# Placing Kirkman's Schoolgirls and Quantum Spin Pairs on the Fano Plane: A Rainbow of Four Primary Colors, A Harmony of Fifteen Tones

## TL;DR

This paper explores the connections between a classical recreational problem, quantum spin operators, and finite geometries like the Fano Plane, providing new constructions and visualizations in design and signal processing.

## Contribution

It introduces a novel approach linking recreational mathematics, quantum information, and finite geometry to generate new designs and patterns.

## Key findings

- Established a correspondence between recreational problem and quantum spin operators.
- Developed an algorithm to generate alternative designs using four basic objects.
- Created color and acoustic patterns corresponding to geometric and algebraic structures.

## Abstract

A recreational problem from nearly two centuries ago has featured prominently in recent times in the mathematics of designs, codes, and signal processing. The number 15 that is central to the problem coincidentally features in areas of physics, especially in today's field of quantum information, as the number of basic operators of two quantum spins ("qubits"). This affords a 1:1 correspondence that we exploit to use the well-known Pauli spin or Lie-Clifford algebra of those fifteen operators to provide specific constructions as posed in the recreational problem. An algorithm is set up that, working with four basic objects, generates alternative solutions or designs. The choice of four base colors or four basic chords can thus lead to color diagrams or acoustic patterns that correspond to realizations of each design. The Fano Plane of finite projective geometry involving seven points and lines and the tetrahedral three-dimensional simplex of 15 points are key objects that feature in this study.