Saturday, May 17, 2014

Can you help with a math problem?

Can you help with a math problem?

One of the apparent weaknesses of the paradigm change storyline (see: http:\\ ) that I am advocating is, when I say people need to "begin with tetrahedron", rather than with the XYZ-cube,   that is NOT an easy sell on several counts.  First off, ALL of us are who are able to read these sentences have already been indoctrinated  or educated or initialized into the cube-first orientation.  And there generally is really no going back, completely, for us.  We start with what's given.  So the best WE may be able to do do is perhaps just  ADD tetrahedron to our thinking and nest it somehow  as primary within our engrained cubism.

Secondly,  though,  there is the large matter of calculation and so-called coorelation with rationality.  In our traditional cube-first initializations, we are intensely satisfied  with observing that a stack of 3 blocks wide by 3 blocks long always involves 9 blocks. And if we stack these three layers deep there are  27 blocks which is also  9 plus 9 plus 9, three times nine and also three cubed.  So, the cubic orientation  comes with this rather inherent counting math flowing quite directly from its basic cubic structure.  The patterns are so rational, efficient and useful.  Everybody likes them.

But when when crawl through the opening in the dominant paradigm shield wall and we saunter  over
into the land of tetrahedra, or at least when I do,  truthfully, I feel pretty vulnerable, if not downright weak when thinking about doing equivalent or comparable types of stackings to get  the familiar counting numbers, associative rules and regular math-like patterns.   It's not so easy.

That is what I am asking for help on, if anyone can outline sensible ways of stacking tetrahedra to get  the familiar mathy patterns that we naturally get with stacks of cubes.   Any thoughts?

When I thought about this puzzle  recently, and admitted this gaping insecurity to myself,  my perhaps overly defensive reactions, or, let me call them epiphanies, streamed forward like a torrent, which, in rough draft form sort of sound like the following...

1. On one hand, one tetrahedron defines four of the corners and six of the diagonals of every unit cube, so stacking 3 of those tetrahedra wide by 3 of those tetrahedra long, still involves 9 tetrahedra, etc., but, even I will admit that is contrived and such units wouldn't stand up - be sustainable - by themselves.

2. Similarly,  two tetrahedra set at 90 degrees to one another do define all eight corners and all twelve diagonals of a unit cube and such artifacts would stand up and  could sort of stack like unit cubes. But here we have sort of a curious or excessive redundancy,  or we get to imagine  such a 'two-state' system consisting of a 'left' and a 'right' state, or we could imagine just one tetrahedron in each cube, say, like in a superposition of ~both of the states, or oscillating between both at some high frequency.  Of course this imagery, to me, conjures up parallels with real and imaginary numbers, and also  the if-one-then-not-the-other, on-off binary-Boolean  one-half spin-paired states, as well as the so-called orthogonality  inherent in electro-magnetism. 

3. However, if we really consider a similar type of stacking as comes within 'cube-onics', in the tetrahedron structure what we get is pretty much like a Sierpinski sponge or a process I call "half-doubling".  That is,  we  first take some length, L, and split  it in sixths to form the six edges of a tetrahedron. Then we can take another increment, L, and connect the mid-points (half points) of each of the six edges which then form/define four smaller tetrahedra stacked about a central octahedron "within the containing tetrahedron".  Then, adding another increment, L, that can "halve" each of those smaller tetrahedra in the same type of pattern of containing four smaller sized tetrahedra stacked about a smaller octahedron. And so on.   

It turns out that the volume of each octahedron is equal to one-half the volume of the containing tetrahedron (equal to the sum of the volumes of four forming stacked tetrahedra.   Moreover, if one considers the octahedra as "holes", then adding increments of L, generates a nested system where the area of the tetrahedra at each level stays constant while the contained volume tends toward zero-ish and the number of tetrahedra tend toward a high number at a rate of 4^n.   Starting with a one meter edge-length the Planck length is approached within about 30 halvings and 4^30   is about 1.15 x 10^18 little tiny tetrahedra.  

Now,  adding increments  has some similarity with making quantum changes.  Also, depending on how well one may be able to visualize increments adding in or flying off such  a dynamically nesting structure,  one MAY get some feel for pulsating nested fields within nested fields through use of this device.  This type of implicit tetrahedral "half-doubling" stacking has a curious, but stochastic, uncertain, natural sort of  multiple-state-like appeal. However, it certainly is not easily recognized, if at all like  the deterministic stacking of unit cubes together to get counting and mathy computational patterns familiar to us within the cubic stacking structures and the cubic orientation.

As I considered and perhaps accepted this deficit of tetrahedral structure a few days ago,  I began to provisionally accept that perhaps the different  classes of structures all have different types of purposes, uses and advantages.  Thus, learning, say, "cube-onics" does get us counting and maths and numerical computational and gives us some hints about rationality from a fairly hard-core, deterministic-like perspective.   Yet, these counting number/math  patterns are just not  as readily apparent or available when  looking at tetrahedral stackings. Or, at least I don't yet see them. So, where I have said, "begin with tetrahedron" based on my  belief that our underlying internal tetrahedral analog math is primary and ought to be initialized early during education, I've been mistaken. That's not the way our  education  has worked and it's likely not going to change, at least not for a long, long time.   The central issue, though, is 'both and more', addition of tetrahedral and other frameworks, and migration toward nested mathematical structures.

From this perspective, and with reality being nested fields within nested fields,  XYZ-cubic stackings give a view into rational, deterministic, let's say correlating pretty well with the classical segment of reality.   The tetrahedral structure/math is more directly reflective of states and multiple-states and stochastic, uncertain, let's say, the non-classical, ~quantum-like events and transactions and realms .  

Thus when we start with cube and then nest tetrahedron within it, we get a fairly straightforward view into the classical and non-classical regions.  And, this is just with plain tetrahedra and not yet sliding off into the tactile push-pull, one-half spins of the~binary or ~magnetic tetrahedra. 

Significantly, considering  nested structural coding as inherent, then the difficulties, say, with contaminating food supplies, or non-sustainable eco-economic--environmental situations are related to assuming a non-nested system.  That paradigmatic assumption is not correct and comes with far-reaching negative consequences.   

It is time to migrate to the emerging nested fields within nested fields scientific paradigm.

Thanks in advance for all of your help.

Best regards,
Ralph Frost

With joy you will draw water
from the wells of salvation. Isaiah 12:3

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