Math in a STEM context: using Carbon Chemistry to connect the dots

[kirby urner]

I've been looking at the Allotropes of Carbon as a unifying theme when it comes to developing 2D and 3D graphics on computer as a part of an integrated STEM curriculum.  [1]

Graphene is flat (2D) whereas buckyballs and nanotubes are more 3D -- as is diamond of course.

A customary drawing tool for 2D graphics is the Turtle, as in LOGO.

I'm interested in extending that theme, though with adults I'm pushing the Tractor as an alternative.

Instead of a "Turtle object" we have a "Tractor object plowing the field" (a read / write activity on a 2D surface -- a field being a simple data structure, an N x M matrix of (X, Y) cells. [2]

We're used to XY square grids and XYZ cube grids, however given Carbon as a theme, I envision more hexagons (planar) and 3D analogs e.g. space-filling rhombic dodecahedrons.

How hexagons tile a sphere with pentagons, ala the soccer ball, is where the buckminsterfullerne comes in, as a strategy for dividing spheres. [3]

If anyone else is using carbon chemistry as a unifying STEM theme, around which to integrate various mathematical exercises, I'd be interested to learn about it.

Adding computer graphics to the curriculum give leverage to those wishing to escape the hegemony of scientific calculators.  Break out of the 1980s!

Today it's easy enough to provide each student with a desktop in the cloud e.g. an EC2 instance on Amazon.  That's the direction we're moving in at my school.

Kirby

[1]  Allotropes of Carbon:

https://www.boundless.com/chemistry/textbooks/boundless-chemistry-textbook/nonmetallic-elements-21/carbon-150/allotropes-of-carbon-582-3569/

https://flic.kr/p/wwhcB4

(screen shot)

[2]

http://worldgame.blogspot.com/2015/07/pws-personal-workspace.html 
(system architecture)

https://mail.python.org/pipermail/edu-sig/2015-August/011290.html
(curriculum content)

[3]

http://dividedspheres.com/

(Divided Spheres -- a primer)

[Joseph Austin]

Kirby,

There may be applications of the non-rectangular grid concept to music.

There is an hexagonal arrangement of the notes of the equal-tempered scale called Euler’s Tonnetz,

such that the edges of topological (geometrically degenerate) triangle represent intervals of a minor third (3 semitones), major third (4 semitones), and fifth (7 semitones).  When extended to the 12 notes of the ET chromatic scale, the result is topological torus.

On a somewhat related matter, I’ve toyed briefly with “raster geometry”, where “distance” is a point count along possible paths between “neighbor” nodes.  You can get interesting results for various definitions of “neighbor”, e.g. allow diagonals or not.

Given a square lattice, grids forms on in the vertical-horizontal direction and the diagonal directions are mutually ir-rational.

Joe Austin

--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.

[kirby urner]

On Fri, Aug 7, 2015 at 1:35 PM, Joseph Austin <drtec...@gmail.com> wrote:

Kirby,

There may be applications of the non-rectangular grid concept to music.

There is an hexagonal arrangement of the notes of the equal-tempered scale called Euler’s Tonnetz,

such that the edges of topological (geometrically degenerate) triangle represent intervals of a minor third (3 semitones), major third (4 semitones), and fifth (7 semitones).  When extended to the 12 notes of the ET chromatic scale, the result is topological torus.

Wow, I'd never heard of that.  I did some reading e.g.
http://www.tonalsoft.com/enc/t/tonnetz.aspx

I've gotten most of my music theory 2nd hand from Stu Quimby (happy birthday Stu) formerly with Design Science Toys (Tivoli, NY).  He was great at relating geometry to music for me.  I bet he's heard of this Tonnetz.

On a somewhat related matter, I’ve toyed briefly with “raster geometry”, where “distance” is a point count along possible paths between “neighbor” nodes.  You can get interesting results for various definitions of “neighbor”, e.g. allow diagonals or not.

Given a square lattice, grids forms on in the vertical-horizontal direction and the diagonal directions are mutually ir-rational.

Joe Austin

This notion of "neighbor" having a variable meaning in grids, and rules for iterating through "generations" based on rules about neighbor count, as in Wolfram sequences is an area I've explored (as have many).

So cellular automata (CA) hold out another set of lattice games, e.g. games like Conway's Life are playable in grids of hexagons (so six neighbors -- but then one might also work with a 2nd, 3rd concentric layer...).

I want students to be comfortable with sphere packing studies of a simple ordinary space, same radius spheres nature, what Conway has called a Barlow packing (for crystallographer Barlow). 

Both the square grid S (checkerboard with spheres inter-tangent, each inscribed in a square) and triangular grid T (each sphere inter-tangent with six neighbors vs only four, defining hexagons) are found in cross-section within the same 3D ball packings (HCP / CCP).

Put another way, when stacking oranges at the market one may start with either a square or triangular base and pile upward to an apex.

This is all well-known much-explored territory, but not that accessible given our obsession with everything square and cubic only in the early grades.  When I introduce Vectors, my tendency is to spice it up with talk of Quadrays (see Wikipedia).

Kirby

[Joseph Austin]

Kirby,

Does Quadray have application to relativistic space-time?

I surmise hexagonal coordinates also have application to isometric drafting,

which seems like a similar concept (representing 3D in 2D vs  4D in 3D?)

Another coordinate scheme that I’ve found useful (I’m not sure the mathematical name) is “relative to a moving point”, or differential coordinates used in ALICE programming and similar graphical systems: forward/back, up/down, right/left, yaw, pitch, roll.  Once we get past mathematical point to extended volumes,

we discover “dimensions” of orientation as well as position.

Have you investigated the Geometric Algebra of Hestenes et al?

He seems to recognize quantities and “directions” associated with areas, volumes, etc.

Carbon, of course, is the key to organic chemistry, DNA, and hence life and evolution.

A thorough appreciation of it’s multi-dimensional and connectional possibilities should be essential to science and future scientists.  

Joe Austin

[kirby urner]

On Sun, Aug 16, 2015 at 3:15 PM, Joseph Austin <drtec...@gmail.com> wrote:

Kirby,

Does Quadray have application to relativistic space-time?

I don't think of them that way but others might (several people have worked on these -- I'm a contributor and write about them more than most helping with R&D). 

Quadray coordinates are more a mind game for those trained in XYZ (which is all of us) and wanting to see a different language game applied "just for fun" (we have motives).

In XYZ, we have six spokes, three positive and three negative, that may be variably extended to reach any point in any of eight octants.

We don't use a slot for each spoke in XYZ, just three, and when a spoke is "in the negative direction" we change the sign so (x, -y, z) for example.  All points in 3D space are reached.

With Quadrays we have only four spokes from the center.  Opposites may be drawn but are redundant in terms of the goal:  to map (provide an address for) all points in space.

Quadrays, four rays from the origin through the vertexes (or face centers) of the surrounding tetrahedron, divide space into quadrants, not octants.

In using a slot for each ray, to show its extension (scale), same as in XYZ, we trade away any need for negative numbers in favor of one more slot, hence vectors with (a, b, c, d) for coordinates.

True, at least one of (a, b, c, d) is 0 as we don't extend in one quadrant for any given point (any point is in but quadrant or on a border, so at least one zero coordinate every time). 

XYZ is the same in that three spokes are inactive (not needed) per any point in a specific octant -- but the slots are shared by + and - so double the business through each one.

 

I surmise hexagonal coordinates also have application to isometric drafting,

which seems like a similar concept (representing 3D in 2D vs  4D in 3D?)

A property of Quadrays is if you calibrate them correctly at the beginning, then all the centers of closest packed spheres have purely integer coordinates, which is not to say that all only-integer addresses correspond to such centers.

If we assume (0,0,0,0) is the center of a CCP sphere, then all combinations of (2,1,1,0), of which there are 12, map the centers of the adjacent / tangent spheres and so on.

https://en.wikipedia.org/wiki/Quadray_coordinates

Another wrinkle and brain teaser is it's the edges of the six tetrahedron to which the four quadrays point, that get the nice number of 1 or 2 for length.  Who said the spokes need to be rational?

If you picture four balls (equal size) packed in a tetrahedron, then the diameter of each ball is the same length as the interval between two adjacent centers, call this D.  1D = 2R. 

So you'll find 1 or 2 assigned to these tetrahedron's edges, with (0, 0, 0, 1)  (0, 0, 1, 0) (0, 1, 0, 0) and (1, 0, 0, 0) calibrated accordingly.

That's enough information to allow a one-to-one correspondence of XYZ and Quadray addresses.  There's a formula that takes it either way, i.e. there's isomorphism between Quardays and XYZ.

Another reason this is useful is indeed to demonstrate alternative namespaces.  What's so "3D" about a Tetrahedron and do we really have fewer dimensions, let alone more?  Karl Menger's "geometry of lumps" essay is worth inserting here.

http://science.iit.edu/applied-mathematics/about/about-karl-menger

So in at least one language game, 4D just means Cartesian res extensa, no time, no specific size, just spatial relationships (what we ordinarily call 3D in everyday math).  Adding Time adds energy and actual relatives sizes of event.  The "real world" is that of Angles (as in shapes, with surface and central angles) plus Frequency (energy added with time/size).

Quadrays develop an intuition that a Tetrahedron is a player, not just the Cube, and that's an ulterior motive we have for introducing Quadrays as a kind of side bar at least.

Another coordinate scheme that I’ve found useful (I’m not sure the mathematical name) is “relative to a moving point”, or differential coordinates used in ALICE programming and similar graphical systems: forward/back, up/down, right/left, yaw, pitch, roll.  Once we get past mathematical point to extended volumes,

we discover “dimensions” of orientation as well as position.

Yes, I've used ALICE and one or two of its precursors.

In Kantian terms (philosophy now) we're compelled to conceptualize in volume, seems a priori, but calling this volume "three dimensional" is not so a priori, as we see in the rear view mirror.

When it comes to still versus moving, one could say that's adding Time, plus both are really missing the Z axis i.e. whether a TV screen or a canvas, both are flat (2D) but motion adds time (3D).

So when we say 2D (for flat) and 3D for spatial, we really mean 3D and 4D if thinking in a more Einstein-informed namespace.  Then subtract the Z axis to acknowledge neither still life nor TV show has a "real" Z axis, at we're back to "2D versus 3D" for "manga versus anime".

Limbering up around the various meanings of "Dimension" versus settling on the belief there's just "one true meaning" is a goal in my curriculum.

H.S.M. Coxeter, a one time student of Wittgenstein's (of "language game" fame) in England (contemporary of Turing's of "Turing Machine" fame) is at pains to point out in his 'Regular Polytopes' (pg 119 in Dover Edition) that a tesseract is not a time machine (paraphrase), much as science fiction writers want to conflate the two for plot-driving purposes.

In other words, the extended Euclidean coordinates of nD polytope math, with its nD sphere packings (written about by Conway and Sloane [1]) is NOT the same namespace as Einstein / Minkowsi.

The Great 4D Shakeout that began in the late 1800s left only two meanings standing by mid-1900s:  Einstein's and Coxeter's (using these names to symbolize entire branches in math).

Coxeter is simply underlining that fact in his reproach to science fiction writers, who'd converge them.  He wanted to warn his readers that all his dimensions were equally spatial, none singled out as temporal, which is when physics kicks in.  Pure mathematics likes avoiding a time dimension, preferring the eternal Platonic.

Then in the late 1970s we saw this third namespace pop up using 4D to mean "timelessly conceptual i.e. volumetric" (i.e. Platonic) and that's where Quadrays are rooted, but this namespace is still esoteric in 2015, remote from common use (which explains why I've branded it as Martian Math i.e. why fight that perception, just go with it as in "yes it's alien -- doesn't mean bad or wrong").

 

Have you investigated the Geometric Algebra of Hestenes et al?

Yes.  Clifford Algebra.
 

He seems to recognize quantities and “directions” associated with areas, volumes, etc.

Quadrays are much easier to wrap your head around given only training in XYZ -- could be introduced much earlier I think?

I use them to help with the idea of "vectors" in the naive sense of "spokes from the origin" i.e. "pointers to points in space".

We currently tend to teach Points in XYZ as a topic separate from XYZ vectors, which we get to later. 

I'm more into doing those all together because XYZ pointers are not any less intuitive than points by themselves.

Quadrays and XYZ vectors have the same API pretty much and that both are scalable, addable, have additive inverses, are rotatable (swing around).

In the Clojure version, I use a "protocol" that defines this API independently of the "guts" where the actual implementations occur.

https://github.com/4dsolutions/synmods/blob/master/qrays.clj
 

Carbon, of course, is the key to organic chemistry, DNA, and hence life and evolution.

A thorough appreciation of it’s multi-dimensional and connectional possibilities should be essential to science and future scientists.  

Joe Austin

Yes, multi-dimensional.  I also say "slippery dimensional" to remind students that we have many language games using that word.

In ordinary language, we also use "dimensions" to mean "any quantum of measure" i.e. "the mass of a thing" is another of its many "dimensions".

Rather than say that's being sloppy, I say lets remember that "dimension" is a multi-purpose tool, like a screwdriver is (may by used also to open paint cans).

Kirby

[1]  http://mathworld.wolfram.com/HyperspherePacking.html