How to describe, in Spatial Geometry and/or Physics and/or Mathematics,
an object whose every point is equidistant from every other point?

---

Request for Question Clarification by pinkfreud-ga on 13 Sep 2002 11:12 PDT

Brunews,

Since a clarification statement that you made earlier is no longer
present here, it would be very helpful to researchers if you could
elaborate on your need. As I understand it, you are wanting a
2-dimensional graphic representation which symbolizes the functionally
equidistant interrelationships of users on the World Wide Web. Any
further description you can give would be very useful.

Thanks for a fascinating question! I wish I were up to the task of
answering it. My degree is in English Literature, so about all I can
do in the area of symbolic graphic thingies is to diagram a sentence.
;-)

I am looking forward to seeing more comments and discussion on this.
The mind boggles. The bind woggles. And vice versa.

~pinkfreud

---

Clarification of Question by brunews-ga on 13 Sep 2002 17:52 PDT

pinkfreud: thank you for your interest and for your question:

1- I would like first to compliment and thank omnivorous for the
information he found for his behaviour; I regret that Google Answers
does not allow us to send private email to each other; I would give
omnivorous 5 stars for his graceful answers and conduct.

2- pinkfreud: I don't think a 2-dimensionnal representation is
possible; so, I restate the problem: my work involves medias on the
web and web architecture; I use a lot of graphics to explain the
patterns of relationships on the web; and I have to explain to my
clients that, on the web, each one of their customers is at the same
distance from them; and, at the same time, every one of the customers
is equidistant a from any other one. Hence my question.

I know that such a space can't be conceptualized in 2 or 3 dimensions;
I initially guessed that a description of such an object would exist
in theoritical physics and mathematics, in the vicinity of black
holes, Bose-Einstein condensates (which I do not pretend to
understand), or n-dimensionnal spaces. And I still believe that there
is somewhere an equation which states: "an object x, made from y
points, each point being equidistant from every other one =
???????????"

What is embarrassing in this question is that it is not a theoritical
object: it is a real one where I can every day measure distances
between points. To understand the behaviours of that object, I am
looking for a frame of references, as exotic as it may have to be.

---

Request for Question Clarification by omnivorous-ga on 13 Sep 2002 18:03 PDT

Brunews -- thanks for your kind words.  I felt that it would be best
to withdraw my response in hopes of getting you a more precise answer
and NOT be charged for the early work.

You've caught the Google Researchers imagination, as you can probably
tell.  I'd done a search using the strategy:
n-Simplex + graph + Internet

and found a fairly interesting paper that tries to apply N-simplex
mathematics to resolving Internet router problems, 'Continuous
Geometric Structures for the Internet':
http://mathlab.usc.edu/~bohacek/AMICT/Edmond/talk_to_Srith_group.ppt

Prof. Edmond A. Jonckheere, Department of EE and Center for Applied
Mathematical Sciences, USC concludes in the paper that "A coarse
geometric theory of networks and information flow is possible, but
does not quite amount to 'off the shelves' mathematics."

Sometimes researchers know that there's no answer to a question; in
this case it appears that we're all seeking a better representation of
Internet dynamics.

---

Clarification of Question by brunews-ga on 13 Sep 2002 19:48 PDT

And I thought I was looking for something exotic but simple! Pheeew! I
just read the presentation of Prof. Jonckheerd: pure poetry! What I
mean is that I am discovering that my simple question has been asked
by highly competent people and that there is still not a definite
answer for it, except in exotic Mathematics that I can't understand!

And all this in a space where we are working, playing, communicating
every day. We are there, but we don't know the geography of where we
are!

Omnivorous, I am fascinated by the notion of movement you just
introduced. Not only we don't know the physical representation of all
these equidistant points, but at the same time they can move?

If it is my decision, you can claim the List Price. But it draws me to
another question, and maybe you can help me phrase it before I post
it:

The force that drives the web is not communication (one way
transmission of information) but relationships (two ways transmission,
with equal amount of communication coming from both sides and every
event of the relationship memorized by both parties). So somebody
could help me check the equation where R is the relationship, Ca is
the communication from A, Cb from B, Ma is the memory that A has of Ca
+ Cb and Mb the memory that B has from Cb + Ca. In a perfect
relationship on the web Ca/Cb = 1 and Ma/Mb = 1.

If all these points are in a relationship with other (equidistant, and
all moving) points, the position of each point can be determined by
knowing the Ca/Cb and Ma/Mb they have with those points. Meaning that
the combination "Ca-Cb ? Ma-Mb" acts like some kind of gravity between
those points.

Meaning, and know whe are maybe in cognitive territory: on the web,
how many reltionships can a point have at the same time (a point: a
person, an entity, a corporation); is there  a limit and a mean point
for Ma + Ca for each point? Or for each class of point? (Like: is
there a limit number of dendrites and axones for nerve cells, how much
two-way traffic dendrites and axones can handle, how much memory space
is available in each cell?).

And the magical question a lot of us are asking: how can we direct
traffic in this dynamic topology? Can we produce a topography first?

Am I going somewhere?

In shaping an answer here, we're trying to arrive at a QUESTION: what
is the best physical representation of links between users on the
Internet?

Somewhat jokingly I have to remark that we're in danger of creating
the 'perfect consulting' arrangement here, where answering the first
question creates the next engagement.

Originally we looked at what types of physical problems tried to
resolve the "equidistant" nature of points in space.  We run out of
solutions in 3 dimensions.  But there are a number of related problems
in physics, if the problem is restated slightly.

Note Dave Rusin's math FAQ in the on-line Mathematical Atlas treats
the issue and problem descriptions:
http://www.math-atlas.org/index/spheres.html 
 
Thanks to summers-GA for pointing out that the generalization of
equidistant points is a simplex.  For dimensions beyond 3, n-Simplex
refers to the number of dimensions.  Mathematicians have attempted to
apply simplex theories to Internet modeling, without success so far. 
Edmond A. Jonckheere, of the Department of EE and Center for Applied
Mathematical Sciences, USC
attempts to model router traffic using n-simplex tools but says in his
paper 'Continuous Geometric Structures for the Internet,'
 "A coarse geometric theory of networks and information flow is
possible, but does not quite amount to 'off the shelves' mathematics."
http://mathlab.usc.edu/~bohacek/AMICT/Edmond/talk_to_Srith_group.ppt 

Modeling particle spacing with randomly-distributed points in 3D may
be adequate to describe the Internet because there are:
*  no barriers between the points 
*  a common link (TCP/IP network) 
*  sub-networks are easily linked other segments or network node . .

This would produce an image similar to the animated GIF on the page
"Particle System Example":
http://astronomy.swin.edu.au/~pbourke/modelling/particle/ 
 
The last image implies that Internet users are randomly distributed in
space and though they may be physically close, the Internet path may
still take their e-mail through the AT&T mail server in Naperville.

As an example, last week someone asked GA researchers how to find the
e-mail addresses for a neighborhood?  Though separated by mere meters,
two neighbors are separated by physical link to the network; by ISP;
by firewall; by Internet communities in they participate.  Absent the
assembly of an e-mail list by the homeowners' association, I have no
way of knowing if a neighbor is on-line.

Problems exist trying to establish any models.  It would be perfectly
valid for you to argue that we're all still "equidistant" because at
Internet speeds, a 'ping' on average will get a response faster from
any IP address than it takes me to walk across the street.

Some ways to express this question afresh may be: what have been the
most-promising five mathematical models to show Internet relationships
or Internet performance?

Some search techniques that may prove fruitful would combine the terms
Internet with a physical representation AND a problem-solving
technique, such as:
Internet + "neural networks" + "data mining"
or
Internet + "traffic analysis" + "network monitor"

Finally, the Open Directory project can quickly get you to some
interesting topics of the topology, structure or relationships of the
Internet:
http://dmoz.org/Computers/Internet/Cyberspace/

We'll be waiting for the next iteration of this topic!

Best regards, 
 
Omnivorous-GA

Very challenging and interesting exchanges to reach a very thorough answer.

Your concept of an object each of whose points is equidistant from all
its other points would, I believe, be feasible in zero-dimensional
physics, which is also known as zero-brane (sic) physics.

Quantum physics has, in many ways, brought science right up to the
shimmering edge of spirituality. I recommend two books which deal with
the overlap between the physical and the mystical: "The Dancing Wu Li
Masters," by Gary Zukav, and "The Tao of Physics," by Fritjof Capra.
Neither book is new (both were initially published in the 1970s,) but
a thought does not need to be new to be worth thinking.

Brunews -- my last clarification has been excised after my request to
withdraw the answer.  I had seen all of your notes and suggested
considering the animated GIF image (spherepoints3.gif) on the page
'Modelling particle spacing in 3D':
http://astronomy.swin.edu.au/~pbourke/geometry/spherepoints/
 
The image has the following attributes common to the web:
*  infinite number of randomly-distributed points
*  no barriers between the points
*  a common link (TCP/IP network)
*  if this image represents a 'segment' of the network, it's easily
interlinked to any other segment or network node . . .

Plus, it moves, meaning that your customers will immediately be drawn
to the image and the story you tell!

The form where every point is equidistant from every other point is
called a simplex.  In one dimensin it is a line segment, two
dimensions a triangle, three dimensions a tetrahedron, and for four
dimensions it is called the simplex, or the 4-simplex  So for n
dimensions (n points in this case) we call it a n-simplex.  It can be
represented by a complete graph on n-vertices if you like a two
dimensional projection: that is, draw n points in a circle and connect
every one to every other one.

omnivorous: thank you; there is a very important sentence at the end
of your comment: "Plus, it moves, meaning that your customers will
immediately be drawn
to the image and the story you tell!" This sentence is fascinating,
but I don't  understand how do you go from the previous sentence to
this one? I am missing an essential step.

summers: thank you; omnivorous told me the answer for 1, 2 and 3D, and
you provide a name and a surname; so we are in an n-simplex. Wow! I
have done a quick research and come back with "The Ehrhart polynomial
of a lattice n-simplex" which, I think, is what I am looking for, and
which, of course, I can't understand because I am not a mathematician.

Interesting crossroad. Any advice?.

>*  if this image represents a 'segment' of the network, it's easily
>interlinked to any other segment or network node . . .
> 
>Plus, it moves, meaning that your customers will immediately be drawn
>to the image and the story you tell! <

It's a switch from physical representation of an idea to what we know
about human reactions: movement attracts the human eye.  It means that
anything in type will receive less attention than the rotating image;
you have the aural attention with your presentation so can explain
Internet infrastructure in your own terms -- or spin a tale of your
choice!

Omnivorous: creating the 'perfect consulting' arrangement here, where
answering the first question creates the next engagement" is not a
danger: it's a very exciting way to proceed (as long as the next
engagement leads to a new question on GA). ;)

And thank you to all who gave directions to this topic.

I would say that such an object in N dimensions is described by the
vertices of a regular N-simplex.  In 1D it is the vertices of a
segment.  In 2D it is the vertices of a equilateral triangle.  In 3D
it is the vertices of a regular tetrahedron, in 4D it is the vertices
of a regular hyper-tetrahedron, etc...

Each of these compound "objects" satisfies the condition that every
vertex is equidistant from every other vertex.

Thhank you, Andy22.

Your know what? A few days ago, I had lunch with a friend. And at the
end, expresso time, we were joined by a friend of his who happens to
dabble in physics. The kind of guy who deals in neutrinos and makes a
living with it. And he says that my space is a point: in every point
in one dimension, there is an infinity of equidistant points.

So, I'm torn between an n-simplex (n dimensions) with n-points or a
1-simplex with n-points. Frankly, I don't know which one to chose, but
both are fascinating. :-)

You're very welcome, brunews.  I would like to add another comment to
all this.

The problem of finding the optimal connection of N points in an
M-dimensional space,
where M < N, has been looked at before in many fields of research. 
Two concepts spring to
mind that might be relevant to this question:  distance metric and
meshing.

1 - Metrics:

In order to even ask a question involving the term "equidistant", one
must have some concept
of the meaning of the axes of the space in which the points are
embedded, and a concept of how
to measure distance within that space.  In our case, we are going to
try to define what the axes
mean in our M-dimensional space.  You seem to suggest that your points
are embedded in
a two-dimensional space:

p = ( p.c, p.m ),

where p.c is the communication component and p.m is the memory
component.
(I've no idea what these mean, but we can get back to that at a later
time.  The point
is, it has meaning to *you*.)

The distance metric - the way you're going to measure how far 2 points
are in the
2-dimensional space - will be used to verify when we our graph is in a
"perfect relationship".
The obvious metric to  use is the flat euler metric in M-dimensions
which is computed for 2 points,
a and b, as follows:

d(a,b) = sqrt( (a.1-b.1)^2 + (a.2-b.2)^2 + (a.3-b.3)^3 + ... +
(a.M-b.M)^2 )

and in our case, the metric is just the 2-d flat metric:

d(a,b) = sqrt( (a.c-b.c)^2 + (a.m-b.m)^2 )

(The Euler metric is not the only one we can use.  In special
relativity, a Riemann metric is used
which leads to a curved space.  This may come into play in this
problem.)

If all the points are plotted in 2-d according to how far they are
from a reference point, O, and we
choose the appropriate reference point, then we obtain a cluster of
point in the (c,m) plane.

2. Meshing

Now we might want to try to reconnect them.  That's called _meshing_. 
   And we're lucky because
there is alot of information available for meshing points in a plane. 
Delaunay triangulations have
the property that every connection is as "equidistant" as can be.  
I'm not representing it very
well, here, but that's something we can expand on later.

Keywords to look for are (Delaunay triangulation, incircle condition).

There is one thing that I don't yet know how to factor into all this. 
You seem to imply that the
commmunication from a-> b is different than the communication from
b->a.  This means:

d( a, b) !=  d(b,a)

This implies the network exhibits some kind of anisotropy, or
hysteresis.   I need to think
about this some more.   

Hoping this helps,

Andy22