Fwd: [Math 2.0] Re: An example of your favorite lesson plan?
kirby urner
---------- Forwarded message ----------
From: kirby urner <kirby...@gmail.com>
Date: Wed, Mar 21, 2012 at 10:15 AM
Subject: Re: [Math 2.0] Re: An example of your favorite lesson plan?
To: mathf...@googlegroups.com
> Fortunately, Earthlings and Martians both map to the same sphere of
> diameter D, radius R.
>
The shared Martian-Earthling spheres lend themselves to the sphere
packing in our curriculum.
The trend in nanotechnology is toward the controlled synthesis of
caltrops and such, with changes in crystalline pattern marking the
change from body to legs for example. CCP to HCP is jargon not
currently injected in kid-friendly presentations, with some
exceptions. Those exceptions will soon be the new norm as we continue
with our inside-outing of the curriculum rubber sheeting. Gotta have
sphere packing, for nano.
> Earth unit of volume = R ** 3 cube
>
> Martian unit of volume = D ** 3 tetrahedron
>
What's interesting is how close these two volumes are.
Take rods of length D and build a regular tetrahedron from six of
them. That's your container. The metal sheeting means it holds water
or slurry.
Take rods of length R and build a regular hexahedron (cube) from
twelve of them. That's what Earthlings do. This is their "unit
volume" in contrast.
Which has greater volume, in absolute terms?
The tetrahedron of edges D has 1/3rd the volume of a cube of edges rt2
(2) -- 2nd root of 2.
Earthlings would assign that cube a volume of rt2 (2) to the 3rd power.
The Google search engine is also a calculator: cube: 2.82842712;
tetrahedron: 0.942809042
Ah so. The Martian tetrahedron is actually smaller than the Earthling
one, of 1 (in XYZ units).
((sqrt(2)^3) / 3) * sqrt(9 / 8) = 1 says Google.
sqrt(9/8) is like a currency exchange ratio, taking us from $1
(Martian) to $1 (Earthling).
> Note that both use exponentiation to express their respective shapes
> as they grow at a 3rd power rate relative to edges (no less true of
> either).
>
Once you've switched to Martian currency, there's no need to jump back
and forth.
Imagine "volume bills" instead of "dollar bills". A cube of face
diagonals D is $3.
An octahedron of edges D is $4. The rhombic dodeca, space-filling, of
long face diagonals D has value-volume $6.
Cuboctahedron of edges D: $20.
You might imagine small pocket change pouches containing these shapes,
perhaps made of a uniform density precious material.
The Earthlings never integrated simple shapes and whole number values
this successfully (until the Martian "invasion" in the 1970s).
Earthlings grow up surrounded by an unfriendly vista of irrationally
volumed shapes, with only the cube standing out as rational.
Psychologically, the cube is insecure and XYZ thinkers tend to
over-compensate by thinking Martian Math is "subversive".
Meeting the XYZers half way has meant conforming to their expectations
somewhat. Rad Math / Verboten Math -- these have been labels.
We bring Phi into it pretty early then, reintroducing the irrational
numbers with the golden mean.
There are reasons to value this approach over the more conventional
one, which stresses Pi so much. We look at Pi too, but not so
exclusively.
The NCLB polynomial and polyhedron, promulgated by my Ministry of
Education (HQS Washington High / OPDX), were early efforts to
strengthen STEM with new core standards.
The systems people are still pretty clueless about tetrahedra (because
they're Earthlings) but are among the first to be abducted and
persuaded of Martian Math's utility.
Kirby