Pub Date : 2023-07-20DOI: 10.1080/0025570X.2023.2231836
David J. Gaebler, M. Panaggio, Timothy J. Pennings
Summary A discrete brachistochrone is the fastest piecewise linear ramp between fixed endpoints with a given number of segments. This article introduces a new conceptual framework for discrete brachistochrones, proves their two fundamental symmetry properties, and examines the manner in which they converge to the cycloid (the well-known continuous brachistochrone) as the number of sides tends to infinity.
{"title":"The Discrete Brachistochrone","authors":"David J. Gaebler, M. Panaggio, Timothy J. Pennings","doi":"10.1080/0025570X.2023.2231836","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231836","url":null,"abstract":"Summary A discrete brachistochrone is the fastest piecewise linear ramp between fixed endpoints with a given number of segments. This article introduces a new conceptual framework for discrete brachistochrones, proves their two fundamental symmetry properties, and examines the manner in which they converge to the cycloid (the well-known continuous brachistochrone) as the number of sides tends to infinity.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47386973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-19DOI: 10.1080/0025570X.2023.2231822
E. Demaine, Quanquan C. Liu
Summary What happens when the only goal in a chess game is to capture at least one piece of the opposite side? Can both sides live peacefully in an equilibrium where neither can capture the other’s pieces? In this short paper, we develop a new set of puzzles which we call chess equilibrium puzzles on this premise. We explain the rules of the game, analyze puzzles that have obvious and generalizable solutions, and provide several interesting puzzles for the reader to solve (solutions are provided at the end). Our puzzles provide an exciting twist to the realm of traditional chess puzzles.
{"title":"Chess Equilibrium Puzzles","authors":"E. Demaine, Quanquan C. Liu","doi":"10.1080/0025570X.2023.2231822","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231822","url":null,"abstract":"Summary What happens when the only goal in a chess game is to capture at least one piece of the opposite side? Can both sides live peacefully in an equilibrium where neither can capture the other’s pieces? In this short paper, we develop a new set of puzzles which we call chess equilibrium puzzles on this premise. We explain the rules of the game, analyze puzzles that have obvious and generalizable solutions, and provide several interesting puzzles for the reader to solve (solutions are provided at the end). Our puzzles provide an exciting twist to the realm of traditional chess puzzles.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43254292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-18DOI: 10.1080/0025570X.2023.2231838
Michael McDaniel, Joshua Wierenga
Summary We construct the icosahedral sphere in elliptic geometry in order to explore the structure of some viral capsids. We prove that all sides of triangular faces are altitudes of other triangles. We interpret math properties to match biochemical facts, which points to the possibility of using math to predict biochemistry.
{"title":"Geometry Goes Viral","authors":"Michael McDaniel, Joshua Wierenga","doi":"10.1080/0025570X.2023.2231838","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231838","url":null,"abstract":"Summary We construct the icosahedral sphere in elliptic geometry in order to explore the structure of some viral capsids. We prove that all sides of triangular faces are altitudes of other triangles. We interpret math properties to match biochemical facts, which points to the possibility of using math to predict biochemistry.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48589960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1080/0025570X.2023.2232259
J. Chase, M. Wright
Summary Bacterial growth is used as a simple example of exponential growth, but a population often grows much faster than the average time-to-division suggests. We examine the effect of randomness in the time-to-division of individual bacteria and the aggregate population growth, revealing intricacies that are often overlooked. Specifically, the average time-to-division of individual bacteria does not by itself determine the aggregate population growth. Exponential population growth occurs in realistic scenarios, but the aggregate growth factor depends in nonobvious ways on the underlying splitting distribution.
{"title":"Bacterial Growth: Not So Simple","authors":"J. Chase, M. Wright","doi":"10.1080/0025570X.2023.2232259","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2232259","url":null,"abstract":"Summary Bacterial growth is used as a simple example of exponential growth, but a population often grows much faster than the average time-to-division suggests. We examine the effect of randomness in the time-to-division of individual bacteria and the aggregate population growth, revealing intricacies that are often overlooked. Specifically, the average time-to-division of individual bacteria does not by itself determine the aggregate population growth. Exponential population growth occurs in realistic scenarios, but the aggregate growth factor depends in nonobvious ways on the underlying splitting distribution.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44036666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-17DOI: 10.1080/0025570X.2023.2231337
K. Sarma, Pijush Pratim Sarmah
Rewriting a nested radical in this form is commonly known as denesting. Denesting a nested radical is often very difficult. Many papers have been written on simplifying radicals. For example, consider the papers by Landau [3], Osler [4], and Zippel [6], and the references contained therein. The great Indian mathematician Srinivasa Ramanujan Aiyangar also contributed a lot in this direction. In his unique way, Ramanujan observed several striking relationships among certain nested radicals. Here are some examples (note that the final example uses the standard notation for a continued fraction):
{"title":"Evaluating Ramanujan’s Nested Radicals: A Sequential Approach","authors":"K. Sarma, Pijush Pratim Sarmah","doi":"10.1080/0025570X.2023.2231337","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231337","url":null,"abstract":"Rewriting a nested radical in this form is commonly known as denesting. Denesting a nested radical is often very difficult. Many papers have been written on simplifying radicals. For example, consider the papers by Landau [3], Osler [4], and Zippel [6], and the references contained therein. The great Indian mathematician Srinivasa Ramanujan Aiyangar also contributed a lot in this direction. In his unique way, Ramanujan observed several striking relationships among certain nested radicals. Here are some examples (note that the final example uses the standard notation for a continued fraction):","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46601093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-27DOI: 10.1080/0025570X.2023.2206280
J. Rosenhouse
and
和
{"title":"Letter from the Editor","authors":"J. Rosenhouse","doi":"10.1080/0025570X.2023.2206280","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2206280","url":null,"abstract":"and","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42979258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-24DOI: 10.1080/0025570X.2023.2203054
M. Frantz
Summary Because prolonged weightlessness has detrimental effects on human physiology, spaceflight experts envision artificial gravity, or pseudogravity, for long-duration space missions and artificial space habitats. This pseudogravity is generated by a constant rotation of the living space, typically a cylinder rotating about its axis. One aspect of this environment that could be disorienting or possibly even dangerous is the behavior of objects whose motion is initiated by being dropped, thrown, or struck, thereafter moving free of any external force (we ignore air resistance). We investigate the resulting pseudogravity ballistic trajectories in a frame of reference that rotates with the cylinder. The surprising and interesting results are greatly clarified by the use of parametric equations in both rectangular and polar coordinates, along with the usual formulations of velocity, acceleration, and curvature.
{"title":"Pseudogravity Ballistic Trajectories","authors":"M. Frantz","doi":"10.1080/0025570X.2023.2203054","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2203054","url":null,"abstract":"Summary Because prolonged weightlessness has detrimental effects on human physiology, spaceflight experts envision artificial gravity, or pseudogravity, for long-duration space missions and artificial space habitats. This pseudogravity is generated by a constant rotation of the living space, typically a cylinder rotating about its axis. One aspect of this environment that could be disorienting or possibly even dangerous is the behavior of objects whose motion is initiated by being dropped, thrown, or struck, thereafter moving free of any external force (we ignore air resistance). We investigate the resulting pseudogravity ballistic trajectories in a frame of reference that rotates with the cylinder. The surprising and interesting results are greatly clarified by the use of parametric equations in both rectangular and polar coordinates, along with the usual formulations of velocity, acceleration, and curvature.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43518782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-24DOI: 10.1080/0025570X.2023.2203052
Z. Ibragimov, Bogdan D. Suceavă
Summary Ptolemy’s theorem is a classical result obtained in the late Greek-Roman period, whose first application was to provide computational support to a geocentric cosmological model. This model’s most important achievement was that it explained the apparent movement of celestial bodies to a subjective observer on the Earth. What makes Ptolemy’s theorem a very interesting case in the history of mathematics is that the Euclidean concept of a Ptolemaic configuration can be investigated in the geometry of general metric spaces, in a situation very similar to the triangle inequality. To complement the historical narrative, in the final part of our paper we introduce a new norm, related to the Euclidean, Chebyshev, and Manhattan norms, and we investigate its properties in relation with other norms, hoping to illustrate how this fundamental configuration traversed Euclidean geometry, complex geometry, and analysis, transformational geometry, to become an interesting classification criterion in metric geometry.
{"title":"Ptolemy Through the Centuries","authors":"Z. Ibragimov, Bogdan D. Suceavă","doi":"10.1080/0025570X.2023.2203052","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2203052","url":null,"abstract":"Summary Ptolemy’s theorem is a classical result obtained in the late Greek-Roman period, whose first application was to provide computational support to a geocentric cosmological model. This model’s most important achievement was that it explained the apparent movement of celestial bodies to a subjective observer on the Earth. What makes Ptolemy’s theorem a very interesting case in the history of mathematics is that the Euclidean concept of a Ptolemaic configuration can be investigated in the geometry of general metric spaces, in a situation very similar to the triangle inequality. To complement the historical narrative, in the final part of our paper we introduce a new norm, related to the Euclidean, Chebyshev, and Manhattan norms, and we investigate its properties in relation with other norms, hoping to illustrate how this fundamental configuration traversed Euclidean geometry, complex geometry, and analysis, transformational geometry, to become an interesting classification criterion in metric geometry.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43519290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-23DOI: 10.1080/0025570X.2023.2199676
D. Kalman
Summary In undergraduate mathematics classes, the most common discrete version of logistic growth is defined by the difference equation . While this is a natural analog of the logistic differential equation, and while in many cases it produces results similar to those of the continuous model, it can also give rise to chaotic behavior. This paper derives in a natural way an alternative discrete logistic model, defined by the Verhulst difference equation, with several noteworthy properties. For example the Verhulst equation has closed form solutions given by continuous logistic curves and never leads to chaotic behavior. Our development of the Verhulst equation also provides a beautiful example of the formulation-application-refinement cycle of mathematical modeling. For these and other reasons, the Verhulst equation deserves a place in the undergraduate curriculum alongside the more familiar logistic difference equation given above.
{"title":"Verhulst Discrete Logistic Growth","authors":"D. Kalman","doi":"10.1080/0025570X.2023.2199676","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2199676","url":null,"abstract":"Summary In undergraduate mathematics classes, the most common discrete version of logistic growth is defined by the difference equation . While this is a natural analog of the logistic differential equation, and while in many cases it produces results similar to those of the continuous model, it can also give rise to chaotic behavior. This paper derives in a natural way an alternative discrete logistic model, defined by the Verhulst difference equation, with several noteworthy properties. For example the Verhulst equation has closed form solutions given by continuous logistic curves and never leads to chaotic behavior. Our development of the Verhulst equation also provides a beautiful example of the formulation-application-refinement cycle of mathematical modeling. For these and other reasons, the Verhulst equation deserves a place in the undergraduate curriculum alongside the more familiar logistic difference equation given above.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47788394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-22DOI: 10.1080/0025570X.2023.2204789
Hans Humenberger
Summary We generalize the concept of the Bevan point and the Bevan circle to a special sort of quadrilateral, so-called bicentric quadrilaterals, which have—like triangles—both an incenter and a circumcenter. As with triangles, the Bevan point V is the reflection of the incenter I over the circumcenter O. There are three other known points on the straight line through V, I, O, thus giving at least six collinear points on this straight line. We also deal with special homotheties, giving primarily synthetic and elementary proofs.
{"title":"On Six Collinear Points in Bicentric Quadrilaterals","authors":"Hans Humenberger","doi":"10.1080/0025570X.2023.2204789","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2204789","url":null,"abstract":"Summary We generalize the concept of the Bevan point and the Bevan circle to a special sort of quadrilateral, so-called bicentric quadrilaterals, which have—like triangles—both an incenter and a circumcenter. As with triangles, the Bevan point V is the reflection of the incenter I over the circumcenter O. There are three other known points on the straight line through V, I, O, thus giving at least six collinear points on this straight line. We also deal with special homotheties, giving primarily synthetic and elementary proofs.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43175975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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