# Category Archives: Space and Geometry

## Topological Automata in Hyperbolic Tilings

Hierarchies of Spaces: Building From the Bottom Up One of the major challenges in building a system that can increase in complexity as it runs is figuring out how to transfer complex structures in a lower level space into simple … Continue reading

## Valency (video)

These are clips of how the particle systems described in previous posts move in space. It’s all very raw, but I wanted to get something out there. The larger structures suffer from convergence problems creating oscillations, twisting, foldings, and other … Continue reading

## Connecting the Dots With Differential Geometry

Particle-Based Structure Formation In order to build a complex and dynamic spatial system, there need be entities operating at multiple temporal and spatial scales.  As mentioned in the previous post, the extremes of the spatial scale are local and global … Continue reading

## Spatial Processes and Spatial Hashing

Spatial Processes There are four broad categories of spatial processes Purely Global Global <-> Local Local <-> Local Purely Local and two general categories of spatial primitives Field Particle Of course, the spatial primitives can be organized into composite structures … Continue reading

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## Transversion and the Torus

As a first test of the efficacy of the new Transversion Automaton design, I set it up to carve out a simple torus.  Since a torus only has two loops whose product generates the surface, it’s easy to model simply … Continue reading

## Hyperbolic Tilings Again

I’m revisiting a lot of the techniques I’ve implemented for working with hyperbolic geometry as I’m rebuilding the Transversion Automaton in order to make the simpler and clearer.  The most basic structure that underlies the topology of the surfaces generated … Continue reading

## A Transversion Machine Redux

Thoughts on the Original Design The motivation for constructing a transversion-based spatial automaton was to construct a surface in 3D Euclidean space from as little local geometric information as possible.  It was the automaton’s role to expand the information into … Continue reading

## Hyperbolic Translations

One of the crucial issues in the construction of the implementation of the transversion automaton is how it interpolate its geometric relationships (transversions and rigid body motions) through space to generate a surface in 3D.  The input data is pentagonal … Continue reading

## Geometric Algebra: The Meet Operation

Background The Meet operation in Geometric Algebra (GA) calculates the intersection of two subspaces. In set theoretical terms, the intersection is given by the relationship where M is the intersection (or Meet). Frequently in GA papers and books, you’ll find … Continue reading

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