3D printing

Recently the Design Office of the University of Oxford acquired a 3D printer, and they offered free print-outs during the period of setting up their printer. This was a perfect opportunity for me to bring one of the abstract spherical functions appearing in my work to life. The function is closely related to the icosahedron, a regular polyhedron, in the sense that it has an icosahedral rotational symmetry. A more detailed description of the function as well as its relation to quantum mechanical entanglement can be found further below.

icosahedron in mathematica icosahedron on table

In the first step of the printing process, the three-dimensional object was printed layer-by-layer by depositing a liquid acrylic polymer. Openings and voids were filled with wax to ensure stability. In the second step, the redundant wax was melted away in an oven. Both steps consume a huge amount of time even with modern 3D printers. For my object (5cm x 5cm x 5cm) the first step took 10 hours, and the second 4 hours!

icosahedron in wax icosahedron after wax has been melted away

I wasn't able to witness the creation of my object in person, but the great people from the Design Office sent me the two amazing pictures above which show the object after the printing and de-waxing process. In the left picture the acrylic body is covered in wax, and in the right picture the wax has been removed. The video below shows the model as well as the software used for the creation and analysis of the polygonal wire model needed by the printer.

 

Mathematical and physical background

The function describing the surface of the 3D object has a relatively simple form. In spherical coordinates the inclination is labeled by θ ∈ [0,π] and the azimuth labeled by φ ∈ [0,2π). We can then define the complex-valued function f(θ,φ) as:

main equation

The function plotted is the squared absolute of the above, i.e.

squared absolute of f

Why is this function relevant? it appears in my work on quantum entanglement. Entanglement is a crucial resource for quantum information tasks such as quantum computation, quantum teleportation or quantum communication. I have been searching for the maximally entangled quantum states of certain sets of states, specifically permutation-symmetric quantum states. As I showed in a publication, the underlying theory allows such states to be visualized by spherical functions, with the shared property that the "volume" of all these functions is the same. The above object is the presumably maximally entangled symmetric quantum state of 12 qubits, and if I were to print out any other symmetric 12 qubit state with a 3D printer, it should consume precisely the same amount of liquid acrylic polymer. (provided that no scaling takes place) The amount of entanglement is determined by the global maximum of the spherical function or, in other words, by the "highest mountain" on the object. The smaller the global maximum is, the more entangled the quantum state is. Thus it becomes intuitively clear that the above object is the maximally entangled 12 qubit symmetric state: A small "distortion" would change the heights of the twenty identical global maxima, with some of them increasing while others decrease. However, only the global maximum decides the entanglement, and its increase would lead to a decrease of the entanglement.