Fourth Dimension (4D)

Previous Topic Next Topic
 
classic Classic list List threaded Threaded
15 messages Options
Reply | Threaded
Open this post in threaded view
|

Fourth Dimension (4D)

kirby urner-4
"""
Even within the mathematical community, the fourth dimension maintained
an aspect of mystery and impossibility for many years. To the general public,
the fourth dimension often suggests science fiction full of paranormal
phenomena,
or sometimes,  Einstein's theory of relativity: "the fourth dimension
is time,  isn't it?"
However, this is confusing mathematical questions with those of physics.
We will return to this briefly later. Let us first try to grasp the
fourth dimension
as Schläfli did, as a pure creation of the mind!
"""

http://dimensions-math.org/Dim_CH3_E.htm

Above are notes to an award-winning series on geometry, currently
showing on community / public cable TV in Portland, Oregon.

The shows are also viewable on line, so those teachers with
privileges may choose to project some of them in class.  They're
pretty short, under 30 minutes each.

While on this topic of philosophy in mathematics **, I'd like to
register my skepticism that "existence claims" for hyperspace
are important to the mathematics.

The writers of the show assert that these higher dimensional spaces
"actually exist" and use the analogy we always get, inheriting from
Edwin A. Abbott's book Flatland:  if two dimensional beings might
think about three dimensional shapes (which we know from our own
experience are real), then why couldn't four dimensional beings see
us as being "trapped" in just three dimensions.

That sounds poetic, but it's hardly what anyone would call "scientific"
reasoning is it?

I make some more skeptical noises, about the same thing, on this
other math-related list, for those interested in more nuances:

"""
As a database guy, my bias is to think of n-tuple addressing as just a
fancy way to store/retrieve records, like URLs for storing/retrieving
web pages (there's a reason it's called hypertext).  Sure, you can
come up with fancy analogies to polyhedra, but I'm not sure this
"really exists" argument is important to the mathematics.  Seems more
like mysticism (or even organized religion).
"""
http://mail.python.org/pipermail/edu-sig/2010-August/010061.html

Start a new thread?  Any math teacher views?

Kirby

Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Joe Niederberger
No hypercubes until you've eaten all your tetrahedra, kids.
And no pushing them off into your napkin either! ;-)

Joe N

- - ----- Original Message -----
From: "kirby urner" <[hidden email]>

> As a database guy, my bias is to think of n-tuple addressing as just a
> fancy way to store/retrieve records, like URLs for storing/retrieving
> web pages (there's a reason it's called hypertext).  Sure, you can
> come up with fancy analogies to polyhedra, but I'm not sure this
> "really exists" argument is important to the mathematics.  Seems more
> like mysticism (or even organized religion).
> """
> http://mail.python.org/pipermail/edu-sig/2010-August/010061.html
>
> Start a new thread?  Any math teacher views?
>
> Kirby

 
Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

kirby urner-4
On Sun, Aug 1, 2010 at 2:12 PM, Joe Niederberger
<[hidden email]> wrote:
> No hypercubes until you've eaten all your tetrahedra, kids.
> And no pushing them off into your napkin either! ;-)
>
> Joe N
>

On pg. 119 of 'Regular Polytopes' (in my Dover edition), Donald Coxeter
takes on those science fiction writers who seem to deliberately
confuse 4d = 3d + time, with the tesseract (aka hypercube).

And I quote:

"""
Little, if anything, is gained by representing the fourth Euclidean
dimension as time.  In fact, this idea, so attractively developed
by H. G. Wells in The Time Machine, has led such authors as
J.W. Dunne (An Experiment with Time) into a serious misconception
of the theory of Relativity.  Minkowski's geometry of space-time
is *not* Euclidean, and consquently has no connection with the
present investigation.
"""

"A tesseract is not a time machine" might be a fun way of saying it.

Thesis:

Edwin Abbott's 'Flatland' opened a wormhole in the literature for
a brand of officially sanctioned mysticism.  Shamanism lives on in
mathematics, among those who reputedly are in touch with "higher
dimensions".  Hyperspace is one of those "gee whiz" memes
used for recruiting tomorrow's priesthood.

>From the same page:

"""
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.
"""

Note that we already have at least two distinct meanings of
4D.  The 4th Euclidean dimension, and Time in a physical
theory.  One might write Coxeter.4d versus Einstein.4d to
distinguish the two "namespaces" (aka language games).

Kirby

Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Joe Niederberger
What exactly are you against?
Interest in dimension > 3 or science *fiction*?
What exactly do you find objectionable about flatland
other than the fact that it investiagtes higher dimensions in an
entertaining way? You think the world can only stand
one definition of 4D?

As far as J.W. Dunne, I wasn't familiar with him or his essay,
but on quickly scanning the wikipedia article it looks like the
sorts of speculations he made were founded on the notion of
block time, which is pretty much a standard conception of
relativistic space-time, the non-euclidean nature notwithstanding.

Read a bit about "block time" and various arguments one way
or the other and then we can continue.
(Relativity actually can make block time look more likely,
or at least it has been argued that way.)
http://plato.stanford.edu/entries/spacetime-bebecome/
http://plato.stanford.edu/entries/time/

Better yet, check out the "Godel Universe"  
http://www.daviddarling.info/encyclopedia/G/Godel_universe.html
(and many other links can be found...)
http://plato.stanford.edu/entries/time-travel-phys/

This is an actual solution to GR that cannot be ruled out based on
current physics. You seem to think the fact that space-time
is not ecuclidean would rule out such entertainment?

Or perhaps we should just stamp it out by fiat.

By the way, the wikipedia article on Dunne mentions Aquinas'
take on eternity, which actually sounds a similar to Pierre-Simon
Laplace. (And also by the way, I went to catholic grade school
with a descendant of Laplace... ;-)  

Joe N
"We Still Might"


- ----- Original Message -----
From: "kirby urner" <[hidden email]>
To: "Math-teach Teach" <[hidden email]>
Sent: Sunday, August 01, 2010 8:46 PM
Subject: Re: Fourth Dimension (4D)


> On Sun, Aug 1, 2010 at 2:12 PM, Joe Niederberger
> <[hidden email]> wrote:
> > No hypercubes until you've eaten all your tetrahedra, kids.
> > And no pushing them off into your napkin either! ;-)
> >
> > Joe N
> >
>
> On pg. 119 of 'Regular Polytopes' (in my Dover edition), Donald Coxeter
> takes on those science fiction writers who seem to deliberately
> confuse 4d = 3d + time, with the tesseract (aka hypercube).
>
> And I quote:
>
> """
> Little, if anything, is gained by representing the fourth Euclidean
> dimension as time.  In fact, this idea, so attractively developed
> by H. G. Wells in The Time Machine, has led such authors as
> J.W. Dunne (An Experiment with Time) into a serious misconception
> of the theory of Relativity.  Minkowski's geometry of space-time
> is *not* Euclidean, and consquently has no connection with the
> present investigation.
> """
>
> "A tesseract is not a time machine" might be a fun way of saying it.
>
> Thesis:
>
> Edwin Abbott's 'Flatland' opened a wormhole in the literature for
> a brand of officially sanctioned mysticism.  Shamanism lives on in
> mathematics, among those who reputedly are in touch with "higher
> dimensions".  Hyperspace is one of those "gee whiz" memes
> used for recruiting tomorrow's priesthood.
>
> >From the same page:
>
> """
> 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.
> """
>
> Note that we already have at least two distinct meanings of
> 4D.  The 4th Euclidean dimension, and Time in a physical
> theory.  One might write Coxeter.4d versus Einstein.4d to
> distinguish the two "namespaces" (aka language games).
>
> Kirby
Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

kirby urner-4
On Mon, Aug 2, 2010 at 8:53 AM, Joe Niederberger
<[hidden email]> wrote:
> What exactly are you against?
> Interest in dimension > 3 or science *fiction*?

A theme here (for me) has been taking advantage of
gullibility and/or deliberately promoting superstitions,
something lots of mathematicians decry (so I feel
I'm in good company in that respect).

This connects to the determinism thread as well, in
the sense that the lay public (which includes funders
and investors) may be tempted (hoodwinked?) into
believing some mathematically reliable model of
"the economy" is just around the corner, if only we'd
just pump another billion or two into the research budget.

Their gullibility traces to age-old dogmas regarding
causal determinism, with not enough caveats regarding
say butterfly effects, which kick in at a practical level
(and practical is what matters *a lot* in the real world
- -- academic debates aside) way above any quantum
level, where the uncertainly principle prevails (along
with unpredictable beta decay and whatever "true
randomness" we might want to posit, as distinct from
"unpredictable yet determined").

I used predicting the weather as a good example.
There's simply no prospect of nailing down all the
butterflies, getting the precision needed, *even if
there were* a purely deterministic computation to
be made.  So our time horizon is limited, and
promising more / better measurements of sufficient
"just around the corner" is somewhat meaningless
hype, in light of what dynamical systems have
shown us about their sensitivity to initial conditions.
So lets be honest with the public.

> What exactly do you find objectionable about flatland
> other than the fact that it investiagtes higher dimensions in an
> entertaining way? You think the world can only stand
> one definition of 4D?

I don't find the idea of "two dimensional beings" to be
credible.  Whenever the imaginative illustrators show
us those beings, we're looking from "above" and "outside"
their two-dimensional existence (see xkcd comic below).
This idea that we're "trapped in three dimensions" should
work both ways i.e. it's just as inconceivable to imagine
only two dimensions as to imagine a "fourth perpendicular".

I'm not disputing any of the math, which is well established
and with practical applications, just being critical of the
mystique and the hype, which seems as unquestioned
as it is ingrained.

A more enlightened approach would be to study the
"fourth dimension" concept more critically and with more
sensitivity to its multiple meanings.  I have a vested
interest in this because of a third meaning of 4D
(as distinct from either Einstein's or Coxeter's).
Hence my '4D vs. 4D vs. 4D' discussion (e.g. in that
meeting with Dr. Livio & Co., written up in one of my
blogs (I'll provide a link later)).

A good source for this more enlightened approach
(one of several) would be:

'The Fourth Dimension and Non-Euclidean Geometry
in Modern Art' by Linda Dalrymple Henderson,
Princeton University Press, 1983.

>
> As far as J.W. Dunne, I wasn't familiar with him or his essay,
> but on quickly scanning the wikipedia article it looks like the
> sorts of speculations he made were founded on the notion of
> block time, which is pretty much a standard conception of
> relativistic space-time, the non-euclidean nature notwithstanding.
>

I'm not familiar either.  That was a quote from Donald Coxeter,
as you saw.  The point being:  n-dimensional Euclidean geometry
(what he was into) and relativistic space-time (a physical theory)
represent two different traditions (lineages).

Not enough is done to disambiguate these in pop culture.

Instead, talk of "higher dimensions" seems to be used rather
cynically by some to encourage awe without much understanding.
There's a lot of hype around hyperdimensions.  This has been
going on since the 1800s, with people somewhat mindlessly
repeating the 'Flatland' memes, rather unquestioningly.

> Read a bit about "block time" and various arguments one way
> or the other and then we can continue.
> (Relativity actually can make block time look more likely,
> or at least it has been argued that way.)
> http://plato.stanford.edu/entries/spacetime-bebecome/
> http://plato.stanford.edu/entries/time/
>

It's not about whether a given physics is true.  Is about
disambiguating.  Coxeter style geometry is *not* non-Euclidean,
it's more extended Euclidean (as he says in the previously
cited quote).

> Better yet, check out the "Godel Universe"
> http://www.daviddarling.info/encyclopedia/G/Godel_universe.html
> (and many other links can be found...)
> http://plato.stanford.edu/entries/time-travel-phys/
>
> This is an actual solution to GR that cannot be ruled out based on
> current physics. You seem to think the fact that space-time
> is not ecuclidean would rule out such entertainment?
>

No, that's not what I think.  I hope I've added some clarity
to my critique.  I'm arguing against the superficial "gee whiz"
analogy put forth by Abbott as too superficial.  I think a
more philosophically astute set of curriculum segments
would raise the level of discussion, and at the same time
make room for yet another spin on '4D'.

The idea of a "fourth perpendicular" is on the face of it
nonsense, and I think better explanations might be
supplied than Abbotts (a social satire, not a formal
mathematical work -- yet it gets leaned on so heavily).

The "projective" metaphor is just that (a metaphor).

"Shining a light" through a multi-axis data structure
to get these 120-cell and 600-cell objects is an
exercise in encoding into and decoding from a data
structure no?

One need not believe in "higher dimensional beings"
or make existence claims regarding "the reality" of
these extrasensory worlds which ordinary mortals
cannot contact -- but which an inner circle of high
priests and gifted amateurs sometimes can.  Where's
CICOP / CSI on that one?

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

> Or perhaps we should just stamp it out by fiat.
>
> By the way, the wikipedia article on Dunne mentions Aquinas'
> take on eternity, which actually sounds a similar to Pierre-Simon
> Laplace. (And also by the way, I went to catholic grade school
> with a descendant of Laplace... ;-)
>
> Joe N
> "We Still Might"
>

Talk of "higher dimensions" still seems the officially approved
way to inject a religious sense into otherwise hard-nosed
science through the back door.  I'm not against a religious
sense.  I'm just into investigating back doors and how they're
used.

Exhibits:
http://www.bibliotecapleyades.net/ciencia/ciencia_hyperspace02.htm
http://www.springerlink.com/content/hm21773110t27324/  (Springer!)
http://xkcd.com/721/  (funny)
http://sv-se.facebook.com/topic.php?uid=2337214044&topic=2627  (on and
on it goes)

Beyond all of the above, the 3rd meaning of 4D that I want to
introduce begins by registering skepticism that we have to look at
"height, width and depth" as each conceptually atomic, as if you
could subtract any one of these and be left with anything
conceptual.

In both Kant's writings and Descarte's, the idea of "res extensa" is
somewhat irreducibly volumetric and/or spatial.

But if it's "irreducible" then what's so "three-ish" about it.  A
long discussion, but we never get around to having it with the
debate so stuck in these old 1800s ruts, i.e. the popular mind
is too dumbed down to even broach the subject.

Ending with another quote:

"""
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....
All conceptual consideration is inherently four-dimensional.
Thus the primitive is a priori four-dimensional, always based
on the four planes of reference of the tetrahedron. There
can never be 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.
(527.702, 527.712)
"""

This is one more philosophy among many, but deserves a
place in the sun.  It should be OK (not verboten) to register
skepticism in the face of the prevailing dogmas regarding
the dimension concept, which is endlessly being messed
with.  Lets keep messing with it (do we have a choice?)
but raise the level of awareness around doing so.

Kirby

>
> - ----- Original Message -----
> From: "kirby urner" <[hidden email]>
> To: "Math-teach Teach" <[hidden email]>
> Sent: Sunday, August 01, 2010 8:46 PM
> Subject: Re: Fourth Dimension (4D)
>
>
>> On Sun, Aug 1, 2010 at 2:12 PM, Joe Niederberger
>> <[hidden email]> wrote:
>> > No hypercubes until you've eaten all your tetrahedra, kids.
>> > And no pushing them off into your napkin either! ;-)
>> >
>> > Joe N
>> >
>>
>> On pg. 119 of 'Regular Polytopes' (in my Dover edition), Donald Coxeter
>> takes on those science fiction writers who seem to deliberately
>> confuse 4d = 3d + time, with the tesseract (aka hypercube).
>>
>> And I quote:
>>
>> """
>> Little, if anything, is gained by representing the fourth Euclidean
>> dimension as time.  In fact, this idea, so attractively developed
>> by H. G. Wells in The Time Machine, has led such authors as
>> J.W. Dunne (An Experiment with Time) into a serious misconception
>> of the theory of Relativity.  Minkowski's geometry of space-time
>> is *not* Euclidean, and consquently has no connection with the
>> present investigation.
>> """
>>
>> "A tesseract is not a time machine" might be a fun way of saying it.
>>
>> Thesis:
>>
>> Edwin Abbott's 'Flatland' opened a wormhole in the literature for
>> a brand of officially sanctioned mysticism.  Shamanism lives on in
>> mathematics, among those who reputedly are in touch with "higher
>> dimensions".  Hyperspace is one of those "gee whiz" memes
>> used for recruiting tomorrow's priesthood.
>>
>> >From the same page:
>>
>> """
>> 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.
>> """
>>
>> Note that we already have at least two distinct meanings of
>> 4D.  The 4th Euclidean dimension, and Time in a physical
>> theory.  One might write Coxeter.4d versus Einstein.4d to
>> distinguish the two "namespaces" (aka language games).
>>
>> Kirby
Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Joe Niederberger

- ----- Original Message -----
From: "kirby urner" <[hidden email]>

> On Mon, Aug 2, 2010 at 8:53 AM, Joe Niederberger
> <[hidden email]> wrote:
> > What exactly do you find objectionable about flatland
> > other than the fact that it investiagtes higher dimensions in an
> > entertaining way?
 
> I don't find the idea of "two dimensional beings" to be
> credible.  

You're serious. What else keeps you up at night?


> The idea of a "fourth perpendicular" is on the face of it
> nonsense, and I think better explanations might be
> supplied than Abbotts (a social satire, not a formal
> mathematical work -- yet it gets leaned on so heavily).

Negative quantities? Preposterous. How can a quantity be
less than nothing? Square root of negative numbers?
Might as well call them absurd numbers! (They did!)


> Note that we already have at least two distinct meanings of
> 4D. The 4th Euclidean dimension, and Time in a physical
> theory. One might write Coxeter.4d versus Einstein.4d to
> distinguish the two "namespaces" (aka language games).

I don't think they are distinct meanings of "dimension".
The meaning of  "dimension" is the same. The difference lies
elsewhere. By the way the Minkowskian spacetime for S.R.
is better called a pseudo-euclidean - so as to keep it distinct
from the curved "non-eucldian" geometries. Just terminology.

Joe N
"We Still Might"


Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

kirby urner-4
On Mon, Aug 2, 2010 at 7:08 PM, Joe Niederberger
<[hidden email]> wrote:

>
> - ----- Original Message -----
> From: "kirby urner" <[hidden email]>
>
>> On Mon, Aug 2, 2010 at 8:53 AM, Joe Niederberger
>> <[hidden email]> wrote:
>> > What exactly do you find objectionable about flatland
>> > other than the fact that it investiagtes higher dimensions in an
>> > entertaining way?
>
>> I don't find the idea of "two dimensional beings" to be
>> credible.
>
> You're serious. What else keeps you up at night?
>

You're not, obviously.

It's not the "being" part, it's the claim to be able to confine
thinking to a "flat land", as if drawing a square and a triangle
(perhaps with talk balloons") constituted thinking in "2D".

Admittedly it's a philosophical discussion, and we don't
necessarily encourage that among children (would they be
less compliant, less defenseless against BS?).

Quoting Kant from Wikipedia:

"""
In 1783, Kant wrote: "That everywhere space (which is not
itself the boundary of another space) has three dimensions
and that space in general cannot have more dimensions
is based on the proposition that not more than three lines
can intersect at right angles in one point. This proposition
cannot at all be shown from concepts, but rests immediately
on intuition and indeed on pure intuition a priori because
it is apodictically (demonstrably) certain."
"""

I'd say that view is "certain" w/r to a specific language game
that puts a lot of emphasis on right angles.  When you
consider that the tetrahedron is the minimum polyhedron,
you're confronted with an irreducible fourness (four facets,
four vertexes) as much as a threeness.  That's the
philosophy expressed in the concluding quote of my
last post (also from Wikipedia -- not that that makes it
"true").

Speaking of 4D on Wikipedia, here's a little coordinate
system I'm introducing in my Martian Math classes these
days.  You get four unique coordinates per point, which
seems wasteful at first, but then you don't need any
negatives, so you have enough "left over" numbers to
populate an entire other space.  Also, four rays from
the origin just seems more economical than six, as
we divide space into four quadrants instead of eight
octants.  No wonder the Martians use it sometimes,
even to design space ships:

http://en.wikipedia.org/wiki/Quadray_coordinates
http://www.4dsolutions.net/satacad/martianmath/mm29.html


>
>> The idea of a "fourth perpendicular" is on the face of it
>> nonsense, and I think better explanations might be
>> supplied than Abbotts (a social satire, not a formal
>> mathematical work -- yet it gets leaned on so heavily).
>
> Negative quantities? Preposterous. How can a quantity be
> less than nothing? Square root of negative numbers?
> Might as well call them absurd numbers! (They did!)
>

The usual spurious arguments that pass for educated opinion.
Kids are mostly defenseless against them, and as adults,
don't see any point in bucking the establishment.

Yet the history of mathematics is replete with such challenges,
with the prevailing wisdom sometimes but a thin veneer over a
maelstrom of counter-arguments.

Many tenuous victories, some possibly with a short half life
(as in the sciences, mathematical paradigms break apart
sometimes -- any good titles on this that you'd care to
recommend?).

Anyway, coming up with new ways of looking (gestalts) is
valuable.  Complex numbers became far more acceptable
once the Argand Plane made it's debut, although some
other guy thought of that first (if that matters).

I think we should do better than mock those with countering
intuitions, especially if in the business of teaching math.

What other gestalts might we encourage, w/r to hyper-
dimensionality?  Could the idea hypertext (the space of
web pages) help with hyperspace (pages = vertexes,
links = edges)?

>
>> Note that we already have at least two distinct meanings of
>> 4D. The 4th Euclidean dimension, and Time in a physical
>> theory. One might write Coxeter.4d versus Einstein.4d to
>> distinguish the two "namespaces" (aka language games).
>
> I don't think they are distinct meanings of "dimension".

I do, most definitely.

> The meaning of  "dimension" is the same. The difference lies

I disagree.  Meaning stems from usage.  "Dimension" is also
used with referenced to physical measures, e.g. mass.

> elsewhere. By the way the Minkowskian spacetime for S.R.
> is better called a pseudo-euclidean - so as to keep it distinct
> from the curved "non-eucldian" geometries. Just terminology.
>

... and probably the terminus of this thread, unless we have some
more serious contributors.  Oh well, I tried.  Here's the URL for
the 4D vs. 4D vs. 4D blog post I mentioned:

http://coffeeshopsnet.blogspot.com/2009/02/glass-bead-game.html

Kirby

> Joe N
> "We Still Might"
Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Joe Niederberger

- ----- Original Message -----
From: "kirby urner" <[hidden email]>
>> The idea of a "fourth perpendicular" is on the face of it
>> nonsense, and I think better explanations might be
>> supplied than Abbotts (a social satire, not a formal
>> mathematical work -- yet it gets leaned on so heavily).

[Joe N responded]
> Negative quantities? Preposterous. How can a quantity be
> less than nothing? Square root of negative numbers?
> Might as well call them absurd numbers! (They did!)
>
[Kirby]
> The usual spurious arguments that pass for educated opinion.

Hardly - read about negative and imaginary numbers - they
were thought to be nonsense exactly because they didn't
seem physically possible. Very mcuh the same thing.


>>> Note that we already have at least two distinct meanings of
>>> 4D. The 4th Euclidean dimension, and Time in a physical
>>> theory. One might write Coxeter.4d versus Einstein.4d to
>>> distinguish the two "namespaces" (aka language games).
>>
>> I don't think they are distinct meanings of "dimension".

> I do, most definitely.

>> The meaning of "dimension" is the same. The difference lies
>> elsewhere

> I disagree.  Meaning stems from usage.  "Dimension" is also
> used with referenced to physical measures, e.g. mass.

So - are you are saying Einstein.4d is a different menaing of
dimension because its related to physical theory? I hope not,
because forget about the physics and you still have 4D
Minkowski space, just mathemetics. So Minkowsi.4d
is not Coxeter.4d (which I presuming is Euclidean 4d)
but the term "dimension" has the sme meaning in both.

On the other hand, if that's not what you meant, then please
explain how "dimension" in Minkowsi.4d means something
different than the same term "dimension" in Coxeter.4d.

Joe N
Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

kirby urner-4
On Tue, Aug 3, 2010 at 8:23 AM, Joe Niederberger
<[hidden email]> wrote:

>
> - ----- Original Message -----
> From: "kirby urner" <[hidden email]>
>>> The idea of a "fourth perpendicular" is on the face of it
>>> nonsense, and I think better explanations might be
>>> supplied than Abbotts (a social satire, not a formal
>>> mathematical work -- yet it gets leaned on so heavily).
>
> [Joe N responded]
>> Negative quantities? Preposterous. How can a quantity be
>> less than nothing? Square root of negative numbers?
>> Might as well call them absurd numbers! (They did!)
>>
> [Kirby]
>> The usual spurious arguments that pass for educated opinion.
>
> Hardly - read about negative and imaginary numbers - they
> were thought to be nonsense exactly because they didn't
> seem physically possible. Very mcuh the same thing.
>

Yet people had ordinary physical interpretations right away.

Look at any early grade school textbook when negative
numbers get introduced.  "Below zero" on a thermometer.

Owing money (having a negative balance).

With complex numbers, quaternions, other mathematical
objects, an easy graphical device is often essential to
widespread adoption, in addition to lexical rules of the
road.

The imaginary numbers had a lot of pressure behind them
(historical inertia, momentum) thanks to the need for
n solutions to an nth degree polynomial. By this time,
they're deeply ingrained in the woodwork, whereas
quaternions didn't take to such a degree.

Quaternions are used in computer graphics some, as
more computationally efficient than rotation matrices.
Then there's Clifford Algebra, tracing back to Grassmann.

nD euclidean geometry is a lot about permuting symmetry
groups and "projecting" homomorphisms in "3D" where
some information is lost or distorted (isomorphisms are
"not lossy" in contrast).

The data structure may give a distance of 1 between all
the nodes (say tangent equi-radius spheres specified by
24-tuples), yet there's no way to build such a creature
(not literally).

So we've got a metaphor going.

Instead of asserting hyper-dimensional polytopes "really exist"
(meaning what?) we might simply say they're metaphorical and
the analogies inhere in the algorithms, what it means to
"spin" for example (or to turn on an "axis").

Hansen was sorta making fun of "math appreciation" as
"not really math", and yet a critical investigation into the
"dimension" concept and how it keeps changing and
shifting over time, or how it gets involved in different
language games, is probably worth some focus.

Better to lift these metaphors into conscious awareness
for evaluation, lest we keep passing on all this flat lander
stuff uncritically and unexamined.

Cantor is another author who did some critical writing
about "dimension", although this material is rarely quoted
or examined (and I don't have my citations handy this time,
though I have in other archives).

Cantor plays around with this "dimension = number of elements
in the address" meme, calls it into question. People say space
is "three d" because the Cartesian apparatus uses 3-tuple
addressing.  That's used as a rationale (even when the axes
are skewed i.e. not perpendicular).

For example, the computer's video display, which portrays depth,
perspective, is isomorphic to a patch of computer memory
that's addressed purely linearly, i.e. one coordinate takes the
place of three in XYZ (think of a raster line, a zig-zag).  So is
space "really" one-dimensional then, by the same token?
Of course Cantor wasn't talking about computers, though
one could say he anticipated some of the thinking that went
into them.

Karl Menger, dimension theorist, is who I cite more often
these days, for his "geometry of lumps" proposal.

'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.

Let's have a non-Euclidean definition of points, lines,
planes such that they're all "lump-like" and therefore
are not distinguished by dimension number.  How
might that work?  We might link this proposal to
that 3rd meaning of 4D (cited above) i.e. all "lumps"
are primitively modeled as a simplex (simplest shape),
whether points, lines or planes or what have you.

Everything has an "inside" by definition, and is minimally
a tetrahedron (topologically speaking).  Spheres, in contrast,
are modeled as multi-faceted, having many small triangular
facets or windows.

This is closer to how computer graphics actually works
(it's discrete, not continuous), so this "geometry of lumps"
is perhaps suitable for use in computer science, which
tends to be both discrete and energy aware (i.e. energy
efficiency matters).

>
>>>> Note that we already have at least two distinct meanings of
>>>> 4D. The 4th Euclidean dimension, and Time in a physical
>>>> theory. One might write Coxeter.4d versus Einstein.4d to
>>>> distinguish the two "namespaces" (aka language games).
>>>
>>> I don't think they are distinct meanings of "dimension".
>
>> I do, most definitely.
>
>>> The meaning of "dimension" is the same. The difference lies
>>> elsewhere
>
>> I disagree.  Meaning stems from usage.  "Dimension" is also
>> used with referenced to physical measures, e.g. mass.
>
> So - are you are saying Einstein.4d is a different menaing of
> dimension because its related to physical theory? I hope not,
> because forget about the physics and you still have 4D
> Minkowski space, just mathemetics. So Minkowsi.4d
> is not Coxeter.4d (which I presuming is Euclidean 4d)
> but the term "dimension" has the sme meaning in both.
>

I think when you change how dimension is used, you change
the spin and therefore the meaning.  Linking meaning to use,
regardless of what the dictionary says, is what we learn to do
in philosophy, thx to Wittgenstein et al.

In Einstein.4d, the "time" dimension is actually treated
differently, algebraically, stands out from x, y, and z as
distinct.  Disguise the space and time dimensions with
four random letters, and you'll be able to distinguish them
anyway, because of how space and time coordinates are
treated.

In the case of Coxeter.4d (a so-called Euclidean namespace
- -- even though Euclid would not have used it), all the dimensions
are "spatial" in terms of treatment.  No axis has primacy as
being mcore "temporal" than the others.  The Minkowski
namespace has "spacelike" and "timelike" vectors in it.

The need to disambiguate seems clear:
http://en.wikipedia.org/wiki/Hyperspace

> On the other hand, if that's not what you meant, then please
> explain how "dimension" in Minkowsi.4d means something
> different than the same term "dimension" in Coxeter.4d.
>

Saying conceptual space is "4D" because the four rays from
the tetrahedron (as distinct from the Cartesian six from the
octahedron) give a minimal signature, is yet more alien and
remote from these other two meanings (the Wikipedia page
on Dimension makes no mention of this usage -- still too
esoteric in 2010).

We could probably quibble endlessly as to whether "dimension"
is always the same concept no matter how used.  Fractional
dimensions were recently injected into the mix as well.

I think it's easier, at least in this computer science context,
to invoke the concept of "namespace" and to suggest that
definitions are local, not global, i.e. Coxeter.4d and Einstein.4d
have distinct meanings of 4d and of dimension in general.

(Many computer languages, such as J, or the Python module
numpy, have a native hyperdimensional array structure.  Is a
polytope a relational data structure with an n-tuple primary key?
One also needs the idea of distance, but that's computed from
the keys, so the relational structure just needs to record the
faces, from which edges may be distilled.  So to "project in
3d" is a kind of data mining then? (see the "data warehouse"
meaning of "dimension" in Wikipedia).

Many ways to go I'm sure.  Math need not be stagnant.

As long as we have room in the sun for 4d in connection with
tetrahedral mensuration (per Martian Math etc.), I think we'll
enjoy some critical mass gravitas, i.e. not radiate away (leak
away) the accumulated inertia (a kind of gravity well, once
you get into it -- makes enough sense to cohere).

Kirby

> Joe N
>

------- End of Forwarded Message

Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Michael Dougherty-2
In reply to this post by kirby urner-4
I'm going to be totally dull here.

To me, the number of dimensions of a system or state of being is just the number of variables needed to describe it.  Usually we like to think of how many real variables are needed, though in grad school you find these crazy linear algebra and functional analysis guys who want to use complex variables.  (SVC guys decide to go further still.)

It is fun to think of having another dimension to play with, like when in Star Trek II Spock says the enemy is thinking very 2-dimensionally, and Kirk points upward to show where he wants to then take the Enterprise.

Complex variables were kind of like that: give more freedom to solve a problem and then bring it back down to the "real" context, or realize it lives better in the complex variables and just stay there.

But "dimension" and similar ideas ("degrees of freedom" for instance) can be rationalized to mean roughly the same thing.  Context and relationships among them (as in Minkowski 4-space) can make this simple idea much more rich.  If I recall, the physics guys used to wow everyone by talking about 10-11-dimensional superstring theory.  Yeah, whatever.

- --Mike D.

- --Mike D.

Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Timothy Golden
In reply to this post by kirby urner-4
> I'm going to be totally dull here.
>
> To me, the number of dimensions of a system or state
> of being is just the number of variables needed to
> describe it.  Usually we like to think of how many
> real variables are needed, though in grad school you
> find these crazy linear algebra and functional
> analysis guys who want to use complex variables.
>  (SVC guys decide to go further still.)

Right On Michael. There is one correction that we can actually make, but the Fullerites don't care for it.

The ray is more fundamental than the line. The ray is unidirectional. The real line is bidirectional, and so all of our talk of dimensional quality (revolving around the real line as you have carefully stated) is not so fundamental as it could be.

The simplex is a general dimensional form. Using rays from the center of a simplex outward to its vertices we can address n dimensional space via the n+1 verticed simplex.

The sum of these rays will yield the origin. It happens that the real line itself can be composed of this simplex geometry. The two verticed simplex constructs the real line out of two rays whose sum yields zero. Now we are getting fundamental. Beneath the two verticed simplex lies a one verticed simplex whose behaviors exactly match the behaviors of time, for time is unidirectional and zero dimensional. I can happily expound on this but don't wish to belabor this post.

Since we have now found a more fundamental construction, then the very word dimensional as tied to the real line can and should be challenged, though the usage can be carried on without conflict, except at the zero dimensional context, where the one-signed system can still provide algebra even while its geometry is nearly nonexistent. Also, because we freely project 1D objects in 2D spaces or higher, then the same can be done with the solitary ray, even while its representation is zero dimensional. This projective stance is fairly new for me.

Above is a fairly geometrical description that brings about polysign numbers. In addition to the vector space qualities there is an arithmetic product, yielding ring behaved algebra in any dimension. Shortly thereafter support for structured spacetime ensues via the progression
   P1 P2 P3 | P4 P5 ...
 where a natural breakpoint in product behavior is observed in P4, where Pn are the n verticed simplex, or the n-signed numbers. Spacetime is structured
   P1 P2 P3
and so the isotropic claims of old deserve to be challenged. Unidirectional time is enough to expose this glaring error, yet few can go here.

 - Tim

>
> It is fun to think of having another dimension to
> play with, like when in Star Trek II Spock says the
> enemy is thinking very 2-dimensionally, and Kirk
> points upward to show where he wants to then take the
> Enterprise.
>
> Complex variables were kind of like that: give more
> freedom to solve a problem and then bring it back
> down to the "real" context, or realize it lives
> better in the complex variables and just stay there.
>
> But "dimension" and similar ideas ("degrees of
> freedom" for instance) can be rationalized to mean
> roughly the same thing.  Context and relationships
> among them (as in Minkowski 4-space) can make this
> simple idea much more rich.  If I recall, the physics
> guys used to wow everyone by talking about
> 10-11-dimensional superstring theory.  Yeah,
> whatever.
>
> - --Mike D.
>
> - --Mike D.
Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Dave L. Renfro
In reply to this post by kirby urner-4
Michael Dougherty wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=7143838

> But "dimension" and similar ideas ("degrees of freedom"
> for instance) can be rationalized to mean roughly the
> same thing. Context and relationships among them (as in
> Minkowski 4-space) can make this simple idea much more
> rich. If I recall, the physics guys used to wow everyone
> by talking about 10-11-dimensional superstring theory.
> Yeah, whatever.

What about the quantum mechanics people and their Hilbert
spaces (or the more inclusive "Fock space" [1]) of countable
infinite dimensions? Of course, here we're only talking
about Schauder bases [2]. Under the usual linear algebra
notion of a basis (Hamel basis), every infinite dimensional
normed linear space (the bread-and-butter object of study
in the huge area of math known as functional analysis)
has cardinality at least continuum (uncountable, and then
some if you're not assuming the continuum hypothesis) [3].

Then there are the various fractional dimensional notions
(somewhat misnamed, because any non-negative real number
is possible, not just the non-negative rational numbers),
and the less well known (but extremely important in many
math applications) distinct dimensional notions defined
by "Hausdorff dimension functions" that allow for uncountably
many levels within the "dimension 0" realm, uncountably
many levels lying above any specified positive real
number r while simultaneously lying  below any real
number s such that r < s, and uncountably many levels
lying above any real number fractional dimension level.

[1] http://en.wikipedia.org/wiki/Fock_space

[2] http://en.wikipedia.org/wiki/Schauder_basis

[3] For example, the space of all continuous functions
f: [0,1] --> R under the sup norm, often denoted by
C[0,1], has (Hamel) dimension continuum (the cardinality
of the real numbers).

Here's something related to some other posts in this
thread that I posted in sci.math on 21 January 2009:

- ----------------------------------------------------------
- ----------------------------------------------------------

Jon Slaughter wrote (in part):

> You can't naturally visualize anything past 3D.
> It is impossible!!! Why!?!?! Because nothing in
> the natural world exists in anything but 3D.
> (as of yet and even if it does no one has definitely
> found it)

I don't see how this proves it is impossible to
visualize higher dimensions. In fact, I've read
about a few reported cases of individuals having
a limited ability to visualize four dimensional space.
I'm not positive right now, but I think one of these
was William J. Sidis (see [1]), at least I think it
was mentioned in Amy Wallace's book about Sidis
(see [2]). From what I recall (but this could have
been something I read about someone else, possibly
H. P. Manning), he had a limited ability to visualize
four dimensions when young (early teens), but this
ability faded away as he grew into adulthood.

By the way, some of Greg Egan's science fiction involves
the ability of super-advanced human-like entities being
able to visualize higher dimensions. One story I've read
where this plays a fairly large role is Egan's short story,
"Wang's Carpets".

[1] http://en.wikipedia.org/wiki/William_James_Sidis

[2] http://www.amazon.com/dp/0525244042

- ----------------------------------------------------------
- ----------------------------------------------------------

Dave L. Renfro

------- End of Forwarded Message

Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

kirby urner-4
In reply to this post by Timothy Golden
On Tue, Aug 3, 2010 at 6:23 PM, Timothy Golden <[hidden email]> wrote:

>> I'm going to be totally dull here.
>>
>> To me, the number of dimensions of a system or state
>> of being is just the number of variables needed to
>> describe it.  Usually we like to think of how many
>> real variables are needed, though in grad school you
>> find these crazy linear algebra and functional
>> analysis guys who want to use complex variables.
>>  (SVC guys decide to go further still.)
>
> Right On Michael. There is one correction that we can actually make, but the Fullerites don't care for it.
>
> The ray is more fundamental than the line. The ray is unidirectional. The real line is bidirectional, and so all of our talk of dimensional quality (revolving around the real line as you have carefully stated) is not so fundamental as it could be.
>
> The simplex is a general dimensional form. Using rays from the center of a simplex outward to its vertices we can address n dimensional space via the n+1 verticed simplex.
>

Dunno about the Fullerites having a problem with your correction, as
using the rays from the center of a simplex outward is the basis for
so-called Quadray Coodinates, a brand of "simplicial coordinates"
(i.e. based on the simplex), and these are useful for appreciating an
alternative vector algebra.

I wrote 'em up for Wikipedia and have them implemented in Python for
use in what I call "Martian Math" these days (a combination of
speculative fiction and contemporary skills, e.g. we use a computer
language to teach about "math objects" such as vectors, polynomials,
polyhedra or whatever (various types of number etc.)).

The Fullerites invest in an alternative model of 2nd and 3rd powering,
using a triangle and tetrahedron in place of a square and cube, so "2
to the 3rd power" is not synonymous with "2 cubed", although the
numeric answer is the same (just not the geometric interpretation).

The goal is not to displace conventional thinking here (as if that
were even possible), but to illuminate the possibility of logical
alternatives.

Given this alternative "tetrahedroning" versus conventional "cubing",
the standard volumes table is thereby altered, such that the regular
tetrahedron (and/or fractional parts thereof) becomes the common
denominator for other shapes, e.g. an octahedron of volume 4 (the
complementary space-filler) and a rhombic dodecahedron of volume 6
(also a space-filler on its own).  The cuboctahedron comprised of 12
balls around 1 in the closest packing conformation, has a volume of
20, with 42, 92, 162... balls in successive layers (with said number
of layers relating to the "frequency" of said cuboctahedron).

Related pictures:
http://www.4dsolutions.net/satacad/martianmath/mm15.html
http://www.4dsolutions.net/satacad/martianmath/mm23.html

This easy whole number volumes set is suitable for sharing at the
middle school level, as I was doing earlier today in my classroom.

Volumes Table:
http://www.4dsolutions.net/satacad/martianmath/mm14.html

Where the Fullerites seem most contrarian is in not seeing this as n+1
basis rays mapping an n-dimensional space.  Rather, what's ordinarily
conceived as "three dimensional" per that quote from Kant earlier in
this thread [1], is renamed "four dimensional" thanks to the four rays
of the tetrahedron (a, b, c, d).

However this 4D space is purely conceptual, setting the stage for
abstract spatial geometry, but is underspecified in terms of singling
out energy events.

3-tuples are actually insufficient to single out a specific event or
happening, as we need to specify the location and orientation of the
XYZ apparatus itself (at the center of the sun? x-axis pointing
where?) and then give an idea of when on what time-line, which adds to
the specifications.  The task of actually referencing events was never
so simple as some textbooks might have us believe, eh?

A vocabulary that developed from these seed concepts (e.g. "4D
tetrahedron") distinguishes "angle" (pure shape) from "frequency"
which latter endows abstract templates with energetic features (e.g.
bigger or smaller than a breadbox).  This "angle vs. frequency"
distinction analogizes with the notion of "class type" (blueprint)
versus "instance object" (special case) in object oriented
programming.

The dimension number increases from 4 minimum (template tetrahedron)
through these additional "energy dimensions" (4D++) used to specify
actual objects and events somewhere in memory (the shared record).

Philosophies come and go of course, as do "purely mathematical"
languages.  Making some room in the sun for yet another use of
"dimension" is a fairly routine operation by this time, given how many
language games already involve it in one way or another.  I'd say the
Wikipedia page on "Dimension" could use an update, maybe next to that
quote from Kant.[2]

The multiplicity of namespaces (more mentioned below) is potentially a
source of confusion, as some seek one single "monolithic meaning" for
all these key terms (e.g. "4D"), apparently unaware that the Tower of
Babel was never completed (and can't be in principle -- was a moral of
that tale).  We celebrate diversity in mathematics (as in philosophy),
not some totalitarian hive mind.

Kirby

[1] http://www.mathforum.org/kb/message.jspa?messageID=7143092&tstart=0

"""
In 1783, Kant wrote: "That everywhere space (which is not
itself the boundary of another space) has three dimensions
and that space in general cannot have more dimensions
is based on the proposition that not more than three lines
can intersect at right angles in one point. This proposition
cannot at all be shown from concepts, but rests immediately
on intuition and indeed on pure intuition a priori because
it is apodictically (demonstrably) certain."
"""

[2]

"""
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....
All conceptual consideration is inherently four-dimensional.
Thus the primitive is a priori four-dimensional, always based
on the four planes of reference of the tetrahedron. There
can never be 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.
(527.702, 527.712)
"""

------- End of Forwarded Message

Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Joe Niederberger
In reply to this post by kirby urner-4

- ----- Original Message -----
From: "kirby urner" <[hidden email]>
> > [Kirby]
> >> The usual spurious arguments that pass for educated opinion.
> >

[Joe N]

> > Hardly - read about negative and imaginary numbers - they
> > were thought to be nonsense exactly because they didn't
> > seem physically possible. Very mcuh the same thing.
> >
>
> Yet people had ordinary physical interpretations right away.
>
> Look at any early grade school textbook when negative
> numbers get introduced.  "Below zero" on a thermometer.
>
> Owing money (having a negative balance).
etc. etc.

Beside the point - the examples were well known, yet some
highly educated people still couldn't accept "negative numbers".
Here's Carnot in 1813 (he was not uneducated, even in 1813):

"To propose that an isolated negative quantity is less than 0
is to envelop thw science of mathematics ... in a labyrinth of
paradoxes each more bizarre than the other: to say that it is
nothing but an opposed quantity is to say nothing at all..."

To Carnot, it was his concept of what "quantity" and number
*should* be, that prevented him from accepting "negative quantites"
or "negative numbers". (Along similar lines people proposed
treating the number as "quantity" and the sign as "quality",
or the number as "noun" and the sign as "adjective".)

Ultimately, the people who *wanted* to think of numbers that way
lost out - either they died out, or found other careers, or keep silent
for the most part and put their public utterances into more normative
terms.

Your whole line of argumentation strikes me the same way - witness
your remarks below - to me it seems like the same sort of thinking
that wants to tame negative signs by demoting them to "adjectives".

[Kirby]
> Instead of asserting hyper-dimensional polytopes "really exist"
> (meaning what?) we might simply say they're metaphorical and
> the analogies inhere in the algorithms, what it means to
> "spin" for example (or to turn on an "axis").

My guess is most people who talk about dim > 3 in mathematical
settings don't worry too much about physical interpretations unless
that is specifically the subject - and in that case its still a question
that
is up for grabs, Kant notwithstanding.

[Kirby]
> In Einstein.4d, the "time" dimension is actually treated
> differently, algebraically, stands out from x, y, and z as
> distinct.  Disguise the space and time dimensions with
> four random letters, and you'll be able to distinguish them
> anyway, because of how space and time coordinates are
> treated.

I'm thinking of both Minkowski spacetime and Euclidean
4D as primarily vector spaces. The dimension is the number
of basis vectors needed. Has nothing to do with the bilinear form.
(Note also, one can change basis, there is no single prefered
"timelike" direction.)

Joe N





------- End of Forwarded Message

Reply | Threaded
Open this post in threaded view
|

Re: Fourth Dimension (4D)

Kirby Urner-5
In reply to this post by kirby urner-4
Date: Thu, 05 Aug 2010 12:09:25 EDT
From: Timothy Golden
To: [hidden email]
[ Reposting from:
http://mathforum.org/kb/plaintext.jspa?messageID=7145949
]

====

You are coming along, Kirby. I did check out your links.
The triangular format in the first link is very important.
This happens to be the nonredundant format of the
electromagnetic tensor, and the format of the polysign
progression.

Yes, I understand that your own quadray representation is
consistent with the simplex coordinate system. I link to
your site from my site
   http://bandtechnology.com/PolySigned 

The fact that you have a vector space consistent with
existing Euclidean goemetry goes in conflict with Fuller's
insistence on challenging the Euclidean version.

====

That's an interesting comment Tim, however may I suggest
that you consider Fuller as role modeling beginning with
ordinary Kantian space (per that Wikipedia quote
elsewhere in this thread, which Kant still considers
"three dimensional" per standard hand-waving) and
"rebranding it" such that we're not duty-bound to call
it "Euclidean" if we don't feel like it.

Like, how alien do one's definitions and/or maxims need
to be to get out from under that "patent" as it were?  
Not that "Euclideanism" is copy protected in any way, more
it's jealously guarded by an unofficial license or meme
cloud -- and yet other civilizations and ethnicities
might want to make their own claims to this space, sans
this particular identification.  Yes, it's a kind of
intellectual property type issue, almost like "who owns
the idea of banking?" -- safe to say not all banks charge
interest, may instead take a percent of profits if the
loan proves profitable, with a socially responsible track
record for repute.

Fuller's stance against the prevailing "zero, one, two,
three" launchpad, using successive right angles to
add dimensions, is somewhat polemical, as he allies
himself with Democritus contra Euclid -- also with Newton
and Euler, i.e. he's not trying to come off as some "me
against the world" type (he has his "Facebook friends").

By calling it (Kantian space) 4D, and at one time even
renaming the cuboctahedron to "Dymaxion" (which he took
back, regretting the hubris), he is creating a space in
the sun for a "geometry of thinking" that's not
necessarily Euclid's (nor "Euclidean"), yet remains
appreciative of Euclid's results and uses them whenever
convenient and proper.  The Pythagorean Theorem gets used
as well, remembering it may be proved with triangles,
not just squares, per this Martian Math web page:

http://www.4dsolutions.net/satacad/martianmath/mm15.html

As for myself, I've recently allied myself with Aristotle
and his apologists on the issue of whether he was wrong
about pyramids filling space.  His 'On the Heavens' is  
conventionally interpreted to include some claim that
regular tetrahedra (all the same size) fill space without
gaps, which they don't, which you'd think he'd have known.

The defenders of Aristotle call this a "straw man" argument
as the attackers are simply setting up an indefensible
position only to knock it down easily -- a kind of "cheap
shot" in other words.

http://mathforum.org/kb/thread.jspa?threadID=2084375&tstart=0

Furthermore, I link Aristotle's correctness about the
pyramid directly to the four space-filler tetrahedra we
know about from D.M.Y. Sommerville in his 1923 paper on
the subject.  He has only the four:  Mite, Rite, Bite
and 1/4 Rite (using Fuller's terminology), the Mite
building both the Rite and the Bite, so being considered
a "most primitive" in Fuller's writings.

====
When these principles are take general dimensionally it is
exposed that the real line itself is one of these simplex
coordinate systems. It is likewise true that the Fullerite
mentality does not care to consider the general
dimensional condition, and instead insists upon remaining
in three dimensions. This is a sad thing, for the time
representation of the single verticed simplex has gone
overlooked by the real valued thinking era. The simplex
can get us out of this mindset, but this realization will
require a more general treatment. Simply come down in
dimension to the plane and we see a three rayed coordinate
system. In 1D this is a two rayed coordinate system, and
then down beneath here is the one rayed system of
unidirectional zero dimensional time; the grail whose buzz
is all about now.

 - Tim
====

Once conceptual Kantian space is rebranded 4D, yet still
a theater for beach-drawn theorems / constructions ala
Euclid (spherical version, approximately flat when zoomed
in), there's a need to bridge to physics and talk about
"energy dimensions" ala least action per time frame or
hf (= E).  Even if you're not interested in physics,
there's a need to share the road when it comes to
"dimensions", with "time" seeming one to all agree upon.
Fuller has passages on time as the only dimension, i.e.
a time tunnel is a precondition for even just an imaginary
space, a theater for change.  

These are not entirely new philosophical arguments or
themes, however in recasting his thinking in an inventive
language, complete with tetrahedral mensuration and
alternative models of 2nd and 3rd powering, he's provided
some added leverage to our "question authority" types,
giving them an edge where needed sometimes, i.e. some
companies languish simply because of stuck-in-the-mud
attitudes whereas a quick shot of beyond-the-cube thinking
might be just the "fresh blood" that'd snap 'em out of
their intellectual torpor.  We shall see.

Kirby

------- End of Forwarded Message