Mathematics

This paper states and proves a generalization of the well-known Desargues involution theorem from plane projective geometry.

After our book---Mathematics and Music: Composition, Perception, and Performance, 2nd edition, CRC Press, 2020---was published, we found a striking example of the importance of the Tonnetz for analyzing the harmonic structure of The Beatles' song, "In My Life." Our Tonnetz analysis will illustrate the highly structured geometric logic underlying the numerous chord progressions in the song. Spectrograms provide a way for us to visualize chordal harmonics and their connection with voice leading. We shall also describe the interesting harmonic rhythms of the song's chord progressions. A lot of this harmonic rhythm lends itself well to a geometric description.

This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.

In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based course and some linear algebra; the appendix utilizes abstract algebra.

This paper discusses a problem that consists of n "lighthouses" which are circles with radius 1, placed around a common center, equidistant at n units away from the placement center. Consecutive lighthouses are separated by the same angle: 360 ∘ /n which we denote as α . Each lighthouse "illuminates" facing towards the placement center with the same angle α , also called "Illumination Angle" in this case. As for the light source itself, there are two variations: a single point light source at the center of each lighthouse and point light sources on the arc seen by the illumination angle for each lighthouse. The problem: what is the total dark (not illuminated) area for a given number of lighthouses, and as the number of lighthouses approach infinity? We show that by definition of the problem, neighbor lighthouses do not overlap or be tangent to each other. We propose a solution for the center point light source case and discuss several small cases of n for the arc light source case.

We use the data of tenured and tenure-track faculty at ten public and private math departments of various tiered rankings in the United States, as a case study to demonstrate the statistical and mathematical relationships among several variables, e.g., the number of publications and citations, the rank of professorship and AMS fellow status. At first we do an exploratory data analysis of the math departments. Then various statistical tools, including regression, artificial neural network, and unsupervised learning, are applied and the results obtained from different methods are compared. We conclude that with more advanced models, it may be possible to design an automatic promotion algorithm that has the potential to be fairer, more efficient and more consistent than human approach.

This paper introduces a new equation for rewriting two unit fractions to another two unit fractions. This equation is useful for optimizing the elements of an Egyptian Fraction. Parity of the elements of the Egyptian Fractions are also considered. And lastly, the statement that all rational numbers can be represented as Egyptian Fraction is re-established.

In this article we generalize the classic "farm pen" optimization problem from a first course in calculus in a handful of different ways. We describe the solution to an n -dimensional rectangular variant, and then study the situation when the pens are either regular polygons or platonic solids.

In this paper, we present a novel method to draw a circle tangent to three given circles lying on a plane. Using the analytic geometry and inversion (reflection) theorems, the center and radius of the inversion circle are obtained. Inside any one of the three given circles, a circle of the similar radius and concentric with its own corresponding original circle is drawn.The tangent circle to these three similar circles is obtained. Then the inverted circles of the three similar circles and the tangent circle regarding an obtainable point and a computable power of inversion (reflection) constant are obtained. These circles (three inverted circles and an inverted tangent circle)will be tangent together.Just,we obtain another reflection point and power of inversion so that those three reflected circles (inversions of three similar circles) can be reflections of three original circles, respectively. In such a case,the reflected circle tangent to three reflected circles regarding same new inversion system will be tangent to the three original ones. This circle is our desirable circle. A drawing algorithm is also given for drawing desirable circle by straightedge and compass. A survey of conformal mapping theory and inversion in higher dimensions is also accomplished. Although, Laguerre transformation might be used for solution of this problem, but we do not make use of this method. Our novelty is just for drawing a circle tangent to three given circles applying a tangent circle to three identical circles concentric with three given ones and then inverting them as original ones by compass and straightedge not any thing else.

Drawing inspiration from both the classical Guerino Mazzola's symmetry-based model for first-species counterpoint (one note against one note) and Johann Joseph Fux's "Gradus ad Parnassum", we propose an extension for second-species (two notes against one note).