Math Creatures in Hyperspace

Functions map between input and output spaces, with each space possessing its own dimensionality determined by the number of independent variables or values.

The term “hyperdimensional functions” intentionally provokes mathematicians because strictly speaking, functions themselves lack dimensionality—only their inputs and outputs possess dimensional properties.

Parametric functions transform arbitrary parameter ranges into geometric curves and surfaces by treating inputs as lengths rather than spatial coordinates.

UV mapping in computer graphics transforms coordinate values into color values (RGB), providing an alternative visualization method for parametric functions.

Higher-dimensional objects emerge through the principle that an infinite stack of slices from dimension N creates dimension N+1.

Adding time as a third input variable (t) alongside spatial inputs (u,v) creates spacetime representation, allowing functions to evolve and be explored temporally.

The parameter system allows adding unlimited inputs while visualizing only single slices where parameters are held constant, treating each parameter as a tunable control knob.

With numerous parameters creating combinatorially explosive spaces, strategic navigation becomes essential since exhaustive exploration proves impossible.

The biological analogy runs deep: parameters function like genes of mathematical creatures, with evolution exploring different parameter sets just as natural selection explores genetic combinations.

Neural networks are functions that can construct any conceivable function given sufficient parameters, making them universal function approximators.