Does Time Smoothen Space? Implications for Space-Time Representation

Abstract

The continuous nature of space and time is a fundamental tenet of many scientific endeavours. That digital representation imposes granularity is well recognized but whether it is possible to address space completely remains unanswered. Part 1 argues that Hales’ proof of Keppler’s conjecture on the packing of hard spheres suggests the answer to be ‘no’, providing examples of why this matters in GIS generally and spatio-temporal GIS in particular. Part 2 seeks to resolve the dichotomy between continuous and granular space, showing how a continuous space may be emergent over a random graph. However, its projection into 3D/4D imposes granularity. Perhaps surprisingly, representing space and time as locally conjugate may be key to addressing a ‘smooth’ spatial continuum. This insight leads to the suggestion of Face Centered Cubic Packing as a space-time topology but also raises further questions for spatio-temporal representation.

Does Time Smoothen Space? Implications for Space-Time Representation

“…The most efficient 3D packing structure is hexagonal [74], such that each higher dimension places nodes at the center of the dual for the mesh in the dimension below it, which is also the center for the interstitial space. Expressed formally (Equation ( 8)):…”