Metatopes: a novel and simple mathematical construct for statistical ontology

(~400 words)

Definition

A metatope is a mathematical object constructed by a layering of spaces. For any space in a metatope, points can be plotted in it. These points in the space correspond to either (1) datapoints from a dataset with a commensurate dimensionality as that space or (2) spaces that contain either (1) or (2) (potentially ad infinitum).

A directed graph can be constructed from any metatope. The spaces within the metatope map to nodes and the connections from a space to the spaces plotted within it map to the vertices. A “highest space” can generally be identified, and that is the root of the metatope’s graph. The degree of a metatope is the maximum distance in the directed graph starting from the root. The degree counts the number of layerings in the metatope: a metatope of degree-$0$ is just a normal space, and its graph is just a solitary node with no vertices. If there is a root space in the metatope and all the other spaces connect upwards to exactly 1 space, then the graph of the metatope is a rooted tree.

For instance, consider a dataset of faces, both normal and cyclopean, parameterized by a robust set of features such as interocular distance, nose length, cheekbone position, etc. All the normal faces can be plotted in one space, and all the cyclopean faces can be plotted in one space, but these two spaces are incommensurate because there are several dimensions that are not shared (most saliently from the list above: interocular distance). Both spaces, however, can enter as points in a higher-level space that would have one dimension called “number of eyes.” This would be a degree-$1$ metatope, and its graph would have two nodes connecting upwards to the root node, for a total of three nodes.

Note that the structure of any metatope depends entirely on the relevant set of datasets (actual or theoretical) and the reckonings made in how to combine them, and that multiple metatopes can often be combined into one metatope.

Motivation

My motivation is to develop a mathematical construct similar to the mere space that can be applied rigorously even when dimensionalities are incommensurate. I have often encountered conversations where “the space of x’s” is tossed around even when the x’s obviously or probably differ in the number of dimensions from one x to another x, such as “the space of conscious experiences.” I hope that the construct operates as an intuition pump and leads to interesting insights, from myself or others.