Subject: Re: Continuing the story - another stab at an IETF mission statement

From: Harald Tveit Alvestrand <[email-masked]>
Date: Tue, 09 Mar 2004 19:25:39 -0800

--On 18. februar 2004 18:06 +0000 Tom Petch <[email-masked]> 
wrote:

> I find your definition of the Internet delightfully ambiguous.  I was
> taught that the Internet (as opposed to an internet or the internet) was
> the public network accessible through public IPv4 addresses (this predates
> IPv6) ie the Internet ceased at a firewall or other such IP level gateway.
>
> Reading your definition, I cannot tell where you stand; are firewalls and
> networks behind them included in IETF mission or not?

Tom,
reviewing, I cannot tell whether I answered this one or not....
I was definitely intending to include them, since IMHO they are connected 
to the internet (see "both core and edge networks", "host to host"). If you 
can suggest words to make this clearer, I'd appreciate it!

                      Harald

---

From: Paul Vixie <[email-masked]>
Date: 10 Mar 2004 04:07:27 +0000

> > taught that the Internet (as opposed to an internet or the internet) ...

been there, done that, sold the t-shirt:

	>> But what *IS* the internet?
	> It's the largest equivalence class in the reflexive transitive
	> symmetric closure of the relationship "can be reached by an IP
	> packet from".		--Seth Breidbart
-- 
Paul Vixie

---

From: Einar Stefferud <[email-masked]>
Date: Tue, 9 Mar 2004 21:07:01 -0800

At 19:25 -0800 3/9/04, Harald Tveit Alvestrand wrote:
>--On 18. februar 2004 18:06 +0000 Tom Petch <[email-masked]> wrote:
>
>I was definitely intending to include them, since IMHO they are connected to the internet (see "both core and edge networks", "host to host"). If you can suggest words to make this clearer, I'd appreciate it!
>
>                     Harald

Just to throw a wrench into your discussion, The Internet just happens to be a 
Manifold (which literally means a bunch of pipes all connected together, such 
that entering at the end of any pipe, you can traverse the manifold and get to 
the open end of every other connected pipe in the manifold. 

Every  manifold pipe can be extended, so it is not possible to define the ends 
in any rigid way. Extension can be with PPP over dial-up, or a NAT router, or 
even a printer or word of mouth, or CD/DVD/TAPE/Postal-Service/etc/et al, 
ad nauseum.  

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.  

Apparently, what people call "the edge of the Internet" consists of an 
imaginary canvas stretched over the ends of all those manifold pipes with 
an imaginary elastic sheet of imaginary fabric.  But this only forms an edge 
if the Internet exists only in a two dimensional plane.  And even then, 
I have problems imagining all those spokes as making an edge.

Actually, they are referring to all those ends of all the manifold pipes, 
in that when attached to an end, the attachment is said to be made at the 
edge. I have big problems trying to imagine this as an edge (or a surface). 
So, I have tried to stop using those terms as they get in the way of thinking 
about various aspects of the Internet.  Not that I really understand much 
more than this about spherical geometry. 

I just wanted to toss this into the mix while all y'all are trying to decide 
what this thing called the Internet actually is.  I notice that all y'all 
have not settled on much of any agreement.  Reminds me of the 8 blind wise 
men trying to discover what an elephant is by each exploring a different part 
with their hands.  So far, I do not know anyone who claims to have touched 
its edge with their hands.

So, I just want to suggest that some of you out there who do understand 
spherical geometry might discover some things from looking it the net with 
spherical geometry glasses.  I only know that certain aspects seem to fit 
better than might be expected. 

For one thing, a manifold has no center, and indeed, an internet has no center. 

From my management consultant background, this has been an important realization, because, without the existence of a center, there is no logical place to put a control center to enable central control.  We like to say that the internet is controlled from its edges, by which I expect they mean it is controlled from its manifold pipe endpoints. 

Also, I note that from any endpoint a user can, and generally does, create a 
personal private network of (sometimes) collegial correspondents that is 
controlled by it owner.  Those networks are centrally "controlled" with 
address lists in address books and routing tables, and such. 

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

---

From: "Robert G. Brown" <[email-masked]>
Date: Wed, 10 Mar 2004 09:27:46 -0500 (EST)

> 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.

> look different from this point of view, hopefully yielding some useful new 

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:[email-masked]

---

From: Einar Stefferud <[email-masked]>
Date: Thu, 11 Mar 2004 00:00:34 -0800

Sorry to have disturbed you Robert;-(...  

But, I have to wonder what this URL is about, given your comment about geometry 
being not spherical. 

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

---

From: "Robert G. Brown" <[email-masked]>
Date: Thu, 11 Mar 2004 08:20:17 -0500 (EST)

> 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 

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