STEM for Art History students

[kirby urner]

I'm back at the code school this Monday night. 

I'm working on two projects:

(A) poll those present regarding any experience with crypto-currencies, bitcoin in particular

(B) continue communicating with Sigurd Andersen so cites our quadrays at the bottom of his:
http://www.systalk.org/math/dimensionalities/

By "working on" I also mean "blogging about" in this context.

(A)   http://coffeeshopsnet.blogspot.com/2016/05/business-model-in-bitcoin.html

(B)   http://coffeeshopsnet.blogspot.com/2016/05/4d-meme.html

I'm hoping (B) will help art school teachers make sense of 20th century intellectual currents, still strong, as described in:

https://mitpress.mit.edu/books/fourth-dimension-and-non-euclidean-geometry-modern-art

Andrius, if you have a chance to look at (B), maybe you can critique its clarity. 

I'm using the most basic of linear algebra concepts, talking about vector algebras.  I don't want art and architecture students to think they need fancy math degrees to understand the basic basics of their own heritage.

Kirby

[Andrius Kulikauskas]

Hi Kirby,

These art schools are for college aged students, yes?

As regards your quadrays at
http://coffeeshopsnet.blogspot.lt/2016/05/4d-meme.html

I think that the picture
https://upload.wikimedia.org/wikipedia/commons/9/99/Quadray.gif
is what I find most clarifying. It made me realize that the origin is
at the center of the tetrahedron. Maybe you could edit the picture to
include a (0,0,0,0) in blue. And maybe use two more distinct colors
like red and black rather than blue and black. And make the red lines
extend beyond the tetrahedron. It would actually be nice to see the
four points (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2). Then it would
be clear that we're seeing a coordinate system, not (just) a tetrahedron.

I find your previous blog post much more clear:
http://coffeeshopsnet.blogspot.lt/2016/04/quadray-coordinates.html

Your most recent blog post is deep into Kirby's world in a way that
reads to me like math on acid. :) Which is to say I find it absolutely
unclear unless I assume that I already know it all. :) Certainly poetic,
funky and fun if you are already in that world, which I imagine would be
the case in person with you.

Clarity-wise I imagine that you could pick and organize your main
points, such as:
* I want to teach the power of coordinate systems without having to make
use of negative numbers. (By analogy: I want to teach a language in
which all sentences are positive and never negative.)
* I want to impart the intelligence that comes from working with more
than one coordinate system. (By analogy: you look at language
completely differently when you can speak more than one language.
Indeed, you can control your mind when you can choose your language.)
* I want to inspire the imagination by showing how 2-dimensional
pictures can illustrate 3-dimensional coordinates which can represent
4-dimensional experiences. And by constructing 3-dimensional models
that can also illustrate 4-dimensional experiences. (By analogy:
language can evoke people and worlds who you know and even inspire you
to care about those who you never thought you could even imagine.)
* I want to empower you to make up your own coordinate system which best
suits your purpose and preferences. (By analogy: you can think in your
own private language and construct your own world which you can then
share in other languages, too.)
* And perhaps: I want to motivate the concept of negative numbers by
showing how it compares to possible alternatives. (By analogy: I want
to show when, how and why you might use negative statements thoughtfully.)

Then make sure your main points appear at the very beginning, at the
very end, and show up in the middle relating to something somehow. Then
I believe your reader will feel secure that they know consciously what
you wanted them to know, and they will give you total liberty to be
Kirby in Kirby World.

Just thinking about this, you might consider doing Euclidean coordinates
with six positive axes: x and a, y and b, z and c, for example. Just
to show that you don't need to use negative numbers if you don't want
to. Just say left and right, up and down, front and back. But at a
certain point your students may see the value of using 4 positive
coordinates instead of 6 positive coordinates. And then that they could
go down to 3 coordinate axes if they introduced plus and minus.

So your art students have enough options to create their own coordinate
systems. And each dimension could mean different things like time,
flavor, musical frequency, friends on Facebook, etc. And likewise
different objects such as regular polygons and Platonic solids can be
constructed and have meaning and be the basis for games. For example,
even in 2 dimensions if you have multiple lines on a page, then you can
do conversions from one description to another, making for
"equalities". The lines don't even have to come from a common origin.
Each line can have its own origin. There can be practical applications
like learning how to space letters evenly when you draw words.

Thank you so much for involving me in your thinking.

Kirby, thank you for your letter on the Catalan and Archimedean
polyhedrons. I'm very much into duality. I will try to make that
evident in my map of math. Thank you for being so loving and supportive
of Andrius and Andrius World.

I mention you and Math Future briefly in my very long introductory post
at John Baez's Azimuth Project:
https://forum.azimuthproject.org/discussion/1688/introduction-andrius-kulikauskas

I'll bring up some thoughts that may be relevant for you and your art
teachers and art students. I'm thinking through a talk that I will
propose for an aesthetics conference here in Vilnius, Lithuania. I want
to talk about mathematical beauty. So I'm wondering what I can say
about that. But I'm thinking it would be good to use my work with the
Mandelbrot set as an example.

I think my main idea is that, from a mathematician's point of view, (at
least my own), the Mandelbrot set is not beautiful. It's not beautiful
because in math beauty is not what you see, but what you imagine. And I
can't imagine the Mandelbrot set. It's just lots of noise. There's no
melody. Whereas I can imagine an equilateral triangle and so it stands
out amongst all of the triangles. I can play with it in my mind, watch
it dance around. I can hum that tune.

What's truly beautiful is Galois theory where you have this assurance
that you can play around with a group of dynamic actions and that will
correspond to a polynomial and its solutions. That is amazingly
beautiful. There is order in the universe. Or the dualities that you
notice with the Platonic solids. Or the fact that there can only be
those solids and no others, that what matters are the number of edges
and vertices and faces and whether they satisfy Euler's formula. That
makes it seem like we have signs of a Why out there some where.

But I will go through steps to show what can make the Mandelbrot set
beautiful mathematically, step by step. The kinds of steps I wrote
about that this is not some accidental set. And that it relates to lots
of key things in math. And that (perhaps) every single point is
actually encoding something meaningful, so that the whole complex plane
is an analysis of all of the possible behaviors of automata. And
furthermore if the intricate structure of the Mandelbrot set was
visually displaying how those behaviors are related. Something like
that would be awesomely beautiful.

I will write more about that. But I'm curious if that's an interesting
subject for your world.

Thank you, Kirby,

You are beautiful!

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

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[kirby urner]

On Tue, May 17, 2016 at 1:40 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

Hi Kirby,

These art schools are for college aged students, yes?

Yes, typically, or older.  Pacific Northwest College of Art (PNCA) is my paradigm art school, but many are like it.

Here's an album from a lecture I went to there:

https://www.flickr.com/photos/kirbyurner/albums/72157617749101670

 

As regards your quadrays at
http://coffeeshopsnet.blogspot.lt/2016/05/4d-meme.html

I think that the picture
https://upload.wikimedia.org/wikipedia/commons/9/99/Quadray.gif
is what I find most clarifying.  It made me realize that the origin is at the center of the tetrahedron.  Maybe you could edit the picture to include a (0,0,0,0) in blue.  And maybe use two more distinct colors like red and black rather than blue and black.  And make the red lines extend beyond the tetrahedron.  It would actually be nice to see the four points (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2).  Then it would be clear that we're seeing a coordinate system, not (just) a tetrahedron.

I'm using that graphic as somewhat iconic as it appears in some other places on the web and therefore serves an anchoring function simply in being repetitious.

However, for more / different / clarifying graphics I recommend:

http://grunch.net/synergetics/quadintro.html  (check Fig 8. especially)

What we really need are more anime i.e. cartoons / animations.  I have some storyboards prepared.

 

I find your previous blog post much more clear:
http://coffeeshopsnet.blogspot.lt/2016/04/quadray-coordinates.html

Your most recent blog post is deep into Kirby's world in a way that reads to me like math on acid. :)  Which is to say I find it absolutely unclear unless I assume that I already know it all. :) Certainly poetic, funky and fun if you are already in that world, which I imagine would be the case in person with you.

Interesting.  The traveling play about Bucky, The History and Mystery of Universe, by D.W. Jacobs, has its psychedelic aspects.

https://flic.kr/p/5qznLu
https://flic.kr/p/5uv4SE

Opening night in Portland: 

https://www.flickr.com/photos/kirbyurner/albums/72157608139268797

Later I gave an IEEE lecture at Portland Center Stage (on the night Obama got elected in 2008).

http://worldgame.blogspot.com/2008/11/ieee-presentation.html

Fuller is indeed associated with that era in US history, when the hippies were experimenting with LSD, as were the academics (Timothy Leary et al).

However Bucky himself managed to escape the fate of a cult leader, though his critics did manage to saddle him with "disciples", which does sound rather cultish.

Keep in mind that Quadrays were invented and evolved by a peer group, not by "just me".  I have a stash of white papers on the topic (not exactly white, more colorful).  I drew from many thinkers, including but not limited to Gerald de Jong, David Chako, Tom Ace and Russell Chu. Amy Edmondson an inspiration.  Applewhite offered a lot of support, including going to bat for me and trying to get me a Guggenheim that time.

 

Clarity-wise I imagine that you could pick and organize your main points, such as:
* I want to teach the power of coordinate systems without having to make use of negative numbers.  (By analogy: I want to teach a language in which all sentences are positive and never negative.)

We may use them, just they're redundant when for rays span space in linear combination, without any need of their mirror / negative reflections.

XYZ without its negative arms is what I call "amputated XYZ" and is helpless to reach all octants outside of (+ + +) unless we permit at least one more vector to reach out the "back side".

One more basis vector would obviate the need for flipping 180, i.e. multiplication by -1.  But in the Cartesian system we prefer to introduce -i, -j, and -k, creating what I call a "jack".

Q-rays are only 4-spoked (caltrop) versus 6-spoked (jack) where "spoke" means "ray" i.e. 1/2 a number line.
 

* I want to impart the intelligence that comes from working with more than one coordinate system.  (By analogy: you look at language completely differently when you can speak more than one language.  Indeed, you can control your mind when you can choose your language.)
* I want to inspire the imagination by showing how 2-dimensional pictures can illustrate 3-dimensional coordinates which can represent 4-dimensional experiences.  And by constructing 3-dimensional models that can also illustrate 4-dimensional experiences.  (By analogy: language can evoke people and worlds who you know and even inspire you to care about those who you never thought you could even imagine.)

I'm wanting art history students to understand that when Fuller, an architect, used 4D in his explorations, he didn't mean either what Coxeter meant, or what Einstein meant, these latter two meanings being distinct from one another with or without Fuller's meaning.

For Fuller, the tetrahedron is the genesis icon for volume in that it shows "container" using the most primitive Platonic (fewest features) available.

In showing three XYZ number lines re-arranged to instead form 3 of the six edges of an infinite tetrahedron, I'm suggesting how volume might be seen as 4D, given a tetrahedron says "4" more loudly than "3".

We understand that in XYZ shoptalk (a namespace), volume is 3D, not a problem.  Lets just be clear that at least on 20th Century genius did not consider himself bound by that convention and his reasoning is quite good.

Adding quadrays as a language game helps solidify the position that 4D in Fuller's sense has a trajectory forward.  When we say IVM.4D versus XYZ.3D, we want art students to understand what we mean.

Likewise Einstein.4D and Coxeter.4D are distinct.

In every case, I'm using Namespace.keyword as my notation.  I'm saying Namepace contains / contextualized the keyword that comes after.  Coxeter.4D is extended Euclidean geometry with four mutual orthogonals.  Einstein.4D introduces a time dimension as a complex (imaginary) variable.  These are distinct namespaces.

Fuller's namespace is yet different again.

Art history students, following the topic of non-Euclidean geometry in the 20th century, need to grasp all three of these meanings.  The first two are well known but Fuller's Synergetics was not published until the late 1970s and it took awhile to digest.

 

* I want to empower you to make up your own coordinate system which best suits your purpose and preferences.  (By analogy: you can think in your own private language and construct your own world which you can then share in other languages, too.)
* And perhaps: I want to motivate the concept of negative numbers by showing how it compares to possible alternatives.  (By analogy: I want to show when, how and why you might use negative statements thoughtfully.)

I want art history students to understand the world into which they were born.  That means tracing the intellectual currents of recent history.  4D as a meme, is critical.
 

Then make sure your main points appear at the very beginning, at the very end, and show up in the middle relating to something somehow.  Then I believe your reader will feel secure that they know consciously what you wanted them to know, and they will give you total liberty to be Kirby in Kirby World.

I'm initiating them into contemporary philosophy.  It's not Kirby World in particular but I'm happy to have a front row seat.

We're talking about cultural memes.  People who have never heard of Kirby will have heard of Synergetics and geodesic domes.

 

Just thinking about this, you might consider doing Euclidean coordinates with six positive axes:  x and a, y and b, z and c, for example.  Just to show that you don't need to use negative numbers if you don't want to.  Just say left and right, up and down, front and back.  But at a certain point your students may see the value of using 4 positive coordinates instead of 6 positive coordinates.  And then that they could go down to 3 coordinate axes if they introduced plus and minus.

I do want students to appreciate the 1-to-1 correspondence between the canonical representation of vectors expressed in both coordinate systems, XYZ and IVM.

IVM = isotropic vector matrix = scaffolding when you keep the uniform-length segments twixt the centers of CCP balls.  Alexander Graham Bell called it a Kite.

 

So your art students have enough options to create their own coordinate systems.  And each dimension could mean different things like time, flavor, musical frequency, friends on Facebook, etc.  And likewise different objects such as regular polygons and Platonic solids can be constructed and have meaning and be the basis for games.  For example, even in 2 dimensions if you have multiple lines on a page, then you can do conversions from one description to another, making for "equalities".  The lines don't even have to come from a common origin.  Each line can have its own origin.  There can be practical applications like learning how to space letters evenly when you draw words.

Thank you so much for involving me in your thinking.

Kirby, thank you for your letter on the Catalan and Archimedean polyhedrons.  I'm very much into duality.  I will try to make that evident in my map of math.  Thank you for being so loving and supportive of Andrius and Andrius World.

I'm glad we both have worlds.

 

I mention you and Math Future briefly in my very long introductory post at John Baez's Azimuth Project:
https://forum.azimuthproject.org/discussion/1688/introduction-andrius-kulikauskas

I'll bring up some thoughts that may be relevant for you and your art teachers and art students.  I'm thinking through a talk that I will propose for an aesthetics conference here in Vilnius, Lithuania.  I want to talk about mathematical beauty.  So I'm wondering what I can say about that.  But I'm thinking it would be good to use my work with the Mandelbrot set as an example.

I have been to a EuroPython in Lithuania, in Vilnius.  It was indeed beautiful.

http://worldgame.blogspot.com/2007/07/slow-food-nation.html

 

I think my main idea is that, from a mathematician's point of view, (at least my own), the Mandelbrot set is not beautiful. It's not beautiful because in math beauty is not what you see, but what you imagine.  And I can't imagine the Mandelbrot set.  It's just lots of noise.  There's no melody.  Whereas I can imagine an equilateral triangle and so it stands out amongst all of the triangles.  I can play with it in my mind, watch it dance around. I can hum that tune.

Have you watched any Mandelbulb videos?

Another world you might enjoy:

https://youtu.be/tsuYZg8k-Zc

 

What's truly beautiful is Galois theory where you have this assurance that you can play around with a group of dynamic actions and that will correspond to a polynomial and its solutions.  That is amazingly beautiful.  There is order in the universe.  Or the dualities that you notice with the Platonic solids.  Or the fact that there can only be those solids and no others, that what matters are the number of edges and vertices and faces and whether they satisfy Euler's formula.  That makes it seem like we have signs of a Why out there some where.

Yes, the Platonics are a good beginning.  That's where to anchor the concept of "dual" as the set is closed under that operation (although in group theory, unary operations are not generally considered).
 

But I will go through steps to show what can make the Mandelbrot set beautiful mathematically, step by step.  The kinds of steps I wrote about that this is not some accidental set.  And that it relates to lots of key things in math.  And that (perhaps) every single point is actually encoding something meaningful, so that the whole complex plane is an analysis of all of the possible behaviors of automata.  And furthermore if the intricate structure of the Mandelbrot set was visually displaying how those behaviors are related.  Something like that would be awesomely beautiful.

Mandelbrots have long been a feature of my experimental curriculum, such as I've had an opportunity to field test it.

I even opened for Mandelbrot once, explaining fractals while he watched from the wings.  He said I'd done a good job.

You'll find the Mandelbrot Set one of the Four Focal Points in this blog post:

http://mybizmo.blogspot.com/2006/09/focal-points.html

Related: 

http://worldgame.blogspot.com/2014/10/recent-lesson-plans.html

 

I will write more about that.  But I'm curious if that's an interesting subject for your world.

Very much so.

 

Thank you, Kirby,

You are beautiful!

I've lost some weight since Vilnius.  :-D

Happy Birthday to me!  May 17.  58.

Kirby

 

[kirby urner]

On Tue, May 17, 2016 at 6:29 PM, kirby urner <kirby...@gmail.com> wrote:

<< SNIP >>

 

I'm wanting art history students to understand that when Fuller, an architect, used 4D in his explorations, he didn't mean either what Coxeter meant, or what Einstein meant, these latter two meanings being distinct from one another with or without Fuller's meaning.

For example, in light of my elucidations, they should have no trouble understanding these passages, quoting directly from Synergetics:

527.701 In synergetics primitive means systemic conceptuality independent of size. (Compare Sec. 1033.60.)

527.702 Geometers and "schooled" people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states --"one-dimensional," "two-dimensional," and "three-dimensional" -- which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of one another nor independently of all the inherent characteristics of all systems and of all systems' inherent complex of interrelationships with Scenario Universe.

<< snip >>

527.712 All conceptual consideration is inherently four-dimensional. Thus the primitive is a priori four-dimensional, being always comprised of the four planes of reference of the tetrahedron. There can never be any less than four primitive dimensions. Any one of the stars or point-to-able "points" is a system-ultratunable, tunable, or infratunable but inherently four-dimensional.

Kirby

[Joseph Austin]

On May 17, 2016, at 4:40 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

It would actually be nice to see the four points (2,0,0,0), (0,2,0,0), (0,0,2,0), (0,0,0,2).  Then it would be clear that we're seeing a coordinate system, not (just) a tetrahedron.

I find your previous blog post much more clear:
http://coffeeshopsnet.blogspot.lt/2016/04/quadray-coordinates.html

Kirby, 

I agree with Andrius that the earlier discussion in the reference is clearer.

To convince myself that every point in space corresponds to 3 positive coordinates in quadray space: 

Instead of considering the tetrahedron itself, I'm considering the four planes formed by pairs of axes, that is, the planes of the blue-blue-black triangles in the "quadray coordinates" diagrams of the reference above. 

I image extending those planes to form boundaries of four concave regions.  The planes, and the edges (axes) jointing them, are at obtuse angles to each other, but nevertheless it is clear that any point will be in one of the four regions and any point in a concave region has orthogonal projections on each edge (i.e. axis) of the surface.

Perhaps this could be clarified by drawing several tetrahedra at equal radial distances, and shading one quadrant of the space and including perpendiculars (coordinate projections) for some "arbitrary" point.

BTW, this reminds me that once upon a time I knew about a four-coordinate system for doing 3D computer graphics, which as I recall was related to conformal mapping and the complex sphere somehow. Perhaps someone recognizes what it is I'm thinking of.  (I'm "recalling" less and less these days.)

Joe Austin

[kirby urner]

On Wed, May 18, 2016 at 7:15 AM, Joseph Austin <drtec...@gmail.com> wrote:

 

Kirby, 

I agree with Andrius that the earlier discussion in the reference is clearer.

To convince myself that every point in space corresponds to 3 positive coordinates in quadray space: 

Instead of considering the tetrahedron itself, I'm considering the four planes formed by pairs of axes, that is, the planes of the blue-blue-black triangles in the "quadray coordinates" diagrams of the reference above. 

I image extending those planes to form boundaries of four concave regions.  The planes, and the edges (axes) jointing them, are at obtuse angles to each other, but nevertheless it is clear that any point will be in one of the four regions and any point in a concave region has orthogonal projections on each edge (i.e. axis) of the surface.

Yes, this is a fine visualization. 

From XYZ experience, we're used to the "jack" of six rays dividing space into eight octants.  In one of those octants, x,y,z are all positive (+ + +) whereas in the seven other octants, at least one of the coordinates is negative.  The octant kitty-corner to (+ + +) is (- - -).  We may label all eight octants with permutations of three plus and minus signs.

Be that as it may, a given point is always in one of those octants *or* right on an axis or plane between octants, where some or all the coordinates become 0. 

That's XYZ.

In Q-rays or quadrays, we get a "caltrop" of four rays dividing space into four quadrants.  In each one of those quadrants, one of the quadrays is not needed whereas the other three provide positive coordinates just like in (+ + +).  The three basis vectors are simply spread wider apart than in XYZ, which latter vectors come to a corner at 90-90-90.  Q-rays are uniformly at 109.47 degrees to one another (assuming their tetrahedron framework is regular).

The unneeded ray is 0, so in one quadrant we have points (+ + + 0).  In another quadrant we have (+ + 0 +) and so on.  A point right on a plane between quadrants will have two 0s e.g. (0 0 + +) whereas points on an actual ray will have the form (+ 0 0 0) i.e. only.  The origin is (0,0,0,0).

What's important to see here is each point in space has just one canonical representation in both XYZ and Q-rays.  There's an isomorphism between the two.  Every point in space has a unique representation in either coordinate system.  Going back and forth between the two systems simply requires some initial stipulations, such as shared origin i.e. (0,0,0) is the same point as (0,0,0,0) etc.

 

Perhaps this could be clarified by drawing several tetrahedra at equal radial distances, and shading one quadrant of the space and including perpendiculars (coordinate projections) for some "arbitrary" point.

I've come up with another way of looking at XYZ that translates well to Quadrays.

Think of having three currencies like dollars, euros and yen.  Or lets use the $ symbol and think in terms of x-dollars, y-dollars, and z-dollars, represented as $x $y and $z.

You need to associate a "cost" with every point in space, which is equal to how much of each currency it would take to move you the necessary distance in each one of those three directions to reach that point. 

To get to (4, 5, 10), floating out there in the first octant (+ + +) someplace, you'd spend exactly ($4, $5, $10) of your three currencies.

In order to get to an octant with negative coordinates, you could go into debt or in any case currencies go negative.  (-$4, -$10, 0) would be a point on the XY plane.

Quadrays are just like that except you have four currencies ($A, $B, $C, $D).  Once again, there's a unique amount that will get you to any point, simply by "riding each vector" in its given direction.

In both cases, we many demonstrate "reaching a point" using "tip-to-tail" vector addition.  ($4, 0, 0) in XYZ means you've paid to move 4 in the +x direction.  You then go $5 distance along Y and $10 along Z and you're at the point you want to be. 

The "cost" is unique. 

Ditto with Q-rays:  always three of the four rays are sufficient as you're always in just one of the four quadrants.  Tip-to-tail vector addition is exactly the same.
 

BTW, this reminds me that once upon a time I knew about a four-coordinate system for doing 3D computer graphics, which as I recall was related to conformal mapping and the complex sphere somehow. Perhaps someone recognizes what it is I'm thinking of.  (I'm "recalling" less and less these days.)

Joe Austin

There's a 4-coordinate approach to XYZ that's associated with foreshortening to gain perspective.

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

Another related topic, this resource looks interesting:

http://link.springer.com/article/10.1007%2Fs00006-014-0439-3

Kirby

[Joseph Austin]

On May 18, 2016, at 12:42 PM, kirby urner <kirby...@gmail.com> wrote:

There's a 4-coordinate approach to XYZ that's associated with foreshortening to gain perspective.

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

Another related topic, this resource looks interesting:

http://link.springer.com/article/10.1007%2Fs00006-014-0439-3

Kirby

Kirby,

Those "ring a bell."

"Homogeneous coordinates" is what I was referring to.

The representation I'm thinking of is actually used in some graphics systems

as the internal representation of points--as best I can recall, I first encountered it in ALICE.

Here is another reference that goes into "practical" detail.

http://www.gamedev.net/page/resources/_/technical/graphics-programming-and-theory/the-total-beginners-guide-to-3d-graphics-theory-r3402

BTW, they also represent surfaces by breaking them into triangles.

(Another area that uses the triangle approach is the "Finite Element Method" of Structural Analysis.)

Finally, I found a paper "ALICE: Lessons Learned Building 3D systems"

that may be of interest to artist-programmers.

The recurrent theme: adapt the system to let the user express things the way the user thinks, don't force the user to adapt to the way mathematicians and programmers think.

Examples: move Forward or Left (from moving object's perspective), not X Y Z  or  even Quadray directions.

Turn 1/4 (of a revolution), not 90º or π/2 radians.

http://repository.cmu.edu/cgi/viewcontent.cgi?article=1197&context=hcii

[kirby urner]

Kirby,

Those "ring a bell."

"Homogeneous coordinates" is what I was referring to.

The representation I'm thinking of is actually used in some graphics systems

as the internal representation of points--as best I can recall, I first encountered it in ALICE.

Yes I've used Alice.  I like that whole "Alice" meme and just last night
viewed the preview for the upcoming Disney version directed by
Tim Burton.

I mentioned earlier my playing the computer game version:

http://bit.ly/1RbMlNb

The Alice you're speaking of (more like Scratch) does not strongly allude
to the Lewis Carroll version as I recall, however I think adding these
connect-the-dots links post hoc is apropos.
 

Here is another reference that goes into "practical" detail.

http://www.gamedev.net/page/resources/_/technical/graphics-programming-and-theory/the-total-beginners-guide-to-3d-graphics-theory-r3402

BTW, they also represent surfaces by breaking them into triangles.

Fortunately, a lot of these algorithms are "under the hood" when using
such tools as POV-Ray, the starting point in my curriculum.

One is free to focus on using XYZ or Q-ray coordinates directly, with
perspective completely taken care of. 

Just define the focus and observer viewpoint (a minimum axis) and
you're ready to render!  Like you were saying earlier, every sphere,
cylinder and wafer (slice) is volumetric, so we needed concern
about points, lines and planes being of different "dimension number"
-- dimension theorist Karl Menger's move in his Geometry of Lumps
approach.

'Modern Geometry and the Theory of Relativity', in
Albert Einstein: Philosopher-Scientist , The Library of Living Philosophers VII,
edited by P. A. Schilpp, Evanston, Illinois, pp. 459-474.

I focus on "string substitution" much as I think a Kosan-type MathPiper / CAS
would do, when reducing an expression to its canonical form.

I start with Madlibs (at a young age -- so they're scatalogical sometimes). 

Then later we're interpolating into like VRML worlds, if anyone remembers:

http://www.4dsolutions.net/ocn/worlds/

For more about scatalogical MadLibs and 3D worlds, see slides 25- 27 in:

http://www.4dsolutions.net/presentations/connectingthedots.pdf

Same idea (substitution into boilerplate) occurs when writing POV-Ray
scene description language file, .pov extension files.

Hand-coding from scratch every time is crazy, why not let cylinders, spheres
and wafers render themselves, using canned syntax with variable substitution?  

POV-Ray is an open source still life renderer, but with looping and variables,
anime become possible.  Delta t (time span) = time for action (units mvd).

povray.org (home base for POV-Ray, see what professionals do with it!)
www.4dsolutions.net/ocn/numeracy0.html  (show's my approach for
Oregon Curriculum Network, 1990s)

Thanks for your additional links.

Kirby

[kirby urner]

On Thu, May 19, 2016 at 1:59 PM, kirby urner <kirby...@gmail.com> wrote:

 

The Alice you're speaking of (more like Scratch) does not strongly allude
to the Lewis Carroll version as I recall, however I think adding these
connect-the-dots links post hoc is apropos.

I think a critic might say Alice.org is indeed connecting to Alice in Wonderland

in having a blond girl holding a world in her hand, no doubt a virtual world.  

I'd call that "weak allusion" in that I see no effort to hook to the mad hatter or

time-aware hare (iconic "white rabbit"), much less Cheshire cat:

http://www.alice.org/index.php?page=what_is_alice/what_is_alice

I don't think those characters could still be under copyright, so the reason

for avoiding too-close ties was probably to keep the aesthetics more neutral,

not tied too closely to any particular work of fiction.

I recall the NCTM logo used to be an octet-truss but that might have seemed

too much a bias or endorsement of a specific aesthetic.  Going with the infinity

symbol instead (NCTM switched logos) seemed a lot more neutral, less definite,
fuzzier. 

When working to represent a broad cross-section, the fuzzier the better sometimes.

Kirby