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Spandrelized Tilings
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 3) |
| Halaman | : | 199-217 |
| Abstrak | : | We discuss a method to generate a new tiling from an edge-to-edge tiling by regular polygons. We place tangent disks at the vertices and then glue the interstitial areas, known as spandrels, onto the disks to create new tiles. We apply this method in the Euclidean plane, the sphere and the hyperbolic plane to obtain a variety of new tilings. |
The Birational Geometry of Ceva’s Theorem
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 5) |
| Halaman | : | 430-442 |
| Abstrak | : | In this article we study Ceva’s theorem and its higher-dimensional extensions from the perspective of algebraic and projective geometry. First, we situate the theorem within the study of algebraic surfaces by relating it to the defining equation of a del Pezzo surface of degree six inside the product of three projective lines. Second, by interpreting (higher-dimensional analogues of) Ceva’s theorem in terms of projections from projective spaces, we recast these results as matrix completion problems. We use these ideas to offer proofs of some higher-dimensional analogues of Ceva’s theorem. This article is written with a nonspecialist audience in mind and we hope that some useful context is provided in the form of remarks in the sections on surfaces for students of algebraic geometry. |
The Pivotal Set of a Boolean Function
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 5) |
| Halaman | : | 414-429 |
| Abstrak | : | We define the pivotal set of a Boolean function and we prove a fundamental inequality on its expected size, when the inputs are independent random coins of parameter p. We give two complete proofs of this inequality. Along the way, we obtain the classical Margulis–Russo formula. We give a short proof of the classical Hoeffding inequality for i.i.d. Bernoulli random variables, and we use it to derive more complex deviations inequalities associated to the pivotal set. We finally follow Talagrand’s footsteps and we discuss a beautiful inequality that he proved in the uniform case. |
Algebraic Relations Between Sine and Cosine Values
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 5) |
| Halaman | : | 403-413 |
| Abstrak | : | The aim of this article is to show that any algebraic relation over ¯? between the values of the trigonometric functions sine and cosine at algebraic points can be derived from the Pythagorean identity and the angle addition formulas. This result is obtained as a consequence of the Lindemann–Weierstrass theorem. |
A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 5) |
| Halaman | : | 383-402 |
| Abstrak | : | The Catalan numbers ???????? count the number of subdivisions of a polygon into m triangles, and it is well known that their generating series is a solution to a particular quadratic equation. Analogously, the hyper-Catalan numbers ???????? count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type ????), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation. A layering of this series by numbers of faces yields a remarkable factorization that reveals the Geode, a mysterious array that appears to underlie Catalan numerics. |
SPOILERS: Solutions and Explanations—The Mathematics of Crossword Puzzles: In Celebration of Karen Hunger Parshall
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 6) |
| Halaman | : | 607-616 |
| Abstrak | : | Readers are advised not to jump directly to the following explanations of the puzzles but first to try each of them on their own. By doing so, one’s appreciation will be enhanced for each of the puzzles themselves, as well as for what their constructors have managed to accomplish. |
Hidden Figures Revealed
| Pengarang | : | Ranthony A. Clark |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 6) |
| Halaman | : | 575-591 |
| Abstrak | : | Nearly 200 mathematicians who identify as Black have earned degrees from The Ohio State University. These individuals have gone on to have prolific careers in academia, industry, and education. They include university presidents, authors, and lawyers—yet their stories have largely remained hidden. This article highlights “Hidden Figures Revealed,” the first comprehensive study of Black mathematicians at a single U.S. institution. We introduce this project, and focus on the historical question: what factors contributed to the rise of seven Black mathematicians earning doctoral degrees in mathematics from 1963 to 1984 at Ohio State, and why did this progress abruptly halt for nearly forty years? |
Spreading the Wealth: Eugene P. Northrop’s Advancement of Mathematics and Science at Home and Abroad
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 6) |
| Halaman | : | 564-574 |
| Abstrak | : | In 1934, Eugene P. Northrop finished his Ph.D. in mathematics at Yale. Competing for scarce jobs, he settled for a position at the private boarding school, Hotchkiss, where he stayed until 1943, when he took up a post at the College of the University of Chicago. Several years spent as an education consultant for the NSF and the Ford Foundation’s Fund for the Advancement of Education positioned Northrop to move from the world of American liberal arts education to a job as the Ford Foundation’s field representative in Turkey. This paper considers Northrop’s work at the University of Chicago and his contributions to the development of university science in Turkey through his position at the Ford Foundation. |
George Bruce Halsted’s Non-Euclidean Pilgrimage to Hungary in 1896
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 6) |
| Halaman | : | 554-563 |
| Abstrak | : | With their limited first publications and slow recognition of their foundational works in non-Euclidean geometry, János Bolyai and Nikolai Lobachevski never received credit proportional to their achievements. George Bruce Halsted, while a mathematics professor at the University of Texas in the 1890s, took on the cause and was at the forefront of international efforts to see that they had their place in history. Even if Halsted did not achieve as much as he set out to do, his visit to Bolyai’s homeland in Hungary proved to be the reinforcement from outside that local caretakers of Bolyai’s legacy welcomed. |
Shooting for the Moon: Benjamin Peirce’s Ambitious 19th-century Efforts to Elevate American Mathematics Through Astronomy
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 6) |
| Halaman | : | 541-553 |
| Abstrak | : | Over fifty years as a mathematics professor at Harvard, Benjamin Peirce worked to elevate mathematical sciences in the US by championing educational reform, promoting research-level publication, and strategizing about scientific funding. He used his roles in the Nautical Almanac Office and the US Coast Survey to train and employ a generation of mathematical astronomers. Peirce further capitalized on the public enthusiasm generated by The Great Comet in 1843, the discovery of Neptune in 1846, observations of Saturn’s rings in the 1850s, and a number of total solar eclipses in the 1860s and 70s. Consistent and intentional advocacy from Benjamin Peirce sought to elevate the profile of American mathematics through astronomical work. This shaped how mathematics developed within the broader context of the establishment of scientific infrastructure in mid-nineteenth-century America. |