Some Current Research Interests:
The history of Mathematics as it relates to classical teaching models. Integration of historical methods in the instruction of mathematics in a truly classical way, integrating the Quadrivium in particular. I have regularly engineer prompts for Chat-GPT with the intention of pushing its boundaries in mathematics and music theory.
Artificial intelligence and its influence on education, the future of educational methods generally, and the distinction between human learning and machine learning. I’m particularly interested in exploring the limitations of AI regarding non-stopping-time and not necessarily linear creative endeavors that are pseudo-algorithmic such as musical composition and poetry.
Mathematics and Musical Tuning; this is a technical enough subject for students to latch on to logically but also qualitative enough for a real debate to have the potential to take place. This can become a microcosm of real life with a nonlinear aspect, definitions to be sorted, with moral and visceral components that have been argued through the ages.
Sequences in the Collatz conjecture converge to the Pythagorean Spiral of fifth for large numbers. This is audible when octave-corrected. I hypothesize that there may be a way to leverage this to dismiss untested intervals beyond a certain threshold.
Beothius and Nicomachus lay out various divisions of the number into categories that differ from what we use today. In these differences there are often new and interesting patterns to rediscover. Several times I’ve found hints at algebraic factorization theorems such as the foil Algorithm for perfect squares, the coefficient of the binomial expansion theorem, observation touching on the roots of unity, and discrete convolution like computations to create integer segments of arbitrary length for geometric sequences with fractional common ratios.
Visualizing rational approximation of irrational numbers via alternate spacial layouts. On the Archimedean spiral in particular certain almost-cyclic behaviors emerge at different scales. These partially match up with continued fractions and p-adic expansions at times.. I suspect a potentially helpful connection for the Collatz conjecture may be in here somewhere.
Some of my most influential reads professionally:
In no particular order
GEB (Hofstadter)
Flatland (Abbott)
From Music to Mathematics (Roberts)
Arithmetic (Nicomachus)
Temperament (Isacoff)
Scholasticism (Pieper
Music in the 16th Century (Palaska)
Fundamentals of Music (Boethius)
Consolation of Philosophy (Boethius)
How Equal Temperament Ruined Harmony (Duffin)
Boethian Number Theory
On the Sensation of Tone (Helmholtz)
Orthodoxy (Chesterton)
The Discarded Image (Lewis)
The Abolition of Man (Lewis)
Mathematics for Human Flourishing (Su)
Prime Obsession (Derbyshire)
A History of Mathematics (Katz)
Harmonics (Ptolemy)
The Scientific Method in Ptolemy’s Harmonics (Barker)
Some amazing YouTube content creators that helped expose me to new ideas
In no particular order
3B1B
Standup Maths
Steve Mould
Veritasium
Smarter Every Day
Lex Fiedman
Welch Labs
Adam Neely (early stuff)
Early music Sources
My own Channel start up: Quadrivial Matters (coming soon)