the interpreter in the REPL:

In [40]: import qrays

In [41]: v1 = qrays.Qvector((0,1,0,0))

In [42]: v0 = qrays.Qvector((1,0,0,0))

In [43]: qrays.angle(v1,v0)

Out[43]: 109.4712206325

In [44]: v1.length()

Out[44]: 0.6123724356957945

In [45]: v1.xyz

Out[45]: (-0.35355339059327373, -0.35355339059327373, 0.35355339059327373)

In [46]: v0.xyz

Out[46]: (0.35355339059327373, 0.35355339059327373, 0.35355339059327373)

...

In [55]: v3 = v0 + v1

In [56]: v3

Out[56]: Qvector (0.9999999999999998, 0.9999999999999998, 0.0, 0.0)

In [58]: v3.length()

Out[58]: 0.7071067811865474

I break the rule of having the basis vectors default to length one, as
it's more important that the tetrahedron defining the four q-rays
has edges of 2R or 1D (R for Radius, D for Diameter of CCP ball). 

Once could say q-rays are unit length in a scale we discard in favor
of this other one, wherein tetrahedron edges are unit 1D (as is the
tetrahedron's volume 1V).

Here's the code:

https://github.com/4dsolutions/Python5/blob/master/qrays.py

Kirby