Quoting from Synergetics (a philosophy):
1073.11 Since unity is plural and, at minimum, two, the additive twoness of systemic independence of the individual system's spinnability's two axial poles, the latter's additive twoness must be added to something, which thinkable somethingness is the inherent systemic multiplicative twoness of all systems' congruent concave-convex inside-outness: this additive-two-plus-multiplicative-two fourness inherently produces the interrelationship 2 + 2 + 2 sixness (threefold twoness) of all minimum structural-system comprehendibility.On math-teach, I came up with a kind of game, developed in part to generate appreciation for the different meanings of these operators (+ and * i.e. addition and multiplication). There's the abstract algebra topic of groups, rings and fields, which is customarily not broached until college, except in a cursory manner that goes by pretty fast.
1073.12 All systems are conceptually differentiated out of Universe.
System + environment = Universe
Universe - system = environment
1073.13 The environment is dual, consisting of the macro and micro (outsideness and insideness). Ergo, a fourth twoness of all prime structural systems is synergetically accountable as2 + 2 + 2 + 2 = 8.
However, as we import more segments to our alternative track, newly lightened by not having as much heavy calculus, we'll find the time for more of this material, which is actually somewhat fun and intuitive for many a demographic (depends on how taught, I favor some cartoons, animations, even claymations). The idea of an entire track sounds ambitious at this point, as we're still looking for single course pilots to pop up around the state, on reservations or wherever.
On Synergeo, I'm linking to Fuller's omni-directional halo concept, which is perhaps most brilliantly defined in No More Secondhand God. The topic is Descartes' Angular Deficit, the difference between 360 around each vertex, and a chordal polyhedron connecting the same dots. One tetrahedron's worth of angle needs to be taken away. Then, once you get such a system, most simply a tetrahedron, you have the ability to expand it through the jitterbug phases. At the icosahedral and cuboctahedral phases, the number sequence 1, 12, 42, 92, 162... kicks in, as generated by the above 2 * P * F**2 + 2, with P = 5. Per Donald Coxeter, this exquisite result was worth memorializing. He wrote a professional paper about it, in his inimitable style.
The above shape-governing formula might be likened to some seed molecule placed in a dendrimer, anchoring its growth. However the more formal visualization of said formula involves sphere packing, and that's something we'd like to talk about more, not because they form a practical applied material, but because the mathematics opens doors. You might want to move to hyper-dimensional sphere packing, which has practical applications in optimizing communications channels, while keeping conversations separate.
This is rich cooking, lots of spice. The flavor is still esoteric for 2010 as no one is much interested in 2 * P * F**2 + 2, especially as written, in some non-standard Pythonic notation (** is the exponentiation operator).
Speaking of Python, I was on edu-sig today chatting with Bill and company about a fine page of exercises, drawn from many walks of life. These are like "cave paintings" in their relative simplicity, yet project some useful ideas. I jumped in on the Caesar Code exercise in particular, partly just to reiterate some earlier work, exercise my skills with a dictionary. Bill wanted uppercase letters and punctuation to make it through unsubstituted, and my version achieves that.
Last night Dave Koski went over some of his more recent discoveries. I find them interesting and relevant. We've both been cogitating on a lot of the same themes. phi/radical(2) has come up again, in connection with a scaled-up S-module (one might say). More later on this.
My editorial on the NYT op-ed piece is not supposed to sound too alarmist, yet I am looking for ways to register some serious concerns. Bill Gates and I might be on the same side this time, in thinking more radical reforms are needed. The trick is to not have teachers get completely defensive in reaction, as if any wish to institute change were an implied criticism of the work of dedicated professionals. On the contrary, I expect these professionals themselves to agree: that change is truly necessary. This is a partnership opportunity, not an attempt to vilify an entire profession.
Finally, my posting to edu-sig revisits the feud in computer science, twixt a lambda calculus wing and other factions. I'm worried that discrete math is the can we keep kicking down the road, when instead we need to be developing a new track that looks a lot more like Bill's. The high school aged would ride these rails, but so could adults, including their teachers. Other curriculum writers, such as Edward Cherlin, go further back into childhood. We're somewhat spread out along this railroad and its many switching yards.
I anticipate lots of new positions will open up if we invest in this future. But will needed engineering occur if the focus is on some soap opera between contentious camps? Perhaps this is not an either / or proposition?
The same questions might be asked around just about any low level conflict I suppose. Sometimes competition helps whereas other times it drives away skittish investors, not sure what to think, but disliking the sense of acrimony?
"If the people are feuding, will they forget to grow food and stockpile for the future?" That might be the more civilizational question. The Native American lore in this region chronicles times when people tried to take short cuts and paid a high price. We'd rather learn from our mistakes than repeat them. Perhaps more philosophers will get involved? More Chiefs? Chieftesses?
On the home front, Quinn stopped by with Nick, also Gideon, also Rose.
