{"title":"Corks","authors":"Selman Akbulut","doi":"arxiv-2406.15369","DOIUrl":"https://doi.org/arxiv-2406.15369","url":null,"abstract":"Remarks relating the various notions of corks.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528413","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}
People typically consider only European mathematics as orthodox, often intentionally or unintentionally overlooking the existence of mathematics from non-European societies. Inspired by Maria Ascher's two well-known papers on sand drawings in Oceania and Africa, this paper focuses on the strong link between modern mathematics and the mathematics behind the sand drawings. Beginning with a comparison of the geography, history, and the cultural context of sand drawings in Oceania and Africa, we will examine shared geometric features of European graph theory and Indigenous sand drawings, including continuity, cyclicity, and symmetry. The paper will also delve into the origin of graph theory, exploring whether the famous European mathematician Leonhard Euler, who published his solution to the Konigsberg bridge problem in 1736, was the true inventor of graph theory. The potential for incorporating sand drawings into the school curriculum is highlighted at the end. Overall, this paper aims to make readers realise the importance of ethnomathematics studies and appreciate the intelligence of Indigenous people.
{"title":"Comparative Study of Sand Drawings in Oceania and Africa","authors":"Linbin Wang, Rowena Ball, Hongzhang Xu","doi":"arxiv-2404.04798","DOIUrl":"https://doi.org/arxiv-2404.04798","url":null,"abstract":"People typically consider only European mathematics as orthodox, often\u0000intentionally or unintentionally overlooking the existence of mathematics from\u0000non-European societies. Inspired by Maria Ascher's two well-known papers on\u0000sand drawings in Oceania and Africa, this paper focuses on the strong link\u0000between modern mathematics and the mathematics behind the sand drawings.\u0000Beginning with a comparison of the geography, history, and the cultural context\u0000of sand drawings in Oceania and Africa, we will examine shared geometric\u0000features of European graph theory and Indigenous sand drawings, including\u0000continuity, cyclicity, and symmetry. The paper will also delve into the origin\u0000of graph theory, exploring whether the famous European mathematician Leonhard\u0000Euler, who published his solution to the Konigsberg bridge problem in 1736, was\u0000the true inventor of graph theory. The potential for incorporating sand\u0000drawings into the school curriculum is highlighted at the end. Overall, this\u0000paper aims to make readers realise the importance of ethnomathematics studies\u0000and appreciate the intelligence of Indigenous people.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574004","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}
We consider the angle in mathematics and arrive at a conclusion that there are two concepts on the issue. One is a descriptive geometrical one, while the other is from functional analysis. They are somewhat different, allow for different options, and both are legitimate and in use. Their difference may cause certain confusions. While the `geometrical angle' allows for different choice of units, the `functional angle' is a purely dimensionless one, being related to the angle in radians. We consider possible options to resolve the problem as it concerns the units.
{"title":"A dual concept of the angle in mathematics and practice","authors":"Savely G. Karshenboim","doi":"arxiv-2404.08560","DOIUrl":"https://doi.org/arxiv-2404.08560","url":null,"abstract":"We consider the angle in mathematics and arrive at a conclusion that there\u0000are two concepts on the issue. One is a descriptive geometrical one, while the\u0000other is from functional analysis. They are somewhat different, allow for\u0000different options, and both are legitimate and in use. Their difference may\u0000cause certain confusions. While the `geometrical angle' allows for different\u0000choice of units, the `functional angle' is a purely dimensionless one, being\u0000related to the angle in radians. We consider possible options to resolve the\u0000problem as it concerns the units.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574406","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}
An age-old controversy in mathematics concerns the necessity and the possibility of constructive proofs. The controversy has been rekindled by recent advances which demonstrate the feasibility of a fully constructive mathematics. This nontechnical article discusses the motivating ideas behind the constructive approach to mathematics and the implications of constructive mathematics for the history of mathematics.
{"title":"Constructive Mathematics","authors":"Mark Mandelkern","doi":"arxiv-2404.05743","DOIUrl":"https://doi.org/arxiv-2404.05743","url":null,"abstract":"An age-old controversy in mathematics concerns the necessity and the\u0000possibility of constructive proofs. The controversy has been rekindled by\u0000recent advances which demonstrate the feasibility of a fully constructive\u0000mathematics. This nontechnical article discusses the motivating ideas behind\u0000the constructive approach to mathematics and the implications of constructive\u0000mathematics for the history of mathematics.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573912","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}
The exposition in Euclid's Elements contains an obvious gap (seemingly unnoticed by most commentators): he often compares not just angles, but *groups* of angles, and at the same time he avoids summing angles (and considering angles greater than $pi$), and does not say what such a comparison of groups could mean. We discuss the problem and suggest a possible interpretation that could make Euclid's exposition consistent.
{"title":"Comparing angles in Euclid's Elements","authors":"Alexander Shen","doi":"arxiv-2404.02272","DOIUrl":"https://doi.org/arxiv-2404.02272","url":null,"abstract":"The exposition in Euclid's Elements contains an obvious gap (seemingly\u0000unnoticed by most commentators): he often compares not just angles, but\u0000*groups* of angles, and at the same time he avoids summing angles (and\u0000considering angles greater than $pi$), and does not say what such a comparison\u0000of groups could mean. We discuss the problem and suggest a possible\u0000interpretation that could make Euclid's exposition consistent.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"57 12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573913","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}
I Gusti Ayu Putu Arya Wulandari, I Putu Ade Andre Payadnya, Kadek Rahayu Puspadewi, Sompob Saelee
The field of ethnomathematics holds significance in the pursuit of comprehending how students can grasp, express, manipulate, and ultimately apply mathematical concepts. However, ethnomathematics is also considered a complex concept in Asian countries such as Indonesia and Thailand, which can pose challenges as it needs to be comprehensively understood. This research aims to fill the gap by understanding the cross-cultural perspective of mathematics educators in Indonesia and Thailand. The participants were lecturers, teachers, and pre-service teachers. Data was gathered through questionnaires and interviews. The analytical approach involved were data reduction, data presentation, drawing conclusions or verification, and data validity. Positive responses were indicated by mathematics educators with the average scores of respondents in Indonesia at 4.77 and Thailand at 4.57. This research concludes the importance of integrating ethnomathematics in education, which is closely tied to cultural development, emphasizing the crucial role of employing comprehensive strategies in its implementation.
{"title":"The Significance of Ethnomathematics Learning: A Cross-Cultural Perspectives Between Indonesian and Thailand Educators","authors":"I Gusti Ayu Putu Arya Wulandari, I Putu Ade Andre Payadnya, Kadek Rahayu Puspadewi, Sompob Saelee","doi":"arxiv-2404.01648","DOIUrl":"https://doi.org/arxiv-2404.01648","url":null,"abstract":"The field of ethnomathematics holds significance in the pursuit of\u0000comprehending how students can grasp, express, manipulate, and ultimately apply\u0000mathematical concepts. However, ethnomathematics is also considered a complex\u0000concept in Asian countries such as Indonesia and Thailand, which can pose\u0000challenges as it needs to be comprehensively understood. This research aims to\u0000fill the gap by understanding the cross-cultural perspective of mathematics\u0000educators in Indonesia and Thailand. The participants were lecturers, teachers,\u0000and pre-service teachers. Data was gathered through questionnaires and\u0000interviews. The analytical approach involved were data reduction, data\u0000presentation, drawing conclusions or verification, and data validity. Positive\u0000responses were indicated by mathematics educators with the average scores of\u0000respondents in Indonesia at 4.77 and Thailand at 4.57. This research concludes\u0000the importance of integrating ethnomathematics in education, which is closely\u0000tied to cultural development, emphasizing the crucial role of employing\u0000comprehensive strategies in its implementation.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"120 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573998","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}
The Brioschi formula expresses the Gaussian curvature $K$ in terms of the functions $E, F$ and $G$ in local coordinates of a surface $S$. This implies the Gauss' theorema egregium, which says that the Gaussian curvature just depends on angles, distances, and their rates of change. In most of the textbooks, the Gauss' theorema egregium was proved as a corollary to the derivation of the Gauss equations, a set of equations expressing $EK, FK$ and $GK$ in terms of the Christoffel symbols. The Christoffel symbols can be expressed in terms of $E$, $F$ and $G$. In principle, one can derive the Brioschi formula from the Gauss equations after some tedious calculations. In this note, we give a direct elementary proof of the Brioschi formula without using Christoffel symbols. The key to the proof are properties of matrices and determinants.
{"title":"The Brioschi Formula for the Gaussian Curvature","authors":"Lee-Peng Teo","doi":"arxiv-2404.00835","DOIUrl":"https://doi.org/arxiv-2404.00835","url":null,"abstract":"The Brioschi formula expresses the Gaussian curvature $K$ in terms of the\u0000functions $E, F$ and $G$ in local coordinates of a surface $S$. This implies\u0000the Gauss' theorema egregium, which says that the Gaussian curvature just\u0000depends on angles, distances, and their rates of change. In most of the textbooks, the Gauss' theorema egregium was proved as a\u0000corollary to the derivation of the Gauss equations, a set of equations\u0000expressing $EK, FK$ and $GK$ in terms of the Christoffel symbols. The\u0000Christoffel symbols can be expressed in terms of $E$, $F$ and $G$. In\u0000principle, one can derive the Brioschi formula from the Gauss equations after\u0000some tedious calculations. In this note, we give a direct elementary proof of the Brioschi formula\u0000without using Christoffel symbols. The key to the proof are properties of\u0000matrices and determinants.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574000","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}
In 1738, the King of Naples and future King of Spain, Carlos III, commissioned the Spanish military engineer Roque Joaqu'in de Alcubierre to begin the excavations of the ruins of the ancient Roman city of Pompeii and its surroundings, buried by the terrible explosion of Vesuvius in AD 79. Since that time, archaeologists have brought to light wonderful treasures found in the among ruins. Among them, the Sator Square is one of the most peculiar, apparently simple but mysterious. Supernatural and medicinal powers have been attributed to this object and its use was widespread during the Middle Age. Studies to explain its origin and meaning have been varied. There are theories that relate it to religion, the occult, medicine and music. However, no explanation has been convincing beyond pseudo-scientific sensationalism. In this study, the author intends to eliminate the mystical character of the Sator Square and suggests considering it as a simple palindrome or a game of words with certain symmetrical properties. However, these properties are not exclusive to the Sator Suare but are present in various mathematical and geometric objects.
1738 年，那不勒斯国王和未来的西班牙国王卡洛斯三世委托西班牙军事工程师 Roque Joaqu'in de Alcubierre 开始发掘公元 79 年被可怕的维苏威火山爆发掩埋的古罗马城市庞贝及其周围地区的废墟。从那时起，考古学家们在废墟中发现了许多奇珍异宝。其中，萨托广场是最奇特的一个，看似简单，实则神秘。人们认为它具有超自然的药力，在中世纪被广泛使用。对其起源和含义的研究多种多样，有的理论认为它与宗教、神秘学、医学和音乐有关。然而，除了伪科学的哗众取宠之外，还没有令人信服的解释。在本研究中，作者打算消除萨托方格的神秘性，并建议将其视为一种简单的回文或具有某些对称特性的文字游戏。然而，这些特性并非萨托方块独有，而是存在于各种数学和几何物体中。
{"title":"Some mathematical and geometrical interpretations of the Sator Square","authors":"Paul Dario Toasa Caiza","doi":"arxiv-2404.01048","DOIUrl":"https://doi.org/arxiv-2404.01048","url":null,"abstract":"In 1738, the King of Naples and future King of Spain, Carlos III,\u0000commissioned the Spanish military engineer Roque Joaqu'in de Alcubierre to\u0000begin the excavations of the ruins of the ancient Roman city of Pompeii and its\u0000surroundings, buried by the terrible explosion of Vesuvius in AD 79. Since that\u0000time, archaeologists have brought to light wonderful treasures found in the\u0000among ruins. Among them, the Sator Square is one of the most peculiar,\u0000apparently simple but mysterious. Supernatural and medicinal powers have been\u0000attributed to this object and its use was widespread during the Middle Age.\u0000Studies to explain its origin and meaning have been varied. There are theories\u0000that relate it to religion, the occult, medicine and music. However, no\u0000explanation has been convincing beyond pseudo-scientific sensationalism. In\u0000this study, the author intends to eliminate the mystical character of the Sator\u0000Square and suggests considering it as a simple palindrome or a game of words\u0000with certain symmetrical properties. However, these properties are not\u0000exclusive to the Sator Suare but are present in various mathematical and\u0000geometric objects.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574006","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}
During the querelle des infiniment petits, Leibniz wrote several texts to justify using Differential calculus among Parisian savants. However, only three were published. Among these publications, ''Sentiment de Monsieur Leibnitz'' had a peculiar destiny. Although we are aware of the manuscript (Gotha FB A 448--449, Bl. 41--42), it is only recently that we have identified a copy of its impression in the British Library catalogue. This copy was printed in 1706 together with writings by other mathematicians united in the defence of the new calculus -- Joseph Saurin, Jacob Hermann and the Bernoulli brothers. Recently published epistolary exchanges indicate that Jean-Paul Bignon, at the time director of the Royal Academy of Sciences, in order to calm down the institution, had prohibited this publication and confiscated the prints.This article examine the epistemological and institutional issues at stake in ''Sentiment de Monsieur Leibnitz''.
{"title":"The Peculiar Destiny of Sentiment de Monsieur Leibnitz (May 1705 -- March 1706)","authors":"Sandra BellaAHP-PReST","doi":"arxiv-2403.20052","DOIUrl":"https://doi.org/arxiv-2403.20052","url":null,"abstract":"During the querelle des infiniment petits, Leibniz wrote several texts to\u0000justify using Differential calculus among Parisian savants. However, only three\u0000were published. Among these publications, ''Sentiment de Monsieur Leibnitz''\u0000had a peculiar destiny. Although we are aware of the manuscript (Gotha FB A\u0000448--449, Bl. 41--42), it is only recently that we have identified a copy of\u0000its impression in the British Library catalogue. This copy was printed in 1706\u0000together with writings by other mathematicians united in the defence of the new\u0000calculus -- Joseph Saurin, Jacob Hermann and the Bernoulli brothers. Recently\u0000published epistolary exchanges indicate that Jean-Paul Bignon, at the time\u0000director of the Royal Academy of Sciences, in order to calm down the\u0000institution, had prohibited this publication and confiscated the prints.This\u0000article examine the epistemological and institutional issues at stake in\u0000''Sentiment de Monsieur Leibnitz''.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140602962","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}
Consider the rectangular triangle with sides with length 1 and 1, then the oblique side has length square root of 2. Now construct on top of the oblique side, a new rectangular triangle with the oblique side as rectangle side and a second rectangle side of length 1. Continue this process indefinitely, what you get is called "the spiral of Theodorus". Now the question is: Can there be two hypothenusa (oblique sides) which lie on the same line? Apparently there can't. A proof of this proposition was given in 1958, but to our knowledge no other proofs are available. Since we had no access to the journal, we wanted to prove it again.
{"title":"In Theodorus' Spiral no two hypothenusa lie on the same line","authors":"Frederik Stouten","doi":"arxiv-2403.20207","DOIUrl":"https://doi.org/arxiv-2403.20207","url":null,"abstract":"Consider the rectangular triangle with sides with length 1 and 1, then the\u0000oblique side has length square root of 2. Now construct on top of the oblique\u0000side, a new rectangular triangle with the oblique side as rectangle side and a\u0000second rectangle side of length 1. Continue this process indefinitely, what you\u0000get is called \"the spiral of Theodorus\". Now the question is: Can there be two\u0000hypothenusa (oblique sides) which lie on the same line? Apparently there can't.\u0000A proof of this proposition was given in 1958, but to our knowledge no other\u0000proofs are available. Since we had no access to the journal, we wanted to prove\u0000it again.","PeriodicalId":501462,"journal":{"name":"arXiv - MATH - History and Overview","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140573997","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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