This mobile depicts a few of the atomic orbitals of hydrogen, more suggestively known as electron clouds, which describe the probability of finding an atom’s electron at some distance from the nucleus (shown as small black beads), which varies in 3D space throughout a spherical volume centered about the nucleus. Their shapes are determined by the unique configurations of their quantum numbers (n,l,m), which describe the electron’s energy level (the principal quantum number n), the orbital’s shape (the azimuthal quantum number l), and the suborbitals available in that energy level (m the magnetic quantum number). From the top, these are the orbitals (3, 2, 0), (3, 1, 0), (4, 2, 0), (2, 1, 0), (4, 3, 0), and, (4, 1, 0).
Like much of my work, this piece evolved out of a creative exercise I assigned myself. In January of this year, I began making simple Mylar hydrogen orbital ornaments by stippling 2D cross-sections of these orbitals with Sharpie onto die-cut Mylar disks, folding the disks in half, and then sewing them into a sphere with precut holes along the folds. While I found these ornaments attractive and loved the way they appear illuminated when viewed in direct sunlight, I wanted to find a more powerful sculptural method using the same materials that would better do these orbitals justice. My solution was to cut Mylar disks of varying radii to be mounted on a rod and separated by spacers and then airbrush the disks using a series of die-cut cardstock airbrush masks in order to build the volumetric gradients from transparent 2D layers. Cutting templates were drafted in Adobe Illustrator, cut using a Silhouette Curio (hobby CNC die-cutting machine), and I used Copic markers in conjunction with the proprietary Copic airbrush attachment and an air compressor to paint them. As each disk was completed, it was mounted on the rod, and the completed orbitals were suspended from a hanger made of Plexiglass.
These are, to my knowledge, the first sculptural models of these orbitals to acknowledge both sides of the wave-particle duality. Like the plastic models made by several science education supply companies, I have color-coded the regions of my orbitals to show what can be thought of as the “crests” and “troughs” of 3D probability waves that pulsate back and forth between the regions they occupy. However, these models give no information about the probability distributions contained within them. As for the particle side of things, Studio Nebula in Japan has illustrated these probability distributions as discrete particles laser etched in acrylic blocks, but they do not differentiate between the positive and negative regions.
This is not my final answer to the problem of how to sculpt these orbitals, and I have a few ideas using different media I wish to investigate in the future. However, I intend to continue developing and refining the sculptural technique I have invented for this particular approach and finding wider applications for it in math and physics.
From September 2017 to February 2018, I primarily focused on producing dozens of sculptural “sketches” depicting as broad a range of mathematical topics as I could using die-cut Mylar plastic film as my main ingredient. This piece depicts the first six terms of the harmonic series as complex sinusoids illustratrated using embroidery floss threaded through three dimensional paths made by holes cut in Mylar disks. The disks have square central holes to prevent them from rotating out of place along the square rod, and they are separated by clear acrylic pony beads. It is one from a series of eleven prototypes made in September 2017 that depicted waves, Laplace, and Fourier transforms which were based on an older paper experiment inspired by an illustration in Hecht’s “Optics”. I particularly like the 4D aspect of these, as the disks taken together are effectively the frames of an animation showing points moving in orbital resonance.
To me, the most moving part of these sculptures is not the finished product but rather the process of creating them. I was fascinated by watching the embroidery floss waves slowly trace out their paths as I worked, intertwining in patterns that seemed to go from chaotic to orderly from one disk to the next. As my applied art stance leads me to concern myself with matters of product design, my plan for these sculptures is to create do-it-yourself kits containing the die-cut Mylar disks, as this would enable me to share this intimate mathematical experience with others. I deliberately made them using supplies easily available at craft stores to help make them more affordable for students.
The Real and Imaginary Parts of Complex Functions
These sculptures were made of die-cut colored cardstock that was then mounted on acrylic rod with spacers to separate each layer. The two different colors show the real and imaginary parts of different complex functions over one branch cut.
Fourier Transform and Gaussian Distribution Cards
These die-cut accordion cards depict the individual terms of the Fourier transform of sine and different standard deviations of a Gaussian distribution on each page. They are available for sale in the shop.
Asterism is a sculpture depicting the stars of the Big Dipper as they actually appear in three dimensional space. When viewed head-on from the front, they align to form the familiar pattern we see here on earth, but as the viewer walks around the sculpture and changes his or her viewing angle, they appear to dissolve into a seemingly random, arbitrary configuration.
The 2D (x,y) coordinates were first taken from an image of the Big Dipper, which were then plotted as 3D points in Geogebra with their distances from earth as their z-coordinates. A cube was then drawn around the points, then line segments drawn connecting different legs of the cube so that they passed through the center of the points in order to anchor them to the frame.
These sculptures were created as a part of a self-directed “summer internship” graciously funded by the University of Tennessee, Knoxville, physics department, in whose lobby they now live. Inspired by an example in the Mathematica documentation for the SliceContourPlot3D function, they depict 3D surfaces as their individual level curves painted onto Plexiglass plates mounted on metal rods with spacers separating them. The functions were first plotted in Mathematica, exported to Adobe Illustrator for editing, and then etched into Plexiglass using a laser cutter. The masking film covering the area to be painted was then removed, lightly sanded, and primed with clear gesso. The color schemes were designed to be colorblind accessible so that, for a colorblind student, the colors will appear to have the same spacing as for someone with normal color vision. Once these colors were chosen, swatches were printed off using a photo printer and taken to a home improvement store so paint samples matching the swatches could be made. After being painted and given a protective clear coat, the edges of each plate were sanded, the masking film finally removed, and the plates then mounted on the rods in the proper order.