Presented by Bronna Butler
Dodecahedral Trajectory’s video transcript
About Bronna Butler
Hi, my name is Bronna Butler. I have been a professional artist for the past 23 years. I have a bachelor’s degree in art and a master’s degree in accounting/finance, from the University of Missouri. I work in stained glass, oil and pastel paints, graphite, stainless steel and recently, a bit of electronics. I have designed and painted over 900 square feet of stained-glass windows in various buildings. I designed and managed the fabrication and installation of a 20 ft tall stainless-steel monument with 4 colored titanium photomontage panels, and I have painted hundreds of commissioned large and small oil and pastel paintings, the largest single oil painting was 8 X 12 feet, which I painted standing on scaffolding.
Since 2015, my muse has been mathematics. My current work focuses on recreational mathematics, puzzles, enigmas, and optical illusions, as well as mathematics and science in general.
Mathemalchemy adventure begins
Beginning in February 2020, I have had the wonderful opportunity to collaborate with a team of 23 mathematical artists and artistic mathematicians who are creating a large multimedia art installation, called Mathemalchemy. It celebrates the creativity and beauty of mathematics. It is the vision of mathematician and physicist Ingrid Daubechies. With the help of artist Dominique Ehrmann, Ingrid has guided the collaboration on the large installation for the past ten months, via Zoom. After its completion in 2021, it will be installed in various museums and other places, and then be permanently installed in Duke University’s planned new Mathematics building. The installation includes a 7.5-foot-tall stainless-steel lighthouse which will stand on a 2-foot base. I fabricated its stained-glass dodecahedron beacon.
The stainless-steel lighthouse was designed by physicist Sabetta Matsumoto and it will sit on one end of the installation.
As a functioning real lighthouse would, the dodecahedron beacon will have Fresnel lenses to focus the light, in this case from an LED, shown here on our workbench.
The glass dodecahedron will fit on top of the lenses and LED.
A inspiring discovery about the dodecahedron
In September, Ingrid introduced the group to an article in “Quanta” magazine regarding the discovery of something new about the dodecahedron. “American Mathematical Monthly” had published mathematicians Jayadev Athreya’s and David Aulicino’s proof that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.
Jayadev Athreya explains the new discovery about Dodecahedrons in Numberphile.
This can be shown with the unfolded net/pattern of the dodecahedron’s pentagonal faces. Our lighthouse subgroup was unanimous, the installation’s dodecahedron beacon should “reflect” this discovery.
I downloaded the net shown in the Quanta article to use as a pattern. 
Still black and white net I lightened the areas of the pentagonal faces on one side of the line. 
I made patterns from the net, drew them on stained glass and cut the glass pieces shown here on my light table. 
Then I constructed a paper model which was interesting visually. 
The asterisks on this image of the glass pieces show how the top of the line will meet at the starting vertex at the bottom of the net, when the dodecahedron is built.
Athreya and Aulicino, joined by mathematician Patrick Hooper, determined that an infinite number of such paths exist on the dodecahedron. Their paper, published in May 2020 by “Experimental Mathematics”, shows that these paths can be divided into 31 natural families.
The beacon will be 9 ½ feet higher than the viewer. So, I wanted to emphasize the path. To do that, I put crimson glass below the path and amber glass above the path. This is the vertex where our trajectory begins. It passes across the faces of the dodecahedron. See, this is the copper foiled path. It just misses this vertex, goes below it. Comes around, comes back….back to the original vertex without passing through any of the other vertices.
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