Richard Schmoetten, Jake E. Palmer, Jacques D. Fleuriot
This contribution reports on the continued formalisation of an axiomatic system for Minkowski spacetime (as used in the study of Special Relativity) which is closer in spirit to Hilbert's axiomatic approach to Euclidean geometry than to the vector space approach employed by Minkowski. We present a brief overview of the axioms as well as of a formalisation of theorems relating to linear order. Proofs and excerpts of Isabelle/Isar scripts are discussed, with a focus on the use of symmetry and reasoning"without loss of generality".
{"title":"Formalising Geometric Axioms for Minkowski Spacetime and Without-Loss-of-Generality Theorems","authors":"Richard Schmoetten, Jake E. Palmer, Jacques D. Fleuriot","doi":"10.4204/EPTCS.352.13","DOIUrl":"https://doi.org/10.4204/EPTCS.352.13","url":null,"abstract":"This contribution reports on the continued formalisation of an axiomatic system for Minkowski spacetime (as used in the study of Special Relativity) which is closer in spirit to Hilbert's axiomatic approach to Euclidean geometry than to the vector space approach employed by Minkowski. We present a brief overview of the axioms as well as of a formalisation of theorems relating to linear order. Proofs and excerpts of Isabelle/Isar scripts are discussed, with a focus on the use of symmetry and reasoning\"without loss of generality\".","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131319931","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}
This paper shows a cut along a crease on an origami sheet makes simple modeling of popular traditional basic folds such as a squash fold in computational origami. The cut operation can be applied to other classical folds and significantly simplify their modeling and subsequent implementation in the context of computational origami.
{"title":"A New Modeling of Classical Folds in Computational Origami","authors":"T. Ida, Hidekazu Takahashi","doi":"10.4204/EPTCS.352.5","DOIUrl":"https://doi.org/10.4204/EPTCS.352.5","url":null,"abstract":"This paper shows a cut along a crease on an origami sheet makes simple modeling of popular traditional basic folds such as a squash fold in computational origami. The cut operation can be applied to other classical folds and significantly simplify their modeling and subsequent implementation in the context of computational origami.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"100 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121474850","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 report on a new, simple, modular, and flexible approach for automated generation of illustrations for (readable) synthetic geometry proofs. The underlying proofs are generated using the Larus automated prover for coherent logic, and corresponding illustrations are generated in the GCLC language. Animated illustrations are also supported.
{"title":"Automated Generation of Illustrations for Synthetic Geometry Proofs","authors":"Predrag Janičić, Julien Narboux","doi":"10.4204/EPTCS.352.9","DOIUrl":"https://doi.org/10.4204/EPTCS.352.9","url":null,"abstract":"We report on a new, simple, modular, and flexible approach for automated generation of illustrations for (readable) synthetic geometry proofs. The underlying proofs are generated using the Larus automated prover for coherent logic, and corresponding illustrations are generated in the GCLC language. Animated illustrations are also supported.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"2257 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130225229","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 report on several scenarios of using automated theorem proving software in university education. In particular, we focus on using the Theorema system in a software-enhanced logic-course for students in computer science or artificial intelligence. The purpose of using logic-software in our teaching is not to teach students the proper use of a particular piece of software. In contrast, we try to employ certain software in order to spark students' motivation and to support their understanding of logic principles they are supposed to understand after having passed the course. In a sense, we try to let the software act as a logic-tutor, the software is not an additional subject we teach.
{"title":"Automated Theorem Proving in the Classroom","authors":"W. Windsteiger","doi":"10.4204/EPTCS.352.6","DOIUrl":"https://doi.org/10.4204/EPTCS.352.6","url":null,"abstract":"We report on several scenarios of using automated theorem proving software in university education. In particular, we focus on using the Theorema system in a software-enhanced logic-course for students in computer science or artificial intelligence. The purpose of using logic-software in our teaching is not to teach students the proper use of a particular piece of software. In contrast, we try to employ certain software in order to spark students' motivation and to support their understanding of logic principles they are supposed to understand after having passed the course. In a sense, we try to let the software act as a logic-tutor, the software is not an additional subject we teach.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127922092","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}
Understanding geometric relationships with little mathematical knowledge can be challenging for today's students and teachers. A new toolset is introduced that is able to create a proof without words by combining the benefits of the Geometric Deduction Database method (to obtain a readable proof of a geometric statement) and the GeoGebra framework (that makes it possible to export these data as an online applet in a simple way).
{"title":"Online Generation of Proofs Without Words","authors":"Alexander Thaller, Z. Kovács","doi":"10.4204/EPTCS.352.10","DOIUrl":"https://doi.org/10.4204/EPTCS.352.10","url":null,"abstract":"Understanding geometric relationships with little mathematical knowledge can be challenging for today's students and teachers. A new toolset is introduced that is able to create a proof without words by combining the benefits of the Geometric Deduction Database method (to obtain a readable proof of a geometric statement) and the GeoGebra framework (that makes it possible to export these data as an online applet in a simple way).","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128351941","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}
A minimally rigid graph, also called Laman graph, models a planar framework which is rigid for a general choice of distances between its vertices. In other words, there are finitely many ways, up to isometries, to realize such a graph in the plane. Using ideas from algebraic and tropical geometry, we derive a recursive formula for the number of such realizations. Combining computational results with the construction of new rigid graphs via gluing techniques, we can give a new lower bound on the maximal possible number of realizations for graphs with a given number of vertices.
{"title":"Realizations of Rigid Graphs","authors":"C. Koutschan","doi":"10.4204/EPTCS.352.2","DOIUrl":"https://doi.org/10.4204/EPTCS.352.2","url":null,"abstract":"A minimally rigid graph, also called Laman graph, models a planar framework which is rigid for a general choice of distances between its vertices. In other words, there are finitely many ways, up to isometries, to realize such a graph in the plane. Using ideas from algebraic and tropical geometry, we derive a recursive formula for the number of such realizations. Combining computational results with the construction of new rigid graphs via gluing techniques, we can give a new lower bound on the maximal possible number of realizations for graphs with a given number of vertices.","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"385 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123630486","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 : 2014-07-09DOI: 10.1007/978-3-319-21362-0_6
Shuichi Moritsugu
{"title":"Integrated Circumradius and Area Formulae for Cyclic Pentagons and Hexagons","authors":"Shuichi Moritsugu","doi":"10.1007/978-3-319-21362-0_6","DOIUrl":"https://doi.org/10.1007/978-3-319-21362-0_6","url":null,"abstract":"","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125408923","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 : 2014-07-09DOI: 10.1007/978-3-319-21362-0_3
J. Davenport, M. England
{"title":"Recent Advances in Real Geometric Reasoning","authors":"J. Davenport, M. England","doi":"10.1007/978-3-319-21362-0_3","DOIUrl":"https://doi.org/10.1007/978-3-319-21362-0_3","url":null,"abstract":"","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126344600","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 : 2014-07-09DOI: 10.1007/978-3-319-21362-0_7
P. Pech
{"title":"Extension of Simson-Wallace Theorem on Skew Quadrilaterals and Further Properties","authors":"P. Pech","doi":"10.1007/978-3-319-21362-0_7","DOIUrl":"https://doi.org/10.1007/978-3-319-21362-0_7","url":null,"abstract":"","PeriodicalId":127390,"journal":{"name":"Automated Deduction in Geometry","volume":"583 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132725651","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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