Don't Know Much About Geometry...

As the UPS commercial says, when you're the shipping manager, you can't think outside the box. Your job is to think about the box. Barely emerging from the throes of a move, I've been looking at a lot of boxes lately. I'm struck by the importance of physical geometry in our daily lives.



It is the geometry of triangles that stabilizes bridges. It is geometry that transforms a flat piece of cardboard into a box (and corrugated cardboard uses triangles for stability too). Geometry figures prevalently in our perception of beauty (the proportions of the golden rectangle being but one example). In fact, beauty is largely judged by proportion, symmetry, and alignment, which are all derived from geometry.



A cardboard box derives its strength from the arrangement of vertices in physical space. Several patents on box designs embody ways to fold/configure the geometry over time (for storage and use). Geometry affects our lives at the microscopic level as well (with macroscopic implications), whether in the DNA double helix, protein folding, or the atomic lattice of semiconductor substrates.



So why is geometry the poor step sister of sexier sciences? (The Hoberman Sphere shows how sexy geometry can be.) Perhaps our inability to visualize physical geometries beyond three dimensions causes us to abandon this useful science when tackling n-dimensional problems. I recall a story called "The Blind Geometer" from Isaac Asimov's Science Fiction magazine over 20 years ago. Some bad guys wanted to kidnap a blind scientist for his ability to perceive multi-dimensional spaces.



My admittedly limited knowledge of network design tells me that engineers focus on network topology, not geometry per se. Let's consider the eight vertices of a cube as nodes in a network. From a network standpoint, you'd be concerned with the interconnections between the nodes, their proximity, latency, capacity, content, etc. But can any network metric be constructed or analyzed based on the "geometry" of the nodes? What does "geometry" even mean in a virtual space where the rules of physical geometry don't apply? Network geometry surely has little to do with the physical location of the nodes themselves.Instead, a network's "geometry" might derive from some other attribute of its nodes, or even from a network attribute unrelated to the nodes themselves. And that is really my point. The "box-ness" of a box does not derive solely from its vertices' locations in physical space. A box's usefulness derives from the overall configuration of all its vertices and the planes between them, plus the physical laws that prevent my housewares from passing through carboard and the ability of physical matter to transmit forces and distribute stresses. Even though we can't easily visualize 4-, 5-,...n-dimensional spaces, we can choose a subset of three dimensions and plot them on XYZ axes to aid in analysis.



What is the relationship between physical geometry and network topologies? Is there one? If not, should there be one? Can visualization in geometric space assist in network optimization? Can geometry engender a hitherto undiscovered property of a network? It will take someone more knowledgeable than me to properly formulate, explore, and answer these questions.



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What is the relationship between physical geometry and network topologies?