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 >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@xxxxxxxxxxxx