Juan's "fractal dimension" of math formulas

[Maria Droujkova]

This essay of Juan Castaneda, @mathguide on Twitter, is a follow-up to a conversation we had at the "Math language and accessibility" Math 2.0 weekly meeting in July: http://sdmath.blogspot.com/2009/07/is-there-connection-between.html Several people weren't sure what Juan meant by dimensions of formulas, and wanted to know more. Here it is! An interesting take, for sure!

A quote: "To read text, we only need the basic linear connection from each letter to the next one, and from each word to the next one. Any text document can be considered as a sequence of characters, however long it may be.
On the other hand, when we look at images in the real world, like homes, people, faces, mountains, trees, animals, and so on, we process this visual information in a very different way. We see color, shades of color, light, texture, and a multitude of details that can only make sense when we consider them embedded in the full three-dimensional space around us. However, our retina is pretty much a flat surface, and our brains have to imagine the three-dimensional world based on the two-dimensional information our flat retina collects from the incoming light. So, the raw material our brain uses to process visual information is nearly two-dimensional in nature. When looking at an image, if we consider a little part of it, there is no such thing as “the next pixel,” because that could be located above, or below, or to the right, or to the left, or in any diagonal direction. Often we can find linear patterns inside some images but the whole image is fully two-dimensional.
So, where does this basic assumption about dimensions leave the written representation of mathematical expressions?"

I liked the diagram of the quadratic formula Juan made. This formula was my first, very fondly remembered, experience with programming, back when I was twelve and first got my hands on a programmable calculator (heavier than my laptop is now), and parsing it as a one-dimensional sequence of steps was a huge challenge. For a computer, formulas or pictures can be one-dimensional strings of 0s and 1s, but humans hardly work this way.

I am still not sure we can formalize this idea of "dimensions" rigorously. It's an intriguing idea, though.


Cheers,
Maria Droujkova
http://www.naturalmath.com

Make math your own, to make your own math.

[Edward Cherlin]

On Sun, Aug 16, 2009 at 6:52 AM, Maria Droujkova<drou...@gmail.com> wrote:
> This essay of Juan Castaneda, @mathguide on Twitter, is a follow-up to a
> conversation we had at the "Math language and accessibility" Math 2.0 weekly
> meeting in July:
> http://sdmath.blogspot.com/2009/07/is-there-connection-between.html Several
> people weren't sure what Juan meant by dimensions of formulas, and wanted to
> know more. Here it is! An interesting take, for sure!
>
> A quote: "To read text, we only need the basic linear connection from each
> letter to the next one, and from each word to the next one. Any text
> document can be considered as a sequence of characters, however long it may
> be.

> So, where does this basic assumption about dimensions leave the written
> representation of mathematical expressions?"

The essential form of a program is not the linear text, but the parse
tree. You can apply the standard definitions of fractal dimension
directly to trees. I can provide a Turtle Art tree structure for the
quadratic formula or what you like. Here is one for the definition of
a parabola as the locus of a point (large points) equidistant from a
given point (colored circles) and a given line (colored lines).
Program and result images attached.

> Cheers,
> Maria Droujkova
> http://www.naturalmath.com
>
> Make math your own, to make your own math.

--
Silent Thunder (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) is my name
And Children are my nation.
The Cosmos is my dwelling place, The Truth my destination.
http://earthtreasury.org/worknet (Edward Mokurai Cherlin)

[kirby urner]

What I'm finding useful in mathematics is this concept of "namespace"
which might be explicitly tied to the notion of "language game".

Maths (British use the plural a lot more) are not monolithic.

In some of the namespaces (sandboxes) I play in, the word "dimension"
has a different meaning than in Cartesian or Euclidean namespaces.

"Fractional dimensions" as a concept is specific to some branches of
math (dynamical systems, chaos math, fractals etc.), hasn't spread to
all branches.

I like to cite 'Regular Polytopes' by the great H.S.M Coxeter, page
119, where he drives a wedge between "dimension" as meant by a
physicist doing work in special or general relativity ala Einstin,
versus "dimension" as meant by a mathematician such as himself,
working in extended Euclidean geometry.

Science fiction writers like to confuse namespaces deliberately as a
way of helping us suspend disbelief. Abbott's 'Flatland' provides a
favorite set of mind tricks of this kind, copied by countless
imitators.

In science fiction, a "tesseract" (4D cube) might double as a "time
machine" (because "time is the fourth dimension" in Minkowski's
metric). That's a deliberate mixing of two namespaces which Coxeter
pries apart in the fond hope of rescuing his geometry from the
dumbing-down effects of Hollywood screenwriting.

Kirby

[ebujak]

Kirby,

Your last post is brilliant! ... fractional dimensions, regular
polytopes, 'Flatland', ...

We can all share in the "blame" for reusing and abusing the same word
to mean different things depending on the context. We all do it. All
fields of study and employment do it regularly. :)

Is time the 4th dimension?
http://www.flickr.com/photos/ebujak/3813825162/in/set-72157621833783647/
http://science.psu.edu/alert/math10-2005.htm

On Aug 16, 7:51 pm, kirby urner <kirby.ur...@gmail.com> wrote:
> What I'm finding useful in mathematics is this concept of "namespace"
> which might be explicitly tied to the notion of "language game".
>
> Maths (British use the plural a lot more) are not monolithic.
>
> In some of the namespaces (sandboxes) I play in, the word "dimension"
> has a different meaning than in Cartesian or Euclidean namespaces.
>
> "Fractional dimensions" as a concept is specific to some branches of
> math (dynamical systems, chaos math, fractals etc.), hasn't spread to
> all branches.
>
> I like to cite 'Regular Polytopes' by the great H.S.M Coxeter, page
> 119, where he drives a wedge between "dimension" as meant by a
> physicist doing work in special or general relativity ala Einstin,
> versus "dimension" as meant by a mathematician such as himself,
> working in extended Euclidean geometry.
>
> Science fiction writers like to confuse namespaces deliberately as a
> way of helping us suspend disbelief.  Abbott's 'Flatland' provides a
> favorite set of mind tricks of this kind, copied by countless
> imitators.
>
> In science fiction, a "tesseract" (4D cube) might double as a "time
> machine" (because "time is the fourth dimension" in Minkowski's
> metric).  That's a deliberate mixing of two namespaces which Coxeter
> pries apart in the fond hope of rescuing his geometry from the
> dumbing-down effects of Hollywood screenwriting.
>
> Kirby
>

> On Sun, Aug 16, 2009 at 4:27 PM, Edward Cherlin<echer...@gmail.com> wrote:

[Edward Cherlin]

On Sun, Aug 16, 2009 at 4:51 PM, kirby urner<kirby...@gmail.com> wrote:
>
> What I'm finding useful in mathematics is this concept of "namespace"
> which might be explicitly tied to the notion of "language game".
>
> Maths (British use the plural a lot more) are not monolithic.
>
> In some of the namespaces (sandboxes) I play in, the word "dimension"
> has a different meaning than in Cartesian or Euclidean namespaces.
>
> "Fractional dimensions" as a concept is specific to some branches of
> math (dynamical systems, chaos math, fractals etc.), hasn't spread to
> all branches.

They share the concept of Hausdorff dimension in the usual treatments,

http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

but there are several other measures of dimension.

http://en.wikipedia.org/wiki/Fractal_dimension
http://en.wikipedia.org/wiki/Hausdorff_dimension
http://en.wikipedia.org/wiki/Minkowski-Bouligand_dimension
http://en.wikipedia.org/wiki/Packing_dimension
http://en.wikipedia.org/wiki/Minkowski_dimension
http://en.wikipedia.org/wiki/Lebesgue_covering_dimension

> I like to cite 'Regular Polytopes' by the great H.S.M Coxeter, page
> 119, where he drives a wedge between "dimension" as meant by a

> physicist doing work in special or general relativity ala Einstein,

[Edward Cherlin]

On Sun, Aug 16, 2009 at 5:48 PM, ebujak<Edward...@hotmail.com> wrote:
>
> Kirby,
>
> Your last post is brilliant! ... fractional dimensions, regular
> polytopes, 'Flatland', ...
>
> We can all share in the "blame" for reusing and abusing the same word
> to mean different things depending on the context.  We all do it.  All
> fields of study and employment do it regularly.  :)
>
> Is time the 4th dimension?
> http://www.flickr.com/photos/ebujak/3813825162/in/set-72157621833783647/
> http://science.psu.edu/alert/math10-2005.htm

It can be _a_ 4th dimension, if you like, but geometrically it is
quite different from spatial dimensions. Minkowski spacetime is not a
conventional 4D metric space. It uses the distance element
x^2+y^2+z^2+iw^2 or equivalently x^2+y^2+z^2-w^2. Spacelike (d>0),
lightlike (d=0), and timelike (d<0) intervals have quite different
properties.

One of the consequences of this is that as you move faster through
space, you move slower through time, so that your 4-velocity is
constant. (v^2 +(sqrt(1-v^2)^2 = 1)

[kirby urner]

Thanks for liking my post ebujak.

I've taken to using "dot notation" to signify which namespace e.g.
Coxeter.4D versus say Einstein.4D, following a convention found in
several computer languages. It's a technique for "disambiguation".

Another meaning of 4D is from this "geometry of lumps" I use when
looking at ray-tracing, computer graphics 'n stuff. In this
namespace, points, lines and planes all have the same dimensional
characteristics as polyhedra, are not infinitely small nor infinitely
anything really, are just like lumps of clay (in ray tracing, you want
light to bounce off stuff, so it makes sense not to have anything
"dimensionless" or 0D).

Dimension theorist Karl Menger wrote an essay on this although he
didn't have this meaning for 4D which comes from another source, a
somewhat esoteric geometry language in which the tetrahedron is the
icon for volume, for space. He considered his proposal
"non-Euclidean" i.e. here's another way of escaping the Euclidean
namespace other than by jiggering with the 5th postulate.

Because the tetrahedron points in four directions (going by vertices
or facets), we call that "four directional" or "four dimensional".
Picture an origin (0,0,0,0) with four radials carving all of space
into four quadrants. The four radii go to the four corners of a
regular tetrahedron and have addresses (1,0,0,0) (0,1,0,0) (0,0,1,0)
(0,0,0,1). We call these "quadray coordinates" and the thing to
remember is we're working in ordinary volume, same as XYZ does, just
calling it 4D.

In this little grammar (language game) 4D "lumps" are purely
conceptual, have yet to acquire "secondary characteristics" associated
with time and space (energy characteristics).

The spatiotemporal is considered an added dimension called "frequency"
(relates to energetic), whereas shape alone, irrespective of
"existence in the world" is characterized purely by angles
(prefrequency).

One might diagram as follows:

4D angles timeless
------------------------------------------------------------
4D+ frequency time-size

Like I said, it's esoteric and not used in many schools, but there's
plenty of resource material out there and it's fun to challenge
students to think in several different ways, to keep their minds
nimble and flexible.

Mathematics is not about learning "one right way" of talking pushing
out all the other ways, but of learning various internally consistent
(i.e. logical) systems for organizing one's generic concepts i.e. it's
not like you can't also learn the more traditional namespaces as well.

Kirby

[Juan]

Maria,

Thank you for writing a message here about my last blog post, and also
for inviting me to this group.
On whether we can rigorously formalize my idea about assigning some
sort of dimensional measure to written mathematical expressions, I am
not sure either but I think it is an interesting topic.
Glad you liked my diagram of the quadratic formula.
Juan

On Aug 16, 6:52 am, Maria Droujkova <droujk...@gmail.com> wrote:
> This essay of Juan Castaneda, @mathguide on Twitter, is a follow-up to a
> conversation we had at the "Math language and accessibility" Math 2.0 weekly
> meeting in July:http://sdmath.blogspot.com/2009/07/is-there-connection-between.html
>
>

[Maria Droujkova]

Klein bottle without self-intersections is a Euclid.4d object then, I gather?

I was playing with Origami the other day, and ran into Robert Lang's version (http://www.kleinbottle.com/Origami%20Klein%20Bottle.htm) on Clifford Stoll's site. I met Clifford this spring at the Great Circles convention at Berkeley, where he made a great appearance. I also found this Modular Origami Klein bottle version, made of 500(!!!) units by Michal Kosmulski http://hektor.umcs.lublin.pl/~mikosmul/origami/misc.html

Paper folding communities and networks are very strong on the web, which adds to the attractiveness of the area for Math 2.0. Not only is it a great pathway into relatively advanced math, but it's possible to make bridges to friendly, already-developed communities of practice, with their own social math objects. For example, Klein bottles.

So, a question to Kirby and everybody - what are different Name.Dimensions of modular Klein bottles? "Modularity" does change dimension - for programming, formula parsing, or geometry.

Cheers,
Maria Droujkova
http://www.naturalmath.com

Make math your own, to make your own math.

[Juan]

On Aug 16, 4:27 pm, Edward Cherlin <echer...@gmail.com> wrote:
>
> The essential form of a program is not the linear text, but the parse
> tree. You can apply the standard definitions of fractal dimension
> directly to trees. I can provide a Turtle Art tree structure for the
> quadratic formula or what you like.
>

Edward,

How do the standard definitions of dimension apply directly to trees?
And, to what kind of trees? A tree with a finite number of nodes is
not a fractal object. The dimension (Hausdorff or Packing) is going to
come up equal to one.
Also, the written representation of a mathematical expression is more
than its parse tree. It is the actual writing. It also has standard
dimension equal to one by the way (as a geometrical subset of the
plane). It's all made up of a finite number of lines (curves or
straight segments) of finite length, each one continuous, and smooth
except possibly at the corners (a finite number of them). Could you
please explain how the standard definitions of dimension can give any
number higher than one for the writings we do for math expressions?
Assuming the lines have no thickness. Otherwise the dimension would be
2. Or it would be any number we decide to assume the marking of the
pen on the paper has. I am going with the traditional geometrical
convention that circles, parabolas, straight lines, and graphs of
differentiable functions are objects of dimension one. That leaves any
standard writing at dimension one, not 1 < d < 2.
Oh, and, the images you attached of the parabola program look great by
the way. Thank you.

Juan

[kirby urner]

On Mon, Aug 17, 2009 at 4:25 AM, Maria Droujkova<drou...@gmail.com> wrote:
> Klein bottle without self-intersections is a Euclid.4d object then, I
> gather?
>

Yes. Or Coxeter.4d. I don't insist on using a proper name or
anything, just need a way of disambiguating concepts. Klein.4d ==
Coxeter.4d. Actually, I don't know that Euclid the guy even had our
"dimension" concept per se, might have looked at us funny if we asked
him if space was "three dimensional".

The game "three dimensional" comes from holding up a kleenex box
(e.g.) and going "height, width and depth". Then you have to pretend
(I'd say pretend to help keep the mind loose) that these dimensions
can be separated i.e. you can have width but no depth, height and
depth but no width etc. Having precisely 90 degree angles is
important somehow.

If a kid challenges the teacher and goes, what if the angle is 89
degrees, could it still be a "Y axis" relative to the "X axis" i.e.
"height versus width" (Y is conventionally "up" so "high" (sounding a
tad sarcastic here)) then the teacher has a variety of responses, most
of which involve quieting the keep and not allowing subversion.

My analysis is math comes with a "music of authority" and if you don't
like the language game they're teaching you (e.g. <<dot on the board>>
"this is a dimensionless point, except it's not *really* a point
because real points are dimensionless, meaning you'll never see them,
not even with the world's most powerful microscope"), if a kid has a
problem with that, then math become a turn-off, because a counter
discourse is not permitted (is verboten).

But then we get to "higher math" and teachers like me who go "yeah,
that 'dimensionless point' stuff is just one way of talking logically
but other imaginative humans have invented these other ways, so take
this 'geometry of lumps' for example..." But most kids don't ever get
to "higher math" like this.

> I was playing with Origami the other day, and ran into Robert Lang's version
> (http://www.kleinbottle.com/Origami%20Klein%20Bottle.htm) on Clifford
> Stoll's site. I met Clifford this spring at the Great Circles convention at
> Berkeley, where he made a great appearance. I also found this Modular
> Origami Klein bottle version, made of 500(!!!) units by Michal Kosmulski
> http://hektor.umcs.lublin.pl/~mikosmul/origami/misc.html
>

I keep encountering the Klein Bottle through the art of Paul Laffoley.
He has ways of combining great figures and ideas from the 1900s and
mixing them into surrealist "machines" or "designs". His Klein Bottle
pictures have to do with taming vegetation, perhaps Goethe's
prototypical plant (the root plant object in some evolutioniary
scheme) to make dwellings. He shows us a Klein Bottle house made out
of vegetation. Wild stuff. He's come to Portland a couple times and
given 3-5 hour lectures with slides, both of which I lapped up eagerly
and think about a lot to this day.

> Paper folding communities and networks are very strong on the web, which
> adds to the attractiveness of the area for Math 2.0. Not only is it a great
> pathway into relatively advanced math, but it's possible to make bridges to
> friendly, already-developed communities of practice, with their own social
> math objects. For example, Klein bottles.
>

Yes, there's some superstar origami guy everyone talks about and whom
I finally got to see at O'Reilly's OSCON (Tim O'Reilly himself in the
audience). Enlisting the computer has made a huge difference, to
where now they can fold absolutely amazing things from single sheets
of paper, like cuckoo clocks with elaborate detail. Amazing.

> So, a question to Kirby and everybody - what are different Name.Dimensions
> of modular Klein bottles? "Modularity" does change dimension - for
> programming, formula parsing, or geometry.
>

I don't know that namespace well enough to know if there's a "right
answer". In higher math, we design namespaces so sometimes there's no
right answer until someone develops a new branch that has one.

My philosophy blog has this meeting with me and Mario Livio ('Is God a
Mathematician?') talking about 4D vs. 4D vs. 4D, a fun way of
reminding readers about these different namespaces.

As a teacher, I come on stage with a kind of esoteric background
(Laffoley showed at these Esozone conferences, sponsored by Portland's
"techno-occult" community).

This 4D with the tetrahedron as anchoring concept (simplest enclosure,
with "spheres" relegated to "complex" because we're dealing in simple
sticks, like toothpicks, and spheres take boxes and boxes of
toothpicks, have lots of windows (holes) instead of just six
(tetrahedron has six windows)) is truly avante-gard to the point of
bizarre, but since it's also a source for geodesic domes, other
practical stuff, I'm able to "ground it" in everyday experience.

It doesn't hurt (in terms of keeping student interest) that I'm able
to call it "math for spy kids" or "spy kid" math (you've seen those
books Dragonology, Wizardology, Spyology maybe...), for reasons I
could go into. A lot of them have seen the movies e.g.
http://www.imdb.com/title/tt0227538/

Kirby

> Cheers,
> Maria Droujkova
> http://www.naturalmath.com
>
> Make math your own, to make your own math.
>

Or 'Make your own math, to make math your own."

[BRADFORD HANSEN-SMITH]

Kirby, you said

"Like I said, it's esoteric and not used in many schools, but there's plenty of resource material out there and it's fun to challenge students to think in several different ways, to keep their minds nimble and flexible."

It is not only fun, but it is essential to get them to think through and beyond what we already know to what we do not. Here is another nimbleness. Holding a paper circle, understanding it is the compression of a sphere, we can say it is a triunity showing three circles. Counting the properties there are three circles and two edges. When folded in half we count 10 half circles. A different logic than constructing from a circle image two semi circles. There is no conflict except in word definition and what is meant by one half. There are an infinite number of diameters in a circle, everyone folds a different diameter, all generalized to one line. Either way there has to be twice as many half circles as there are diameters, except when folding the circle disk in half where you get 10 half circles. (With thirty kids each folding a different diameter, there are three hundred folded half circles) Given the circle is both Whole and part you can consistently have as many half circles as you want without conflict. It is possible we accept one diameter line of symmetry and then project other symmetries and dimensions from that generalization. When you talk about 4D. and the obvious properties of 3D space, why do we assume the fourth D is projection of the same; is it the problem of being consistent to the words and the numbers?  Experiential understanding of 3D tells me that the fourth aspect is interaction, movement that takes place between the 3Ds generating 7 possible combinations: back to 10 again. Interaction (movement) is not time & space bound, but uses both for forming expression.  This brings us back to the terahedron, a spherical patttern of 10. Through movement in multiples the tetrahedron is the pattern base for the other regular polyhedra. This is demonstratedable, yet seems in conflict with classifying regualr solids having different number properties. Interaction, experience,  is often at odds with closed loop static concepts. Logic seems to be the currency of the day; value being relative depending on what country you visit. Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com --- On Mon, 8/17/09, kirby urner <kirby...@gmail.com> wrote:

[kirby urner]

In my language game with 4D (as distinct from the Coxeter or Einstein
ones), not invented by me (when I say "my" it's like my car in the
parking lot, which I didn't design or build, just drive around (Chevy
Malibu)) we don't say space is 3D.

We say the tetrahedron is the most primitive topological "bird cage"
(might hold a bird, has an inside, like Tweety Bird the canary).

But the tetrahedron looks like an arrow head and has four point, four
faces. Since we don't preach about heightwidthdepth as three
separations of the integral (Kantian?) experience of space, we don't
have that "threeness" except the tetrahedron has 6 edges i.e. two
3-vector zig-zags which might be used to calibrate the interior (XYZ
style).

So conceptual space (ordinary XYZ space, the space of ordinary
polyhedra) is 4D not 3D. I explain this is a different namespace,
pioneered by a Medal of Freedom winner with 42 college degrees, world
famous, yet studying his math is considered 'verboten' in North
American universities, under the thumb of unimaginative dweebs <<laugh
track, children laughing loudly hearing a teacher talk this way>>.

Except conceptual space has no energy investments i.e. there's a
difference between a conceptual 4d cube and one build out of metal,
modeled in cardboard, marble, used for a shack. All these are *added*
dimensions having to do with energy, which is where the concept of
Frequency comes in.

Kirby

[Edward Cherlin]

On Mon, Aug 17, 2009 at 5:38 AM, Juan<here...@gmail.com> wrote:
>
> On Aug 16, 4:27 pm, Edward Cherlin <echer...@gmail.com> wrote:
>>
>> The essential form of a program is not the linear text, but the parse
>> tree. You can apply the standard definitions of fractal dimension
>> directly to trees. I can provide a Turtle Art tree structure for the
>> quadratic formula or what you like.
>
> Edward,
>
> How do the standard definitions of dimension apply directly to trees?
> And, to what kind of trees? A tree with a finite number of nodes is
> not a fractal object. The dimension (Hausdorff or Packing) is going to
> come up equal to one.

Look at the finite branching structures at

http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

where cauliflower, for example, is listed with dimension 2.33= (ln 13)/ln 3.

The idea is to look at the branching ratio, and then compute the
dimension of an infinite fractal with the same branching ratio, such
as the universal covering space of the original structure.

> Also, the written representation of a mathematical expression is more
> than its parse tree. It is the actual writing. It also has standard
> dimension equal to one by the way (as a geometrical subset of the
> plane).

It has dimension 0 as a finite set with the discrete topology, and up
to 2 when you include matrices, subscripts and superscripts, and
'expression \over expression' structure, again assuming infinite
extension. It would be trivial to extend to arbitrarily high dimension
if we could comprehend such structures. For example, in the J
programing language, we can say

2 3 4 $ i. 24

and have it displayed as two planes of three rows and four columns.

0 1 2 3
4 5 6 7
8 9 10 11

12 13 14 15
16 17 18 19
20 21 22 23

This is only because we use planar displays. With a 3D display, we
could line up the numbers in space.

> It's all made up of a finite number of lines (curves or
> straight segments) of finite length, each one continuous, and smooth
> except possibly at the corners (a finite number of them). Could you
> please explain how the standard definitions of dimension can give any
> number higher than one for the writings we do for math expressions?
> Assuming the lines have no thickness. Otherwise the dimension would be
> 2. Or it would be any number we decide to assume the marking of the
> pen on the paper has. I am going with the traditional geometrical
> convention that circles, parabolas, straight lines, and graphs of
> differentiable functions are objects of dimension one.

And thus ruling out fractals entirely a priori. I am going with the
later tradition (Borel, Lebesgue, Hausdorff, Besicovich, Minkowski,
Bouligand, Kolmogorov, Julia, Fatou, von Koch, Sierpinski, Renyi,
Mandelbrot, Penrose, Conway...)

> That leaves any
> standard writing at dimension one, not  1 < d < 2.

Again, see the fractal examples page, where space-filling curves have
dimension 2, and any dimension between 1 and 2 is possible in the
plane. Curves can have full measure in any finite-dimensional space.
There are space-filling curves in Hilbert space with infinite fractal
dimension.

> Oh, and, the images you attached of the parabola program look great by
> the way. Thank you.

The point was the branching and subroutine structure.

Arithmetic is conventionally represented with a branching factor of 2
between 0D nodes (scalars). In APL-like languages, it is possible to
have higher dimension nodes (arrays), and a higher branching factor
(_nested_ arrays and arrays of functions). My father, an actuary,
routinely worked with 6D mortality tables. Lisp-like languages can
also have arbitrarily large branching factors. In more conventional
programming languages, one can have higher branching factors using
case statements.

> Juan

[Juan]

Dear Edward, please read below:

On Aug 17, 11:13 am, Edward Cherlin <echer...@gmail.com> wrote:

> On Mon, Aug 17, 2009 at 5:38 AM, Juan<herest...@gmail.com> wrote:
>
> > On Aug 16, 4:27 pm, Edward Cherlin <echer...@gmail.com> wrote:
>
> >> You can apply the standard definitions of fractal dimension
> >> directly to trees.
>

> > Edward,
>
> > How do the standard definitions of dimension apply directly to trees?
> > And, to what kind of trees? A tree with a finite number of nodes is
> > not a fractal object. The dimension (Hausdorff or Packing) is going to
> > come up equal to one.
>
> Look at the finite branching structures at
>
> http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
>
> where cauliflower, for example, is listed with dimension 2.33= (ln 13)/ln 3.
>
> The idea is to look at the branching ratio, and then compute the
> dimension of an infinite fractal with the same branching ratio, such
> as the universal covering space of the original structure.
>

That is an idea. Now, just to clarify:

1) In the example of the cauliflower, the tree is very regular. As
they say:
"Every branch carries around 13 branches 3 times smaller."
That is hardly the case for the parse tree of a mathematical
expression; most come as irregular trees. What would the "branching
ratio" be? It changes from one node to the next.

2) Notice the part where they specify "3 times smaller." Taking the
size of each symbol into consideration is going to be important.
However, that information is in the writing of the math expression,
not in its parse tree.

3) There are many math expressions with the same parse tree structure
but they look nothing similar to each other written out on paper (or
displayed on the screen). For example: (a+b)/c, on one hand; and ab+c,
on the other. They both have the same parse tree structure: the root
node connects to two nodes, the second of those is terminal, while the
first connects to two terminal ones. However, the usual fraction form
of (a+b)/c, with the sum up there in the numerator, then the
horizontal fraction bar in the middle, and the variable c down there
in the denominator, it looks to me "more 2-dimensional" (whatever that
means, yet to be determined) than ab+c, which is usually written out
as a line of plain text. So, while it might make sense for some
purposes to define a measure on math expressions based only on their
parse trees, that would not be what I have in mind when I talk about
their "dimension," because I want to assign a "dimensionality measure"
of 1 to the expression ab+c, but I want to assign some number bigger
than 1 to the expression (a+b)/c [when written with a horizontal
fraction bar].

4) By the way, I have to come up with a much better name than
"dimension" because while my concept is somehow related to the
Hausdorff dimension, it is not it. At least it is by no means apparent
to me at this point that it should be the same. So, borrowing from
Kirby's terminology, I hereby make myself into a namespace, and will
write juan.dimension to refer to my concept, while using the word
"dimension" in the very specific sense of Hausdorff dimension.

5) One way to get around the possible (and statistically predominant)
irregularity of the parse trees would be to replace each terminal node
with a copy of the whole tree, ad infinitum, thus generating a fractal
object, as you propose. You would still need to choose a size ratio
for the successive, smaller copies. While this is certainly possible,
I am not sure it fits my interest in the handwriting of math
expressions because, from my perspective it looks unnecessarily
abstract, and somehow arbitrary.

6) It might work better taking the tree-to-handwriting-analogy a
little bit literally. It would amount, in the case of the quadratic
formula, to replacing every occurrence of a, b, and c, by a whole new
(smaller) copy of the quadratic formula itself, over and over, from 1
to infinity. That would generate a fractal object with a tree-like
structure but it would not be a tree because it would not be
connected. Generating this fractal object based on the whole, original
handwritten formula, captures much more of it than doing it from its
parse tree. I will think about this more. Thank you for your
suggestion.

> > Also, the written representation of a mathematical expression is more
> > than its parse tree. It is the actual writing. It also has standard
> > dimension equal to one by the way (as a geometrical subset of the
> > plane).
>

> It has dimension 0 as a finite set with the discrete topology,

No, I am not talking about the discrete topology.

> and up
> to 2 when you include matrices, subscripts and superscripts, and
> 'expression \over expression' structure, again assuming infinite
> extension.

I see in the previous phrase that you totally get my intuitive concept
(which is obviously intuitive for you, too). My point is that such
concept is not "visible" through a direct, verbatim application of the
Hausdorff dimension concept to the set of points in the plane that
made up the written representation of the math expression in question.
Not unless you use such a set to generate a fractal object, and then
that fractal can have a Hausdorff dimension bigger than 1 but not so
the original expression.

> It would be trivial to extend to arbitrarily high dimension
> if we could comprehend such structures. For example, in the J
> programing language, we can say
>
> 2 3 4 $ i. 24
>
> and have it displayed as two planes of three rows and four columns.
>
> 0   1   2   3
> 4   5   6   7
> 8   9  10  11
>
> 12 13 14 15
> 16 17 18 19
> 20 21 22 23
>
> This is only because we use planar displays. With a 3D display, we
> could line up the numbers in space.
>

Yes. I am talking about planar displays, nothing else.
And, no, the lines have still Hausdorff dimension one even when we
write matrices. We sometimes think of matrices as "two-dimensional"
but that is just the juan.dimension concept I am talking about. Even
though we fill up a rectangular grid with numbers, each numeral is
made up of a few smooth lines, and smooth lines have Hausdorff
dimension one. So the whole written representation of the matrix is a
finite set of smooth lines in the plane, and therefore it has
Hausdorff dimension one. Here the keywords are "smooth" (infinitely
differentiable), and "finite" (no aggregating of an infinite number of
segments inside a finite area).

> > It's all made up of a finite number of lines (curves or
> > straight segments) of finite length, each one continuous, and smooth
> > except possibly at the corners (a finite number of them). Could you
> > please explain how the standard definitions of dimension can give any
> > number higher than one for the writings we do for math expressions?
> > Assuming the lines have no thickness. Otherwise the dimension would be
> > 2. Or it would be any number we decide to assume the marking of the
> > pen on the paper has. I am going with the traditional geometrical
> > convention that circles, parabolas, straight lines, and graphs of
> > differentiable functions are objects of dimension one.
>

> And thus ruling out fractals entirely a priori. I am going with the
> later tradition (Borel, Lebesgue, Hausdorff, Besicovich, Minkowski,
> Bouligand, Kolmogorov, Julia, Fatou, von Koch, Sierpinski, Renyi,
> Mandelbrot, Penrose, Conway...)
>

I am not ruling out fractals. Fractals do exist. But they each have
their very specific definition. And the fact of the matter is that
math formulas people commonly write on flat displays (including
formulas that describe or define fractals) are not fractal objects. I
do not think any of the mathematicians you list above would say that
the quadratic formula (considered as the set of points in the plane we
highlight to make it look like the quadratic formula), none of them
would say such a set of points constitutes a fractal object. Again, I
am not ruling out fractals but I am not mistaking non-fractal objects
for fractal ones.

> > That leaves any
> > standard writing at dimension one, not  1 < d < 2.
>
> Again, see the fractal examples page, where space-filling curves have
> dimension 2, and any dimension between 1 and 2 is possible in the
> plane. Curves can have full measure in any finite-dimensional space.
> There are space-filling curves in Hilbert space with infinite fractal
> dimension.
>

Yes, you are right. Curves of every dimension between 1 and 2 are
possible in the plane but again, the curves we use to write
mathematical expressions are smooth, infinitely differentiable, not
infinitely squiggy. I am talking about how people write in reality.
Such lines are not fractals, are they? The only reason I use the word
"fractal" in my article is because I am asking about some sort of
measure for the written form of math expressions, a measure that
somehow may have an intuitive relation to the formal concept of
dimension but with the goal that some expressions would have non-
integer numbers assigned to them by that measure. I used the word
"fractal" to mean non-integer but I am not pretending that handwriting
is fractal. I am not interested in talking about fractals per se just
because they exist, and they are a huge, active field of mathematical
research, no. The point of fractals, for me, in relation to
handwriting, is that they show many examples of non-integer values of
Hausdorff dimension. Do we agree, or do we not, in that the vast
majority of human handwriting cannot be formally considered as fractal
objects, fractal subsets of the plane? Because that is my main
interest in the article: handwriting. Both "fractals," and "dimension"
come as analogies for an idea in development, the idea of thinking
about some math handwriting as linear, and some other math handwriting
as "more two-dimensional," intuitively, whatever that may end up
formally meaning. The analogies themselves, whether appealing, or
helpful, or not; do not make a non-fractal object into a fractal one,
do they?

> > Oh, and, the images you attached of the parabola program look great by
> > the way. Thank you.
>
> The point was the branching and subroutine structure.
>
> Arithmetic is conventionally represented with a branching factor of 2
> between 0D nodes (scalars). In APL-like languages, it is possible to
> have higher dimension nodes (arrays), and a higher branching factor
> (_nested_ arrays and arrays of functions). My father, an actuary,
> routinely worked with 6D mortality tables. Lisp-like languages can
> also have arbitrarily large branching factors. In more conventional
> programming languages, one can have higher branching factors using
> case statements.
>

We are O.K. with the branching structure, and the branching factors,
too. As you remarked before, we usually use flat displays, so, likely,
we will not need to go above two dimensions in our considerations.

Thanks again for your valuable comments.
Juan

[Edward Cherlin]

We appear to be in violent agreement. ^_^ It's a pleasure doing math with you.

Please note that the theory of fractals includes structures that do
not have constant branching patterns. I will not go into detail here,
since it is easy to find. Suffice it to say that there are several
possibilities, and in some of them an object can have a range of
dimensions. None of these theories represent "the truth" about
particular structures, just different points of view. It is possible
to treat math formulas in many of those points of view.

I am interested in some of them that apparently do not suit your
interests. That's fine with me. Where would math be if we all had to
do it the same way?

[Juan]

On Aug 19, 12:54 pm, Edward Cherlin <echer...@gmail.com> wrote:
> We appear to be in violent agreement. ^_^ It's a pleasure doing math with you.
>

Dear Edward,

It is pleasure doing math with you, too.

> Please note that the theory of fractals includes structures that do
> not have constant branching patterns. I will not go into detail here,
> since it is easy to find.

I believe you. I bet there are plenty of those in the fractal world.
Not surprising. Are there some standard ways to generate a fractal
tree from any given rooted tree?

> Suffice it to say that there are several
> possibilities, and in some of them an object can have a range of
> dimensions. None of these theories represent "the truth" about
> particular structures, just different points of view. It is possible
> to treat math formulas in many of those points of view.
>

Yes, I agree on the multiplicity of possibilities. In my article I am
interested in a very particular, very specific point of view for
looking at math expressions in a certain way. However, at this point I
do not have a way of formally describing it. Since I do not know how
to explain it, in a sense you could say I do not know what I am
talking about. However, I have some intuition about it, some sort of a
gut feeling. From this intuition, it is much easier to tell what is
not it than it is to tell what it is, if that makes sense. It is kind
of having a mental image of a house and its surroundings but not the
address. Since I do not know how to get there, it seems like I am
lost. Of course there are many other houses in the same city, and many
ways to go in different directions. Still I have my idea, and (at the
risk of being repetitive) this idea assigns a juan.dimension > 1 to
the expression (a+b)/c, while assigning a juan.dimension = 1 to the
expression ab+c, even when both expressions have an equivalent parse
tree. That is why I know my idea has nothing to do with any measure
that uses only the parse tree of a math expression as the starting
point.
Please do not get me wrong. I hold no pretense of talking about "the
true dimension" of a math formula. I do not even believe there is such
a thing. So, I agree with you in what you said above.
Now, thank you very much for mentioning matrices (in passing, in one
paragraph of a previous post). I had not even considered matrices
until I read that part of your comment. After thinking about it for a
while, it became clear to me that this cloudy idea of "dimension" that
I have, assigns a juan.dimension = 2 to all mxn matrices, as long as
both m and n are greater than 1. However, I have a juan.dimension < 2
for the quadratic formula. So, there is something special about
matrices in this nebulous idea of mine, and having that specific
reference point might help to clarify the muddy waters to some extent.

> I am interested in some of them that apparently do not suit your
> interests. That's fine with me. Where would math be if we all had to
> do it the same way?
>

Hey, that is fine with me, too. Please feel free to elaborate on the
details of those possibilities you are interested in when considering
a "dimensional measure" for math expressions. So far I only have a
vague idea about one of the possibilities you mention, that is the one
of taking the parse tree of a math formula, and using it to generate a
fractal tree in some way, and then assign to the math formula the
Hausdorff dimension of the fractal tree generated from the parse tree.
Did I get it right?

Thanks again for your feedback.
Juan

[Maria Droujkova]

On Fri, Aug 21, 2009 at 7:20 AM, Juan <here...@gmail.com> wrote:

Please do not get me wrong. I hold no pretense of talking about "the
true dimension" of a math formula. I do not even believe there is such
a thing. So, I agree with you in what you said above.
Now, thank you very much for mentioning matrices (in passing, in one
paragraph of a previous post). I had not even considered matrices
until I read that part of your comment. After thinking about it for a
while, it became clear to me that this cloudy idea of "dimension" that
I have, assigns a juan.dimension = 2 to all mxn matrices, as long as
both m and n are greater than 1. However, I have a juan.dimension < 2
for the quadratic formula. So, there is something special about
matrices in this nebulous idea of mine, and having that specific
reference point might help to clarify the muddy waters to some extent.

Juan,

I see juan.dimension as a psychological or maybe pedagogical idea. It describes how people perceive formulas. From that point of view, a matrix, or a grid, to use John Mason's and other British researchers' word for potentially infinite cell structures, seems two-dimensional because it strongly underlies the idea of co-variation between two variables or array members. However, a grid may have more "fractality" to it, for example, if you fill each cell with quadratic formulas depending on row/column parameters.

I hope ideas that include both math concepts and pedagogy, such as co-variation, can help to formalize dimension.

Another take, more esoteric, would be to look at how human eyes track different math entities, or even which areas of the brain (visual vs. textual) react - which is dicey, because it changes between novices and experts.



Cheers,
Maria Droujkova

Make math your own, to make your own math.

http://www.naturalmath.com social math site

[Juan]

Maria,

Thank you for referring to the juan.dimension idea as a pedagogical or
psychological one. It helps putting it in perspective, and it gives a
good context to work on it.
As far as a pedagogical tool, I have only shown the parse trees of
math expressions to a few students who find it hard applying the
PEMDAS rules because they somehow have a strong tendency to want to
evaluate math expressions by going always symbol by symbol in a left-
to-right order.
At this point the idea of juan.dimension is definitely a psychological
one. I am not sure if I can formalize it but I think matrices can
help. You are right, matrices with math formulas in them may have a
juan.dimension < 2. I think what you refer to as "fractality" is the
same concept.
One question: the concept of "co-variation" you mention, is it related
to statistical covariance, and correlation? Or is it a separate
concept that arises in a field other than statistics?
I do not know how far the following ideas can go into formalizing the
concept of "fractality" or juan.dimension for written math expressions
but here they are, for whatever they are worth:
Starting with a written math expression, we can consider each one of
its symbols, individually.
Then we can consider, for each symbol, the smallest closed box
{a<=x<=b; c<=y<=d} that contains that symbol.
Next, we project each box onto the X axis, and also onto the Y axis.
In this way each individual symbol gets associated with two intervals,
a horizontal one, and a vertical one. We can think of these intervals
as the "horizontal range," and the "vertical range" of the symbol.
Next, we define three matrices:
Matrix A has the information from the parse tree of the math
expression. This is independent of the written representation, and it
may introduce items for operations that lack a symbol in the written
form of the expression, like multiplication in ab+c. We can say the
entry (i,j) for symbols i and j equals -1, 0, or 1 depending on
whether there is a direct dependency between the symbols. For example,
in ab+c, the entry for (+,c) would be 1, the entry for (c,+) would be
-1, and the entry for (a,b) would be 0, while that for (a,*) would be
-1.
Matrix B has the information on the intersections of the vertical
intervals. This matrix does not have extra items for any operations
not shown in writing. It only has entries for the vertical ranges of
the written symbols themselves, not for any implied operation. We
could make it a binary matrix, but it would probably be a good idea to
consider not only empty/non-empty intersections but whether or not an
interval is wholly contained in another one, as the vertical range of
a "minus" sign is contained in the vertical range of the letter to the
right of it. The information in this matrix may not be quite as useful
for all symbols. For example, the symbol for an integral has a long
vertical range, encompassing nearly all symbols to the right of it,
except maybe for the upper and lower limits, or for exponents that
might apply to the whole result of the integral.
Matrix C has the information on the intersections (and/or subset) of
the horizontal intervals.
We can look at expressions that are typically written out in a line of
text, and note that (for the symbols actually written out) all their
direct parse-tree-connections are a subset of the connections given by
the intersection matrix of their vertical ranges.
Then we can look at how these three matrices behave/interact in the
case of plain numerical matrices, and then start looking at other
examples of expressions like the type of algebraic fractions found in
rational functions, to see how much their 3 matrices resemble those of
typographically linear expressions, or those of numerical matrices,
and in which ways they are different.
I have nothing concrete yet but I think this direction looks
promising.

Have a good day.
Juan

[Edward Cherlin]

On Fri, Aug 21, 2009 at 4:20 AM, Juan<here...@gmail.com> wrote:
>
> On Aug 19, 12:54 pm, Edward Cherlin <echer...@gmail.com> wrote:
>> We appear to be in violent agreement. ^_^ It's a pleasure doing math with you.
>
> Dear Edward,
>
> It is pleasure doing math with you, too.
>
>> Please note that the theory of fractals includes structures that do
>> not have constant branching patterns. I will not go into detail here,
>> since it is easy to find.
>
> I believe you. I bet there are plenty of those in the fractal world.
> Not surprising. Are there some standard ways to generate a fractal
> tree from any given rooted tree?

Indeed. The simplest is to replicate the entire tree at each leaf
node, but this is math. You are allowed to generalize in any way you
like, as long as it results in interesting problems or insights. For
example, you could replicate the tree at _every_ node, leaf or not.
You can replicate the tree more than once at a node. You can view
these trees in several different topologies, including embeddings into
metric spaces.

For structures containing loops, the universal covering space is
frequently of great interest. You could loop from a leaf to the root
of the tree, rather than making copies, or from a leaf to somewhere
else in the tree. This leads into the theory of recursive functions,
including mutual recursion.

>> Suffice it to say that there are several
>> possibilities, and in some of them an object can have a range of
>> dimensions. None of these theories represent "the truth" about
>> particular structures, just different points of view. It is possible
>> to treat math formulas in many of those points of view.
>
> Yes, I agree on the multiplicity of possibilities. In my article I am
> interested in a very particular, very specific point of view for
> looking at math expressions in a certain way. However, at this point I
> do not have a way of formally describing it. Since I do not know how
> to explain it, in a sense you could say I do not know what I am
> talking about. However, I have some intuition about it, some sort of a
> gut feeling. From this intuition, it is much easier to tell what is
> not it than it is to tell what it is, if that makes sense. It is kind
> of having a mental image of a house and its surroundings but not the
> address. Since I do not know how to get there, it seems like I am
> lost. Of course there are many other houses in the same city, and many
> ways to go in different directions. Still I have my idea, and (at the
> risk of being repetitive) this idea assigns a juan.dimension > 1 to
> the expression (a+b)/c, while assigning a juan.dimension = 1 to the
> expression ab+c, even when both expressions have an equivalent parse
> tree. That is why I know my idea has nothing to do with any measure
> that uses only the parse tree of a math expression as the starting
> point.

You are exploring a multi-dimensional space of ideas. It is usual to
feel a bit lost at first. For example, not long ago string theorists
in physics worked out that the space of possible string theories could
be described with 208 dimensions, and we have tools to examine only
tiny corners of this space. %-[

> Please do not get me wrong. I hold no pretense of talking about "the
> true dimension" of a math formula. I do not even believe there is such
> a thing. So, I agree with you in what you said above.
> Now, thank you very much for mentioning matrices (in passing, in one
> paragraph of a previous post). I had not even considered matrices
> until I read that part of your comment. After thinking about it for a
> while, it became clear to me that this cloudy idea of "dimension" that
> I have, assigns a juan.dimension = 2 to all mxn matrices, as long as
> both m and n are greater than 1. However, I have a juan.dimension < 2
> for the quadratic formula. So, there is something special about
> matrices in this nebulous idea of mine, and having that specific
> reference point might help to clarify the muddy waters to some extent.

Again, there are other options. In APL-like programming languages, it
is very important that one can have arrays dimensions of lengths 1 or
0. It is also important to be able to convert singletons of length 1
in all dimensions to scalars of dimension 0. The shape of an array is
always an array of dimension 1, which can have 0 elements in the case
of a scalar. YMMV.

>> I am interested in some of them that apparently do not suit your
>> interests. That's fine with me. Where would math be if we all had to
>> do it the same way?
>>
>
> Hey, that is fine with me, too. Please feel free to elaborate on the
> details of those possibilities you are interested in when considering
> a "dimensional measure" for math expressions. So far I only have a
> vague idea about one of the possibilities you mention, that is the one
> of taking the parse tree of a math formula, and using it to generate a
> fractal tree in some way, and then assign to the math formula the
> Hausdorff dimension of the fractal tree generated from the parse tree.
> Did I get it right?

Exactly.

> Thanks again for your feedback.

A pleasure, as always.

> Juan
> >
>



--
Edward Mokurai Cherlin
Silent Thunder (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) is my name, and
Children are
my nation. The Cosmos is my dwelling place, the Truth my destination.
http://earthtreasury.org/

[Edward Cherlin]

On Tue, Aug 25, 2009 at 10:36 PM, Juan<here...@gmail.com> wrote:
>
> Maria,

>
> I do not know how far the following ideas can go into formalizing the
> concept of "fractality" or juan.dimension for written math expressions
> but here they are, for whatever they are worth:
> Starting with a written math expression, we can consider each one of
> its symbols, individually.
> Then we can consider, for each symbol, the smallest closed box
> {a<=x<=b; c<=y<=d} that contains that symbol.
> Next, we project each box onto the X axis, and also onto the Y axis.
> In this way each individual symbol gets associated with two intervals,
> a horizontal one, and a vertical one. We can think of these intervals
> as the "horizontal range," and the "vertical range" of the symbol.

This seems to be very close to the internal model of formulas in TeX,
using nested hboxes and vboxes. Built-up forms, including matrices,
integrals, and the horizontal line for division, have boxes attached
or embedded for subsidiary expressions.

> Next, we define three matrices:
> Matrix A has the information from the parse tree of the math
> expression.

> Matrix B has the information on the intersections of the vertical
> intervals.

> Matrix C has the information on the intersections (and/or subset) of
> the horizontal intervals.

I don't understand where intersections would occur. The model I am
using is strictly nested.

> We can look at expressions that are typically written out in a line of
> text, and note that (for the symbols actually written out) all their
> direct parse-tree-connections are a subset of the connections given by
> the intersection matrix of their vertical ranges.

Now I am confused about whether we are talking about text, as in
programming or TeX, or 2D math formulas.

--
Edward Mokurai Cherlin
Silent Thunder (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) is my name, and
Children are

[kirby urner]

On Fri, Aug 28, 2009 at 10:45 AM, Edward Cherlin<eche...@gmail.com> wrote:

<< SNIP >>

> You are exploring a multi-dimensional space of ideas. It is usual to
> feel a bit lost at first. For example, not long ago string theorists
> in physics worked out that the space of possible string theories could
> be described with 208 dimensions, and we have tools to examine only
> tiny corners of this space. %-[
>

<< >>

>

> Again, there are other options. In APL-like programming languages, it
> is very important that one can have arrays dimensions of lengths 1 or
> 0. It is also important to be able to convert singletons of length 1
> in all dimensions to scalars of dimension 0. The shape of an array is
> always an array of dimension 1, which can have 0 elements in the case
> of a scalar. YMMV.
>

In my curriculum writing, the J language (by Iversons, Hui) is great
for doing "multi-dimensional" in an intuitive "data store" sense that
dispenses with any need to "visualize" a fourth orthogonal.

Once you get the the cube (XYZ), you just have an "row of cubes" (next
dimension), "square of cubes" (next dimension), "cube of cubes" (next
dimension) and so on. Nothing crazy-making like in Coxeter's 'Regular
Polytopes' where he implies we should doff our hats to weird shaman
types who know what "fourth orthogonal" means without blowing smoke.

That was a trap, a pitfall, the way we tell it today (snared a lot of
otherwise intelligent, functional human beings).

Kirby

[Juan]

On Aug 28, 10:54 am, Edward Cherlin <echer...@gmail.com> wrote:

>
> This seems to be very close to the internal model of formulas in TeX,
> using nested hboxes and vboxes. Built-up forms, including matrices,
> integrals, and the horizontal line for division, have boxes attached
> or embedded for subsidiary expressions.

It is very likely. I do not know TeX but I have seen those container
boxes of different sizes in some Math editors.

>
> I don't understand where intersections would occur. The model I am
> using is strictly nested.
>

Intersections occur, for example:
1) In a summation formula, let's say, the sum of the first n odd
numbers equals the square of n. The summation is indicated by the
uppercase Greek letter Sigma; the stopping value of the summation
index is represented by the letter n. I know there are other options
but I always place the n on top of the Sigma; and the initial value of
the index "i = 1" below the Sigma. So, when assigning boxes to each
symbol, there we have five symbols: n, Sigma, i, =, and 1. The
horizontal range of the Sigma is an interval that contains the
horizontal ranges of all the other four symbols, at least in the way I
write that expression. I have seen some books, and some professors who
write it differently, with the n and the "i = 1" kind of "hanging to
the right" of the Sigma, not directly above or below it. In the case
of those writings, the intersections would be empty. Yes, a math
editor handles the n, i, =, and 1 as sub-boxes nested in the big box
of the whole thing but I am not considering that model at all. In the
boxes I am considering, the box that encloses the Sigma does not
contain any of the other symbols, only its projection on the X axis
may or may not contain the projections of the boxes that contain n, or
i, =, or 1; depending on how the expression is actually written out.

2) In the quadratic formula, the square root symbol goes over the
whole discriminant, b^2 - 4ac. In the usual math editor model (TeX ?)
there is a box for the square root, and the box for the subtraction is
nested within the box for the square root. In that model the nested
boxes tend to follow the hierarchy of the parse tree whenever
possible. In my model, the fact that the box of the square root
contains the boxes of all the symbols in the discriminant is -while
not quite a "coincidence"- just a consequence of the way the symbols
get written out but definitely not a necessary part of the model's
design. So, instead of talking about nested boxes, I only talk about
intersections of the boxes' vertical and horizontal ranges.

>
> Now I am confused about whether we are talking about text, as in
> programming or TeX, or 2D math formulas.
>

O.K., well, I am talking about 2D math formulas in the sense of say,
you attend a math class where the professor is writing formulas on the
blackboard, then, when the blackboard is full, and before the
professor erases it, you take a picture of it. That kind of math
expressions are the ones I am talking about. At least in principle,
you can consider them independently of any computer language, or any
digital representation. You just look at them on a board, a piece of
paper, a page in a book, and you idealize the lines as smooth curves
in the XY plane.

There is always a space in between two successive letters of a typed
word. Now I am talking about typed text, by a printer, or a
typewriter. So the container boxes need to include a little white
space frame around the symbol. I was thinking that they would be the
minimum box containing the symbol but no. Considering typed text
having juan.dimension = 1, then we have to assume a standard minimum
"buffer distance" between two contiguous symbols, and add half of that
distance to each side around each box.
Once we do that, and assuming a rational width, and rational height
for all the boxes, we can look for a "common denominator" length, that
will evenly divide the heights and the widths of all the boxes from
all the symbols. In other words, we can assume there is a maximum
tiling square that can fill up the box of every single symbol, by
tiling them all with that same little basic square.
So, with this new tool, of that basic "unit" square tiling all the
boxes, probably we can get a way to define juan.dimension without
looking at the intersections, not even at the parse tree. The minimum
box that contains the whole expression (with its proper "buffer
distance" around) will be entirely subdivided into basic squares all
of the same length. Each symbol with its buffer distance will be
contained in a box, and that box will contain an integer number of
basic squares, as tiles, neatly arranged in rows and columns within
that box.
The next step is making the whole thing into a black-and-white
subdivision. Each individual container box (one per symbol) gets
painted black. All the remaining white space left in the big box that
contains the whole expression is painted white. Now the little basic
squares give us a shape made up of black squares of the same size.
Doing the above process with a line of text gives us pretty much a
long, horizontal rectangle, all black. Solid black. For this we only
need to make sure we standardize the height of the boxes so all
letters in the same type-font-size have the same height. Well, the
spaces interrupt the line, so the rectangle is not a single rectangle
but a sequence of horizontal, long rectangles of variable length. A
single word gives a solid rectangle.
The next step is to tile the whole XY plane with an infinite number of
copies of the big rectangle, the one containing the whole expression.
We just repeat it, one copy next to the other, to the right, left,
above, and below.
Next, we count how many copies (the minimum number) we need
horizontally, and how many copies we need vertically, so that they
make a perfect square. Then we replicate, with the bigger squares, the
tiling pattern of the small, basic squares. We put "big-squares-with-
some-black-in-them" whenever we had a tiny little basic square colored
black in the original structure; and we leave completely white the big
squares that correspond to white spaces from the original expression.
In this way we build a larger "copy" of the original expression but
with some "fractality" already built-in. Then we repeat the process to
the next, bigger, scale, and so on and on ad infinitum. At the
beginning there is no evident fractal structure zooming into the
formula but it becomes apparent when zooming out.
At this point we can start counting black area and white area
contained within a big area of standard shape around the origin,
whether squares or circles. Chances are there will be a limit value
for some ratio when the appropriate parameter goes to infinity.

I apologize for the coarse description. I know it may be considered
imprecise at this point but I think it is a very close approximation
to what I was looking for. Now I only need to focus on what exact
ratio are we talking about so that lines of plain text will result in
a number very close to one, while numerical matrices will result in a
number closer to two. If need be, we could define the character blank
[space] as a solid "letter," and fill in "its box" with black. this is
sort of artificial but in that way we would get exactly 1 for any line
of plain text. I do not know how that convention would affect math
expressions but we'll see. I will try to come up with an example, and
a picture of it. It will be much more clear (or somewhat clear) that
way.

O.K., sorry for the delay in responding. Thank you very much for your
feedback, your questions, and for the answers to my questions you
provided in your previous post.

Juan

[Juan]

On Aug 28, 12:08 pm, kirby urner <kirby.ur...@gmail.com> wrote:

>  Nothing crazy-making like in Coxeter's 'Regular
> Polytopes' where he implies we should doff our hats to weird shaman
> types who know what "fourth orthogonal" means without blowing smoke.
>
> That was a trap, a pitfall, the way we tell it today (snared a lot of
> otherwise intelligent, functional human beings).
>
> Kirby

Kirby,

Please excuse my ignorance but I am afraid I do not have the slightest
idea of what you are talking about here, with terms like "crazy-
making," ""should," "doff our hats," "shaman types," "trap,"
"pitfall," "the way we tell it today," "snared" and such. I am really
sorry but really, all those terms do not make any sense to me. I am at
a loss trying to figure out what could you possible mean by all that.
With all due respect, it sounds almost like the discourse of some
opposition party in some third world country run by a dictator. I
understand you are passionate about this topic, and I respect that. I
do not want to get into any controversy, however, just out of respect
for you, and your ideas, I have to ask for some clarification because
I may very well lack the necessary background to be able to understand
your perspective, and what is the issue, let alone taking any side, or
opinion about it because I am drawing a blank here. It is obvious to
me you are saying something that is important to you. I just cannot
figure out what it is. Thank you for your patience, and understanding.

Juan

[Maria Droujkova]

On Sun, Sep 6, 2009 at 4:58 AM, Juan <here...@gmail.com> wrote:

On Aug 28, 12:08 pm, kirby urner <kirby.ur...@gmail.com> wrote:

>  Nothing crazy-making like in Coxeter's 'Regular
> Polytopes' where he implies we should doff our hats to weird shaman
> types who know what "fourth orthogonal" means without blowing smoke.
>
> That was a trap, a pitfall, the way we tell it today (snared a lot of
> otherwise intelligent, functional human beings).
>
> Kirby

Kirby,

Please excuse my ignorance but I am afraid I do not have the slightest
idea of what you are talking about here, with terms like "crazy-
making," ""should," "doff our hats," "shaman types," "trap,"
"pitfall," "the way we tell it today," "snared" and such. I am really
sorry but really, all those terms do not make any sense to me.

I mostly think of it as poetry, in the sense of "beauty over utility" values. Here is what I understand from these paragraphs, though, as far as the utility goes. I am writing it here to check with Kirby if my understanding makes sense to him. I'd be curious.

"Nothing causing people to lose themselves in futile pursuits separating them from the community of practice, like Coxeter's "Regular Polytops," where he implies that exotic notions such as "fourth orthogonal" should become highly respected. In the opinion of Kirby's working group, that approach was a pitfal. It attracted many intelligent people not otherwise prone to engaging in unproductive pursuits, and kept them occupied for too long."

Personally, I find it easy to fall in love with obscure results. Some of us should do it, going deep into topics that don't seem fruitful at all, and bringing back an occasional discovery to be treasured by the rest of their community. However, the ability to connect with many people within communities pushes research and development to higher quality and efficiency. Someone working on an esoteric branch with a tiny minority of others must constantly question the decision, because of how difficult that journey will be - maybe too difficult to gain anything whatsoever.

Of course, pursuits of majorities have to be questioned all the time as well, but probably on different grounds.

[kirby urner]

On Sun, Sep 6, 2009 at 1:58 AM, Juan <here...@gmail.com> wrote:

On Aug 28, 12:08 pm, kirby urner <kirby.ur...@gmail.com> wrote:

>  Nothing crazy-making like in Coxeter's 'Regular
> Polytopes' where he implies we should doff our hats to weird shaman
> types who know what "fourth orthogonal" means without blowing smoke.
>
> That was a trap, a pitfall, the way we tell it today (snared a lot of
> otherwise intelligent, functional human beings).
>
> Kirby

Kirby,

Please excuse my ignorance but I am afraid I do not have the slightest
idea of what you are talking about here, with terms like "crazy-
making," ""should," "doff our hats," "shaman types," "trap,"
"pitfall," "the way we tell it today," "snared" and such. I am really

Your ignorance is excused.

From page 119 of 'Regular Polytopes' (H.S.M. Coxeter, first printing 1963 I'm pretty sure):

"Only one or two people have ever attained the ability to visualize hyper-solids as simply and naturally as we ordinary mortals visualize solids; but a certain facility in that direction may be acquired by contemplating the analogy between one and two dimensions, then two and three, and so (by a kind of extrapolation) three and four."

This seems a clear allusion to Abbott's 'Flatland", more a comedy of manners than serious philosophy -- but it's been picked up and endlessly recycled e.g. in those 'What the Bleep' movies (you probably dunno what I'm talking about, filmed here in Portland, lots of 'Flatland' cartoons, especially in the 2nd version:   ).

You can see where this "one or two people" would prompt me to think of them as endowed with superpowers.  You'll find this meme echoed all over the Internet, fun to get students into sleuthing it out i.e. "look for web sites where people claim to 'see' a fourth perpendicular and report back" (use a buzzbot if you like, a kind of web crawler).
 

sorry but really, all those terms do not make any sense to me. I am at
a loss trying to figure out what could you possible mean by all that.

I should have given my source and page number originally.

He goes on to discuss the "fourth dimension as Time" meme, which he is careful to dissect away from his extended Euclidean approach (all dimensions spatial). 

That takes us back to my use of dot notation for disambiguation, i.e. coxeter.4d and einstein.4d point into different namespaces and its only somewhat weak science fiction (or even strong) that confuses these two for reasons of plot fabrication e.g. entering a tesseract (coxeter.4d) to discover it's a time machine (einstein.4d -- although as a patent office clerk, he was more into clocks, another kind of time machine).
 

With all due respect, it sounds almost like the discourse of some
opposition party in some third world country run by a dictator. I

Yes, Python Nation is a fledgling dictatorship.
 

understand you are passionate about this topic, and I respect that. I
do not want to get into any controversy, however, just out of respect
for you, and your ideas, I have to ask for some clarification because
I may very well lack the necessary background to be able to understand
your perspective, and what is the issue, let alone taking any side, or
opinion about it because I am drawing a blank here. It is obvious to
me you are saying something that is important to you. I just cannot
figure out what it is. Thank you for your patience, and understanding.

Juan

Then there's yet a third namespace I'm calling fuller.4d, which starts with the tetrahedron ab initio as the Kantian spatio-temporal beginning (not the sphere because deceptively not simple, and not the invisible dot of 0 dimension). 

Actually, the tetrahedron (4 directional, no ability to make "height, width, and depth" stand alone) is 4D all by itself, and then if you want a "time axis" you could call that "the fifth dimension" and make enough sense to keep the audience happy (I've done it many times, this namespace coheres, isn't as fuzzy as its detractors would wish).

4D  angle shape
=============
time frequency

Key:  angles alone define shape without reference to time/size e.g. a regular tetrahedron has just such angles no matter relative energy involvement, which we relegate to the "frequency" dimension (which has a time arrow).

Anyway, it's one of those philosophies for the ages. As a simple mathematics suitable for high school aged kids, I've taught this stuff as adjunct faculty for Portland State, using math lab facilities at Oregon Graduate Insitute and PSU itself.  I assure you the syllabus is intact, easily shared over the web, and my students, albeit self selected guinea pigs, often report satisfaction later in life, given our 'Beyond Flatland' approach.

Kirby

[kirby urner]

I mostly think of it as poetry, in the sense of "beauty over utility" values. Here is what I understand from these paragraphs, though, as far as the utility goes. I am writing it here to check with Kirby if my understanding makes sense to him. I'd be curious.
 

"Nothing causing people to lose themselves in futile pursuits separating them from the community of practice, like Coxeter's "Regular Polytops," where he implies that exotic notions such as "fourth orthogonal" should become highly respected. In the opinion of Kirby's working group, that approach was a pitfal. It attracted many intelligent people not otherwise prone to engaging in unproductive pursuits, and kept them occupied for too long."

That's pretty good Maria. 

However I really have no problem with people working with polytopes in the tradition of Coxeter (why should I?), track Polyhedra (an inner circle of polytope scholars), am a fan of Russell Towle et al. **

The language game resemblance between polytopes (as data structures) and polyhedra of the type studied by the Greeks (and during the Renaissance) is a strong one.  Check out these cool drawings!:

http://bibliodyssey.blogspot.com/2009/09/geometric-landscape.html  (share!  students love this stuff)

 

Personally, I find it easy to fall in love with obscure results. Some of us should do it, going deep into topics that don't seem fruitful at all, and bringing back an occasional discovery to be treasured by the rest of their community. However, the ability to connect with many people within communities pushes research and development to higher quality and efficiency. Someone working on an esoteric branch with a tiny minority of others must constantly question the decision, because of how difficult that journey will be - maybe too difficult to gain anything whatsoever.

fuller.4d is far more esoteric and less understood than coxeter.4d, so I can hardly be accused of trying to discourage "futile pursuits". 

It's just here in Portland (Oregon) we send a small contingent to college already well versed in this "alien geometry" in which points, lines, planes are all like "lumps of clay" and are said to be "four dimensional" simply because the tetrahedron (our "paradigm lump") is conceptually irreducible in an almost Kantian sense. 

Most math professors are inexplicably clueless about this stuff, but may recognize the name Karl Menger as the "geometry of lumps" guy.

Playing in this sandbox in no way ruins a student from mastering the usual XYZ shop talk (coxeter.4d), nor are Einstein's theories out of range (einstein.4d), so I'm claiming a more enchanced (better, superior) math education is emergent in our neck of the woods (Silicon Forest, where you'd expect to find cutting edge thinking).

Of course, pursuits of majorities have to be questioned all the time as well, but probably on different grounds.

An additional feature of this geometry of lumps is a unit volume regular tetrahedron that divides evenly (as a whole number) into the related cube, octahedron, dodecahedron (rhombic) and cuboctahedron i.e. unlike the Platonic Five, which have no "set in stone" size relationship (by pure convention, edges set equal sometimes), i.e. this is a "maze" (deprecated term) i.e. a "nested" arrangement of concentric shapes.

http://www.4dsolutions.net/ocn/cp4e.html shows this nesting arrangement in an animated GIF (used elsewhere as well).

The mnemonic value of having 1, 3, 4, 6, 20 for the above shapes, combined with a modular dissection language that's highly streamlined (B,A,T modules of volume 1/24, mites, sytes, kites), means that our elementary age students are rocketing ahead of most others in Lower48 USA, still stuck in their calculator-based flatland approaches.

The above language maps easily to more conventional nomenclature i.e. our Mite (space-filling) is the trirectangular tetrahedron depicted on page 71 of 'Regular Polytopes' with numbered vertices.

We export our curriculum writing directly to schools elsewhere as well, e.g. the color poster series showing a lot of the above information was published in Singapore.

http://snb.nl.sg/det_4980220.aspx  (note categorization:  philosophy)

If you really wanna understand all this better, I recommend my Coffee Shops Network blog as the place to start, right at the beginning, when Dr. Mario Livio visits ('Is God a Mathematician?').  There's information on 4D vs. 4D vs. 4D.  I also took this show on the road to a hacker convention in Chicago near O'Hare in March. 

Hackers are spreading this geometry independently of math teachers, using Python + VPython for example.  Here in Oregon, math teachers sometimes go to hackers for training, as in Silicon Forest, we celebrate our geeks.

Kirby Urner
Chief Marketing Officer (CMO)
Coffee Shops Network
http://coffeeshopsnet.blogspot.com/

Cheers,
Maria Droujkova
http://www.naturalmath.com

Make math your own, to make your own math.

** Wolfram wrote an appreciative good bye when he passed away suddenly, referred Juan to his when he asked me about writing a visualization tool to show students what's meant by four dimensional space etc.
 

http://blog.wolfram.com/2008/10/10/russell-towle-1949-2008/

[kirby urner]

This seems a clear allusion to Abbott's 'Flatland", more a comedy of manners than serious philosophy -- but it's been picked up and endlessly recycled e.g. in those 'What the Bleep' movies (you probably dunno what I'm talking about, filmed here in Portland, lots of 'Flatland' cartoons, especially in the 2nd version:   ).

Sorry, forgot my citation:

http://www.imdb.com/title/tt0499596/

You'll find I poke fun at "hypercross dogmatists" elsewhere in my writings (go google?).

Given the prevailing orthodoxy, and the root meaning of "ortho" dox, I think such ribbing is in order.

Kirby

[Bradford Hansen-Smith]

For those interested: Another monthly post about exploration of center off-center circle folding. http://Wholemovement.blogspot.com/

Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com

wholemovement.blogspot.com/