The video introduces the concept of “math creatures” or “manals,” complex and colorful mathematical objects defined entirely by mathematical functions. These creatures live in a mathematical zoo that viewers can explore online. The creator emphasizes that functions, which map inputs to outputs, can describe anything in the world, and by extending functions to have many inputs and outputs—beyond the usual three dimensions—we enter the realm of hyperspace. Although functions themselves do not have dimensionality, their inputs and outputs do, and visualizing functions with high-dimensional inputs and outputs is the main challenge addressed.

Starting with the basics, the video explains how simple functions with one input and one output can be visualized as a 2D graph, such as the sine function. Adding more outputs, like a second output for cosine, transforms the visualization into parametric functions, where outputs become coordinates in 2D or 3D space. For example, a function with three outputs can be visualized as a 3D curve. Adding more inputs increases the dimensionality of the input space, such as moving from a line (one input) to a plane (two inputs), which can be visualized as a surface in 3D space by mapping inputs to outputs.

The video then explores combining multiple outputs, including spatial coordinates (XYZ) and color values (RGB and transparency A), to create richly detailed visualizations that exist in higher-dimensional output spaces. By adding a third input dimension, such as time (T), the functions become dynamic, evolving over time and allowing viewers to explore slices of these high-dimensional objects sequentially. This slicing technique helps make sense of complex, multi-dimensional functions by visualizing one slice at a time, much like scrubbing through frames of a video.

A key part of the project is a web-based tool that allows users to create, manipulate, and visualize these hyperdimensional functions interactively. The tool supports multiple inputs and outputs, letting users write JavaScript code to define functions with parameters that act like control knobs. Users can explore vast parameter spaces, generating intricate shapes like shells or jellyfish, and even save and share their creations. The tool also supports advanced mathematical constructs like matrices, complex numbers, and fractals, enabling the creation of highly complex and computationally intensive visualizations.

Finally, the video touches on the broader implications and motivations behind the project, including the desire to visualize neural networks as functions with many inputs and outputs. Neural networks, being universal function approximators, fit naturally into this framework. The creator demonstrates simple neural network models within the tool and encourages viewers to experiment with different architectures and parameters. The video concludes with an invitation to explore these mathematical spaces, contribute insights, and engage with the community, highlighting the beauty and power of functions to describe and create complex, colorful worlds.