Hypertoons 2017

Synergetics has taken various twists and turns since the 1970s, when Macmillan came out with the second volume, designed to interleave with the first, and including an index. In that second volume, we got a more complete basis for working with the five-fold symmetric shapes already within the concentric hierarchy, namely the T, E and S modules.

Picking up on that thread, we have since amplified the 120 T-module volume 5 rhombic triacontahedron, of radius just a tad under 1 (0.99948...), to another copy weighing in at 7.5 and sharing a set of tips with the rhombic dodecahedron (RD), of volume 6.

The 120 E-module volume 5+ RT, on the other hand, of radius precisely 1, scales up by Φ (linearly) to give the 15√2 volumed RT, dubbed SuperRT, wherein the Jitterbug Icosahedron of edges 2R, embeds as long diagonals, with a volume of 5√2 Φ².

The ratio of said Jitterbug Icosahedron to its related larger-volumed cuboctahedron (CO) of equal edges is defined by what David Koski and I are calling the S-factor i.e. CO/Icosa = S/E = (2√2)/Φ²  or (2√2)Φ^-2 or about 1.080363.

Starting with any Icosahedron, one application of the S-factor nets the volume of the corresponding CO with same edge lengths, the "Jitterbug" relationship.

Two applications of 1/S-factor (the reciprocal), on the other hand, nets the volume of the corresponding CO with faces flush to the same containing octahedron, the "skew" relationship.

In other words, this smaller-volumed CO and Icosa are skew to one another with overlapping facial regions, as shown in Synergetics 2, Figure 988.00.

For example, the Icosahedron inscribed in the Octahedron of volume 4 has S-factor edges.  Two applications of 1/S-factor gives the volume 2.5 CO with edges R, i.e. precisely 1/8th the volume of the 2R 20 volumed JT CO.

CO2.5 * S-factor * S-factor gives the S-factor-edged Icosa inscribed in the volume 4 octa, from whence S-modules are derived.

One more application of the S-factor gives the CO shown below, nestled precisely within the RT5+ at the green points.

In contrast, the CO with √2 edges, shown above is precisely 1/3rd the RT5+ volume.

Finally, David realized another relationship:  S3, the volumetric version of the Synergetics Constant, ordinarily used to take us from cubic units to tetrahedron units, is also the ratio of SuperRT to the Jitterbug CO. S3 = √(9/8).

Put another way, the RT with long diagonals equal to the Icosahedron from which a CO is developed by Jitterbugging, has a volume S3 times that CO's volume.  20 * S3 = 15√2.

We may also introduce the T-factor as the ratio between the RT5+ and RT5, i.e. the E-modules and T-modules rhombic triacontahedrons.  That number is (3√2)/Φ³ or (3√2)Φ^-3. The reciprocal of its 3rd root is the linear scale factor 0.99948... (mentioned above) taking us from the E-mods RT to the T-mods RT.

Koski Paper

Original 2015 paper by David Koski:

[ note typo:  "It has a volume of 5(√2)ø2 = 18.52295." should read "18.512296...". ]

What's an S module?

The canonical octahedron of volume 4 has inscribed within it, faces flush, an icosahedron of smaller volume.

Their volumetric difference, carved into 24 modules, four meeting at each octahedron vertex, comprise the S modules, 12 the mirror image of the other 12.

The S mod's volume is (φ **-5)/2 relative to unit volume tetrahedron of edges equal those of the canonical octahedron.

What's an E module?

Take the Rhombic Triacontahedron (RT) that precisely shrink-wraps a sphere (as does the Rhombic Dodecahedron of volume six), and you will find that in tetravolumes it slightly outweighs the RT of volume precisely 5, with 0.9994... the radius.  Explode this 5+ volume RT into 120 wedges and you'll have your E modules, left and right.  The slightly smaller RT of volume 5 is made from T modules.

Following David's paper, we scale S and E modules up and down, by the golden ratio.

I use the convention that lowercase means "scale down" as in "shrink" and capitalized means "scale up" as in "expand".

Since to scale edges by a linear factor (in this case φ) is to change volume by a 3rd power of that factor, the numbers 3, 6, 9... are used to indicate by how much volume has changed up or down i.e. by φ**3, φ**6 or by φ-3, φ-6.

Examples:
smod6 means Smod * (φ**-6)
Smod6 means Smod * (φ ** 6)

In the Python code below, we're confirming that the concentric hierarchy volumes of the tetrahedron, cube, octahedron, icosahedron, may be expressed in lowest term sums of S and E modules of mix scale.

Related Python code (hit the run button to compute output, appears at bottom):

Factory Girl (movie review)

Andy Warhol fans and detractors already know this story. I was only glancingly familiar with most of his work until recently, when Portland Art Museum unveiled a major retrospective.  That helped me tune in more of his scene, though I didn't catch the name Edie Sedgwick until last night, when I finally saw the dramatization.

My thoughts flashed to Patty Hearst a few times, and her relationship with her own family. I'm not a know-it-all on these families, just we have a lot of windows and telling remarks in the public record, which facilitates discussion of celebrities.  Orson Welles comes to mind.

Edie was an heiress from a Santa Barbara ranch family, transplants from Boston, East Coasters on the Pacific. Hearst Castle is on the same coast.  When I think of Hearst, I often flash on Homer Davenport, his lead political cartoonist in some chapters, and native of Silverton, Oregon.  I have quite a bit about Homer in my blogs owing to my friendship with Gus Frederick, an expert on Homer's life and to some extent times.

Edie big dream was to find herself in New York and to pioneer a freer way of being alive in a city big enough hearted to support such experiments.  She was by all accounts bold, but in falling victim to drug abuse, got derailed.  This was the story of a generation and has not ceased being the core plot of many scenarios.

Folks in my cohort have their own generational window in that I was old enough to have Warhol on my radar, but not adult enough to track the soap operas.  I uncover the history of my own time in my later years, having lived through it in my own day dreamy way, as some kid in Italy or whatever.

A lot of work went into making this a real telling. The filmmakers undertake their task seriously. I'm reminded of Mishima.  In being a dramatization, the script takes many liberties with the facts, many of which remain unknown. This movie is but one possible assembly of an intricate jigsaw puzzle.

Musing

:: screencast & guitar by Curtis Palmer ::

More CSN Esoterica

At the Bird Feeder

Here I am, an Oregon-based math teacher, flapping around the bird feeder looking for cyber-currency to keep my Internet channels alive.

That I had to replace the Android and lost my mycelium wallet bitcoin app is no help, and now I've lost the bitcoin public key. I just need to start over.

Not that I expect anyone has slipped coinage into that particular begging cup, deeply buried and now lost.

OCN (Oregon Curriculum Network) has gotten as far as it has with mainly me as its principal sponsor, though I acknowledge the booster rockets have helped.

These days I'm with Coding with Kids and helping middle-schoolers acclimate to eye-balling source code.

Having the topic be akin to what they're covering in class anyway, helps reinforce both the relevance and potential familiarity of the "learning to code" branch.

Last week was Prime Numbers.  This week is Triangles.

I also teach older adults at night, so sometimes bill myself as "pedagogue by day, andragogue by night".

The more "adult version" of the Triangle class both includes more concepts about triangles, and uses more of the Python language.  I've got that at Github.

Codesters is not a full implementation of core Python, plus includes many __builtins__ that core Python does not have.  Sprites for example. And a stage.

You'd think as CMO of the CSN, I'd be doing that full time, but lets remember this was 2017 and most adults were not informed about American Transcendentalism.  We were still bootstrapping back then.

Paid to Play

Paying students and teachers alike, not necessarily at the same rate given different responsibilities, does not assure that any given school will stay afloat. The true cost that never goes away is "opportunity cost": what you're doing right now is at the cost of everything else you're not doing.

Even if my code school pays a stipend of N ETH per completed task K, who says task K is relevant to one's long term plans? You could have gotten N-2 ETH for completing task T at another school, but guess what, the relevance of T is so much greater, that the -2 delta is more than made up for. You've factored in opportunity cost.  Good for you, that's smart.

The model LMS that meters out credits in the form of some non-cash currency, not legal for all debts, but good towards many things anyway, is premised on the wish to reward and support those doing what we regard as real homework, developing in ways a program promises its enrollees will.  Whether we need a blockchain technology to make it work will depend on many factors.

Those with a GST background will have more responsibilities helping government services figure it out, in collaboration with participating universities.

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:: magnified view ::

Related reading @ Grunch.net
USA OS (from the 1990s)