mathcasting concepts (parking some TV ideas)
kirby
had a few too many, apologies -- KTU ]
====
A mathcast is like a screencast, or an animation (cartoon) about math
(math manga).
Here's a set of mathcasting ideas I thought I'd better publish to the
Internet before some corporation comes along and selfishly copyrights
all this exclusively for itself. This way, the material is better
protected for real humans (corporations == "artificial humans" or
"legalized zombies" as some call them in translation).
1. Setting: A Tropical Island
The main idea is a Polynesian Island or other 'Lost' kind of place,
where the natives have thought deeply in terms of coconuts.
Yes, each coconut is slightly different, but these folks are no less
intelligent than the ancient Greeks (or even the contemporary ones)
and have idealized the unit-radius coconut as a kind of perfect
sphere, never to be found in nature as such, but perhaps to be modeled
using specialized technologies (we'll get to those in future
episodes).
When we do the animations though, it'll be OK to use idealized
coconuts, perhaps with the holes or other markings.
Coconuts may still have their skin and might be green in that case.
Lets leave it to the individual artists how they want to standardize
the look and feel, for consistency and memorability. We may not have
unlimited budget, but lets presume high production values. Anime
shops in Japan might be jumping in here.
2. Core concepts: Triangles and Tetrahedra
The core idea is the natives put three coconuts together to get a
triangle and this is their unit of area. The radius 1 means a
diameter of 2, i.e. 2R = 1D. They measure in D units quite a bit.
Adding a fourth coconut atop the original three gives a tetrahedron of
coconuts: this is their unit of volume.
In a "hypertoon" (a kind of cartoon), the key frames may also be
branching off points to related scenarios (I've published samples to
Youtube). When looking at a growing tetrahedron of coconuts, starting
with layers 1, 3, 6, 10, 15... one sees the triangular numbers.
This might be a branch point to triangular numbers segments, formed as
the sum of consecutive integers starting from 1 to N, or from 0 to N
if 0-indexed. The 0th triangular number could be zero and so on.
Triangular and tetrahedral numbers appear in columns of Pascal's
Triangle, lets not forget, itself a triangle, plus Pascal's
Tetrahedron is likewise well defined.
In model-view-controller architecture, the viewer would have some
control over the sequence used to investigate the model.
3. Relative Volumes
Now you may not have seen where the National Council of Teachers of
Mathematics has included a lesson plan on Tetrahedral Kites. The
purpose of this unit is to explore "non-traditional volumes" by which
is meant a unit volume tetrahedron. Regular tetrahedra don't fill
space, although an irregular one does easily, but in complement with
an octahedron of the same edge length. As the NCTM lesson plan makes
clear, this octahedron will have a volume of 4 relative to the
tetrahedron of volume 1.
As the "three sided pyramid" of coconuts grows, we see a volume
counter on the side, which is calibrated to the total volume of the
Matrix. By "matrix" we mean the imaginary (or drawn) skeleton of
edges connecting adjacent coconut centers -- the centers of coconuts
which touch or "kiss" one another. A coconut completely embedded in
the matrix is surrounded by 12 others, kissing it in 12 places.
The volume counter starts at 1. With a next layer of 6 coconuts, one
gets a tetrahedron of three coconuts along a side. We call this
2-frequency (two intervals, 2 diameters) and its Matrix has a total
volume of 8. A 3-frequency tetrahedron has a volume of 27, a
4-frequency one of volume 64 and so on. Volume is frequency to the
3rd power. Our natives have a consistent 3rd power volume concept,
based on their tetrahedron.
4. Island Features
They've completely mapped their island's surface in terms of
triangles. Computer graphics which show a skeletal version, a mesh of
triangles, with lots of topography. We could use a volcano towards
the center of the island. Many educational segments will be made
here, so some richness in features will be welcome.
There's a swiftly flowing mountain river we will use to study
electricity generation (without a dam). The night sky will give us
constellations and navigation segments. The geography of the solar
system (astronomy) will be a part of our inventory.
5. Tetrahedral Mensuration
It's this unit tetrahedron that's especially important though. We
should not dilute that. These cartoons may be students' first exposure
to the idea and we want a certain elegance in presentation. The
rhombic dodecahedra embracing each coconut, the centers of their
diamond faces being the 12 kissing points between adjacent coconuts,
will serve as a basis for many a lesson plan.
That their volume is six will become quite ingrained, as it is for the
natives. Its 12 short face diagonals form a cube of volume 3 (one half
of six) whereas the long diagonals form that octahedron we talked
about, of volume 4. Keeping these easy whole number volumes front and
center is part of the point. Understanding this native civilization
depends on keeping a handle on this information, as various puzzles
materialize, in need of solving (like in Uru by Cyan of Spokane).
6. Frequency and Volume
12 coconuts around a nuclear one: if building that tetrahedral
pyramid, the embedded coconuts have 12 around them, defining the
Matrix shape known as the cuboctahedron (8 triangular facets, 6
square). Its volume will be 20 in native units. Its 24
surface edges are all length D. Its spokes to the center, from each
vertex, are also length D. Around the layer of 12, may be packed
another layer of 42 coconuts, then 92, 162... each time, the
frequency (number of intervals) goes up by 1, and the volume is 20 * F
* F * F or F to the 3rd power. For the tetrahedron, the volume is 1 *
F * F * F. For the octahedron its 4 * F * F * F. So you see how
these natives think of volume.
7. Looking Ahead
OK that's enough to provide a basis. I haven't suggested any
characters yet. We plan to have some strong female role models doing
a lot of the engineering, maybe some whale riding. This isn't all
about Robinson Crusoe or some marooned European. We may have some
outsiders show up later, with some awkward cubical mensuration system,
to be used for contrast, but we won't need them right away.
Those of you who know chemistry realize we're setting the stage with
an important lattice, usable to map diamond, various crystalline
compounds. Those of you into architecture know that our Matrix is
likewise the octet truss. The NCTM lesson plan in Tetrahedral Kites
is called that for a reason: because Alexander Graham Bell studied
this Matrix in the early 1900s, and called his tetrahedral-octahedral
constructions Kites.
In other words, we're getting ready for a rich set of educational
segments, all connecting back to this island subculture and its
talented natives. Don't think of them as scientifically illiterate or
"behind" in any way. As the season progresses, it will be clear that
we're dealing with a high technology civilization, with under-ocean
aspects.
Kirby Urner
Oregon Curriculum Network
Portland Oregon
Partial list of sponsors:
4dsolutions.net
flextegrity.com
Related Links:
http://www.4dsolutions.net/ocn/numeracy3.html
http://www.4dsolutions.net/ocn/urner.html
http://www.bfi.org/our_programs/bfi_community/synergetics/synergetics...
http://worldgame.blogspot.com/2010/02/update-from-pauling-campus.html
http://wikieducator.org/Martian_Math
Edward Cherlin
an idea is subject to copyright.
There are excellent graphic coconuts in Super Mario: Sunshine.
You can construct a tetrahedron from two lines of four coconuts, plus
two rectangles of 2 × 3 coconuts. This is an excellent puzzle.
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Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/
kirby urner
> Ideas cannot be copyrighted. The textual or graphic representation of
> an idea is subject to copyright.
>
> There are excellent graphic coconuts in Super Mario: Sunshine.
>
Coconuts in action:
http://www.hawaiianjewelryandgift.com/images/Green%20Coconut.jpg (real coconuts)
http://www.lotusspace.com/VE-ghost.jpg (12 around 1 -- matrix of volume 20)
http://www.4dsolutions.net/ocn/graphics/4spheres.gif (classic 4
coconuts, unit volume)
http://en.wikipedia.org/wiki/File:Close-packed_spheres.jpg
(4-frequency, matrix volume 64)
http://www.complang.tuwien.ac.at/schani/supermag/scaling/6_long_tetrahedron.jpg
(6-frequency, internals removed)
http://perfectperiodictable.com/Images/tetrahedron%20stack.jpg
(7-frequency "coconuts", volume 343)
http://hobbit.ict.griffith.edu.au/~anthony//graphics/polyhedra/
(still cool, after all these years)
http://wlym.com/~animations/harmonies/BookII/pictures/Rhomic%20Dodecahedron.jpg
(volumes 6, 3 (red), 4 (green))
http://www.grunch.net/synergetics/hierarchy/vols.gif (in a tropical paradise).
> You can construct a tetrahedron from two lines of four coconuts, plus
> two rectangles of 2 × 3 coconuts. This is an excellent puzzle.
>
Along the lines of (spoiler / hint):
http://www.rwgrayprojects.com/synergetics/s04/figs/f1701.html
http://www.rwgrayprojects.com/synergetics/s05/figs/f2708.html
http://www.rwgrayprojects.com/synergetics/s01/figs/f00103.html
kirby
This essay doesn't yet explicitly link to this thread, but it
does mention this thread's tropical island:
"""
One might easily imagine a tropical island paradise where
the natives pack together three coconuts (as idealized
equal-radius spheres) and call that a unit of area, pack
together four coconuts (as a tetrahedron) and call that
a unit of volume. Why not study this island culture then,
explore the consequences of this more primitive topological
beginning? Might this civilization turn out to be more
advanced than our own? Or perhaps we should look at
Martian Math? Here in Oregon, we're looking at both
of these storyboards as useful backdrops for future
mathcasts (animations, cartoons).
"""
[ http://mathforum.org/kb/thread.jspa?threadID=2047973&tstart=0 ]
Here's a blog post mentioning the idea of a series:
"""
More recently, I've been storyboarding a TV series
involving a tropical paradise. Coconuts count as money,
with the thermodynamics of "open system Earth"
spelled out in some segments (lots of computations
with energy). Think of Bill Nye the Science Guy as
an influence.
These same coconuts are also stand-ins for idealized
spheres, forming the basis for triangulations (three
coconuts) of the island's topography, and
tetrahedralizations (four coconuts) of various spatial
designs (such as kites).
These tropical island animations (anime) will be
standalone and/or dropped into contextualizing
shows. Think of Sesame Street as another influence.
I call them "mathcasts" because of their somewhat
high level of mathematical content. We're doing
a lot with polyhedra for example.
"""
[ http://mybizmo.blogspot.com/2010/02/exchange-with-japan.html ]
Kirby