A Cylindrical Galton Board at the Galton Board's 150th Anniversary

TL;DR

This paper revisits Galton's 1875 apparatus by extending it to a cylindrical form, linking classical normal distribution demonstrations with circular statistics and exploring new behaviors in a wrapped lattice setting.

Contribution

It introduces a cylindrical Galton board model that connects traditional normal distribution demonstrations with circular statistics, providing new insights and physical realizations.

Findings

Cylindrical lattice leads to height-dependent behavior not seen in planar designs.

Wrapped binomial and wrapped normal distributions emerge from the geometry.

Practical construction and pedagogical implications are discussed.

Abstract

The Galton board is a well known device for showing how repeated Bernoulli trials on a triangular lattice produce an approximately normal distribution. Marking the 150th anniversary of Galton's 1875 construction, this paper revisits the original apparatus and extends it to a cylindrical setting in which the peg lattice is wrapped around a cylinder. This creates angular periodicity and leads to height dependent behaviour that does not arise in the classical planar design. The cylindrical form links Galton's demonstration of variation and the emergence of the normal distribution with modern ideas in circular statistics, giving a physical realisation of binomial random walks on a circular linear product space. We distinguish cases where the wrapped lattice covers only an arc from those that span the full circumference, and show how these geometries lead to wrapped binomial and wrapped normal behaviour. We describe the construction of our physical model, discuss practical issues for replication, and analyse its statistical and pedagogical properties as a modern reinterpretation of Galton's work.