Hexagonal Pyramid

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 hexagonal pyramid: zeros sit at its 7 vertices, and poles at its 7 face directions. The surface rises where the function's size grows, so poles become spikes and zeros become pits. The result has a large double spike under the hexagonal base, a ring of 6 spikes above the equator over the triangles, 6 pits around the base corners, and a double pit at the apex.

The other pieces in this series come from solids with many equal faces. This one mixes them: one hexagon among six triangles, and one apex where six edges meet among corners where three meet. The spikes and pits come in two sizes as a result. The pyramid is also self-dual: its polar dual is the same pyramid, upside down, so the pattern of spikes is the pattern of pits, flipped.

The math

• The recipe (R2): a zero at each vertex, of order equal to the number of edges meeting there, and a pole at each face direction, of order equal to the face's number of sides.

• Here the raw orders are 6 at the apex and 3 at each base corner, and 6 at the hexagon and 3 at each triangle. They share a factor of 3, and dividing it out gives the function itself: a double zero at the apex, 6 simple zeros at the base corners, a double pole under the base and 6 simple poles over the triangles. Degree 8.

• With the apex at the north pole of the sphere, the function is f(z) = C · (z⁶ − b⁶) / (z² (z⁶ + t⁶)), where b and t are the positions of a base corner and a triangle's pole in the plane (real numbers, for a corner on the real axis). The double zero at the apex sits at infinity, where f falls off like 1/z².

• The relief has the pyramid's 12 symmetries, mirrors included: 6 rotations about the vertical axis and 6 mirror planes. The phase (its colour in a domain-colouring picture) is less symmetric: a sixth of a turn about the axis turns it by a third of a full turn.

• 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 14 features, the 80 mm size shows everything.

• Whole model: one piece for each size.

• Cut halves: the model is cut through its centre along a vertical 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 the double spike under the base, 9 mm from its tip. That spike lies in the cut plane, so the hole runs through both halves and lines up when they are glued. The ornament hangs with the base up and the apex's pit at the bottom. 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.

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.