storyboarding Martian Math cartoons
kirby urner
of my readers, who are struggling to get clear on what's in my
Notes for Teachers in Martian Math. [0][1]
Having talked it over with Glenn Stockton, a veteran of a certain
dam project in Paige, Arizona, I'm thinking in terms of a chasm
and a dam, with concrete being poured from above. A gantry
crane bridges the chasm, riding railroad tracks on each side, and
buckets of concrete get lowered to the waiting crew, lifting from
each side of the chasm.
Where the plot thickens is we have Martians on one side of
the chasm, Earthlings on the other. I'm going ahead with some
ETs in the picture because (a) this is a cartoon and (b) we want
to get across the idea of something "alien" going on, on the
Martian side of the chasm.
The Martians, you see, do not share the cube-based model
of N x N x N. When they have three edges of identical length
and want to show off the corresponding volume, they build a
six-edge "tepee" (i.e. tetrahedral tent) with all edges N. That's
their model of "N to the 3rd power" (and so the word "cubed"
is inappropriate). When they multiply three numbers a, b, c
to define a volume, they use three spokes from a common
origin at 60 degrees to one another, of lengths a, b, c. These
vectors define a fourth facet (triangle ABC) for a total enclosed
volume of a x b x c. This same lump of clay may be reshaped
into a hexahedron or brick, but its edges won't be a, b and c.
On the Earthling side, XYZ rules, along with our ancient
concept of dimension. We say "height width and depth"
come as three independent dimensions, each of which
has 90 degree relationships with the other two. 1 x 1 x 1
is represented, not as a tetrahedron with edges 1, but as
a hexahedron with edges 1 and all 90 degree angles -- as
a cube in other words.
The two skeletal coordinate systems we want to study go
on either side of the chasm. The XYZ Earthlings have their
unit volume cubes, a tessellation of space. The Martians
have something more like the Alexander Graham Bell
octet-truss (see Notes for Teachers [2] and the related
course page [3]). The rods (edges) are twice as long as
the cube people are using, and the regular tetrahedron
so built is what Martians call their "unit of volume".
So which container actually holds more concrete? It's
a fairly close outcome. It turns out that the Martians see
the XYZ unit cube as 1.06066 units, i.e. as a tad more
concrete. The Earthlings see the same thing.
The puzzle we're trying to solve, for our students, is the
fact of a cube, on the Martian side, with edges radical(2),
that weighs in with a volume of 3.
That's strictly verboten when you think how sqrt(2) to
the 3rd power is certainly not 3. What the Martians are
doing must be sloppy or fuzzy -- or so it first seems.
But what's actually true about the Martians is they don't
think of cubes a 3rd powering models. The 90 degree
angle was a convention, related to Earthling superstitions
around "dimension", and a consistent / logical mathematics
might put tetrahedra in the limelight, versus the cube, as a
primitive model of 3rd powering.
Why go to all this trouble? What current state standards
encourage any such brain twisting and how might students
benefit?
Well, for one thing, we want to call attention to the
importance of axioms and definitions to these cultural
enterprises, these mathematical language games. All
mathematics is "ethno" mathematics, i.e. we don't get
free from a set of assumptions that embed in a
cultural matrix. Any given mathematics is not
"culture-free", nor even "culture neutral".
We also want to unlock and demystify the Fuller syllabus
itself, which, combined with 'The Pound Era' by Hugh Kenner,
several other contributions, makes for some fascinating
literature. Students learn a lot in a little time, as they
trace the connections to so many other writers and artists,
other thinkers.
So is this a math class or a literature class? Why should
these two be considered far apart instead of close together?
Think of 'Alice in Wonderland' and 'Flatland', both pregnant
with logico-mathematical content.
Martian Math, in being literary, influenced by H.G. Wells, as
well as Orson, is not without its hard little puzzles. Figuring
out about tetrahedral mensuration is precisely one of those
little prizes one gets for sticking with a liberal arts approach.
Using narrative, storytelling, animation, to set the stage
and get concepts across, is not really a new idea. Indeed,
developing these new computer animations could represent
a paying gig for many a talented artist / designer / engineer,
ala the 'Sesame Street' model (clips coming in from all
corners).
In a more philosophical context, we're dealing with an ideal
instance of a "duckrabbit" i.e. one of those gestalt switches
that both leaves everything the way it was, and changes
everything. Switching from the Earthling to the Martian
perspective doesn't invalidate either one, yet exposes a
degree of freedom one might have not appreciated otherwise.
This makes Wittgenstein's stuff come alive too, not just
the spatial geometry. Karl Menger's "geometry of lumps"
is also a dot worth connecting. More on that in my blogs.[4]
Kirby
[0] http://groups.yahoo.com/group/synergeo/message/62799
[1] http://www.4dsolutions.net/satacad/martianmath/teacher_notes.html
[2] ibid
[3] http://www.4dsolutions.net/satacad/martianmath/mm24.html
[4] http://coffeeshopsnet.blogspot.com/2009/03/res-extensa.html
For further reading:
http://mathforum.org/kb/message.jspa?messageID=7143092&tstart=15
Maria Droujkova
Kirby, I would like to ask for a picture at least every paragraph!
I bet this will increase the number of people who can relate to this (or who read beyond the first sentence) by several orders of magnitude.
Cheers,
Maria Droujkova
Make math your own, to make your own math.
kirby urner
Even without animations, still pictures, like a storyboard, might move
mountains, in terms of cluing a readership.
Over on the Synergeo list, I think we've achieved consensus on how to
get the idea across, of two ethnicities, one using a cube for 3rd
powering (our familiar XYZ model), another using a tetrahedron
(somewhat alien, considered strange).
Those would be "Earthlings" and "Martians" respectively.
In this storyboard, they're collaborating on building a dam across a
chasm, and getting clear on what's meant by "one unit of concrete" is
part of the story. Turns out we'll need a conversion constant and
this exercise helps us figure out what it is.
It's a lot like liters versus quarts.
To set up a relationship between the two coordinate systems, we say
both sides have the unit-radius balls.
The Martians pack four of them together for their unit tetrahedron
with edges D (ball diameter), whereas the Earthlings take that radius
(R = 1/2 D), and build a cube out of it, with vertexes (0,0,0) (1,0,0)
(0,1,0) (0,0,1) (1,0,1) (0,1,1) (1,1,0) (1,1,1).
[ The Martian tetrahedron may be described with coordinates (1,0,0,0)
(0,1,0,0) (0,0,1,0) (0,0,0,1) -- but it's not necessary to delve into
so-called "quadray coordinates" (see Wikipedia) in order to understand
the core idea: a different model of 3rd powering (i.e. n x n x n is
always depicted as a tetrahedron, not a cube). ]
I'm thinking students could help develop this scenario as a math
project (I'm also hoping to get some professional studios interested
-- more advanced students, one could say).
I've created a new web page about this story (pictures load slowly
because they're actually much bigger):
http://www.4dsolutions.net/satacad/martianmath/storyboard.html
Of course some teachers will ask why go to all this trouble to explain
such an obscure and esoteric business as "Martian Math". Who cares
about a "unit volume tetrahedron" anyway? Just some guy named Kirby
in Portland, Oregon? So what?
I refer them to my "heuristics for teachers" at the Wikieducator site,
where it becomes clearer how fluency with spatial geometry might
dramatically improve once one has learned to see in the "Martian way"
(a different set of gestalts, mathematically valid).
There really is a whole subculture (ethnicity) behind this alternative
approach, and a literature. I think it's well worth the effort to
make the leap and to understand that mathematics allows for such
diversity, even at this basic level.
Kirby
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Maria Droujkova
You are so right Maria.
Even without animations, still pictures, like a storyboard, might move
mountains, in terms of cluing a readership.
Over on the Synergeo list, I think we've achieved consensus on how to
get the idea across, of two ethnicities, one using a cube for 3rd
powering (our familiar XYZ model), another using a tetrahedron
(somewhat alien, considered strange).
Those would be "Earthlings" and "Martians" respectively.
In this storyboard, they're collaborating on building a dam across a
chasm, and getting clear on what's meant by "one unit of concrete" is
part of the story. Turns out we'll need a conversion constant and
this exercise helps us figure out what it is.
This can be a nice stop-motion animation project with physical models created ahead of time as a part of the fun. Something for a math club to do over a few weeks?
kirby urner
Yes.
Stop-motion animation is often associated with clay, and clay is a
useful medium for showing volume relationships because you can
deform and reform it while assuming none is added or subtracted
(like conservation of energy).
http://www.youtube.com/results?search_query=claymation
On the Martian side of the chasm, the tetrahedron of edges D (unit
volume) might be reformed into a cube and set next to an Earthling
cube of edges R (a different unit volume).
It will turn out that the Earthling cube is 1.06066... times larger than
the Martian cube (formerly a tetrahedron), or sqrt( 9/8 ). We know
this number as S3 in our literature.
http://en.wikipedia.org/wiki/Synergetics_(Fuller)
The other things to use besides clay in conservation of volume
exercises is of course fluid and/or dried grains or even sand (if the
containers are not water proof).
I've tended to use beans in classroom demos, pouring them from
polyhedron to polyhedron, because cornmeal and smaller things are
harder to clean up after spillage (gets between floor boards).
Here's a picture of some of my polyhedra with beans in them,
taken but days ago at the college:
http://www.flickr.com/photos/17157315@N00/4863914712/
Here's the full set, back when my polyhedra were new -- the
30-faceted rhombic triacontahedron (volume 5) and cuboctahedron
(volume 20) have both disintegrated by this time:
http://wikieducator.org/File:Still_life_sm.jpg
A couple challenges though: making perfectly regular shapes
out of clay would usually require a mold. Just doing it by hand
is too imprecise (in my case at least).
Also: even very precise work in clay is going to make it hard
to show small differences on a small scale.
That's why I think the stop motion animation might need some
annotations and/or voice-over narration.
I like the idea of using blue-colored clay to represent water
and having it pour from one vessel to another -- mixing the
media as it were (having one look like another).
Kirby
>
>
>
> Cheers,
> Maria Droujkova
>
> Make math your own, to make your own math.
>