A more "complete" version of the Pi-theorem: DRAFT

TL;DR

This paper aims to provide a comprehensive characterization of dimensionally invariant relations using independent dimensionless Pi groups, extending the traditional Pi-theorem to a more complete form.

Contribution

It introduces a complete Pi-theorem that fully characterizes dimensionally invariant relations through independent Pi groups, surpassing the traditional partial results.

Findings

Complete characterization of dimensionally invariant relations

Extension of Pi-theorem to a 'full' version

Framework for identifying all independent Pi groups

Abstract

The traditional Pi-theorem tells us that for any dimensionally invariant relation there exists a full set of independent dimensionless "Pi groups" which can be used to nondimensionalise the relation. In this paper, we seek to understand better the structure of dimensionally invariant relations and sets, by giving a complete characterisation of them in terms of independent dimensionless Pi groups. The traditional Pi-theorem only goes part of the way towards achieving such a characterisation. Our characterisation presented here can be viewed as the "complete Pi-theorem".