thoughts about Clifford Algebra

[kirby urner]

Greeting Andrius --

I've been reading more about GA, which I've tackled

before.  Area and volume come directly from doing the

computations.  The multiplication operator between

two of the main elements (bivectors?) consists of

both inner (dot) and outer (wedge) projects.  The
results are glued together somewhat like complex
numbers are glued together:  by plus (+). 

1 + 2i is a complex number, whereas (dot product +
wedge project) is the look of a bivector, sort of, once
simplified to canonical form, however a future MathPiper

does that.

My question is one of interpretation:  we're so used

to area and volume being quadrilateral- and parallelepiped

shaped and I'm wondering why that's mandatory.

Consider an operation called "closing the lid" where

when I give you two vectors fanning out from the

same tail point, I simply connect their tips in a triangle

and call that their "area" (we could say A1 or A-prime).

Similarly, any three vectors that fan out from a terminus,

giving convex and concave aspects, unless flat to a

common plane (volume 0) give a "close the lid" volume

of the tetrahedron thereby defined. 

Just connect the tips in both cases, to close off the
unique area or volume in question. 

So much simpler than all this elaborate construction

to get four and twelve edges respectively, keep it at three

and six edges, right?  Isn't that logically consistent?

The usual objection is this would mess up our simple

volume and area formulas.  Not by a lot though, as

the triangle is 1/2 the quadrilateral's area and the

tetrahedron is 1/6th of the eventual parallelepiped,
or "skew brick" (a normal brick if the vectors are
90-90-90 (a special case)).

This result generalizes i.e. does not require any

specific angles between the vectors.

However, that's not quite what I mean. 

Even with the very same area and volume numbers,
we could map those to the triangles and tetrahedrons
and say, "that's a visual representation of this area
or volume number." 

When we draw two vectors at 60 degrees, and both
are unit in length, then "close the lid" multiplication
give us the unit triangle.  By definition.

"Close the lid" multiplication of three vectors fanning
out with the angles of a regular tetrahedron, gives
the volume V' of that tetrahedron.

Actually, the angles don't matter when it comes to
closing the lid, but 1 x 1 x 1 == 1 is shown most
canonically with a regular tetrahedron.  The three

lengths vary independently of one another, just as

in XYZ.

You might recognize the above as the Synergetics

approach.  Indeed, when I take any three quadray

basis vectors, of the total four stemming from

(0,0,0,0), I expect "close the lid" multiplication to

give me the volume of one quadrant of the canonical
home base tetrahedron (not the whole tetrahedron). 

The angles between any two of them are 109.47,
degrees, like Nitrogen to Hydrogen bonds in NH4.  

Sure enough, that's what I get when I plug the three
lengths into my "give me the volume when I give
you three lengths from a common corner" formula (0.25):

    def test_quadrant(self):
        qA = Qvector((1,0,0,0))
        qB = Qvector((0,1,0,0))
        qC = Qvector((0,0,1,0))
        tet = Tetrahedron(qA.length(), qB.length(), qC.length(),
                (qA-qB).length(), (qA-qC).length(), (qB-qC).length())
        self.assertAlmostEqual(tet.ivm_volume(), 0.25)

The above test passes.  qA, qB and qC all radiate
from a common origin and in computing three vector
subtractions I'm getting the "close the lid" edges (the
edges of the lid that closes off the concavity). 

What I'm calling ivm_volume is simply "close the lid"
volume as described above, using a volume-computing

algorithm that takes six edges for input, of any tetrahedron.

However in Synegetics, we do not simply apply conversion
factors of 1/2 and 1/6 to go from "XYZ" to "IVM" area and
volume. 

Since we have two independent models of area and volume
in the first place ("build a bigger box" versus "close the lid"),
we might as well scale them intelligently vis-a-vis one another
for going back and forth.  We have some degrees of freedom
in how we choose to proceed. 

That's where the R-edged Cube versus D-edged Tetrahedron
come in (2R = D).  Their volumetric relationship is R-cube is
sqrt(9/8) larger than the D-tetrahedron, in terms of pouring
water and getting spillage. 

If each is a model of Unit in its own coordinate system, then
there's the conversion constant we need.  We do something
similar for Area.

So my question w/r to Clifford Algebra is, given this is a

new algebra, not yet widely used in Physics, would this be

a good time to introduce novel models of representation

where we keep the same numbers, do not alter the results,

but allow for a different visual interpretations, by simply

mapping numbers to shapes with this different "close the

lid" model. 

I don't think there'd be algebraic consequences in the
theory itself, just in applications to the real world.

Something to think about.

Kirby

[kirby urner]

On Mon, Jun 13, 2016 at 8:21 AM, kirby urner <kirby...@gmail.com> wrote:

Greeting Andrius --

I've been reading more about GA, which I've tackled

before.  Area and volume come directly from doing the

computations.  The multiplication operator between

two of the main elements (bivectors?) consists of

both inner (dot) and outer (wedge) projects. 

Sorry, I meant to say "products" not "projects".

You'll find parallel ruminations in my on-line journals

this morning, not connecting to Clifford Algebra in

particular, but just to these two concepts of
area-volume more generally (quadrilateral-

parallelepiped based versus triangle-tetrahedron

based).

http://controlroom.blogspot.com/2016/06/putting-lid-on-it.html (putting a lid on it)
http://mybizmo.blogspot.com/2016/06/wheres-origin.html  (more quadrays)

Kirby

[Joseph Austin]

Kirby, Andrius,

This question underscores the notion that units of measure are as significant as numerical quantity.

In physics, the conserved quantity is often the product, often of two or more measurements using different units of measure,

e.g. work measured in newtons x meters.

Through habit of use, we are accustomed to the interpretation of "area" as "square length",

but perhaps it is not so intuitive what a "square second" or "kilogram meter" are.

I think it is simply a matter of convention (or convenience) that we measure areas and volumes in "square" or "cubic" units,

such that the product of the linear units of measure of the components equals the numerical unit of measure of the product.

But since length, area, and volume are different units of measure, I agree there is no mathematical constraint on using tetrahedral or triangular units in place of cubic or square units.

As for application to physics, David Hestenes has applied GA across the spectrum of classical and quantum Physics,

and written a couple books on the subject.  If you are not already familiar with it, I commend to your attention his 2002 Oersted Lecture on the subject.  One of his examples includes the tetrahedral symmetry of the Methane molecule.

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Joe

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[kirby urner]

On Mon, Jun 13, 2016 at 12:20 PM, Joseph Austin <drtec...@gmail.com> wrote:

Kirby, Andrius,

This question underscores the notion that units of measure are as significant as numerical quantity.

In physics, the conserved quantity is often the product, often of two or more measurements using different units of measure,

e.g. work measured in newtons x meters.

Through habit of use, we are accustomed to the interpretation of "area" as "square length",

but perhaps it is not so intuitive what a "square second" or "kilogram meter" are.

Right (pun). We're obsessed with right angles, the prevailing orthodoxy.  Just

closing off the triangle, given the two tail-originating vectors (any two) and
calling that an "area" is too easy apparently, too trivial.  Why would anyone

want to return *that*?

We know when people are saying "x squared" and "y cubed" all around
us, that we're still very much in Rome.  Some may be struggling to gain their
freedom, from this gravitational vortex, but it ain't easy. 

So many true believers frequent the schools, drumming in their "one true
teaching" (a dogma) with a series of not-safe-to-question "thou shalts".

Thou shalt not have a volume six rhombic dodecahedron, if you want

any kind of XYZ credential.

 

I think it is simply a matter of convention (or convenience) that we measure areas and volumes in "square" or "cubic" units, such that the product of the linear units of measure of the components equals the numerical unit of measure of the product.

The question is, at what cost this convenience procured and when are we
asked to comparison shop? 

As a carnival hawker of our slicer-dicer, I'll say our tetrahedron plays with

others so much more adeptly, with claymation studio friends Octahedron
of volume 4, and Cube of volume 3.  It's not just about Rhomy, our
volume 6 "single parent" of this dual-two (they're embedded as long and
short diagonals respectively).

When was XYZ geometry so neighborly and whole-numbered?  Show me

which book.

Fortunately, we're not faced with an either / or decision in that we're as free

to explore this other mathematical branch, wherein 3rd powering is sometimes
innocent of right angles. 

In the 21st Century, we have this new freedom, thanks to pioneering in the 20th.

 

But since length, area, and volume are different units of measure, I agree there is no mathematical constraint on using tetrahedral or triangular units in place of cubic or square units.

That's the next 4D after Euclidean and Non-Euclidean. 

I've switched to a more compact memeplex in my Twitter feed,
given the gross-of-characters limit per tweet:  we're talking
about the Tesseract, the Time Machine, and the Tetrahedron. 
Nice alliteration no?  Sounds like Narnia.

* Tesseract

the hypercube, corners like web pages for nodes, each hyperlinked
by means of edges with an applied metric or distance formula (the
Pythogorean (but on steroids -- could be N-D).

* Time Machine

Adding what physics most cares about:  time passing, whereas for
strict Euclideans the constructions are all instantaneous, so close
together on the sandy beach that we get to neglect two aspects

of spacetime: 

(A) the finite curvature of the Earth, meaning this sandy beach only
approximates an "infinite plane" of science fiction

(B) the finite speed of light, meaning getting around the perimeter
of some triangle is a spatio-temporal affair.

Once (A) and (B) are factored in, we leave behind Euclid's space in
favor of Minkowski's.  We use a Lorentz Interval, not Pythagorean
Distance, for our metric (non-Euclidean).

* Tetrahedron

We sense the planet is "in" a space, with outer space as some
container, like a fish tank, inside of which our Earth floats, in orbit
about the Sun, its moon local in orbit. 

Of course it's silly to think of the universe as having literal walls
around it (as the US and China have both wanted, at different
times in their evolution).  But if it did, if we were to represent this
container, how few such walls could we get away with?  Yes,

that's a leading question (Socratic dialog). 

One might leap to the picture of a regular hexahedron (in English:
a "qyoob"), with the solar system disk seen inside it.  

However the branch of topology known as Graph Theory suggests
another shape, the less feature-rich, more primitive tetrahedron
(fewer edges, fewer nodes).  There's no container that's simpler,
even if one imagines just-a-triangle from some spatial vantage
point. 

Just a triangle is still four points, including the eye of the beholder. 
The points themselves are like balls (space-occupying,
concave-convex).  They may become "infra-tunable" at some

threshold in frequency -- we know that from science.  Such points

are just dots before they vanish, we might say zero-dimensional.

Given how the tetrahedron radiates Fourness (more than Threeness),

in this final 4D namespace we speak of the 4Dness of Volume, which
is nevertheless Euclidean enough to accommodate XYZ no problemo.

If you're vanishing into nothing, zero-dimensional is OK, but the moment

you occupy volume, you're able to contain a tetrahedron, as well as

be contained in one, so you're 4D like everything in this space, which

is purely conceptual, so like the Euclidean, instantaneous, no need

for time.  However we're free to add time.  Just set a clock ticking.

 

As for application to physics, David Hestenes has applied GA across the spectrum of classical and quantum Physics,

and written a couple books on the subject.  If you are not already familiar with it, I commend to your attention his 2002 Oersted Lecture on the subject.  One of his examples includes the tetrahedral symmetry of the Methane molecule.

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Joe

Yes.  At Pycon I met QuantumDoug (an alias) who turned me on to the

existence of this tome:

https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521715954

by Chris Doran

I'm seeing how both tensor and quaternion calculus is embedded in this

Clifford Algebra framework and understand why physicists are intrigued

by it. 

My question for Andrius, and yourself, is along the lines of whether non-cubist
interpretations might apply in some versions, or is GA the sole property of

the "must live in Rome" orthodox?

I know there's a lot of inertia behind a "fourth perpendicular" presentation

of the space, pushing Pythagorean distance to a next level.  In that reading,

the 4Dness of the tetrahedron gets lost, along with its alternative visual
interpretations (IVM-based).

The art colonies are far more aware of the 4D IVM as introduced by Fuller
in Synergetics than are most physics communities, at least in my neck of
the woods. 

That's why I've been circling Art History has having the lighted runways
where it's safe for jets with our 4D markings, to come in for a landing.

The more straitjacketed, who need to "cube" all the time when 3rd powering,

tend to have a short fuse around Synergetics, especially when the cube's

instability and relative awkwardness is pointed out. 

They go on the defensive and start dragging their XYZ "jacks" out on the
runway, not wanting our IVM "caltrop" way of thinking to find footing.

Fuller noted on occasion that art colonies tended to befriend him, whereas
a lot of more established STEM types were more likely taking pot shots
from their silos, concerned to defend some memeplex orthodoxy.

Kirby

[Joseph Austin]

On Jun 13, 2016, at 5:53 PM, kirby urner <kirby...@gmail.com> wrote:

My question for Andrius, and yourself, is along the lines of whether non-cubist 
interpretations might apply in some versions, or is GA the sole property of

the "must live in Rome" orthodox?

I'm somewhat out of my depth, knowing not much more about GA than I do about Quadrays,

but it would be worth exploring whether you can axiomatize a "vector algebra and calculus" of Quadrays.

In Physics, vectors represent quantities that have a "magnitude" and a "direction".

Standard vector analysis represents planes by their normal vector, e.g. angular momentum.

(Of course, no one believe that the "motion" of a spinning thing is in the direction of its axis--it's just a mathematical trick.)

GA gives a representation more like a circumferential "round" vector.

They also use the "vector cross product" to represent interactions between two vector fields, say the Electric and Magnetic fields.

When relativity came along, it turned out that the cross product was not an invariant. GA supposedly remedies this.

The usual reasoning for three dimensions is that is the minimum number needed to be linearly independent.

There is also a symmetry argument for using three, based on the  symmetries of animals and the natural world.

Bilateral symmetry establishes a left-right direction.

The preferred direction of motion of most animals establishes the forward-backward direction.

The "pull" of gravity establishes an up-down direction.

(These are the object-centric directions, along with their corresponding rotations, that we use in ALICE--which may be familiar to artists.)

Is there a corresponding "natural" basis for tetrahedral directions?

I suppose you could make a case based on the bonds of carbon, which is the fundamental scaffolding of organic material.

But,  do tetrahedral directions provide insight into description or classification of organic compounds, or crystal lattices?

The C6-H12-O6 system certainly leaves out a lot of structural information!

My question would be: can I express "the Laws of Physics" in tetrahedral coordinates and use tetrahedral geometry and calculus to do useful calculations?  Physics is no stranger to transforming coordinate systems: we readily transform to Spherical or Cylindrical or Elliptical or Lorentzian or even do Fourier transforms and introduce multi-dimensional spaces.  One might say Physics will choose the coordinates most natural to the system being studied.

As long as the transformations can be defined and computed, and the formulas in the alternative coordinates can be expressed and solved (hopefully, more easily than in cartesian coordinates) we have no problem using alternative coordinates.

But a physicist will want to see equations, or a mathematician, axioms; not just philosophical or aesthetic arguments.

Joe

[kirby urner]

On Mon, Jun 13, 2016 at 6:29 PM, Joseph Austin <drtec...@gmail.com> wrote:

On Jun 13, 2016, at 5:53 PM, kirby urner <kirby...@gmail.com> wrote:

My question for Andrius, and yourself, is along the lines of whether non-cubist 
interpretations might apply in some versions, or is GA the sole property of

the "must live in Rome" orthodox?

I'm somewhat out of my depth, knowing not much more about GA than I do about Quadrays,

but it would be worth exploring whether you can axiomatize a "vector algebra and calculus" of Quadrays.

In Physics, vectors represent quantities that have a "magnitude" and a "direction".

Standard vector analysis represents planes by their normal vector, e.g. angular momentum.

(Of course, no one believe that the "motion" of a spinning thing is in the direction of its axis--it's just a mathematical trick.)

GA gives a representation more like a circumferential "round" vector.

They also use the "vector cross product" to represent interactions between two vector fields, say the Electric and Magnetic fields.

When relativity came along, it turned out that the cross product was not an invariant. GA supposedly remedies this.

The usual reasoning for three dimensions is that is the minimum number needed to be linearly independent.

I'm thinking of a different language game this evening.

Take four XYZ vectors from the center to the four corners
of a regular tetrahedron, expressed in ordinary XYZ
coordinates.  We could express them like this:

a = xyz_vector(x= 0.35355339059327373, y= 0.35355339059327373, z= 0.35355339059327373)
b = xyz_vector(x=-0.35355339059327373, y=-0.35355339059327373, z= 0.35355339059327373)

c = xyz_vector(x=-0.35355339059327373, y= 0.35355339059327373, z=-0.35355339059327373)
d = xyz_vector(x= 0.35355339059327373, y=-0.35355339059327373, z=-0.35355339059327373)

The game is to pick any point at random, and reach it by

stretching (or shrinking) each of these four vectors, then

adding them tip to tail, as in everyday vector addition.

For any given point, you'll need at most only three of them,

but each component gets a spot in a 4-tuple, as all will be
needed to span the surrounding volume.  The quadray data

structure is:  (_, _, _, _).

So here we have the three coordinates per quadrant, like three

per octant with XYZ.  We can say that's why space is 3D if we

want to.  Three of these four basis vectors, correctly scaled,
are sufficient to reach any point -- but sometimes a different
three, depending on where the point is.

In XYZ we let the coordinates go negative, whereas here
there's no need.  The negatives of a, b, c, d, i.e. the four

vectors pointing oppositely, defining a dual tetrahedron,

are not needed and all have positive coordinates anyway.

-(1,0,0,0) = (0,1,1,1)
-(0,1,0,0) = (1,0,1,1)

and so on.

Lets try this experiment, of picking any point at random

and reaching it by stretching (or shrinking) each of these

four vectors spelled out above.

xyz_vector(x=2.0, y=5.0, z=-5.0)
ivm_vector(a=9.899494936611664, b=0.0, c=14.14213562373095, d=9.899494936611664)

xyz_vector(x=1.0, y=9.0, z=-10.0)
ivm_vector(a=14.14213562373095, b=0.0, c=26.870057685088803, d=15.556349186104043)

xyz_vector(x=-8.0, y=-5.0, z=-6.0)
ivm_vector(a=0.0, b=18.384776310850235, c=19.798989873223327, d=15.556349186104043)

I'm using my Python modules to get these.

First I ask for any old point, no need to be

this restrictive in keeping each coordinate

between -10 and 10.

In [178]: z = Vector((randint(-10, 10), randint(-10,10), randint(-10, 10)))

Then I check to see what XYZ point I've got:

In [179]: z

Out[179]: xyz_vector(x=2.0, y=5.0, z=0.0)

Then I ask to see the same point as reached

using three of my linearly independent basis vectors,

none of them made from the others, with only

positive numbers used:

In [180]: z.quadray()

Out[180]: ivm_vector(a=9.899494936611664, b=0.0, c=7.071067811865475, d=2.82842712474619)

These pairs point to the same point in space, just with the

quadrays we never need to reverse direction to go in the direction

opposite to any of our original four. 

We have a positive linear combination of at most three of them,
every time.  Our four coordinates are always positive or zero.

I think it's fine to say "so what, just another language

game".  It's just another data structure for storing locations,

another addressing scheme.  Like spherical coordinates.

Another recording system.

We want to express every point in space as a linear
combination of four basis vectors, just as we do with the
mutually orthogonal ones. 

Only three lengths are needed per any given point,
because in each quadrant, the ray not defining it, not
a border edge, i.e. pointing away from said quadrant,
is not needed to reach the points within. 

We don't give away it's seat at the table though.  We
keep slots for all four.  That's what the number 0 is for,
after all.  A placeholder.

There is also a symmetry argument for using three, based on the  symmetries of animals and the natural world.

Bilateral symmetry establishes a left-right direction.

The preferred direction of motion of most animals establishes the forward-backward direction.

The "pull" of gravity establishes an up-down direction.

(These are the object-centric directions, along with their corresponding rotations, that we use in ALICE--which may be familiar to artists.)

Is there a corresponding "natural" basis for tetrahedral directions?

I suppose you could make a case based on the bonds of carbon, which is the fundamental scaffolding of organic material.

What's interesting about a regular tetrahedron is if
you stand on a surf board (front to back); and hold

out your hands (left to right); you have two edges of

the tetrahedron. 

They're at 90 degrees but do not intersect (feet below,
arms above).  Do we call that skew?

From Wikipedia:

In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron.

Excellent, exactly the construction I was talking about.

With strings tied from each outstretched hand to

the front and back of the board, four additional edges

are added.

So in this picture, we have front-back / left-right /

up-down, same as before.

 

But,  do tetrahedral directions provide insight into description or classification of organic compounds, or crystal lattices?

The C6-H12-O6 system certainly leaves out a lot of structural information!

My question would be: can I express "the Laws of Physics" in tetrahedral coordinates and use tetrahedral geometry and calculus to do useful calculations?  Physics is no stranger to transforming coordinate systems: we readily transform to Spherical or Cylindrical or Elliptical or Lorentzian or even do Fourier transforms and introduce multi-dimensional spaces.  One might say Physics will choose the coordinates most natural to the system being studied.

As long as the transformations can be defined and computed, and the formulas in the alternative coordinates can be expressed and solved (hopefully, more easily than in cartesian coordinates) we have no problem using alternative coordinates.

Given the one-to-one correspondence between XYZ
and Q-ray coordinates, one may always jump from one

to the other.

The way I've size the four basis vectors, the centers of

closest packed spheres in the CCP arrangement, well

known in crystallography, will always have only positive
integer coordinates.  That could prove useful in some

applications.

 

But a physicist will want to see equations, or a mathematician, axioms; not just philosophical or aesthetic arguments.

Joe

Note that definition multiplication of two and three lengths

in terms of "closing the lid" (making the corresponding

triangle or tetrahedron), does not depend on the mechanics

of Q-rays at all.  Quadrays are nowhere mentioned in the

original two-volumed Synergetics.

Rather, we may use software routines to synchronize the

quadrays with this alternative way of doing business.  Some

of the results will prove interesting.  Any tetrahedron, regular

or irregular, with CCP balls for vertexes, will have a whole

number volume.  Dr. Robert Gray proved that one.

Kirby

[kirby urner]

On Tue, Jun 14, 2016 at 12:07 AM, kirby urner <kirby...@gmail.com> wrote:

On Mon, Jun 13, 2016 at 6:29 PM, Joseph Austin <drtec...@gmail.com> wrote:

On Jun 13, 2016, at 5:53 PM, kirby urner <kirby...@gmail.com> wrote:

My question for Andrius, and yourself, is along the lines of whether non-cubist 
interpretations might apply in some versions, or is GA the sole property of

the "must live in Rome" orthodox?

I'm somewhat out of my depth, knowing not much more about GA than I do about Quadrays,

but it would be worth exploring whether you can axiomatize a "vector algebra and calculus" of Quadrays.

In Physics, vectors represent quantities that have a "magnitude" and a "direction".

Standard vector analysis represents planes by their normal vector, e.g. angular momentum.

(Of course, no one believe that the "motion" of a spinning thing is in the direction of its axis--it's just a mathematical trick.)

GA gives a representation more like a circumferential "round" vector.

They also use the "vector cross product" to represent interactions between two vector fields, say the Electric and Magnetic fields.

When relativity came along, it turned out that the cross product was not an invariant. GA supposedly remedies this.

The usual reasoning for three dimensions is that is the minimum number needed to be linearly independent.

I'm thinking of a different language game this evening.

Take four XYZ vectors from the center to the four corners
of a regular tetrahedron, expressed in ordinary XYZ
coordinates.  We could express them like this:

a = xyz_vector(x= 0.35355339059327373, y= 0.35355339059327373, z= 0.35355339059327373)
b = xyz_vector(x=-0.35355339059327373, y=-0.35355339059327373, z= 0.35355339059327373)

c = xyz_vector(x=-0.35355339059327373, y= 0.35355339059327373, z=-0.35355339059327373)
d = xyz_vector(x= 0.35355339059327373, y=-0.35355339059327373, z=-0.35355339059327373)

<< SNIP >>
 

First I ask for any old point, no need to be

this restrictive in keeping each coordinate

between -10 and 10.

In [178]: z = Vector((randint(-10, 10), randint(-10,10), randint(-10, 10)))

Then I check to see what XYZ point I've got:

In [179]: z

Out[179]: xyz_vector(x=2.0, y=5.0, z=0.0)

 

Then I ask to see the same point as reached

using three of my linearly independent basis vectors,

none of them made from the others, with only

positive numbers used:

In [180]: z.quadray()

Out[180]: ivm_vector(a=9.899494936611664, b=0.0, c=7.071067811865475, d=2.82842712474619)

We can check this answer:

In [182]: p = 9.899494936611664 * a + 7.071067811865475 * c + 2.82842712474619 * d

In [183]: p.xyz()

Out[183]: xyz_vector(x=1.9999999999999993, y=4.999999999999999, z=-3.14018491736755e-16)

I'm scaling a, c, and d basis vectors as given above, the ones from the center to

the corners of a tetrahedron, and getting back (2, 5, 0), with the expected floating

point fuzziness.

If we want to say space is 4D because (+, +, +, +) gets us to every point, no negatives needed, 0 a placeholder, that's OK too.  That will help with understanding Synergetics.  It's not a matter of deciding what the one true answer really is.  Given we're dealing with different definitions and axioms, there's no need to worry about any logical contradiction.

Kirby

[kirby urner]

Rather, we may use software routines to synchronize the

quadrays with this alternative way of doing business.  Some

of the results will prove interesting.  Any tetrahedron, regular

or irregular, with CCP balls for vertexes, will have a whole

number volume.  Dr. Robert Gray proved that one.

I thought I'd take a moment to demonstrate any CCP-vertexed

tetrahedron having a whole number volume, starting with the
quadray home base tetrahedron having a volume of 1.  Its

corners are (1,0,0,0) (0,1,0,0) (0,0,1,0) (0,0,0,1) with

corresponding XYZ coordinates given above as a, b, c and d.

Consider 12 vectors pointing from the nuclear sphere at
(0,0,0,0) to the 12 vertexes of 12 spheres closest packed around
it, at the corners of a volume 20 (in the Synergetics namespace)

cuboctahedron.

Here are those 12 vectors in quadray form, each having xyz

equivalent expressions.  We're looking at all permutations of

{2, 1, 1, 0}

i.e.:

In [206]: the_twelve = set(itertools.permutations((2, 0, 1, 1)))

In [207]: the_twelve

Out[207]:

{(0, 1, 1, 2),

(0, 1, 2, 1),

(0, 2, 1, 1),

(1, 0, 1, 2),

(1, 0, 2, 1),

(1, 1, 0, 2),

(1, 1, 2, 0),

(1, 2, 0, 1),

(1, 2, 1, 0),

(2, 0, 1, 1),

(2, 1, 0, 1),

(2, 1, 1, 0)}

Turning these twelve tuples into Q-rays then:

In [208]: the_qrays = [Qvector(p) for p in the_twelve]

In [209]: the_qrays

Out[209]:

[ivm_vector(a=0, b=1, c=1, d=2),

ivm_vector(a=0, b=2, c=1, d=1),

ivm_vector(a=2, b=1, c=0, d=1),

ivm_vector(a=0, b=1, c=2, d=1),

ivm_vector(a=1, b=2, c=1, d=0),

ivm_vector(a=1, b=0, c=2, d=1),

ivm_vector(a=2, b=1, c=1, d=0),

ivm_vector(a=1, b=1, c=2, d=0),

ivm_vector(a=1, b=2, c=0, d=1),

ivm_vector(a=2, b=0, c=1, d=1),

ivm_vector(a=1, b=0, c=1, d=2),

ivm_vector(a=1, b=1, c=0, d=2)]

What I'll do is add any five of these at random while

at the same time scaling them by integers between

1 and 40, which is like adding those five selected
repeatedly.

Other ways of generating random vertexes are very

possible of course. 

The goal is to always add vectors that go tip to tail
from one ball to a neighboring ball in the CCP packing
matrix (the IVM).  We're like hopping from stepping

stone to stepping stone a random number of time.

In [210]: from random import choice

In [211]: def rand_vertex():

     ...:     v = Qvector((0,0,0,0))

     ...:     for i in range(5):

     ...:         v = v + randint(1, 40) * choice(the_qrays)

     ...:     return v

     ...:

Here are the three resulting vectors, all fanning from the origin:

In [214]: A, B, C = rand_vertex(), rand_vertex(), rand_vertex()

In [215]: A

Out[215]: ivm_vector(a=16, b=0, c=17, d=35)

In [216]: B

Out[216]: ivm_vector(a=128, b=0, c=104, d=28)

In [217]: C

Out[217]: ivm_vector(a=9, b=0, c=40, d=47)

They're all in the same quadrant, but not coplanar, so by "closing the lid" we'll get a tetrahedron from them.

Now lets compute that tetrahedron's volume, using an algorithm that accepts the six edge lengths as input:

In [220]: tet = Tetrahedron(A.length(), B.length(), C.length(), (A-B).length(), (B-C).length(), (A-C).length())

In [221]: tet.ivm_volume()

Out[221]: 27185.0

So there we have it, a whole number volume for said tetrahedron.

These vectors all have xyz coordinates as well, and an XYZ volume:

In [224]: A.xyz()

Out[224]: xyz_vector(x=12.020815280171307, y=-0.7071067811865475, z=-12.727922061357855)

In [225]: B.xyz()

Out[225]: xyz_vector(x=18.384776310850235, y=72.12489168102785, z=-1.414213562373095)

In [226]: C.xyz()

Out[226]: xyz_vector(x=5.65685424949238, y=0.7071067811865475, z=-27.577164466275352)

In [227]: tet.xyz_volume()

Out[227]: 25630.2638

The R-edged cube is a little bigger than the D-edged tetrahedron (D = 2R) so the xyz_volume number is a tad less (fewer XYZ cubes).

Allowing for floating point rounding (I do some in the algorithms), the conversion constant is sqrt(9/8):

In [228]: 27185.0/25630.2638

Out[228]: 1.0606601715898063

In [229]: sqrt(9/8)

Out[229]: 1.0606601717798212

Kirby

[Joseph Austin]

On Jun 14, 2016, at 3:07 AM, kirby urner <kirby...@gmail.com> wrote:

Only three lengths are needed per any given point, 
because in each quadrant, the ray not defining it, not
a border edge, i.e. pointing away from said quadrant, 
is not needed to reach the points within.  

We don't give away it's seat at the table though.  We 
keep slots for all four.  That's what the number 0 is for, 
after all.  A placeholder.

But if you DID record the projection of your point on the fourth axis (extended negatively as in xyz),

then the fourth coordinate would not be "zero" but some lineaar combination of the other three.

So your "zero" is not magnitude zero but more like "n/a".

At some point, a computer would need to do some "exception processing" to ignore your artificial zeroes,

which could introduce an addtional complexity that might cancel whatever advantage you claim for IVM.

Joe

[kirby urner]

On Tue, Jun 14, 2016 at 10:03 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Jun 14, 2016, at 3:07 AM, kirby urner <kirby...@gmail.com> wrote:

Only three lengths are needed per any given point, 
because in each quadrant, the ray not defining it, not
a border edge, i.e. pointing away from said quadrant, 
is not needed to reach the points within.  

We don't give away it's seat at the table though.  We 
keep slots for all four.  That's what the number 0 is for, 
after all.  A placeholder.

But if you DID record the projection of your point on the fourth axis (extended negatively as in xyz),

then the fourth coordinate would not be "zero" but some lineaar combination of the other three.

So your "zero" is not magnitude zero but more like "n/a".

I'm not sure you're seeing it yet, maybe.

By "axis" you mean something infinitely long, a line as defined in Euclidean science fiction.

No one has ever experienced such a line, there's no empirical evidence for them, they're not "self evident" in that sense.  Rather, by social contract, we agree to have these infinite lines.  That's what you call an "axis" in this namespace.

Another related "axis" is like the axis of the Earth, very finite in length, about 8000 miles.

In the Quadrays, we don't have any infinite "axes" to start with, only "rays" i.e. half-lines that go from an origin in one direction only.  The idea of "basis vectors" is very finite.  They all have fixed and equal length. 

They go to the vertexes of a regular tetrahedron, from its center, 1/4th of the way from each face center, to the opposite apex. 

https://www.uwgb.edu/dutchs/symmetry/TetrahedronProps.htm

Speaking of tetrahedrons, note that Buckminster Fuller is not cited anywhere at:

https://en.wikipedia.org/wiki/Tetrahedron

not even under pop culture.  To me, these are all dots that inter-connect, Wikipedia doesn't have to do it for me.

On an equilateral triangle (in contrast to a regular tetrahedron) the perpendicular bisectors make a point 1/3 of the distance from each base to the twice-as-distant corresponding apex.

https://www.cs.princeton.edu/courses/archive/fall98/cs341/solutions/f_2_1.jpg  (easy to see with this diagram)

These four rays (hence "quad") divide allspace (all of space) into four quadrants each bordered by three rays.

Inside each quadrant, we play the XYZ game as if in the first octant, with every point reached by adding basis vectors, appropriately stretched.

Note in this Euclidean-like world, no time or mass or momentum is signified.  Considering vectors simply as laser pointers to dots, is par for the course, not my invention.  There is no time variable in the picture.

With these very-like-XYZ quadrays, we know which vectors we mean by positional representation (a, b, c, d), just like we do in (x, y, z).

So when I say a "0 always appears" that's because a star shining in a quadrant of space, cannot be in any of the other three quadrants at the same time. 

One of the four rays is inactive or dormant, vis-a-vis any given star.  There's always a "back stage" vis-a-vis whichever quadrant is "lit up".  The other 3/4s.

In this sense, of four rays from (0, 0, 0, 0), named a, b, c, d, there is no "linear dependence" of any one, on the others.  Take any three and only allow growing and shrinking *in that direction* (no reversing!).  They're helpless to reach the "back yard".

Every point in space is a linear combination of these four rays, with a 0 adding nothing from one of the four players, depending which one is unneeded (a star is in only one quadrant at a time or on a border).

By "linear combination" I mean stretch / shrink and add.  Do not reverse direction as that would be unnecessary.  Just reach out to the star in your quadrant with the three arms you need, neglecting the arm you don't (not right now).

In XYZ we need six arms but call it three because of "axes" (three "axes" intersect).  That's actually more arms than we need.

In this IVM mechanism / apparatus, we need only four arms, not six, and call it four. 

But for any given point, we don't need one of them.  So we use zero.  That's it's purpose, in XYZ too.  But then in XYZ we get to reverse basis vectors and not call that new basis vectors.  We resort to the negation symbol as a permanent feature of the address.  More arms, extra symbol.  Cube less topologically primitive.  Tetrahedron plays well with others.  You don't have to give up your old favorites.  So what's the fuss?  4D IVM geometry is here to stay, not even such a new kid on the block by now.

 

At some point, a computer would need to do some "exception processing" to ignore your artificial zeroes,

which could introduce an addtional complexity that might cancel whatever advantage you claim for IVM.

Joe

I don't see them as artificial. 

I see "amputated XYZ" as addressing only one octant with all positives and zero i.e. the positive numbers have very limited range.  7/8ths of space is not addressed.

The workaround is to allow three more arms and to not call these basis vectors even though they do the important work of bringing 7/8ths of space back into the picture. 

With quadrays, the workaround for the original three rays not hacking it, was to allow only one more arm and to adjust the angles to have more symmetry. 

XYZ all-positive is 90-90-90 but once we stick one more basis ray out the back (not three), then why not relax to 109.47-based symmetry, and make it a game with caltrops instead?  You can always jump back to jacks.

Kirby

[Joseph Austin]

On Jun 14, 2016, at 2:32 PM, kirby urner <kirby...@gmail.com> wrote:

I'm not sure you're seeing it yet, maybe.

Apparently not.

I think I get that you can name the points.

And compute volumes.

But then what?  

What else can you do with those coordinates that is simpler or more useful than cartesian or polar coordinates?

Joe

[kirby urner]

At the moment I'm mostly prying open, and keeping open, the possibility for a third meaning of 4D.

As the curtain rises in 2000 AD we have:

4D as in Tesseract (any N-D polytope, Euclidean)

4D as in Time Machine (any black hole, non-Euclidean)

4D as in Tetrahedron (Synergetics, a hybrid, also timeless)

The 3rd 4D on this list has few champions (more than just me for sure), and as the winner of a Synergetics Explorers Award (with David Koski) I feel some obligation to keep a foot in the door for contestant number #3.

Quadrays were proposed by some VP in Manhattan, by the name of David Chako, whom I met once in person at the base of the World Trade Center, along with P. D. Ouspensky fan Gary Nackenson (another Princeton alum) -- well before those Twin Towers came plunging down.

Besides that brief meetup in meat space, we congregated on a listserv named Synergetics-L, which for a time I loosely moderated, kicking off only the one guy, for going on and on (and on) about the OJ trial. 

We attracted help from one Tom Ace, a brilliant Silicon Valley mathematician who knew how to do integrated circuit routing algorithms.  He came up with the 4x4 rotation matrix, still there in the Python code (he used C or C++).

Tom was a boyhood friend of my first Princeton roommate, Rick Sonnenfeld, tenured professor of Physics @ New Mexico Tech (he studies lightning, his mesa great for that).

I've shown how addressing the CCP is done with all positive integer addresses. 

Could that prove useful?  A puzzle piece.  CCP = FCC = ground zero in crystallography.

The Cartesians count on 7/8ths of their space getting negative numbers in their addresses, thanks to the slave labor contributed by three second class non-basis arms considered Negro or Negative. 

Only one octant is all positive, the best, most privileged neighborhood. 

My 4D civilization is less two-tier in having only positive basis vectors and all positive addresses, regardless of quadrant.  That's a plus in itself.  Talk to neuro-scientists down the hall, if you think negative addressing doesn't matter.

I don't need to prove any utility beyond keeping the 4D IVM alive, as in that context we already have the volume six rhombic dodecahedron, the encasement per each CCP sphere.  Whole number volumes for more shapes, means recruiting more STEM fans at a young age.  Math won't be such a turn-off, given the new streamlining.

Any civilization with our wholesome approach to the CCP, with volume six voronoi cells, will run circles around those with no such crystallography background. 

I'm confident in our geek "world domination" meme (means "self domination" as we're each a world). 

All CEOs with advanced radar already know about GST by now, if connecting with global data at all.  :=D

Kirby