Another Introduction to Tetravolumes

[kirby urner]

Another Introduction to Tetravolumes

by Kirby Urner

(July, 2016)

Here's a new Jupyter Notebook undertaking to compare

the 4D IVM tetravolume measuring system vis-a-vis the

3D XYZ cubic volume system we all learn in school.

https://github.com/4dsolutions/tetravolumes/blob/master/Computing%20Volumes.ipynb

By 4D I don't mean "four mutual perpendiculars" but four

rays from the origin through the corners of the topologically
minimum container, the most primitive "cage" enclosing
volume, a "simplex" of Euclidean dimension: Dim=3. 

Think of the 4D as a "branding mark" signalling a different

approach to 3rd powering.  The Notebook expresses this

difference in a way similar to the way Bucky does in

Synergetics 982.44-47.

Imagine two sticks of varying length with a common origin,

at any angle to start, and imagine an operation that connected

the two tips with a 3rd segment and thereby marked off

the area within. 

This is "closing the lid" in the case of a triangle (Dim=2). 
However, leaving the angle unspecified fails to nail down
the numeric results of the computation.  We need to decide
on a fixed angle to always get the same result.

Fixing the included angle defines a specific triangle (SAS). 

In XYZ multiplication, that included angle is 90 degrees,
upon which, after we "close the lid", we then multiply by 2,
placing two of the resulting triangles on opposing sides
of a shared hypotenuse, thereby defining a rectangle.
 

That's our standard algorithm for area.  |A||B| is a rectangle,

a square if |A| = |B|.  We all know this cold by elementary

school.

In the IVM (different scaffolding), the included angle is
60 degrees and upon closing the lid, we do not proceed
further to double the resulting area.  That internal original

area, of the triangle, is simply defined to be |A| x |B|,
where |A| and |B| are the stick lengths. |A||B| is an equal-

lateral triangle if |A| = |B|.

https://youtu.be/2B1XXV2Eoh8  (my explanation on Youtube)

Moving on to volume (Dim=3)...

In XYZ we start with 3 sticks from a common origin (0,0,0)
fanning out in a mutually orthogonal arrangement.  Closing
the lid begets a right tetrahedron that is only 1/6th of the
total volume we come up with, by forming a rectilinear
parallelepiped from these sticks. |A||B||C| is a brick, maybe

a cube.

In the IVM, three sticks from a common origin fan out

along the imaginary edges of a regular tetrahedron, which

establishes relative direction.  The magnitudes are variable,

so the resulting tetrahedron may not be regular.  The angles

at this corner remain fixed, as they do in the XYZ case.

We simply "close the lid" and call that the volume.  The

numbers stay the same. 2 x 4 x 5 = 40, same as before,
it's just that 40 is tetrahedron-shaped, and obtained from
adding three sticks to the initial three, thereby forming a
six-edged, four facets shape, not a hexahedron as before.[1]

How might we bridge these two operations?

Taking our cue from Bucky, we construct a tetrahedron

from four unit-radius balls, closest-packed, edges 1/2 D

where D = ball diameter.  Saying the edges are 1 and

the volume is 1, makes the cube in which said tetrahedron

inscribes = 3 tetravolumes.  That's a fixed and known

relationship:  the tetrahedron inscribed in a cube as face

diagonals has 1/3rd the volume (this generalizes to any

parallelepided).

However in XYZ this same cube of edges √2 (in R units)
and face diagonals 2, will have a volume of 2.828427...
or √2 √2 √2 i.e. "√2 cubed". 

So there's our conversion constant:  3/2.828.. or √(9/8).

Another way to think about it:  the XYZ scaffolding or

matrix consists of unit cubes of edges R.  It wasn't originally

developed with sphere packing in mind (unlike the IVM,

which is the scaffolding associated with the FCC or CCP).

The unit R (2R = D) gives us R * R * R = 1 or R-cubed
in XYZ (cube shaped), whereas D * D * D givers us 1 in
the IVM (tetrahedron shaped).  It's like converting
between currencies or energy values (joules, calories)
where the R-edged XYZ cube is about 6% bigger than
the D-edged IVM tetrahedron.  The two units of volume

are somewhat close (like the Canadian and American

dollar) but not the same.

If you know the volume in XYZ cubic units, multiply by

√(9/8) to get the same volume in IVM tetrahedron units,
or use √(8/9) to go the other way.

Why bother?  What's the pay-off? 

The tetrahedron of edges D divides thrice into a cube of face
diagonals D (as we've seen) four times into an octahedron
of edges D (its space-filling complement in the IVM), six
times into a rhombic dodecahedron (space-filling CCP ball
encasement), twenty times into a cuboctahedron (from
12-balls-around-1 and connecting corners).

Volumes Table:

Tetrahedron           1

Cube                  3

Octahedron            4

Rhombic Dodecahedron  6

Icosahedron          18.51...

Cuboctahedron        20

2-Frequency Cube     24

The five-fold symmetric Icosahedron, Pentagonal

Dodecahedron, and Rhombic Triacontahedron all fit

in here as well, there incommensurably in most cases. 
The Jitterbug Transformation is used to connect the

Cuboctahedron of edges D to a corresponding

icosahedron of edges D.

Said Icosahedron of volume ~18.51, combined with
its dual, define a rhombic triacontahedron that, if
shrunk down by by 1/Phi (linearly) gives the RT
of 120 E-mods (60 left, 60 right-handed). 

The RT sharing vertexes with the RD of volume 6 has
volume 7.5 exactly (IVM tetravolumes) and shrunk by
2/3 gives the RT of volume 5 exactly, and the 120 T-mods,
same shape as the E-mods but a tad smaller, by about
.001%

http://www.rwgrayprojects.com/synergetics/s09/figs/f86548.html

What Koski does is phi-scale the E-mods to express

volumes as a linear combination of mods of different

size (same shape).  Quoting from Koski's paper:

E module denotations:

e6 = ((√2)/8)ø ̄⁹ = .002325
e3 = ((√2)/8)ø ̄⁶ = .009851
E = ((√2)/8)ø ̄³  = .041731
E3 = ((√2)/8)ø⁰  = .176766
E6 = ((√2)/8)ø³  = .748838

Quoting again from his paper [2]:

A rhombic triacontahedron with a radius of ø¹, is dubbed
the Super RT. The long diagonal of the rhombic face = 2,
which is R.B.Fuller’s edge for the tetrahedron, octahedron,
cuboctahdron or VE, and the resultant icosahedron from
the Jitterbug transformation.

The volume of the Super RT is 15√2 or 21.213203 =
120E3 = 480E + 120e3 [tetravolumes].

The icosahedron with an edge = 2, inscribes within the
Super RT. It has a volume of 5(√2)ø² = 18.52295. It has
an exact E module volume of 100E3 + 20E = 420E + 100e3.
[tetravolumes]

That's about all the mathematics one needs to know, to understand
about tetravolumes.  It's not that hard. 

We also get to think again about foundational matters, such as
what basic assumptions we might vary to produce interesting
new flavors of mathematics.

[1]  Just for fun, lets compute the lengths of the "lid" edges and feed

these numbers to our Pythonic tetrahedron volume computer, same
one as in the Jupyter Notebook.  Given edges a=2, b=4, c=5 I get

d= e= f= for a volume of...

a = 2
b = 4
c = 5
d = 3.4641016151377544
e = 4.58257569495584
f = 4.358898943540673Another Introduction to Tetravolumes

tetra = Tetrahedron(a,b,c,d,e,f)
print("IVM volume of tetra:", tetra.ivm_volume())

IVM volume of tetra: 39.99999999999998  (check)

[2]  http://4dsolutions.net/synergetica/RevisitingS&Emodules.pdf

[kirby urner]

On Mon, Jul 25, 2016 at 7:14 PM, kirby urner <kirby...@gmail.com> wrote:

Another Introduction to Tetravolumes

by Kirby Urner

(July, 2016)

Blog version:

http://worldgame.blogspot.com/2016/07/another-introduction-to-tetravolumes.html

Kirby

[Bradford Hansen-Smith]

Kirby,  I like your demonstration, it works, but is lacking context. If one is not familiar with triangulation or has every drawn a higher frequency triangle grid understanding the pattern base of multiplication that comes from the structural nature of division, there would be little idea beyond this isolated point location and a vector line demo. With two vectors one can “close the lid” to make a triangle, then see by extending the vectors and drawing parallel lines equally spaced to the “lid” form a figure of higher frequency that gives interesting proportional information from which we can make certain number calculations. Lacking context thus becomes a working formula with little visual understanding of why. For you it is simple having an extensive background in both geometry and computational math that allows necessary connections. For a beginning student this is not a place to start. We are taught the difference between static squares and triangles rather than relationships within spherical packing from which we have a choice of isolating any number of different shapes, relationships, symmetries, volumes, what ever part you want, knowing they are separated aspects interconnected within a single ordered spherical context. Without stacking spheres we cannot expect one to really understand the implications of one point and two lines except as free floating abstract concepts.

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[kirby urner]

On Tue, Jul 26, 2016 at 7:45 AM, Bradford Hansen-Smith <wholem...@gmail.com> wrote:

Kirby,  I like your demonstration, it works, but is lacking context. If one is not familiar with triangulation or has every drawn a higher frequency triangle grid understanding the pattern base of multiplication that comes from the structural nature of division, there would be little idea beyond this isolated point location and a vector line demo. With two vectors one can “close the lid” to make a triangle, then see by extending the vectors and drawing parallel lines equally spaced to the “lid” form a figure of higher frequency that gives interesting proportional information from which we can make certain number calculations.

They're the exact same calculations, just a different graphic applies.  2 x 4 x 5 is still 40, just that's a tetrahedron of those dimensions.  I use '4D' to signal the context shift.
 

Lacking context thus becomes a working formula with little visual understanding of why.

The 2-vector triangle case is a preamble for the 3-vector tetrahedron case. 

The volumetric case provides context:  whole number volumes for well-known polys. 

Easier to reason about than how we teach 'em today. 

And it's not either/or. 

XYZ isn't going anywhere, just making room for IVM thinking in addition.

 

For you it is simple having an extensive background in both geometry and computational math that allows necessary connections. For a beginning student this is not a place to start.

I'm just showing that core mathematical innovations in Synergetics are neither trite nor difficult, nor wrong. 

There may have been some lingering doubts on that score.

This is something art history and philosophy teachers might share.   Or American History teachers.  Mathematicians have for some reason been too busy to care about any of this stuff and I'm not expecting the IMU to turn on a dime.  Inertia and all that.

 

We are taught the difference between static squares and triangles rather than relationships within spherical packing from which we have a choice of isolating any number of different shapes, relationships, symmetries, volumes, what ever part you want, knowing they are separated aspects interconnected within a single ordered spherical context. Without stacking spheres we cannot expect one to really understand the implications of one point and two lines except as free floating abstract concepts.

Did you miss the tetrahedron / octet truss part of the essay? 

Did you see the picture of the octet truss?

http://nbviewer.jupyter.org/github/4dsolutions/tetravolumes/blob/master/Computing%20Volumes.ipynb

Kirby

[Bradford Hansen-Smith]

Kirby, no did not miss that part of essay. Did not see any pictures of spheres only the octet truss, cube and and a variety of mites. Visual information is primary which is why I suggested stacking spheres because then students get a chance to discover for themselves the points of connections and vectors that forms a variety of relationships and get a real sense of where the cube and tetrahedron/octahedron as relationships come from. Without some kind of context we tend to think these things just exist within themselves without question and all that is left is to reduce them down to smaller elements, rebuild in various ways connection to other constructions making it more abstract and less experiential. I know with the marvels of technology and the direction we are going, we will need more than what that gives us.

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[kirby urner]

I think my mistake might have been in using the word "Introduction" as it might connote leading a rank beginner by the hand. My target audience is more a West Point physics teacher like Dr. Bob Fuller was (different Fuller).

I'll consider changing the title. Beginners should follow links for two hours and watch the cartoons, then maybe read it again. Definitely helps to know trig. I used angular functions between the lines, as when netting out that 2 x 4 x 5 tetrahedron of volume 40. Thanks for the illuminating feedback.

Kirby

[kirby urner]

I made a small change to focus the target more.

Apropos of that change, I'm appending the body of a posting to that Physics listserv I'm on.

I'm spiraling back through Jupyter Notebooks as a core code school focus again:

http://nbviewer.jupyter.org/github/4dsolutions/Python5/blob/master/Atoms%20in%20Python.ipynb

(code my students have already worked on, we're in week 4 of 5)

Kirby

Appended posting to a Physics listserv for more context:

Here's something I've billed as "Philosophy for Physics Majors".

It reads differently than "Physics for Poets" which I believe Princeton had, when I was an undergrad there (Class of 1980) and philo major.

Primary link:

http://goo.gl/pWoufe  (safe: goes to Jupyter Notebook displayed in Nbviewer)

Actual URL (before Google shortening):

http://nbviewer.jupyter.org/github/4dsolutions/tetravolumes/blob/master/Computing%20Volumes.ipynb

It's basically showing some foundational stuff by varying definitions that "could have been different" (core spatial geometry concepts).

We might not ever explore the branch, not seeing the trailhead, not knowing we even had the choice.

Philosophy is about revealing such hard-to-find pathways sometimes (unknown unknowns).

If wishing to venture down this trail even further, I'd recommend:

http://worldgame.blogspot.com/2016/07/another-introduction-to-tetravolumes.html

(cites this same Notebook)

http://www.4dsolutions.net/synergetica/eja1.html

(not by me, curated by me)

... along with my Martian Math stuff.

Kirby

About Me:

http://worldgame.blogspot.com/2010/07/physics-conference.html

(there's a Flash-based slide show embedded, pictures from an AAPT conference, and if you have the Flash plug-in installed in your web browser, Adobe may let you know if it's not up to date).

[Joseph Austin]

Someone may have mentioned this before, but having just watched Peter's video, i had Pascal's triangle on my mind.

Of course there is also a Pascal's "Pyramid" (actually, tetrahedron) formed is a similar fashion,

with each node (say a sphere) at one level being the some of the adjacent nodes at the previous level, 

with the external edges being all ones,

that gives the trinomial coefficients.

And presumably the concept generalizes to more dimensions, though visually that would be difficult to construct.

So noting that Clifford Algebra generalizes the concept of "vector" product to a multinomial product,

there may well be some essential correspondence between "tetra-volumes" (and faces and edges and vertices)

to physical components of spatially-extended quantities,

if not as a geometric principle, at least as an organizing principle.

Joe Austin

[Bradford Hansen-Smith]

Joe, I was with you until the last line, not knowing the difference between a "geometric principle" and an "organizing principle." Can you speak to the difference?

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Bradford Hansen-Smith
www.wholemovement.com