[math4wisdom] Re: Help create a video about Math 4 Wisdom learning paths
- From: kirby urner <kirby.urner@xxxxxxxxx>
- To: math4wisdom@xxxxxxxxx
- Date: Fri, 3 Mar 2023 08:19:53 -0800
Greetings all --
As soon as I watched the latest M4W YouTube, I felt catalyzed to make a
YouTube myself, one I'd been mulling over in connection with our topic:
Learning Paths, and (complementary) Teaching Techniques (these could also
be for self teaching i.e. for auto-didacts and other polymaths).
The topic of my video is Andragogy, what is it? How is it distinct from
Pedagogy?
https://youtu.be/2MO24W5-ZVw
On Fri, Mar 3, 2023 at 4:23 AM Andrius Kulikauskas <math4wisdom@xxxxxxxxx>
wrote:
Dear John, Thomas, Jon, Kirby and all,
I have uploaded the latest Math 4 Wisdom video, "120°+120°=90° in Dynkin
Diagrams. Teamwork in Creating Learning Paths. (What is Geometry?)"
https://youtu.be/kJYFleyQiGo
Thank you for your help!
From Jon's paper model I now realize that 120°+120°+90°=0° and thus
120°+120°=-90° better says what I mean to say. With Jon I also realize that
I need to emphasize that math lets us think not in two or three or four but
multiple dimensions. I am learning that the Dynkin diagrams are modeling
the assumptions of counting, which is that counting is higly symmetric - we
can simply relabel everything - but that ultimately counting is sequential
and thus chains together dimensions. We'll explore that further, more and
more as a team, in future videos in this series, What is Geometry?
I was having similar thoughts coming from diving into "abstract nonsense"
namely Category Theory some more.
A Learning Path (in a kind of category) is like a series of transformations
ala f: a --> b (turn by 120).
f(a) --> b (turn by 120)
f(b) --> c (another turn by 120)
==
(f * f)(a) --> c (compose f with f)
==
g(a) --> c (turn by -90).
So: (g * f * f)(a) --> a (identity transformation).
Something along those lines, right?
Here I'm using * (asterisk) to mean "compose" (LaTeX \circ) i.e. we do: "f
after f after g" to a, symbolized as (f * f * g)(a), to get back where we
began. Or are use using inverse(g) i.e. ~g?
(g * ~g)(a) --> a. Every "do" (g) has its corresponding "undo" (~g).
All this gets a little tricky in 3D because we know if you start with a
book, cover facing you, title top, and then:
(a) turn book away 90 degrees, around left to right X axis, so cover now
faces up
(b) now turn book 90 degrees clockwise around vertical Z axis (cover still
facing up, top to the right)
You don't end up where you would if you started with the cover facing you,
title at the top, as before and did:
(a) turn book 90 degrees clockwise around vertical Z axis (cover now faces
left, title top)
(b) now turn book away 90 degrees, around left to right X axis (cover now
faces left, title away from you)
https://robotacademy.net.au/lesson/rotations-are-non-commutative-in-3d/
i.e. rotation matrices are non-commutative.
Probably what's unclear to most viewers is how "turning by 90 degrees"
breaks us free into a whole different dimension, whereas a rotation by 120
degrees does not seem to quite achieve escape velocity, even though 120
includes 90 (90 + 30).
They're thinking back to elementary school days, where one clock hand
pointing in any direction = 1 dimension, whereas 2 clock hands with any
non-zero angle between them = 2 dimensions i.e. a clock hand to 12:00 and a
clock hand to 12:01 describe a thin triangle and a triangle is 2D.
Then a 3rd clock hand, pointing outside the plane formed by the first two,
forms a tetrahedral wedge that's 3D.
90 degrees was never special in this animation.
But in this M4W demo, your "clock hands" (straws) are not so much "basis
vectors" as "roots" in a Lie Group are they not? Their rotational
relationships are more metaphoric. These "clock hands" are already "any
dimensional" in some extended Euclidean n-d sense.
When three non-coplanar vectors from a common origin define a triangle with
their three tips, that defines a fourth triangle. Four non-coplanar points
determine four non-coincident triangles.
That's more what Fuller naively meant by "space is 4D". This fourness is
intrinsic to our most primitive idea of "a container". Four vectors from
the origin to four non-coplanar points will span space with only positive
scaling and addition, no need for reflection or rotation.
That's quadrays in a nutshell. Only 4 basis vectors (a, b, c, d) scale and
sum to reach all points in four quadrants of space.
XYZ needs its 3 basis master vectors (x, y, z) and their reflections, the
not-basis slave vectors, (-x, -y, -z) to scale and sum to all eight octants
of space.
Once we have that container, the none, one, two and three arrow assemblies
are seen to occur inside it i.e. 4Dness pervades. That's a very
non-Euclidean way of talking.
In extended Euclidean geometry, ala Donald Coxeter, we can have as many
orthogonals as we like through the origin. Synergetics is not an extended
Euclidean geometry. In Synergetics, adding dimension to 4D (4D+) means
adding energy (= frequency), our bridge to physics talk. Additional
dimensions like time and mass, the dimensions of time-size (time-space,
space-time).
Schema:
4D = eternal = timeless = relates to Mind
=====================================
4D+ = temporal = special case = relates to Brain
So there we have two geometries already (call them Coxeter's Euclidean n-D,
and Bucky's Dymaxion 4-D).
[ When the four planes of a tetrahedron converge to a common origin, you
get a "dymaxion" of four intersecting hexagons:
http://rwgrayprojects.com/synergetics/plates/figs/plate31.html ;]
Then Einstein's geometry, with its non-Euclidean metric, would be yet a 3rd
geometry.
I say all this in connection with the question: What is Geometry?
Different namespaces using 4D:
4D: Coxeter (Lie Groups fit here right?)
4D: Fuller
4D: Einstein
Kirby
Other related posts:
