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@xxxxxxxxxxxx > -- 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