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Re: Continuing the story - another stab at an IETF mission statement
2004-03-11 06:30:21
On Thu, 11 Mar 2004, Einar Stefferud wrote:
Sorry to have disturbed you Robert;-(...
You didn't disturb me. I love this stuff:-)
But, I have to wonder what this URL is about, given your comment about
geometry
being not spherical.
OK, I will explain a bit of history and mathematics. Briefly, since
this is way off topic. Geometry is the study of objects in a space of a
given number of dimensions and a given type. Historically it of course
began with (actually somewhat before) Euclid's famous Elements and plane
geometry as the first axiomatically developed branch of mathematics.
This was so cool that it was damn near turned into a religion for close
to 2000 years.
Axioms are unprovable assumptions upon which the mathematics is based.
They are NOT (as is often asserted) "obvious truths" -- one can choose
the axioms, and different choices (as was discovered by Gauss and
Riemann only a couple hundred years ago) lead to different geometries
(note well the plural) in particular to non-planar geometries. Euclid's
was planar, and plane geometry is one of the few really standard and
universal exposures to mathematics even non-math/science oriented
students are subjected to.
Formally, geometry is distinguished from other forms of axiomatically
developed algebra (as yes, there is an algebra associated with each
geometry, including one called "geometric algebra" which contains things
like Quaternions, Grassmann and Clifford algebras) by virtue of having
a) a "manifold" (the "space" mentioned above) and a "metric" (named a
"Riemannian metric" in honor of Gauss's protege, Riemann, who nominally
discovered non-Euclidean differential geometry although it seems
historically likely that Gauss was already aware of it and supported
Riemann BECAUSE he was on what Gauss already knew to be a fruitful
track). The metric of a geometry on the manifold typically defines its
properties.
There are many ways to vary the manifold, the metric, the axioms. One
can make the manifold 1, 2, 3...N dimensional (where N can be infinite,
and in some branches of mathematics is). The metric is what we would
call the measure of distance -- given two points in the manifold, how
can we compute the distance between them. Note that this isn't
precisely true but I don't want to get into differential geometry and
local/differential definitions of the metric here.
A spherical geometry is the study of figures on a spherical manifold and
is certainly of interest -- it is the geometry we live in on the planet,
after all. It differs from Euclidean plane geometry in that triangles
have >= \pi radians in the sum of its angles (instead of = \pi radians),
two points define TWO line segments, one short, one long, on a so-called
"great circle", and certain special pairs of points (antipodal points)
define and infinite number of line segments and the possibility of
drawing a "biangle". Two distinct lines always intersect twice. And
more.
NONE of this resembles a useful mathematical description of the Internet
except to the (nearly irrelevant) extent that of course the physical
network is physically wrapped around the roughly spherical globe and
that spatial distance and the speed of light create temporal delays in
packet propagation in a predictable manner (note that this describes the
USE of the network, not its geometry per se and certainly not its
connectivity or routing characteristics). As far as your other remarks
about the Internet being a manifold of some sort, well, it is and it
isn't.
A physical information network is a kind of graph, a directed multigraph
or pseudograph. To be completely honest, I don't know that graph theory
is, properly speaking, a geometry, although I went along with the
metaphor in my previous reply. It is generally considered a branch of
discrete mathematics. One often VISUALIZES graphs as being EMBEDDED in
a geometry, e.g. drawn figures consisting of nodes and links on a plane
or other surface, but in many cases the "metric" is discrete and
completely distinct from the underlying manifold metric. In others it
isn't. The travelling salesman problem is one where there is a
correspondance between a spatial metric and the graph metric (a "cost
function" or "distance" assocated with each link). The famous Bridges
problem also required a 2d manifold (the city of Konigsburg) and a rule
that links could not cross in that manifold. A network where physical
propagation and distance-related delays are not being considered is an
example of a graph metric that is divorced from the underlying
visualization manifold (each link is a discrete "hop" and counts the
same in the total "distance").
The dimension of a network or graph is also an issue. Graphs are nearly
always drawn on a 2d planar manifold for visualization purposes, but one
could argue that they are actually very high dimensional objects where
links open out new dimensions, or where links are permitted to cross
because in a higher dimensional manifold they can go "over" and "under"
one another (as can a network, in fact). I'm not really an expert on
graph theory, so I don't know whether or how graph theory is formulated
as a high dimensional geometry -- I do know somebody over in the math
department who is (and who's worked on networks) so maybe I should ask
him.
BTW, if you want to learn the actual mathematics at least at the
introductory level, Wolfram's "MathWorld" site (mathworld.wolfram.com)
is truly awesome -- a great resource. You'll find graph theory under
discrete mathematics, and a variety of geometries discussed under
geometry.
The reason that I picked nits over your original reply (and continue to
know) is because a) you are correct, one CAN gain insight into the
network by studying its underlying mathematics; and b) you have to use
the right mathematics or you'll gain the wrong insight. Geometry is NOT
spherical, spheres ARE geometrical, there are literally infinitely more
non-spherical geometries than there are spherical geometries (and please
let's not get started on infinity:-) and a network is not a geometry at
all in the formal/classical sense, although geometry may be relevant to
studying a network and is certainly useful when it comes to visualizing
it.
rgb
http://www.google.com/search?q=spherical+geometry&sourceid=opera&num=10
Google only found [Results 1 - 10 of about 221,000. Search took 0.24
seconds.]
Maybe we need to reverse our views of things.
If geometry is not spherical, then maybe it is spheres that are geometrical.
Anyway;-)...\You might find some interesting theorems among those 221,000 web
cites. Or, maybe this one asking about ["spherical geometry" + manifolds]
will
be of even more interest. [Results 1 - 10 of about 563. Search took 0.08
second]
http://www.google.com/search?hl=en&lr=&ie=ISO-8859-1&q=%22spherical+geometry%22++%2B+manifolds&btnG=Google+Search
In any case, I was claiming that the Internet is a Manifold in terms of
spherical
Geometry, (or as you like it) Geomety of Spheres; and clearly there are
manifolds
in the concepts of spherical geometry, and I see no difficulty in mapping
your
network onto my manifold. It is a set of pipes all connected together
somehow
such that a packet can enter via one pipe and flow through the manifold to
arrive
at the exit (given the correct address) of any other pipe in the manifold.
This seems to me to describe exactly what you say about the internet,
so I believe we no longer have any disagreement.
Cheers;-)...\Stef
On Tue, 9 Mar 2004, Einar Stefferud wrote:
It might be interesting to view the Internet through the contextual lens
of
spherical geometry concepts which I think fit as well as anything,
contrary
to some of our historical internautical terminology. For example, in
spherical
Geometry, a manifold has no edges, and has no center, while IETF folk
insist
that the Internet has an edge somewhere (just one) but I have not heard
any
claims that it has a surface, or that it has a center.
Not to be picky, but the geometry isn't spherical. In fact, the
geometry of the Internet is a network -- a network IS a geometry
consisting of nodes (locations) connected by links. The mathematics of
a network is called graph theory. The network geometry of the Internet
isn't horribly well ordered or simple and is highly dynamic. It
certainly isn't (hyper)spherical in any dimensionality -- spherical
geometries have certain properties that the network lacks, although of
course there exists a projection of the physical network onto the
physical sphere (the globe) that provides some useful information.
Less than one might think, of course. The network isn't necessarily
simply connected, for example, as I could go upstairs and unplug my
router and create a network fragment disconnected (transiently) from the
rest of the Internet. The metrics are not obviously connected to real
space geometry on any but a very local scale. For example, I am LESS
than two physical miles away from my office at Duke as I type this.
However, I'm 17 network hops away from my desktop there, and traceroute
reveals that the packets go through Atlanta and Raleigh (it can be worse
depending on congestion and dynamic routing -- I've seen as many as 30
hops).
The network geometry is multidimensional and nodal. One can define a
surface (of a simply connected nodal set) -- the union of all nodes with
a single entry/exit route (link). Similarly, it has an interior (all
nodes with multiple links). It has a norm that permits a discrete
measure of distance to be constructed -- the "hop" from one node to
another (the information revealed by traceroute measures a normed
distance between nodes, albeit quite possibly a transient one and one
where physical distance is nearly irrelevant). It even has a center --
one could usefully define it to be the union of all interior nodes that
are a weighted MINIMUM distance, on average, from the entire surface --
the so called "backbone" -- although this isn't a sharp concept and may
not even be all of that useful because of details of the network.
For example, one can generate a variety of renormalized views of the
Internet where nodes are THEMSELVES networks (or the routers/gateways
that isolate them) -- "rgb.private.net" (my home LAN might be one) --
and the relevant network links are ones that connect routers, ignoring
the edge nodes served by the routers. Then there are aggregations of
LANs (such as duke.edu) which may have multiple links as well as LAN
aggregations that have just a single link. Nowadays although one can
still talk about a network "backbone" people also speak of "clouds" and
use other metaphors to more accurately describe the core connectivity.
A lot of this topology is built into both the internet addressing scheme
and the underlying routing schema. "Usually" a surface node has a
single IP number and is part of a IP LAN that is at least reasonably
spatially contiguous. "Usually" interior nodes have multiple IP
numbers. "Usually" routing attempts to dynamically solve a problem in
the topology such as "how to I get a packet from this node to that node
with a minimal number of hops, strictly less than the TTL value, no
loops, no dropped packets". Even here one has to be somewhat fuzzy as
there are multiple protocols in use in layers -- what does one call an
ethernet bridge, for example, and how do you describe entities such as
compute cluster nodes that might have a proprietary non-ethernet non-IP
interface, or various devices that link to nodes. There are even cost
functions that have to be applied, as some of the intermediary links may
charge a de facto "toll" for transit.
Naturally, all of this has been studied extensively by mathematicians
since Euler and the Seven Bridges of Konigsburg (which more or less
invented the subject), and work continues today. Equally naturally, all
of this has been studied by computer scientists and network engineers
from the pre-Internet beginning, and was very intelligently incoded into
the network as we know it today. Their dynamic solution for routing and
addressability may not be theoretically optimal -- I'm not an expert in
graph theory but I'd be surprised if it was -- but it has proven
evolutionarily to be amazingly robust and more than "good enough" at the
scales it has worked with so far. Note that there are plenty of
networks that do NOT scale -- decnet, appletalk, raw ethernet -- and
that TCP/IP is actually one of the greatest human accomplishments of all
time -- a true wonder of the world -- if one looks at it a certain way.
I think that one of the major questions associated with IPv6 is going to
be whether or not that robustness and scalability persists in the
new/extended model. It is not obvious to me that it will, only because
(as a colleague of mine who works in complex systems is wont to say)
"more is different" -- new structures emerge, often nonlinearly, when
you make something bigger and potentially more complex. I'm optimistic
though, and humans are pretty good at fixing things that don't work so
even where problems emerge I expect that we'll fix them. I'm also
optimistic that a lot of the new structures that emerge will be GOOD
ones -- the additional intrinsic complexity will permit us to make
amazing extensions to the network, IF they scale in application.
Surely, some of you will be quite upset about my observations, but I ask
you to
stay cool and just ponder it all for a while to see of things don't start
to
look different from this point of view, hopefully yielding some useful
new
insights.
Enjoy;-)...\Stef
Why would anybody be upset? They are "a" way of viewing the network,
possibly a somewhat projective and naive view, but as you say, it can
still yield certain insights. However, from an engineering perspective
they aren't horribly useful. Check out network/graph theory -- there
are plenty of sites you can google, and some good books on the subject.
Then you'll have a better grasp of the actual underlying mathematics
(which is really quite lovely and can be extended all the way down to
the network of nerves that is generating the HIGHLY nonlinearly
organized impulses that are typing this reply and the network of traces
through which flowing electrons are encoding and processing my typing so
that it can be sent out over a much simpler network (the one we are
discussing) to you.
rgb
--
Robert G. Brown http://www.phy.duke.edu/~rgb/
Duke University Dept. of Physics, Box 90305
Durham, N.C. 27708-0305
Phone: 1-919-660-2567 Fax: 919-660-2525
email:rgb(_at_)phy(_dot_)duke(_dot_)edu
--
Robert G. Brown http://www.phy.duke.edu/~rgb/
Duke University Dept. of Physics, Box 90305
Durham, N.C. 27708-0305
Phone: 1-919-660-2567 Fax: 919-660-2525
email:rgb(_at_)phy(_dot_)duke(_dot_)edu
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