On Dec 22, 3:48 pm, "J" <
jpdem...@rocketmail.com> wrote:
...
> > This isn't a show stopper though, as reassurance is what
> > we're after.
>
> Reassurance of what?
>
What we want to advance ab initio, from the top, is our streamlined
volumes table, wherein usually irrationally volumed shapes (thanks to
conventions) fit inside one another in a more memorable manner, with
lots of whole number volumes we weren't seeing before. This
concentric hierarchy is a core artifact in the curriculum, like part
of a backbone or spine.
We want to reassure teachers that this is fair play, that there's no
"catch" or "fine print" regarding some foundational mathematical rules
that we've broken. Furthermore, there's already a half century of
literature developing these concepts, with more on the way. This is
not about encouraging "fringe thinking" among students, is about
rescuing a languishing spatio-visual sophistication and reconnecting
with valuable heritage.
Reassuring math teachers may well involve getting under the hood such
as we're doing here and exploring the logic or grammar of
multiplication, what it means, how it might be visualized. The unit
volume tetrahedron is not "too good to be true", just goes by a
different set of rules, is permitted.
"Multiplication" is already the term for many binary operations with a
family resemblance. We multiply complex numbers and have
corresponding visualizations on the complex plane. We multiply
matrices, scalars and vectors, polynomials.
..
However, in this usage, we're adhering closely to a primitive 'two
edges make an area' model (one of the oldest), and a 'three edges make
a volume' model (like we do with a rectangular prism or brick).
We traditionally speak in terms of two and three dimensionality.
That's the ballpark we're in, though with some sensible alterations
for the purpose of developing these new language games.
> I wouldn't suggest at all that my remarks show that there is anything wrong with the different approaches. I wouldn't speak of "show stoppers", no.
>
> I think the compact packing of spheres might be your best lead for demonstrating applicability here. It's easy to visualize, easy to show using diagrams and models, obviously relevant to a variety of problems.
Yes I think so too. Six spheres fit around a nuclear sphere to form a
hexagonal packing. Then three spheres nest in the valleys, both above
and below, for a total of 12 balls connecting to that central one.
How those two sets of three, top and bottom, relate to each other (as
triangles pointing the same way or opposite) determines whether we
call this the CCP or HCP, both equally compact.
Packing outwardly from that nuclear ball in the CCP, we get successive
layers of 12, 42, 92, 162... balls, always in growing cuboctahedral
conformation. Here are a few pictures from Peter Pearce's book
'Structure in Nature is a Strategy for Design':
http://www.flickr.com/photos/17157315@N00/4207576595/in/photostream/
http://www.flickr.com/photos/17157315@N00/4208340720/in/photostream/
http://www.flickr.com/photos/17157315@N00/4207576271/in/photostream/
http://www.flickr.com/photos/17157315@N00/4208341378/in/photostream/
Here's that same sequence from the Encyclopedia of Integer Sequences.
Note the vast literature, including a pointer to K. Urner's web site
(mine) in the Links section.
Back to our volumes table, that 1-Frequency cuboctahedron (one
interval, 12 balls around a nuclear ball) has a volume of 20. So
simple! That's relative to the tetrahedron formed by taking four
balls and compacting them together, and connecting their centers --
our unit.
We may describe the volume of the growing cuboctahedron as 20 times
the 3rd power of F, where F is the number of intervals along any
edge. We also study the ball counts in successive layers (1, 12, 42,
92, 162...) an integer sequence, when introducing computer programming
concepts. There's a long literature behind doing this per 'The Book
of Numbers' by Conway and Guy, and 'Gnomon' by Midhat Gazale.
Also, here's another segue to Wittgenstein and RFM if we need it, as
he uses rule-generated sequences in many examples (including in the
PI) when investigating what it means to "understand" in terms of
continuing to follow some rule. "Now I can go on" means I know what
to write after 162.
Donald Coxeter (famous geometer, once of student of Wittgenstein'
s)
writes about this very rule by the way. From his recent biography by
Siobahn Roberts, 'King of Infinite Space' we get:
"""
Coxeter told Fuller how impressed he was with his formula -- on the
cubic close-packing of balls. And he later took pleasure in proving
it, noting in his diary one day in September 1970: "I saw how to prove
Bucky Fuller's formula," and publishing it in a paper, "Polyhedral
Numbers." Of course more than anything, Coxeter fell in love with
Fuller's geodesic domes.
"""
> > We wouldn't use Russell-Whitehead'
s Principia to prove 1 +
> > 1 = 2 in an
> > everyday classroom, either.
>
> No. But I also wouldn't accept that PM shows what arithmetic operations "mean".
>
Yes, a subtle difference.
Reassuring has the flavor of showcasing close similarities (family
resemblances) even while making some changes, setting up a kind of
duckrabbit tension between the two approaches (90 degree and 60
degree).
> Those are viable alternatives, yes. "N-triangled"
, aside from being a bit "cute" has the problem, etymologically, of suggesting the number 3 (where etymologically, "squared" does not suggest 4) and possibly being confused with raising to the third power.
>
In a given lesson plan for teachers, we might invoke "n-triangled" and
"n-tetrahedroned" as swap-in terminology for n-squared and n-cubed.
This is to limber them up regarding our visualizations, getting them
to the point of realizing that (n x n) and (n x n x n) have non-
rectilinear interpretations.
Once they've had the necessary insight, then we maybe de-emphasize the
nomenclature. We go back to saying "n-squared" like everyone else,
but now there's an embedded reminder to think of a triangle-based
model instead.
Teacher's who've been through this training or workshop have a new
appreciation for these nuances. We expect them to return to their
rectilinear ways, having sampled our "gypsy math" (back to the ethnic
minority motif) but with new insights, new gestalts -- an important
aspect of "meaning" per the PI (Part 2 especially).
This is not a new pedagogical / andragogical technique: to use some
interim or "training wheels" vocabulary or notation, and then to let
go of it, having had the requisite insights.
For example, many elementary school math curricula introduce negative
numbers by making the minus sign a kind of superscript in the top left
hand corner of the number. That way of notating negative numbers goes
away in later years, is replaced by a non-subscripted subtraction
operator.
> > only to show that "squaring" and "2nd powering" may be
> > logically decoupled, likewise "3rd powering" and "cubing".
>
> I wonder about this way of putting things. And I wonder if it's necessary.
>
> Here's a diagram showing a triangle made up of smaller triangles that are similar. We can calculate the number of smaller triangles that make up the larger by counting the triangles along two of the sides and multiplying.
>
Here's another way of looking at it. We do the multiplication
independently, as a calculation per standard algorithm (developed from
the abacus some centuries ago), with no visualization.
Then we take our completed calculation and map it to a square and
triangle respectively, not to figure out an answer, but to display an
already-obtained result.
The computation might involve any two numbers, not just n x n. Just
extend two vectors along adjacent edges of an n x n grid for these
respective distances (a, b) and connect the tips of your arrows. The
enclosed area (a x b) will now have the correct surface area in terms
of triangular units.
Imagine turning the enclosed area to liquid and pouring it out into
unit triangle receptacles. Maybe the last unit triangle is only
partially filled, as the area is not a whole number.
Imagine tilting a 10 x 10 triangle of 100 units to put the apex at the
bottom and having some fixed amount of liquid fill the figure
vertically, parallel to the opposite edge.
If we started out showing 3 x 5 = 15 as an internal triangle with
unequal edges, then after tilting we'd have this same 15 as our areal
result (the same amount of liquid) but with two edge readouts at 2nd
root of 15 (about 3.83). The remaining trapezoidal region of the 10 x
10 triangle would now be 100 - 15 or 85 units.
We can (and do) play similar liquid redistribution games with squares
and cubes, thereby heightening the analogy.
We end up thinking of any amorphous surface area in terms of the
triangular units we might use to express it. Any volume (any 3D shape
at all) might be expressed in terms of tetrahedral measuring cups.
This mindset then sets the stage for our polyhedral "mixing bowls" and
pouring actual liquid or grain (I use beans) from one receptacle to
the next, per volumes table. Six tetrahedra worth of beans exactly
fill a rhombic dodecahedron per our canonical arrangement. Said
dodecahedra encase our CCP spheres.
http://wikieducator.org/Image:Ve_rd.png (teacher work, VRML browser)
http://wikieducator.org/Image:Ve_rd_ve.png (internal cubocta has
tetravolume 2.5)
http://wikieducator.org/Image:Still_life.jpg ("mixing bowls" -- an
international standard)
> We do something analogous with a square made up of smaller squares. And we take this as a picture of the operations N x N or N-squared.
>
> (No one should dispute any of that.)
>
> Therefore...
>
> (Here's where you meet resistance.)
>
> The operation N x N can be applied in both cases, whether or not we call the cases "the same" in any other sense. What sense is relevant here? Isn't this a matter of wanting to persuade someone to adopt a way of speaking? And is that way of speaking important?
>
We want to motivate a more 60-degree based approach to spatial
geometry more generally. A subculture imbued with this alternative
aesthetic might build houses looking more like this one:
http://www.flickr.com/photos/17157315@N00/4208340102/in/photostream/
(op cit Pearce, not a geodesic dome)
Or this one (done in Google Sketchup by Trevor Blake):
http://www.youtube.com/watch?v=Veli7iwqSuA (also not a dome)
> ("...the four sides of the cube."
>
> "Um, a cube has six sides."
>
> "The top and bottom aren't 'sides', asshat!")
>
> If we say that one is one of many applications of the operation n x n. while the other is a picture of what we mean by "n x n", no one could object!
>
> If we say that we can use the triangle rather than the square as a unit of area, this may arouse suspicion. A reminder that we can readily convert between the units to whatever precision needed (they are only "incommensurable" in a philosophically innocent sense) and check the calculations made using triangles against calculations using squares ought to alleviate that.
>
And we can use a tetrahedron instead of a cube as a unit of volume.
We actually put more emphasis and attention on this last bit, as it's
the volumes table we turn to first.
But then if teachers have philosophical objections, qualms, we might
retreat to the flat two dimensional case and explain how we're able to
work this approach consistently even in flatland.
> Then we have simply a method that is more efficient for some purposes.
>
> > Yes, you're defending the practicality of our rectilinear
> > 90-degree-
> > based conventions.
>
> It isn't just a question of practicality. What I hope to convey is that what the "90-degree-based conventions" show as a picture of arithmetic operations is different. Yes, there is an analogous "input" and "output" with the triangular case, but not all of the same transformations.
>
> "Seeing as" is important here. An n x n square can be seen as n columns or as n rows. And each column (or row) is the same arrangement (squares in a lined up or stacked) making their grouping perspicuous. Being the same in this sense, the connection between multiplication and addition is clarified: n x n is a stack of n height (or in a line, n length), n times ("times" as literal repetition: the same again). And that they are the same is not just a matter of their having the same arrangement (grouping) but their lining up (correlation)
.
>
In the case of our equilateral triangle, you could envision a turtle
walking across the base, with a straight line to the apex defining a
"curtain" (the turtle is drawing a curtain across a triangular
window).
As the turtle traverses its edge, the curtain closes. This isn't an
arcing motion, i.e. the turtle is not moving like a pendulum. It
moves straight across.
If it lurches forward at equal intervals, then the area covered by the
curtain likewise increases at
equal intervals. This is a model of n x m, with n the full length of
the triangle, and m just a portion across the base, up to and
including another n (m < n).
Now you might vary the other edge as well (two turtles).
We've let go of the underlying grid pattern by now (training wheels).
Maybe there's an electronic readout of the area, in terms of unit
triangles, but only one such triangle is shown, off to the side, as
part of the legend or key. You needn't "count triangles". Just do
the multiplication with pen and paper, as usual, and recognize this as
a legitimate display or visualization (like something Tufte might do).
> What corresponds to grouping, to correlation, or to "the same again" with the triangles?
>
You're concerned with reading off the right answer from the grid,
thinking of an orchard of X trees in some triangular arrangement, and
wondering how you'd easily compute how many. The rank and file
system, the array, the phalanx, stacks the same number side by side,
repeatedly, is easy to read.
I'm happy with a "when in Rome" approach, i.e. lets look at the
complementary strengths of both models.
The thing about squares is you may connect opposite corners in either
of two ways to form two triangles (not equilateral)
, and then you're
maximally connected in the sense of having no more adjacent dots to
connect.
Those wire frame computer drawings of bodies, faces, curved surfaces,
usually break down into an all-triangles vista (are "omni-
triangulated"
). That's an aesthetic we're encouraging, one that "goes
with the territory".
Triangles have structural integrity, whereas squares need to be
stabilized. In pure geometry, this doesn't matter as the wire frames
are ethereal. Nevertheless, there's a minimalist aesthetic associated
with triangles and tetrahedra that we want to harp on.
Our claim would be that squares and cubes are not as "with the
grain" (of nature) as triangles and tetrahedra, and this counts for
something when it comes to applications.
To get very general about it, Oregon's bioengineering community would
rather students think of triangles and tetrahedra as emblematic of
"high tech" (organic, natural, green), with the older gridiron
aesthetics (rectilinear) more retro, emblematic of a more backward,
less tech-savvy industry. Those are connotations, aesthetic
considerations -- important to meaning.
When I visited ONAMI the other day, our nanatechnology lab (onami.us),
currently under construction on the HP campus in Corvallis (in a
facility owned by OSU), I noticed the PR materials featured a Zome kit
for assembling a buckyball, all hexagons and pentagons (no right
angles). That's indicative of the direction we're taking: more
spatial geometry in high schools, more computer use (e.g. vZome), and
more emphasis on non-rectilinear patterns in nature.
> Also, with squares, we also have the ability to form rectangles, n x m, which obey the same rule of multiplication and exhibit the other aspects mentioned above. What is an "n x m" triangle, where n does not equal m? And can n x m still be multiplied?
>
> One of the colored graphics you showed previously does show how this works, but it is not really analogous to the "n x n" triangle in the way that the n x m rectangle is analogous to the n x n square.
>
It's not too far different though. We're accepting two or three
inputs in the form of edges, and figuring a resulting area or volume
based on these edges.
Interestingly, the regular tetrahedron'
s three pairs of opposite (non-
touching) edges are at 90 degrees to one another. It's not like the
90 degree angles go away. They just need to share the road more.
> To put it in your terms (?), the "n x n" when speaking of triangles belongs to a different "namespace". Or rather, the relationship between the arithmetic operation and the pictures is not the same here as it is with the square. That's not because the later is the "correct" picture or because the triangles are "sophistry", as your Midwestern interlocutor might have it, but just because the connection between the operation and the square involves more than just the fact that you can count the squares along each edge and multiply them to arrive at the same total as counting all of the squares within the larger square.
>
> He is right to object to "it means the same here", though he offers no clear grounds that I can see.
>
Yes, I agree with "not the same" but then fall back on "family
resemblance" and even "close family resemblance"
.
> But if I've helped you see what would be good grounds for that objection, perhaps by pointing out the differences as well as the similarities, such conflicts can be averted.
>
> I have no objection, provided
>
> > we're free to
> > establish / explore a logically permitted alternative in
> > some
> > lessons. We want to keep using that concentric
> > hierarchy with its
> > volumes table.
>
> Certainly! I have no objections to any of this!
>
That's refreshing. It's an uphill slog to get these curriculum
reforms implemented, despite the 50+ years of literature and a
subculture of scholars such as myself, frustrated by our slow
progress, agitating for an upgrade.
People telling the history of the "math wars" will have a field day
with our story someday.
I've been using my background in the philosophy and religion
departments (Princeton, Rorty, Preller) to seek this bridge to
Wittgenstein and his lineage.
Here's an opportunity to show where philosophy is making a positive
contribution in some practical sense, helping our math teachers co-
develop a more up-to-date and relevant spatial geometry curriculum.
Philosophy of mathematics to the rescue!
Thanks again for your assistance, hope you will continue our dialog.
Maybe others will join in. The Coxeter-Wittgenstei
n connection is of
some interest as well.
Kirby
>
>
> JPDeMouy
>
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