Icosahedral Crown

A complex function you can hold.

This sculpture is a picture of a single mathematical function. A rational function on the Riemann sphere is pinned down by its zeros and poles. Here they come from a regular icosahedron: poles sit at its 12 vertices, and zeros at the midpoints of its 30 edges. The surface rises where the function's size grows, so poles become spikes and zeros become pits. The result is a crown of 12 tall spikes, with a pit in the saddle between every pair of neighbours.

The math

• Function: f(z) = T(z)² / V(z)⁵, built from Klein's classical icosahedral forms: V = z(z¹⁰ − 11z⁵ − 1) for the 12 vertices, and T = z³⁰ − 522z²⁵ − 10005z²⁰ − 10005z¹⁰ + 522z⁵ + 1 for the 30 edge midpoints. One vertex sits at infinity, the north pole of the sphere, which is why V has degree 11 rather than 12.

• 12 poles of order 5 (one for each edge meeting at a vertex) and 30 zeros of order 2: degree 60.

• The relief has the full symmetry of the icosahedron, mirrors included: 120 symmetries, the largest symmetry group there is for a polyhedron.

• The radius at each point of the sphere follows |f|. The constant is fixed so that |f| has geometric mean 1 over the sphere, which puts "sea level" midway between the spikes and the pits.

Files

Two sizes, measured tip to tip: 80 mm and 130 mm. With 42 features, it shows its detail best at 130 mm.

• Whole model: one piece for each size.

• Cut halves: the model is cut through its centre along a symmetry plane, so the two halves are identical. Print the half file twice. Each half lies flat on its cut face.

• Hanging ornament: halves a and b with a 1.5 mm thread hole through one spike, 9 mm from its tip. No spike lies exactly in the cut plane, so the hole runs through a spike next to it, entirely within one half. Thread it, tie a loop, and hang it on a tree.

Assembly

• Every half has a 5.2 mm alignment hole in the middle of its cut face, for an M5 dowel pin (5.00 mm diameter):

• 80 mm: M5 × 10 pin, 6 mm holes;

• 130 mm: M5 × 20 pin, 11 mm holes.

• With the settings below, the pin gives a light friction fit. Glue the pin into one half, then glue the halves together; the pin keeps them aligned. If your printer makes holes tighter or looser, ream the hole lightly. You can also glue the halves without a pin.

• Glue: a budget cyanoacrylate (super glue) for plastics works well with PLA. I used LOOCTOT glue for plastic.

• Ornament hole: run a 1.5 mm drill bit through it by hand in a pin vise to clean it up. Monofilament (fishing line) makes a good hanging loop. Keep glue away from the hole while assembling.

Print settings (tested)

• PLA, 0.4 mm nozzle

• 0.20 mm layers ("QUALITY" profile)

• 15% gyroid infill

• Halves print cut face down. No supports are needed.

Finishing

Minor stringing can occur between the spikes. I remove it with brief passes of hot air from an SMD rework station. Keep the airflow moving and apply heat sparingly: the thin PLA tips soften quickly. If a spike bends slightly, let it cool enough to touch safely, then gently reshape it by hand while it is still pliable and hold it in position until it sets.

Part of Polyhedra as Complex Functions: how this piece is made, an interactive 3D view of its zeros and poles, and the object's own page. Code and data: GitHub.