modified 1998 Jan 04.

Thoughts on dimensions, dimensionality, ... Some of my thoughts, and a lot of other people's thoughts.

David Cary also maintains related pages:

non-physical dimensions; dimensionality as a learning aid for *particular* concepts.

see also "thought-space"

The geek code http://www.geekcode.com/ uses over 32 dimensions (!) is one of the more humorous ways of categorizing humans using many more than 3 axis (dimensions). [FIXME] A more serious method uses only 4 dimensions.

_Cyberspace: First Steps_ by Benedikt "a good article on N-Dimensional visualization" -- recc. Jerry Isdale/Sysop, 72662.410@compuserve.com

"Random Versus Rational Which Is Better for General Compound Screening? " http://www.netsci.org/Science/Screening/feature09.html by S. Stanley Young, Mark Farmen, Andrew Rusinko III asks "What is the inherent dimensionality of chemical space pertinent to drug design?" and gives an estimate of "11 to 18 dimensions". "It would be very useful to have some small number of variables that are pertinent to biological activity. Computational chemists can search for the magic few dimensions or analysis of data might indicate the important few dimensions for a particular type of biological activity. The fact that high dimensional spaces are so large is often termed "the curse of high dimensions." In the drug design situation it means that it is unlikely that a compound good enough to be considered an optimized drug is to be found by screening 10,000 or even 100,000 compounds. Experience supports this assertion." "The many facets of a molecular structure that are pertinent to drug action can only be described in high-dimensional space. Reduction of such space using multi- dimensional scaling ... or principal components ... have been used to simplify the problem of selecting a "diverse" sets of compounds. These methods project points from a high dimensional space into a lower dimensional space, but are most likely an over-simplification."

Date: Mon, 29 Jan 1996 09:18:52 -0700
To: miiko sokka <miiko.sokka at pp.inet.fi>
From: cary at agora.rdrop.com (David Cary)
Subject: Re: 1-2-3-??? dimension?

I'm also interested in dimensions other than 3.

There are some digital signal processing (DPS) theorems that are most easily explained in N-dimensional space (where N is the number of basis functions, or the number of data samples).

In particular, the sphere-packing probem, where you want to pack the most "spheres" as possible in a fixed volume, corresponds exactly to finding the best signal/power ratio given a fixed noise energy. We usually use x1 to indicate the distance in the first dimension, .... x25 to indicate the distance in the orthogonal 25th dimension ... xN to indicate the distance in the Nth direction.

This uses "plain" Euclidean geometry, where the distance between 2 points is

d = sqrt( x1^2 + x2^2 + x3^2 + .... + x25^2 + ... + xN^2).

Special Relativity (I don't know about General Relativity) is most easily explained using 4 dimensions, x,y,z,t,
where t is the "special" fourth dimension known as time.

The "proper spacial distance" between 2 events is
d = sqrt( x^2 + y^2 + z^2 - t^2).

The "proper time" between 2 events is
d = sqrt( t^2 - (x^2 + y^2 + z^2) ).

Fractal theory is concerned with non-integral dimension. For example, in fractal geometry, it makes sense to say Britain's coastline is "1.2 dimensional". Most of the fractals people talk about are embedded in 2D space and have dimensions between 0 and 2, merely because these are the easiest to draw on 2-dimensional computer screens and print out on 2-dimensional paper. There's quite a few people working on fractals embedded in 3D space that have dimension slightly higher than 2.00. However, I know one person who is working on visualizing a fractal that is embedded in 4 dimensions (I forget the exact dimensionality, but it's obviously less than 4.00). I don't know any other work on fractals with dimension greater than 3, but that's more than likely merely due to difficulty in displaying it rather than any mathematical problem.

miiko sokka <miiko.sokka at pp.inet.fi> wrote:
+I would like to know what different dimensions there are according to
+theories. I know something about fibretheory (is it correct?) where is
+said to exist at least 10 dimensions.
+Yes, and I want also to know what theories there are.
+And what theory says that time is the fourth dimension? What are the
+opinions of modern day physicists? How many agrees with who?
+What you say about imaginary dimension?
+Greetings for all, Miiko J. Sokka

(posted and emailed)
Newsgroups: comp.graphics.algorithms
From: jim@kd3bj.ampr.org (Jim Paris)
Subject: Re: 3D shadow of 4D object ?
Sender: news@kd3bj.ampr.org (news)
Organization: The KD3BJ Usenet BBS
Date: Tue, 1 Oct 1996 01:28:37 GMT
Lines: 26

MORTALIS <lobo@iamerica.net> writes:

>> It is an object that extends into the fourth dimension.
>duh! I was wondering what the 4 dimension was? Is it just an
>attribute(that could be anything) or what?

Sorry, I was in a bad mood.  When he refers to a 4D object, he's
referring to an object that extends into the fourth spacial dimension.
It's very hard to imagine; only if you can look at the 3D axes (X, Y, and
Z) and imagine a line that is perpindicular to every single one of them
can you imagine the fourth dimension.

It's usually referred to as W, ie, (X,Y,Z,W).

The original poster was interested in 4D objects and one of the best ways
to study them is to project them into 3D, just like a 3D object can be
projected into the 2D computer screen.  Usually, the 4D object is
projected to 3D, then 2D, for display on a computer screen, which
_really_ causes a loss of data.  One program I've seen, called HyperGeo,
actually let you use 3D glasses so you were only projecting once - it was
very effective.

-blackbob [jim@jtan.com]
From: tao@sonia.math.ucla.edu (Terence Tao)
Newsgroups: sci.physics.relativity
Subject: Re: The Basis of Special Relativity
Date: 3 Mar 1997 19:45:56 GMT
Organization: UCLA Mathematics Department
Lines: 30
Daryl McCullough <daryl@cogentex.com>
>> 2.  Describe what happens within a posited physical space.  This is
>> the objective approach.
>It is also the approach of special relativity.

True, but I think you and Henry are talking at cross purposes as to
the meaning of "space".

Henry seems to think that the only space in which physics can take place
is in the familiar three-dimensional space that we experience at every moment
of our existence.

That, though, is like Plato's prisoners thinking that the only space
which is truly physical is the two-dimensional cave wall on which
they see their shadows.

Usually in physics the "space" that things live in is quite different from
R^3.  In Hamiltonian mechanics phase space is used, which is usually
6N dimensional (N = number of particles).  In relativity four-dimensional
spacetime is the best stage in which to set the theory.  In quantum mechanics
a Hilbert space is used.  In string theory a high-dimensional spacetime
manifold is the setting.  Nobody really knows what the
proper "space" is for quantum gravity yet.  And so forth.

We see only three dimensions at a time, but the space that things _really_
live in could be much, much bigger.


non-integral dimensions (fractals) and "power laws"

J.B.S.Haldane: _On Being the Right Size_ (Oxford University Press, London). DAV: if this is the book I think it is .... very good on scaling laws.

see also computer_graphics_tools.html#fractals

Date: Wed, 21 Feb 1996 00:48:14 -0700
To: Sandra Hawkes <hawkes at unixg.ubc.ca>
From: cary at agora.rdrop.com (David Cary)
Subject: dimensions

Well, <blush> I haven't put on any exhibits; only shown my art to a few friends. The images I create are abstract fractals, not in any way representative of real objects. They're just patterns that "look cool" (or in the words of my friends, "They're eye-candy").

I really enjoyed _The Algorithmic Beauty of Plants_ by Prusinkeiwicz and Lindenmayer. I usually skip the math, look at the extremely pretty color pictures. _The Fractal Geometry of Nature_ by Benoit Mandelbrot is the book that really popularized "fractals", objects with fractional dimensions -- like trees, coastlines, clouds. (I read this years ago, and I've forgotten if it was difficult/easy to read).

David Wright has put some extremely abstract fractals on his page
(I don't understand any of the math, I just look at the pretty pictures).
I've actually met David Wright, and he actually studied under Benoit Mandelbrot.

Web browsers are getting really easy to use once they're installed -- just point and click. But sometimes it takes a *long* time to get them installed :-(

There are some aspects of fractal geometry that are easy to explain to a 6-year-old, but there's also some hairy math that is still a little beyond my grasp.
It's a lot like farming. A child can understand harvesting the wheat, but if you yank the side off the harvester it would be difficult for him to understand all the little gears and gizmos that pull the wheat through the machine. Then there's the concept of wheat hybrids and the gene pool which ... I really don't understand.

Hm. "dimensions for children". I'll have to think about that one. There's a book called _Flatland_, by Abbot, I think... it's a fairy tale that explains "1 dimensional", "2 dimensional", etc.

Were you able to access the
newsgroups ?

>From: Sandra Hawkes <hawkes at unixg.ubc.ca>
>To: David Cary <cary at agora.rdrop.com>
>I've cut back on my screen time.  I just can't sit and read for long.
>Can you recommend a easy to understand book?  Maybe even a book on how
>the dimensions function written for children? That would be most
>helpful.  I have serious problems understanding physics, even basic
>math.  I've read children's books on other adult topics and it worked for
>The stuff about your art sounds really interesting.  Had any exhibits?
>An exhibit catalogue perhaps that I could look at?  I have text only
>capabilities in how I read the newsgroups.  I'm not complaining. Am
>grateful to be able to read them.  And I really don't want to attempt to
>upgrade my computer abilties. Gotta keep things really simple.

HyperSpace: 4 or more physical dimensions

From: gambit16@cris.com
Newsgroups: sci.physics
Subject: REQ: Hyperspace type pages
Date: Mon, 16 Sep 1996 20:05:12 GMT

	I am interested in visiting URLs concerning theoretical
physics, specifically centering on the ideas of high dimensions and
parallel universes, like those discussed in Michio Kaku's Hyperspace.
If you have any such URLs, can you please email me.  Usenet is so
informal.  Thank you for your time
Newsgroups: comp.graphics.algorithms
From: jim@kd3bj.ampr.org (Jim Paris)
Subject: Re: 3D shadow of 4D object ?
Organization: The KD3BJ Usenet BBS
Date: Tue, 1 Oct 1996 00:10:21 GMT
Lines: 16

Yamaha / XYZZ <scriven@CS.ColoState.edu> writes:

>If you want to know more about four physical dimensions, try reading
>"The Boy Who Reversed Himself" by William Sleator, or "The Fourth
>Dimension" (author unknown), or an older book called (I think)
>"FlatLand".  The first two are about four physical dimensions in
>relation to our three.  FlatLand is about three dimensions in relation
>to a two-dimensional creature.

Or try "Hyperspace".  It's a very good book on the subject.

The sept/oct issue of Quantum also has some information on 4D space..

-blackbob [jim@kd3bj.ampr.org]
From: Debbie Foch/LIBRARIAN, 74354,1675
To: John Ratcliffe, 70253,3237
Topic: Fun With Tony
Msg #2761, reply to #2757
Section: HOT! Roswell Video [20]
Forum: UFO
Date: Sat, 1995 Jul 29, 21:56:07

Hi John,
I agree with your comments to Pete. The problem is that I don't think we or scientists in general understand the true nature of the universe and what it means to be hyper or multidimensional. Existence or travel outside of space-time is of course quite foreign to us; however, that doesn't mean it's not possible. Use of non-linear types of communication, outside of space and time, is also foreign to most of us (at least consciously), but is none-the-less valid.

Take care, Debbie

From: "Timothy J. Ebben" <time2@visi.com>
Newsgroups: sci.physics
Subject: Re: EPR Solution 4D-Space!
Date: 18 Jan 1997 15:49:37 GMT
Organization: Ascension Enterprises, Inc.
Lines: 165
X-Newsreader: Microsoft Internet News 4.70.1160
George Penney <gpenney@thezone.net> wrote
> 		     A Solution to the Einstien-Podolsky Paradox.                  This
> is a solution based on 4D Space.If a Flatlander lived in his 2D universe
> which consisted of a flat plane and we intersected it with a circular
> which say was spring loaded so we
>  could make it bigger or smaller at will.Now if we laid this rind flat on
> his plane,he would see a circle,if we now lift the ring out of his plane
> it is inclined 90 degrees to his plane he would see two points (or
> dots),seperated by the dia of the ring
> ..We then rotate the ring(say clockwise),perpendicular to the plane,he
> observe them moving in unsion in his space,but not conected.He would
> conclude they obayed some law(such as a force between them that made them
> move in unsion).He then positons him
> self on one of the dots(lets say one is blue and the other red,he's on
> blue dot).He can't visalize why they move together as he is in
> then make the dia of the ring larger so the dots are further apart,he
> can see both and that they ar
> e moving in unsion.Everytime the blue dot moves so dose the red dot.Then
> slowly increase the dia so it's dia is 6x10^8M in dia(The speed of light
> his universe is 3x10^8M/S,also we can rotate the ring as slowly as we
> like). We now reverse the rotatio
> n of the ring CCW and although he cant observe both the blue dot (which
> is on),and the red dot simaltanesly he just knows that the red dot has
> reversed it's direction.But he reasons how can this be since information
> can't travel between them at gr
> eater than the speed of light?? Note now that the flatlander now has a
> paradox the same as the Einstien-Podolsky paradox.Two pairs are
> communicating information(as QM would predict),but faster than the speed
> light)??.Of course what he dont know is that
>  the pairs are connected in 3D-Space.If he did he can conclude that
> Reletivity and Quantum Mech are not in violation of each other!!!.Thus we
> have a resolution of this paradox.The same would apply if we had this
> paradox in our 3D-Space and concluded that
> the pairs are connected in 4D Space.

>          In working out this solution I also noticed a peculiar property
> partical spin in 4 or higher dimensional spaces.It goes as follows:---
> First lets discuss some aspects
> of an n-dimensional object intersecting an (N-1)-dimensional space. I'll
> this by going back to the flatlander and our 3-space.In the flatlander's
> universe his circle is the same to him as our sphere(Keep this in mind)
> that he can't enter his circle
> without breaking it's circumference(lets assume his circle is not solid
> inside like we would have if we shaded the circle inside).To us in
> we could step inside his circle without breaking the circumference due to
> the fact that we have access to 1
> more dimension than he has.His circle is the same as our 3D hollow
> sphere,we woudn't be able to enter our sphere without breaking it's
> but it could easily be done from  4D-space because in 4-space our
> would be equivalent to their 4D circ
> le.Now if we inter- sect the 2d circle with out sphere perpendicular to
> in the center of his circle passing the sphere down through the plane of
> the circle,if he were inside he would see a dot that would become a small
> circle that would get bigger in
> dia as we continued to pass the sphere through the 2D circle--- (let the
> dia of OUR  sphere = the dia of his circle),midway through our sphere
> form a concetric circle with his own,then start to decrease in size back
> a point and finaly dissapear c
> ompletly!!.This would seem very odd to him as all he is observing are
> sectional pieces of our sphere at any one instant in time.It would be the
> same as if we saw an object suddenly appear in our 3-Space,continualy
> change shape and then dissapear.We
> can get even stranger effects if we intersect irregular shaped geometric
> objects from higher spaces into lower spaces.Of course the flatlander
> visualize our sphere due to his restriction of being confined to his 2D
> universe,however he can construct
> the laws of 3D-Space Geometry based on the cross sections that he seen of
> our 3D objects Similerly we can construct the laws of 4D-Space and
> N-Dimensional Spaces and their Geometries.

>                                 Keepin
> g this in mind let's get to rotation or spin properties of 2D space and
> space,then apply this to higher spaces.If we rotate the Flander's circle
> if HE rotates what he considers to be his SPHERE it can only rotate in
> directions(CC or CCW) or if y
> ou like it can rotate one way or the other[for the terms CC & CCW can
> interchange depending on where you view the rotation from in space].Now
> we in 3-Space take the circle(----- his sphere)and rotate it down-
> ward(around it's dia) into the plane of his
>  2-Space and perpendicular to it we can rotate it in two more directions
> our 3-space!!He would observe two points seperated by the dia of his
> that would be stationary.Again he would not be able to visualize that the
> circle(his sphere) had TWO mo
> re modes of rotation or spin in 3D-Space!From this it follows that a
> in our 3D-Space which has only 2 directions of spin would have MORE than
> directions of spin in 4Space and even more in higher dimensional
> spaces.Like the Flander this goes again
> st our common sense(common sense being that layer of predudice layed down
> prior to the age of 16---I could'nt resist getting that in).Also we would
> not be able to mentaly visualize this.(unless we were Charles Hinton who
> claimed he could  visualize 4D obj
> ects such as Hypercubes and so on).


>               George Penney




one particular theory with more than 4 physical dimensions

From: Mitchell Porter <qix@desire.apana.org.au>
Subject: >H designer universes, spin networks, and string theory

Transhuman Mailing List

[Eugene Leitl]
> if we can tweak our spacetime to cause
> precipitation of an edge-of-chaos high-dimensional CA universe and
> stabilize the access channel long enough to bootstrap, then we can
> fabricate our own Eden.

I've thought a bit about what this would require. Assume that you're
doing this by causing inflation to occur in a very small region of
space, where (somehow) you've controlled the state of the Higgs and
other fundamental fields so as to determine the effective physics
of the new universe.

Since we're used to thinking in terms of atoms - relatively pointlike
nuclei surrounded by fuzzed-out electrons - let's suppose that the
quanta of the new universe will form atom-like structures, although
the electron orbitals will be hyperspheres rather than 3D. (I'll
keep speaking of electrons, nuclei and atoms, although it should be
understood that these are particles in the new universe which behave
somewhat like those in existing matter.)

Hyper-atoms might form as they did in our universe, either shortly
after inflation, or in stars. Either way, what you will want to do
is to tweak their properties so that they have tendencies to form
a very regular crystalline lattice (of d dimensions). The
hyper-atoms in this crystal then form the cells of your cellular
automaton, and the spin states of some electrons can form the cell

The most difficult thing is to keep the "access channel" (which is,
I presume, an umbilical wormhole connecting the baby universe to
ours) open, since the baby universe is, by hypothesis, of a
different effective dimensionality. I guess it just takes more of that
"exotic matter" which you build wormhole rigging out of. As you pass
down the wormhole throat, some of the space dimensions which are
Planck-size in our universe would acquire a wider and wider radius,
until they were on an equal footing with our three large dimensions.
What sort of intermediate structures this would entail, I have no
idea yet.

There's actually a preprint out there on d-dimensional atoms:

The Periodic Table in Flatland

It focuses on the two-dimensional case.

I think there's also something in the appendices to A.K. Dewdney's

[text lost, but Eugene was saying that in some current string
theories space has "27 (?) dimensions"]
> dimension, but most are curled up real small so that we can't see
> them at our scale.

The original string theory (before supersymmetry) had 26 dimensions -
25 space, 1 time. It's not really current (post-1984 superstring theory
has ten spacetime dimensions, and there are some lower-dimensional
models), although it's been suggested that you can get superstrings
out of 26-dimensional string theory.

[Anders Sandberg]
> On Wed, 17 Apr 1996, Eugene Leitl wrote:
> > If fact, if we can tweak our spacetime to cause
> > precipitation of an edge-of-chaos high-dimensional CA universe and
> > stabilize the access channel long enough to bootstrap, then we can
> > fabricate our own Eden.
> I had a similar idea for how life near the Omega Point could create a
> state in a Penrose spin lattice to evolve towards the Omega Point, even
> if the time direction would become ambigious.

I'd like to hear more about this.

[Eugene Leitl]
> Can one understand Penrose spin lattice, I mean is it physics fit for a
> mere human? Have you a ref handy?

 week 46 
 week 55 
abstract gr-qc/9505006

The first two should be expository posts by John Baez from
sci.physics.research, touching on spin networks, while the latter
is a paper in which gravity theorists Carlo Rovelli and Lee Smolin
actually calculate something that might be real using spin networks.

[Anders Sandberg]
> I think Kaufmann wrote about them in a book about knot theory, they
> didn't look that absurd to me. I'm not sure it is physics yet, and most
> people seems to have moved to string theory instead (a pity, I really
> like the idea of the universe as a kind of random graph with a cellular
> automaton rule).

That would be in _Knots and Physics_ (1993). The paper by R&S, cited
above, might be the first time that an actual physical quantity was
computed using spin networks. (The quantity in question was the
quantum of surface area, in their theory of quantum gravity.)

[Eugene Leitl]
> > > Can one understand Penrose spin lattice, I mean is it physics fit for a
> > > mere human? Have you a ref handy?
> >
> > Well, theoretical physicists seem to handle them, but if they are human
> > is still an open question... :-)
> I also, wonder. I once bought a string theory book from Kaku, just for
> the fun of it -- I didn't really expected to understand anything. But
> I didn't understand not a single line of it: not contents, of course
> not the chapters, nothing. Only the conclusion, because there was no tech
> speak in it. This was a really scary experience, this book could
> have been written by an alien. I wish I could do physics properly, <sigh>.

Is this _Introduction to Superstrings_? I tried to teach myself string
theory from that. I didn't quite succeed, but I learnt a lot. You'd
need some idea of quantum field theory to start with though
(recomendation: Feynman's _QED_).

> String theory is not very aesthetical,
> unless one happens to be a musician, may be ;)

Or a mathematician... The sum over topologies, and the way that
strings in 10 dimensions can be described as 10 fields in 2
dimensions, are the two ideas which stick in my mind as elegant.

One of the hot topics right now is membrane theory (or, as people
insist on calling them, p-branes, p-dimensional membranes) - a
membrane being an entity of 2 or more dimensions (point particles are
zero-branes, strings are one-branes). Some people think that the
superstring theories are all approximations to an 11-D membrane
theory, dubbed "M theory".


Other uses of the term "dimension"

??? http://frank.mtsu.edu/~rbombard/RB/Ex/myst.html Flatland

quaternions (and matrix transforms)

quaternions and matrix transforms (rotation matrix)

see also computer_graphics_tools.html#writing

"Quaternions are not a well known mathematical concept; but ... are the primary choice for representing spacecraft orientation. " "the NPS aeronautical model uses quaternions." -- unknown


the Covariant Theory of physics http://infoweb.magi.com/~jgc/ tries to carefully explain some 4D concepts.

conference on spatial information theory http://www.sis.pitt.edu/~cosit97/ "Spatial information theory is the basis for the construction of Geographic Information Systems (GIS), but also necessary for other uses of geographic information and useful for information system design in general."

Non-Linearity of Thought http://www.catalog.com/sft/bobf/nonlinear.html some practical advice for those who think non-linearly.

On the dimensionality of spacetime Max Tegmark's library: dimensions http://www.sns.ias.edu/~max/dimensions.html

see video_game.html#4dtoys

URL: http://rdrop.com/~cary/html/dimension.html

Started: 1997 ? before Sep 16.
Original Author: David Cary.
Current maintainer: David Cary.

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