几何 y 代数 Merry-Go-Round
Statement
Teaching the same calculus class for many semesters has presented me with opportunities to listen, reflect, and approach elementary notions repeatedly from multiple perspectives. Once in a while, I develop certain intuition, understanding, and visceral feeling, which are difficult to put in words, but may eventually be crystallized in other forms.
Artwork
Geometry and Algebra Merry-Go-Round
12.0 x 12.0 x 0.5 in
color prints on vinyl record album
2023
Dot product, cross product, and scalar triple product can be introduced in multivariable calculus from either the geometric or the algebraic vantage point, and then the other meaning is deduced. Though, I sometimes imagine that in a circular classroom, one writes on a cylindrical blackboard, going from geometry to algebra and from algebra to geometry ad infinitum, as if riding a merry-go-round. The iff statement of Massey that cross product exists in and only in dimensions 1, 3, and 7 also fits the ride. As in scalar triple product, the proof of this iff statement mixes the dot and the cross, though geometry and algebra can no longer be separated, but be grouped and turned into number theory. I hope to convey Baum's analogy that the shift from Adams' original proof using secondary and higher operations of ordinary cohomology theory to Adams-Atiyah's simpler K-theory proof using only the primary operations of this extraordinary cohomology theory is comparable to the shift from the geocentric to the simpler heliocentric model of our solar system. Together with the pleasant coincidence of the clutching construction of the earth's tangent bundle with the 720 degrees daily turn of analog clocks, and those of the 2- and 8-fold Bott Periodicities with the arrangements of the planets, this collection of ideas were projected onto 2D vinyl records and their gatefold jacket, from the 3D blackboards, whose color can also be green.
| back cover | front cover |
| inner jacket |
Animation
On a turntable, I tested the vinyl records with the above mathematical graphics printed on their stickers: they do sound like from other planets as the stylus scratches the matte paper. If you would want to get a similar but digitized feeling at your computer, watch the animations below. For best effect, match them with the following singles on YouTube, which themselves satisfy a certain set of symmetries. Though, after a while, it's better to turn off the sound, as math is ultimately done in silence, in solitude, and sometimes in darkness, like on Sides B and D.
| Side A: Fujii Kaze, Kirari | Side B: Grimes, Genesis |
| Side C: TOMOO, Super Ball | Side D: Ryuichi Sakamoto, Merry Christmas Mr. Lawrence |
Each of the animations below turns 6 discrete degrees clockwise per second, synchronized with heartbeats, like the second hand of an analog clock, while our attention moves counterclockwise relative to the disk. They may evoke similar feelings about this clock two hours' drive away.
Side A
Side B
Side C
Side D
Human-Machine Interaction
During the Fall semester of 2023, I put a miniature (4.0 x 4.0 in) edition of these vinyl disks on the bulletin board outside my office. I invited everyone to experience the geometry and algebra merry-go-round by physically turning them about the push pins. Coincidentally, students taking complex analysis that semester would recognize the punctured disks over which an analytic function can be expanded as a Laurent series.
In this artwork, graphics were designed using Adobe Illustrator and LaTeXiT. Animations were made using Adobe Illustrator and Adobe Photoshop. Prints were taken care of on Walmart.com and at the Walmart Store in Hadley, MA. Jacket was manufactured at Freestyle Vinyl. Empty vinyl disks were shipped from Amazon.com.
This work was inspired by the artwork of the album A Tribute to Ryuichi Sakamoto -- To the Moon and Back, influenced by the concert The Planets performed by students at Amherst College, and partially benefitted from the faculty research annual allowance awarded by Amherst College.
Mathematical Footnote
This document unrolls the above circular mathematics to traditional rectangular pages with more annotations.