# Fibonacci, golden ratio, and vector bundles

## TL;DR

This paper explores a family of vector bundles over moduli spaces, connecting algebraic geometry, physics, and number theory by deriving Fibonacci summation formulas involving the golden ratio through multiple computational methods.

## Contribution

It introduces new summation formulas for Fibonacci numbers expressed via the golden ratio, derived from the rank computations of specific vector bundles in algebraic geometry.

## Key findings

- Derived Fibonacci summation formulas involving the golden ratio
- Computed the rank of G_2 vector bundles in three different ways
- Established connections between algebraic geometry and number theory

## Abstract

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional group $G_2$ case of these bundles in three different ways, we derive a family of summation formulas for Fibonacci numbers in terms of the golden ratio.