Deformation Spaces and Static Animations

TL;DR

This paper explores how 3D printing can visualize continuous mathematical deformations through static animations, connecting geometry, dynamics, and chaos theory, and provides a tutorial for creating such visualizations.

Contribution

It introduces the concept of static animations as a novel way to visualize continuous families of mathematical objects using 3D printing, with practical guidance and code.

Findings

Static animations effectively visualize mathematical deformations.

Connections established between 3D printing and various mathematical fields.

Tutorial enables readers to create their own static animations.

Abstract

We study applications of 3D printing to the broad goal of understanding how mathematical objects vary continuously in families. To do so, we model the varying parameter as the vertical axis of a 3D print, introducing the notion of a static animation: a 3D printed object each of whose layers is a member of the continuously deforming family. We survey examples and draw connections to algebraic geometry, complex dynamics, chaos theory, and more. We also include a detailed tutorial (with accompanying code and files) so that the reader can create static animations of their own.