Mathematics Commons^™

A Method For Folding Origami Rectilinear Polygon Extrusions, Everett L. Jin

Journal of Humanistic Mathematics

We present a method for generating crease patterns for folding any rectilinear polygon extrusion with a specified uniform minimum edge-to-edge distance, using a rectangular sheet of paper. We demonstrate that the resulting models are efficient and watertight. In addition, they achieve optimal efficiency, meaning they attain the smallest possible scale factor from the original sheet of paper to the folded rectilinear polygon extrusion. Finally, we provide a code implementation of the method. Our study not only facilitates crease pattern design for origami artists but also offers an innovative, art-driven pedagogical tool for inspiring interest in mathematics.

Manipulating Mathematics, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Anxiety Gets All The Likes, Saniye Nur Ergan

Journal of Humanistic Mathematics

This internet meme (created by me:)) emerges from a simple yet persistent question that underlies much of my work: Why do we continue to treat certain emotions in mathematics as visible, nameable, and research-worthy, while others linger in the background as faint, unarticulated traces?

Being Different Doesn’T Make You Odd, Raffaella Mulas

Journal of Humanistic Mathematics

The number 2 is beautiful and it has many unique properties. It is the only even prime number, and being different from the other prime numbers doesn't make it odd. This is true also for people: being different doesn't make you odd. It makes you beautiful and unique.

Nightmare In The Library, Charles A. Coppin

Journal of Humanistic Mathematics

Students of real analysis and calculus find that the completeness property of the real numbers is difficult to understand, especially, its importance. The word numbers denote real numbers throughout this piece. After all, the real numbers are not any less imaginary than the so-called imaginary numbers. Although, sometimes, it does gain some mention in calculus courses, its presence as a topic is a mere will-o’-the-wisp of bygone days when teachers would often reach deep into the big ideas of calculus as a mainstay of their courses. For the sake of cultural literacy and the development of mathematical maturity, we believe …

Logical Conclusion, Robert Dawson

Journal of Humanistic Mathematics

There is a well-known anecdote about Kurt Gödel, just before his citizenship hearing, telling Einstein that the Constitution was logically flawed. This alternate-history story explored the possible ramifications.

To Those Who Ask "What's The Name Of Your Book, Again?", Marion D. Cohen

Journal of Humanistic Mathematics

No abstract provided.

Missing, James S. Huntley

Journal of Humanistic Mathematics

Fifty-five word stories have achieved a certain currency as a reflective format, perhaps especially in medicine. Many of them turn out to be poems. Here is one such story inviting the reader to consider the nature of the absent, of the 'missing', of zero, and the philosophical underpinnings of mathematics.

Yet No Ramanujan, Db Jonas

Journal of Humanistic Mathematics

The poem Yet No Ramanujan conveys the musings of a layman in his garden, obsessed by the way that mundane things, the small experiences of everyday life, suggest the ways in which, as Baudelaire says, "everything is number." This meditation ends with the recognition that the putative stability of a self, a thinking subject, is after all, however complex and unique, like all matter, a kind of oscillation, a complex product of number.

To The Mysterious Gamma: Euler- Mascheroni Constant, Md Sadikur Rahman

Journal of Humanistic Mathematics

No abstract provided.

Poems For A Calculus Class (After Reading Midlife Calculus And At The Dimensional Border), Kevin Farey

Journal of Humanistic Mathematics

This poetry folder is made up of two short trios. The first begins with “Methods,” a thirteen-line poem by Britt Kaufmann, and continues with two thirteen-line responses to it. The folder’s second half consists of three (non-thirteen-line) responses to a collection of thirteen-line-almost-sonnets by Philip Fried, who had in turn been inspired, during the pandemic, by Edwin Abbot’s Flatland.

Both Kaufmann and Fried are authors of previous JHM poetry folders. Poems from Kaufmann’s book Midlife Calculus appeared in the January 2025 issue and poems from Fried’s series At the Dimensional Border appeared in the July 2024 issue. In addition, …

Plato's God, Vijay Fafat

Journal of Humanistic Mathematics

A poem expressing a thought about recursion in the Platonic Realm, of mathematical archetypes and the dreaming conception of "God"...

The Tool/Weapon Duality Of Mathematics, Alexandre Borovik

Journal of Humanistic Mathematics

This paper outlines the principal moral and ethical issues of mathematics and existential choices a mathematician may face in his professional life. My text is a manifesto; it is a list of declarations rather than a systematic development of the theme. Indeed each section could be expanded into a paper on its own. I hope to write fairly soon a more detailed exposition.

Sumthin: A Context-Dependent African American Vernacular English Word For Variables, Mark Apedo

Journal of Humanistic Mathematics

If mathematics is a language, then African American Vernacular English (AAVE) could hold the key to a fascinating understanding of the evolution of algebra from human speech in North America. The use of the pronoun something (sumthin) to depict an unknown amount of money that varies implies that algebra developed as a logical necessity in human communication. Exploring this phenomenon and leveraging it to demonstrate the concept of algebra could enhance the comprehension of the subject among speakers of AAVE.

The Math Teaching Atlas: Trails, Anchors, And Compass In Action, Ivan Z. Feng

Journal of Humanistic Mathematics

This paper advocates a shift grounded in mathematical reasoning: teach and write math through explicit routeways rather than narratives, anchor new ideas to familiar concepts on a roadmap, and use motivation and concrete examples as a compass and simulation for discovery.

Reminiscences Of My Past Life As A High School Math Teacher, Wayne Nirode

Journal of Humanistic Mathematics

In this article, I reflect on my twenty years as a high school math teacher. I describe four distinct phases of my career in five-year increments. I aim to make public both what makes teaching high school students rewarding and what makes it difficult. Also, my writing illuminates how indelible the first five years were to my development as a teacher. I hope that this brief memoir helps early-career teachers see past the tumultuous beginnings while also inspiring veteran teachers to reflect on their own careers.

Creating Art With The Generalized Pythagorean Theorem, Ángel Colín

Journal of Humanistic Mathematics

The Pythagorean Theorem and its generalizations have been widely studied. In this article we focus on one particular generalization that asserts that the area of a polygon with one side the hypotenuse of a right triangle is the sum of the areas of similar polygons whose corresponding sides are the other two sides [4]. Using this theorem together with interactive software or simply pencil and paper, we create some visually enticing artwork. This activity, we believe, can contribute positively to geometry teaching at all academic levels.

Let's Play Uno! Training Mathematical Thinking With An Inclusive Card Game, Thierry Meyrath, Clara-Ioana Mincu, Antonella Perucca

Journal of Humanistic Mathematics

We explore probability exercises based on the game UNO. Excluding the Wild cards, the deck contains precisely 100 cards, which makes it convenient to express probabilities as percentages. We discuss the mathematical relation between cards that can be played after other cards. We also propose an algebraic exercise that relates several quantities, such as the number of cards left in the losing player’s hand. Finally, we argue that UNO is a very inclusive game: it is logistically easy to play, allows for players of different skill levels, and can be adapted to meet special needs. Beyond being playful exercises, our …

Partnering With Students To Create A Positive Image Of Mathematics, Manmohan Kaur

Journal of Humanistic Mathematics

In a 1989 paper, Barbeau states, “Probably no area of human activity is as afflicted as mathematics with a gap between the public perception of its nature and what its practitioners believe it to be.” Decades later, the situation is much the same. With the technical advances in the contemporary world, mathematics affects our day-to-day activities in essential ways such as e-commerce and credit card security. Mathematical problem solving intrinsically requires creative and critical thinking. However, mathematics is often perceived as uninventive, sterile, static, and removed from human needs and culture. This paper describes how mathematicians can own the image …

Why Read The Classics (Of Mathematical Proofs)?, Cosimo Perini Brogi

Journal of Humanistic Mathematics

Inspired by Italo Calvino’s (almost) homonymous writing, this short essay suggests a formal perspective to complement the typical categorisation of classical results in mathematics on aesthetic bases. The traditional proof that √2 is not a rational number provides a simple example of proof mining. When studied through the tools of logic, that proof has also led to designing new systems for formal proving based on cyclic proofs. This example thus supports the central thesis of this essay that reading the classics of mathematical proofs enables us to acquire a taste for mathematical beauty and can open new directions in mathematics …

A Mathematician Expresses Grief, Richard Delaware

Journal of Humanistic Mathematics

A famous Scottish mathematician of the eighteenth century shares a tragic sorrow.

How (And Why) To Start A Math Circle At The County Jail, Aram Bingham

Journal of Humanistic Mathematics

Math circles are a common form of outreach usually run by professional mathematicians as enrichment for K-12 students or teachers. As a setting to stimulate interest and develop positive socialization around mathematical practice, a regular math circle can also be an effective bridge to higher education in carceral settings for both facilitators and participants. This article describes my journey through starting and running math circles in county jails, with a view towards the expansion of mathematics education as another tool for the dismantling of mass incarceration.

Making Mathematical Art Via The Harmonic And Perturbed Harmonic Functions, Mehmet Pakdemirli

Journal of Humanistic Mathematics

Starting from the definition of a harmonic function series, we define a new series which we call the perturbed harmonic function series. We explore the relationship of the new series with the Weierstrass and Riemann fractal functions as well as its convergence and differentiability properties. We then illustrate the potential of the harmonic functions in generating complex figures that sometimes resemble natural objects, with explicit numerical examples.

A Note On Modeling Prejudice, Alan Lockard, Edwin Harcourt

Journal of Humanistic Mathematics

We model prejudice in the context of an iterated prisoner’s dilemma tournament. Prejudice here is defined as the inability to distinguish amongst members of an identifiable group. We run a computer simulation where agents of one of two groups with defined strategies are randomly matched against each other. The agents either cooperate with or defect against the agent they are matched with. Our focus is on agents who play an unprejudiced version of the Tit-for-Tat strategy (cooperate with any player in the first encounter, and then apply the same strategy [defect or cooperate] that the opposing player played against them …

How Did Cauchy Prove His Integral Theorem? A Historical Reconstruction For Education Research, José Gerardo Piña-Aguirre, Rosa María Farfán Márquez

Journal of Humanistic Mathematics

In his article “Mémoire sur les Intégrales Définies, prises entre des limites imaginaires” of 1825, Cauchy proves that the integral of a complex function f defined on complex numbers is independent of the path of integration if the function f remains finite and continuous along the chosen path. In contemporary literature that analyzes the mathematics of Cauchy, the specific mathematical process that allowed Cauchy to prove this result is not explicitly presented. The purpose of this manuscript is to communicate a symbolic-algebraic derivation that justifies the narrative argument used by Cauchy to support his proof, thus hopefully filling this gap. …

Gold And Silver In Balance: Khayyam’S Mathematical Analysis Of Binary Alloys, Yousef Yassi, Reza Yassi

Journal of Humanistic Mathematics

The history of titration dates back to Archimedes who established that an object submerged in a liquid displaces an amount of liquid whose weight is equal to the buoyant force acting on the object. Since then, many scientists and engineers have tried to optimize his approach or devise new instruments for titration purposes. Omar Khayyam (1048–1123), in addition to designing a hydro-static balance, developed mathematical approaches for titration of binary alloys, that is, alloys made up of only two elements. Here we explore the different versions of Khayyam’s work on titration given by various sources. We also compare the accuracy …

Polygonal Number Similarity, Gunhan Caglayan

Journal of Humanistic Mathematics

This note takes an exploratory approach to define and then visualize the notion of polygonal number similarity between pairs of k-gonal numbers P_k(αn) and P_k(n) , where α is an integer scale factor of 2 or greater.

A Just Balance And The Teaching And Learning Of Mathematics, Marshall Gordon

Journal of Humanistic Mathematics

Society’s response to the question of what is worth sharing and how it best can be provided as conceived by the educational establishment shapes the education of every individual in school. As the classroom experience is a commitment to the future of society, it is good and just for society to support the educational development of each person in order that all can participate productively. In this way a “just balance” is established. In the particular case of educating students in mathematics, however, there is apparently an unjust balance, for despite the considerable time and effort students and teachers in …

Translating Trattato Dell'alcibra Amuchabile, Olympia Nicodemi

Journal of Humanistic Mathematics

We present a detailed look at one small section of a premodern algebra manuscript from circa 1365 C.E. The manuscript, Tratatto dell’Alcibra Amuchabile (Anonimo), is written in the Tuscan vernacular, with no symbols, no punctuation, and wildly inconsistent (pre-printing press) spellings. We present some of the hurdles to translating this “rhetorical algebra” followed by an annotated translation of one of its algebra rules, Rule 5, through which we learn the methods and mindset of algebra that preceded and seeded the breakthroughs in algebra of the 16^th century. We also point to some of the significant differences between algebra then …