The Buildings Gallery: visualising buildings

TL;DR

This paper introduces The Buildings Gallery, an interactive web application that visualizes mathematical buildings associated with groups with BN-pairs, serving as an educational and research tool linking algebra, geometry, and group theory.

Contribution

It presents an algorithmic method to create 3D interactive visualizations of buildings, bridging theoretical mathematics and accessible visualization.

Findings

Developed a visualization algorithm for buildings

Created an interactive web application for educational use

Enabled exploration of buildings associated with BN-pair groups

Abstract

Buildings are beautiful mathematical objects tying a variety of subjects in algebra and geometry together in a very direct sense. They form a natural bridge to visualising more complex principles in group theory. As such they provide an opportunity to talk about the inner workings of mathematics to a broader audience, but the visualisations could also serve as a didactic tool in teaching group and building theory, and we believe they can even inspire future research. We present an algorithmic method to visualise these geometric objects. The main accomplishment is the use of existing theory to produce three dimensional, interactive models of buildings associated to groups with a $BN$-pair. The final product, an interactive web application called The Buildings Gallery, can be found at https://buildings.gallery \citep{Bekker2020}.