Sunday, October 27, 2013

Thursday, October 24, 2013

Book of Rhombs

Glenn Stockton was regaling me with stories this morning, as we hiked around Mt. Tabor, stemming from all the Megalithic Math he's been studying.  He's been devouring Keith Critchlow's new book Time Stands Still.  At one point in our conversation he mentioned finding it very British to hear diamonds described as "rhombs" as if this latter word were so familiar.

Meanwhile, David Koski has been pushing this triangular book covers demo from several angles.  Start with any rhombus really, but some have more interesting properties.  We started with the two book covers being equilateral triangles of edges D, then right triangles with edges D, and now, in this latest video, the long diagonal of the rhomb is D, while the short diagonal is sqrt(2).

This D is the diameter of the unit-radius sphere.

I'd actually written quite a bit about these two rhombs defining a Coupler when placed at 90 degrees, but it took David's nudging for me to finally realize I was again covering this same territory, now with the "triangular book covers and two oppositely flapping pages".  Putting the Coupler at the XYZ origin is a great way to build a bridge to the IVM and Synergetics way of thinking more generally.

In massaging the source code for this demo, I realized that my code for the inadvertent tetrahedron was hard coding around all edges being D except the green and magenta, so needed to fix that for this video to have the right volumes.

Towards the end, I start mentioning the Rite, though it might not be clear that's actually the name of a specific tetrahedron.  The Rite and quarter Rite are both space-filling tetrahedrons.  Aristotle said tetrahedrons fill space and is often criticized on the theory he meant regular tetrahedrons.  However irregular tetrahedrons do fill allspace with identical copies of themselves and without left and right handedness, the Rite is one of these, as is the Mite.

To recap a theme of the last three "triangular book covers" videos:  the flapping triangular page defines two equal volumes, with a 3rd "inadvertent tet", again of equal volume, supplying a space-filling complement to the other two.  Indeed, any two of the three tetrahedrons formed, may be used to build an octahedron (two and two needed), with the third tetrahedron playing the role of the complementary space-filler ala the isotropic vector matrix model, but skewed and/or stretched (same topology).

In this case, starting with the rhombus of the rhombic dodecahedron, when the page is at 90 degrees, all three tetrahedrons are Rites and the octahedron formed by any two is the Coupler, of unit volume in Synergetics.

The rhombic triacontahedron hovers as tantalizingly relevant.  A next video might get into five-fold symmetric space-filling more, David's forte.  The page tip needs to click stop at "4/8" on the way to its vertical at 9/8, where 8/8 is the regular tetrahedron.  Length-determining volumes are the 2nd roots of these fractions.  That's back to when our rhombic book has edges 2 (i.e. D).

Link to source code on Github.

Saturday, October 12, 2013


I spent a lot of time reading Alan Watts as a younger person, none of which time I regret; he was / is a good teacher of what we may legitimately call "Buddhist thought".  For those who don't know, this intellectual guy lived in Sausalito.  The Wikipedia picture shows him in full guru costume, which at the time was a trendy form of rebellion against establishment Western dress.  People were re-balancing their relationship with Asia, especially around the Pacific Rim.

Watts was in turn a student of D. T. Suzuki, a Japanese Zen master, and a lot of the Watts stuff works at translating such words as "satori" as "enlightenment" and so on.  But then what does "enlightenment" even mean in English?  You have the "Age of Enlightenment" which points back to such French luminaries as Voltaire.  You have the several dictionary definitions.  "Enlightening" can mean becoming aware of a more inclusive or elucidating way of looking.  That's a link to Wittgenstein, who baked "ways of looking" into his core "language games based" elucidation.

One has times in life wherein dots connect and circuits flip on.  Epiphanies may be fleeting, hour-long, ongoing themes.  Salvador Dali had some lengthy epiphanies.  He didn't worry, like a Viagra commercial, about an epiphany lasting too long.  In hindsight, surrealism benefited enormously from Dali's willingness to experience "satori" quite a bit.

One of the things the enlightenment literature tends to recommend is maddeningly complex practices of some kind, lots of tedious, repetitious, stupid, boring stuff.  This is no accident.  The mind is more prone to produce breakthroughs when forced into some corner and made to fend for itself.  Koans were / are like this:  puzzling little sayings and mantras designed to produce "aha!" experiences, more than one.  But then just life itself induces these "aha" experiences.  You don't need to go looking for koans.  They're in your face at all times, if you know where to look.

That being said, it's also true that communities need dishes washed, pigs milked, goats tended, fish smoked, or whatever the tasks of a subculture.  Were "enlightenment" to be reserved only for those on vacation or in retirement, that'd be droll.  Busy home owners need "enlightenment" as much as anyone.  An egalitarian flavor enters in, but also in reward for some kind of meekness, or humble submission to "chores" (doing your share of the work, participating in building / sustaining community).  The Buddhists call this Sangha i.e. Community.

Westerners often get bent out of shape by the word "Community" as it rhymes with "Communist", and yet they pay lots of lip service to "Fellowship" and "Church Community" as a good thing. It's disbelief in any God that made Communists a bad thing, but then Buddhism was never attacked in this way, at least not directly.  So Alan Watts could be rebellious and anti-establishment and not-communist at the same time, which was doubly subversive.  I was / am a fan.

Lots of movies use "satori" in that they help the audience experience revelations about things.  The plot twists and turns, and by the end there's a satisfying resolution, or not.  The ending may not be what matters.  Satori is found in films, that's what matters.  No wonder Japanese cartoons (anime) are often so philosophical / spiritual, so Zen in some cases.

The Quakers have "satori" too, which I might talk about another time.  The mode of "expectant waiting" is precisely that cultivated by many a devoted seeker.  To somewhat personalize the provider of insights as "God" (in place of "the muses") is the monotheist mode, but you need not be a "believer" to appreciate the power of intuition.  Kant's obsession with the possibility of synthetic judgments a priori is no less a meditation on whether moral truths might share something with the logically imperative.  You don't need to be a believer in some "God" to experience satori, as any atheist might tell you (whether Communist or not).

Sunday, October 06, 2013

IVM 1-2-3

Given how I wrote the code for these demos, spreadsheet style, with governing globals up top, it wasn't hard to stretch the spine of the book, to make the two book covers make a square, instead of a rhombus.

In the previous "book covers" video, two equilateral triangles lay flat against a plane, with a triangular "page" flapping between them.  In this one, it's two right triangles laying flat, and when the page reaches 90 degrees, the half regular octahedron shows up, each of the complementary tetrahedrons a quarter of same.

Then there's the "inadvertent tet" made from the purple and green rods, others red.  Right when the complementary tets are equal, it turns regular (they're produced together) and the "octet truss" is born (the pure IVM).

Let's be clear though:  the IVM was there last time too, with the equiangular book covers.  The regular tet's complement, the "iceberg tet" is a quarter octahedron, just like the two "iceberg tets" forming here in complement (see Fig. 987.210D).

So this time the inadvertent is the regular tet and both complements are icebergs.  Last time the IVM formed when both the inadvertent tet and one of the complements were icebergs, with the other complement a regular tet.  So two views of the same thing.  A little dance.

Here again, even with the different book covers, you have the option to pair the inadvertent tet with an iceberg (1/4 oct) to get an oblate octahedron of volume 4 + another iceberg to fill space. That's not the focus, but is a consequence of the generalization in the earlier video, that any two of the three may be chosen to build the octahedron, leaving the third tetrahedron to complete the "IVM-like" space-filling matrix.

It's not hard to see that the IVM gets to "waver" in some affine ways (to become "IVM-like").  Picture a layer of squares, like a checkerboard, then another layer above, but with its squares offset to have its corners above the others' centers.  Keep stacking that way, corners over centers, and connect each center to the four corners below.  The distance between layers is just right such that these slanted intra-layer members are also all the same length, the length of the square edges.  That's your IVM.  No shortage of squares.

Now picture the squares "wavering" to become rectangles as the distance between layers also wavers.  All the rods have become stretchy but we're keeping the layers parallel and no rods are disconnected, so the same 12 from every hub.  The familiar topology.

Space Filling Triads of Tetrahedrons

Do we say "tetrahedra" or "tetrahedrons" for the plural?  My spellchecker prefers the latter, but through long habit, I tend to use the "hedra" ending.

Tetrahedrons in the plural is what this video is about.

My technique was to code in the PyCharm IDE by JetBrains, to which I subscribe, while importing the visual package from VPython dot org.  Then I turned on QuickTimePlayer on the Apple Mac Air, which does a decent job of screen recording.

Finally, I pull that recording into iMovie and talked over it, before uploading to YouTube.  These are skills within range of a broad audience and are also increasingly the skills associated with academic studies.

David Koski provided most of the brain power in terms of providing the original insight I'm endeavoring to communicate.

What's somewhat interesting about this video is what's not shown, or what I leave out of the narration.

For example, I don't make it abundantly clear that the "inadvertent tetrahedron" with four red edges, one green and one purple, also has the very same volume as that of the two complements with which it is associated.

These three, the two complements plus the inadvertent tet, are what comprise the space-filling triad.  Any two will assemble an octahedron with two copies of each (for a volume of 4x whatever volume we're at), and the remaining tetrahedron will complete the space-filling, with a volume 1/4 that of the octahedron at all settings.

In the video, I use the term "isotropic vector matrix" somewhat loosely, as it's the topology of this simplicial complex that I'm focused on, whereas clearly not all the rods are the same length, as they are in the pure IVM scaffolding (as they are in the XYZ scaffolding).

In the IVM topology, every vertex has 12 rods emanating therefrom and tetrahedrons combine with their partner octahedrons in a ratio of 2:1 i.e. there are twice as many tetrahedrons.

Do the triangular book covers need to start out as equilateral triangles?  No.  In a future demo, I will start with 45-45-90 degree book covers lying flat to make a square and go through the same transformation.  A triumvirate of space-filling tetrahedra are made that way as well.  Indeed, we can make the pure IVM rather straightforwardly.

The demo I'm showing here does have the pure IVM within range.  When either complement is the regular tetrahedron, the inadvertent tet and complement are the same 1/4 "orange slice" of a regular octahedron (four wedges =  1 octahedron).  David and I call these wedges "icebergs".

In XYZ accounting (cube based), when the page tip is at 90 degrees, the octahedron and tetrahedron have a volume ratio of 4:1, as always, but the volume actually is 4, the tetrahedron 1.

I'm assuming red edges of 2, my value for D, the Diameter of the four unit radius spheres that might pack to create and all-red-edges tetrahedron when their centers were interconnected.

Synergetics accounts this as a model of D to the 3rd power, which is why the volume numbers differ by sqrt(9/8).  When the complements reach their highest volume at 90 degrees, that's sqrt(9/8) more than the regular tetrahedron volume (= that of its iceberg complement).

Wednesday, October 02, 2013

Afghanistan Revisited

Maria A. Beebe is back.

Wanderers are getting an update on the state of Information and Communications Technology (ICT), from someone who has lots of on the ground experience.

Dr. Beebe has met with us before.  Carol (mom) joined us this time as she's tasked with making a public presentation on the current state of Afghanistan.

Cisco has quite a bit of market penetration.  Curriculum writers have clamored for less brand-based training.

She started off by showing us this World Bank video about ICT in the recent past.

Getting university curricula ramped up, as well as providing mobile phones to an increasing number of subscribers are the primary goals.

Afghanistan suffers from a syndrome similar to China's:  in pirating Windows, the ICT population is both stymied by virus prone platforms, and their skills development is being retarded, as they struggle with black box software.

I was curious about the level of censorship of the Internet, e.g. whether opposition groups fighting the US occupation could have their web sites.

Others asked to what extent phone calls were being surveilled.