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

Abstract

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.

Figures (9)

• Figure 1: (left) 4DLO showing a compound of three tesseracts, with student built work just visible in the back. (right) Singing affected the sculpture and its program.
• Figure 2: (a) The fully symmetries of 24-cell; (b) half those, directing the edges; (c) A transitional moment during a sequence showing off the twenty four octahedral cells of the 24-cell, one by one. A similar sequence showed off the twenty four cubes of the three tesseracts sharing these edges. (d) Four mutually perpendicular groups of four rings of length six.
• Figure 3: (left) A spherical octahedron and its stereographic projection. Both are divisions of their space into triangular cells bounded by circular arcs meeting at right angles. (Right) A stereographic projection of a 16-cell, with tetrahedral cells and faces and edges meeting at right angles.
• Figure 4: (left and middle) Two cell-down 24-cells, shown as a compound of three tesseracts and as $T^*$ shown as three-fold rotational symmetries of a tetrahedron. [New pix coming] These models [will be] were built at the seventeenth Gathering for Gardner, Feb. 2026, with Scott Vorthmann.
• Figure 5: (At left, below) A first model of a compound of two compounds of three 16-cells ( "two$\cdot$ three 16-cells") made of playing jacks and automotive tubing; the marked half-circles lie in a $\sqrt 2$-to-1 rectangle (see Fig. \ref{['plans']}). (Above left) Six 16-cells in threes: (far left) vertex-down in red, with vertices $V_8$; $V_{16}^+$ and $V_{16}^-$ in white and blue; (mid-left) a symmetrical arrangement in $V_{24}'$. (Right) A Compound of Two Compounds of Three Sixteen Cells, G4G12 (2016).