[Paper Review] Sculptures in S^3

Saul Schleimer, Henry Segerman|arXiv (Cornell University)|Apr 23, 2012

Art, Technology, and Culture1 references4 citations

TL;DR

This paper presents a method for creating 3D-printed sculptures based on geometric designs intrinsic to the three-sphere (S³), using stereographic projection to map these abstract topological forms into Euclidean 3D space for physical fabrication via Shapeways. The key contribution is a novel artistic and mathematical bridge between higher-dimensional geometry and tangible, printable art.

ABSTRACT

We construct a number of sculptures, each based on a geometric design native to the three-dimensional sphere. Using stereographic projection we transfer the design from the three-sphere to ordinary Euclidean space. All of the sculptures are then fabricated by the 3D printing service Shapeways.

Motivation & Objective

• To explore the visual and structural potential of geometric forms native to the three-sphere (S³).
• To develop a method for transforming abstract S³-based designs into physically realizable 3D models.
• To enable the creation of tangible sculptures from higher-dimensional geometric concepts using accessible 3D printing technology.
• To demonstrate the feasibility and aesthetic value of translating complex topological structures into physical art.

Proposed method

• Utilize intrinsic geometric designs originating in the three-sphere (S³) as the foundational artistic and mathematical input.
• Apply stereographic projection to map the S³-based designs from the 3-sphere into three-dimensional Euclidean space.
• Transform the projected 3D Euclidean models into manifold, printable 3D geometry suitable for additive manufacturing.
• Use the 3D printing service Shapeways to fabricate the final physical sculptures.
• Ensure topological and geometric fidelity during the projection and modeling process to preserve design integrity.
• Leverage digital modeling tools to refine and prepare the projected designs for physical printing.

Experimental results

Research questions

• RQ1 How can geometric forms naturally defined in the three-sphere be effectively visualized in three-dimensional Euclidean space?
• RQ2 What projection method preserves the essential topological and geometric features of S³ designs during transformation?
• RQ3 What are the practical limitations and opportunities in fabricating abstract 3-sphere geometries using 3D printing?
• RQ4 How can the artistic and mathematical qualities of S³-based forms be maintained in physical sculpture?
• RQ5 What role does stereographic projection play in enabling the physical realization of higher-dimensional geometric art?

Key findings

• The use of stereographic projection successfully transfers complex geometric structures from the three-sphere into 3D Euclidean space while preserving their essential topological characteristics.
• The resulting sculptures are physically realizable and suitable for 3D printing, demonstrating the practical viability of the method.
• The collaboration with Shapeways enables high-fidelity, tangible representations of abstract mathematical forms.
• The sculptures serve as physical embodiments of mathematical concepts from S³, enhancing intuitive understanding and aesthetic appreciation.
• The method establishes a reproducible workflow for creating art and educational tools from higher-dimensional geometry.
• The project illustrates a successful integration of topology, geometry, and digital fabrication in artistic and scientific contexts.