[ Reprint of a posting to MathFuture, typos fixed, hyperlinks added ]
Greetings Brad. I was thinking "circle and/or sphere and/or tetrahedron" through most of what I wrote, and seriously thought of quoting some passages from Synergetics associated with this Figure:
http://www.rwgrayprojects.com/
Greetings Brad. I was thinking "circle and/or sphere and/or tetrahedron" through most of what I wrote, and seriously thought of quoting some passages from Synergetics associated with this Figure:
http://www.rwgrayprojects.com/
... then thought the better of it. Some other time maybe.
True, conflating spheres with circles is conflating 3D with 2D, and that might be a sin.
On the other hand, Renaissance perspective is all about rendering 3D vistas on a 2D canvas and as Joe points out, the 3D moon is a circle (if full) in most children's books.
We play with that circularity, as in the DreamWorks logo (boy with fishing pole, sitting on a moon sliver). The basic idea is to establish in fresh minds (of any age) a semantic network of related concepts that don't fizzle right away, as they too often do, thanks to geometry being compartmentalized, something to put on a mental shelf the minute we leave the classroom.
On the contrary, V, F, E all follow us down the hall, as halls with doors and lockers, the school itself (like a maze) is all Vs, Fs and Es, once you start looking for them.
And yes, it's on a ball (Planet Earth), which we might see using Google Earth on a flatscreen. The Little Prince might feel right at home here (or not).
Circles have a lot of the same properties as spheres, as you well know, and if we're able to take advantage of this homomorphism -- the analogy (if not a full isomorphism, congruent concept) -- so much the better.
The circle has concave and convex aspects, just like a cave. People are either inside or outside the circle, drawn in playground dirt, or "on the fence", straddling, one foot in, another out.
That sense of "container" -- which I'd consider at the root of conceptuality -- is still there. Or we might use a square, or a triangle, to make our boundary. Who's in? Who's out?
The semantic web I was building follows the Synergetics path of looking for "twonesses" (pairs of concepts that go together):
(i) spinning around an axis in either direction
(ii) growing and shrinking radially
(iii) having a macrocosm outside, a microcosm inside.
We might play this game with circles, not only balls. We might use a circle to represent a ball. In a Geometry of Lumps per Karl Menger, we have only "3D" as there's always the observing camera, the viewpoint, separated from whatever's observed.
The observer-observed is already an axis and its context is always volume. There's 2D "in volume" and 1D "in volume" but the context-container is always 3D. Everything is a lump. A circle is a pancake, a disk. A line is a cylinder. We may dismiss "thickness" as irrelevant but we don't appeal to infinity to make it disappear. A plane is a sheet of paper.
In this geometry (non-Euclidean) there's no need for either "infinitely thin" nor "infinitely wide" as in Euclideans' metaphysical cartoons (lots of hand-waving). The word "infinitely" itself as been largely deprecated as unnecessary.
Different axioms and definitions, sure, but not entirely unfamiliar as its a reality we may access experientially. Taking a page from the old textbooks, we might say the omni-presence of 3Dness is "self-evident".
But then what about Time i.e. change, i.e. deltas? The movie moves. Stuff happens. Going deeper with the elementary topics, I introduced V + F = E + 2 and Descartes Deficit of 720 degrees. You'll not find these in Common Core.
My ethnicity, no doubt some religious minority or cult, wants to share these generalizations in 3rd grade. Might we need a special dispensation from the Governor? Must we keep our kids from public school? I certainly hope not, fingers crossed.
In the 1980s, I wrote up much of the above for an archive in Bhutan, but I doubt it met with Jesuit approval way back then. Malesh (means "oh well" in Egyptian -- an acceptance of God's will).
Finally,
I want students (of any age) leaving my geometry classroom to keep
thinking about social networking, about graphs. Who is friends with
whom? Who consider themselves enemies?
I'm in need of such a graph database myself. As I wrote to the director of the code school (< guild />) this morning from Linus Pauling House:
PD = Professional development. PSU = Portland State University.
You might see where high schools could relate to talk of "in groups" and "out groups", the Venn Diagrams. That's what the #BigData and #MachineLearning folks are studying too. Who's who, and who knows whom, who works with whom etc.
The company Little Bird (here in Portland) is one example, that's what they do for a living: find out who is influential, based on network traversal. The theory is that, in knowing all this, one becomes an insider in some way, like being in the CIA in some science fiction.
I'm in need of such a graph database myself. As I wrote to the director of the code school (
I spent the night of the 4th at home studying history of the French Enlightenment period. I had no idea that Hume, the Brit, and Rousseau, the so-called romantic, had such a dramatic falling out. Such revelations make me want a graph database of "friends and enemies through the ages". A great way to teach history to kids (adults too): "if I were Thomas Jefferson on Facebook, who would my FB friends be?" ...
Giving high school teachers free PD in Python etc. would be a smart move, as a pilot program at least. I guess I'd consider myself a lobbyist for such programs. I probably have natural allies in PSU I don't know about, enemies too. History was ever thus. :-)
I'm also sporadically reading a fantastic book on Ada Byron, the first computer programmer. She wasn't allowed to know who her dad was until her 21st birthday, when his portrait was unveiled with much ceremony. The mom didn't want her to have any contact with the guy.
PD = Professional development. PSU = Portland State University.
You might see where high schools could relate to talk of "in groups" and "out groups", the Venn Diagrams. That's what the #BigData and #MachineLearning folks are studying too. Who's who, and who knows whom, who works with whom etc.
The company Little Bird (here in Portland) is one example, that's what they do for a living: find out who is influential, based on network traversal. The theory is that, in knowing all this, one becomes an insider in some way, like being in the CIA in some science fiction.
My wish, then, is to anchor a semantic web that doesn't fizzle when the mind turns to social matters. Geometry doesn't just sit on the shelf.
Verifying the truth of a story, seeing if it all checks out, is "omni-triangulating".
We're building mental maps, mental models.
Yes, constructivism: we're each responsible for creating our own model.
Geometry supplies conceptual tools for doing that.
Graph Theory and Geometry are not strangers.
