The {m 4DLO} and other tubing models of $S^3$ symmetry

The {\em Four-dimensional Light Orchestra} or { \em 4DLO} was an interactive sculpture at the National Museum of Mathematics (MoMath) from November 20, 2025 through January 2026, illustrating various sub-symmetries of the 24-cell with colored lights, part of a larger sequence of tubing sculptures aiming to bring to life a few lines of tables appearing in Conway and Smith (2002), reprinted in {\em The Symmetries of Things}. Best of all museum patrons could manipulate {\em 4DLO}’s lighting by singing and making funny noises into a microphone, and they did with gusto. Here we describe some of the technical aspects of this sculpture and its context.

💡 Research Summary

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The paper documents the design, construction, and operation of the Four‑dimensional Light Orchestra (4DLO), an interactive sculpture that was on display at the National Museum of Mathematics (MoMath) from November 2025 to January 2026. The work visualizes selected sub‑symmetries of the regular 4‑polytope known as the 24‑cell ({3,4,3} in Schlӓfli notation) by means of colored LED strips mounted on a physical framework made from flexible tubing. The authors explain how the tubing, cut into circular coils and joined with six‑way connectors, naturally follows great‑circle arcs when projected stereographically from the 4‑sphere S³ onto three‑dimensional space. This property makes the tubing an ideal medium for representing circles that lie on S³, which are the fundamental geometric objects underlying the 24‑cell’s edges and faces.

A substantial portion of the paper is devoted to the algebraic description of the model. Unit quaternions q = a i + b j + c k + d 1 are used to coordinatize points on S³. The vertices of a tesseract are given by V₁₆ = ½(±i ±j ±k ±1), while the vertices of a 16‑cell are V₈ = {±1, ±i, ±j, ±k}. Their union V₂₄ = V₁₆ ∪ V₈ forms a “vertex‑down” 24‑cell; the dual 24‑cell’s vertices are V′₂₄ = (1/√2)(±x ±y) with x, y drawn from {1,i,j,k}. Left‑multiplication (x ↦ qx) and right‑multiplication (x ↦ xq) by a unit quaternion q act as right‑handed and left‑handed Hopf fibrations respectively, sliding every point of S³ along a family of great‑circle “rails”. By choosing q to be purely imaginary, the authors generate rotations about a specific great circle; the rotation angle θ is controlled by the scalar part of q via r = cos θ + sin θ · q. The mapping x ↦ rxr⁻¹ therefore implements a three‑dimensional rotation, with the double‑covering property (r and –r give the same rotation) yielding binary rotation groups G* that lift ordinary 3‑D rotation groups G.

The paper identifies several important binary groups that appear in the 24‑cell’s symmetry structure. The binary tetrahedral group T* consists of the 24 points V₂₄ and encodes the 3‑fold rotations that map edges to edges. The binary octahedral group O* corresponds to the 48 points V₂₄ ∪ V′₂₄ and captures the 4‑fold rotations of a cube. The authors use the notation ±