This chapter looks more closely at Theorem 15.1 and gives more information about the PET that appears in that result. The basic idea of the proof is to remove from the torus Ŝ the singular set, i.e., the places where the PET is not defined. What is left over is isometric to the interior of a convex parallelotope. Section 16.2 analyzes the singular set and Section 16.3 constructs X1. Section 16.4 constructs the second parallelotope based on the action of the PET from Theorem 15.1. The proof of Theorem 16.1 finishes at the end of Section 16.4. Section 16.5 restates the case of Theorems 15.1 and 16.1 that apply to the pinwheel map associated to outer billiards on a polygon without parallel sides. The result is Theorem 16.9. Finally, Section 16.7 shows how Theorems 0.4 and 16.9 match up.
{"title":"The Nature of the Compactification","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.20","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.20","url":null,"abstract":"This chapter looks more closely at Theorem 15.1 and gives more information about the PET that appears in that result. The basic idea of the proof is to remove from the torus Ŝ the singular set, i.e., the places where the PET is not defined. What is left over is isometric to the interior of a convex parallelotope. Section 16.2 analyzes the singular set and Section 16.3 constructs X1. Section 16.4 constructs the second parallelotope based on the action of the PET from Theorem 15.1. The proof of Theorem 16.1 finishes at the end of Section 16.4. Section 16.5 restates the case of Theorems 15.1 and 16.1 that apply to the pinwheel map associated to outer billiards on a polygon without parallel sides. The result is Theorem 16.9. Finally, Section 16.7 shows how Theorems 0.4 and 16.9 match up.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127628344","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 chapter contains the statement and proof of the Vertical Lemma. Section 10.2 uses symmetry to cut down the amount of work that needs to be done. Section 10.3 translates the picture to prove something more symmetric. The Vertical Lemma associates a point at the center of the unit integer square with a point on the east edge of the square. A more symmetric picture is obtained when working with the centers of the east edges rather than the centers of the squares. Section 10.4 presents some alternative formulas for the map Φ: Π → X' and Φ: Π → X, which will help with the calculations. Section 10.5 proves the version of the Vertical Lemma for Λ'. Section 10.6 makes a careful study of the congruences and shows that the Λ'-based result implies the Λ-based result.
{"title":"The Vertical Lemma","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.14","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.14","url":null,"abstract":"This chapter contains the statement and proof of the Vertical Lemma. Section 10.2 uses symmetry to cut down the amount of work that needs to be done. Section 10.3 translates the picture to prove something more symmetric. The Vertical Lemma associates a point at the center of the unit integer square with a point on the east edge of the square. A more symmetric picture is obtained when working with the centers of the east edges rather than the centers of the squares. Section 10.4 presents some alternative formulas for the map Φ: Π → X' and Φ: Π → X, which will help with the calculations. Section 10.5 proves the version of the Vertical Lemma for Λ'. Section 10.6 makes a careful study of the congruences and shows that the Λ'-based result implies the Λ-based result.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122165105","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 chapter fixes some even rational parameter p/q as usual. It shows that the pixelated spacetime slices of capacity 2p are combinatorially equivalent to certain of the tilings from P. Hooper's Truchet tile system [H]. Section 7.2 describes the Truchet tile system. Section 7.3 states the main result, the Truchet Comparison Theorem. One can view the Truchet Comparison Theorem as a computational tool for understanding some of the pixelated spacetime diagrams. Section 7.4 uses the Truchet Comparison Theorem to get more information about the surface Σ(p/q) from Corollary 6.6. Section 7.5 proves a curious result from elementary number theory which underlies the Truchet Comparison Theorem. Section 7.6 puts together the ingredients and proves the Truchet Comparison Theorem.
{"title":"Connection to the Truchet Tile System","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.11","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.11","url":null,"abstract":"This chapter fixes some even rational parameter p/q as usual. It shows that the pixelated spacetime slices of capacity 2p are combinatorially equivalent to certain of the tilings from P. Hooper's Truchet tile system [H]. Section 7.2 describes the Truchet tile system. Section 7.3 states the main result, the Truchet Comparison Theorem. One can view the Truchet Comparison Theorem as a computational tool for understanding some of the pixelated spacetime diagrams. Section 7.4 uses the Truchet Comparison Theorem to get more information about the surface Σ(p/q) from Corollary 6.6. Section 7.5 proves a curious result from elementary number theory which underlies the Truchet Comparison Theorem. Section 7.6 puts together the ingredients and proves the Truchet Comparison Theorem.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121068881","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 : 2019-02-19DOI: 10.1007/978-3-319-92868-5_2
Daniela Anna Selch, M. Scherer
{"title":"Properties of the Model","authors":"Daniela Anna Selch, M. Scherer","doi":"10.1007/978-3-319-92868-5_2","DOIUrl":"https://doi.org/10.1007/978-3-319-92868-5_2","url":null,"abstract":"","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131377574","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 : 2019-02-19DOI: 10.1007/978-3-642-37617-7_17
P. Major
{"title":"Proof of the Main Result","authors":"P. Major","doi":"10.1007/978-3-642-37617-7_17","DOIUrl":"https://doi.org/10.1007/978-3-642-37617-7_17","url":null,"abstract":"","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115150112","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 chapter begins Part 5 of the book. This part is devoted mostly to the study of the distribution of the plaid polygons: their size and number depending on the parameter. Section 21.2 gives a criterion for a point in FX to have a well-defined orbit. Section 21.3 revisits the pixelated spacetime diagrams of capacity 2, and uses them to construct a large supply of plaid polygons having large diameter. The construction in Section 21.3 works one parameter at a time. Section 21.4 takes the limit of our construction relative to a sequence of even rational parameters converging to our irrational parameter. This limiting argument completes the proof. Section 21.5 explains how to associate a plaid path to an infinite orbit. Section 21.6 gives a quick alternate proof of Theorem 21.1, based on results from [S1].
{"title":"Existence of Infinite Orbits","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.25","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.25","url":null,"abstract":"This chapter begins Part 5 of the book. This part is devoted mostly to the study of the distribution of the plaid polygons: their size and number depending on the parameter. Section 21.2 gives a criterion for a point in FX to have a well-defined orbit. Section 21.3 revisits the pixelated spacetime diagrams of capacity 2, and uses them to construct a large supply of plaid polygons having large diameter. The construction in Section 21.3 works one parameter at a time. Section 21.4 takes the limit of our construction relative to a sequence of even rational parameters converging to our irrational parameter. This limiting argument completes the proof. Section 21.5 explains how to associate a plaid path to an infinite orbit. Section 21.6 gives a quick alternate proof of Theorem 21.1, based on results from [S1].","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114368523","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 chapter contains the statement and proof of the Segment Lemma. It fixes a parameter A = p/q and set P = 2A/(1 + A) and continues the notation from Chapter 8. The chapter is organized as follows. Section 9.2 explains the existence and image of the anchor point. Section 9.3 explains how to reinterpret the classifying map from Section 8.4 as a map defined on a certain subset of R 3. The technical result, Lemma 9.4, will be useful for the computations in this chapter. Section 9.4 and 9.5 treats the vertical and horizontal cases of the Segment Lemma, respectively.
{"title":"The Segment Lemma","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.13","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.13","url":null,"abstract":"This chapter contains the statement and proof of the Segment Lemma. It fixes a parameter A = p/q and set P = 2A/(1 + A) and continues the notation from Chapter 8. The chapter is organized as follows. Section 9.2 explains the existence and image of the anchor point. Section 9.3 explains how to reinterpret the classifying map from Section 8.4 as a map defined on a certain subset of R\u0000 3. The technical result, Lemma 9.4, will be useful for the computations in this chapter. Section 9.4 and 9.5 treats the vertical and horizontal cases of the Segment Lemma, respectively.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124987366","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 chapter revisits the idea of pixelated spacetime diagrams considered in Chapter 5. It proves a technical result, the Curve Turning Theorem, which gives a way to understand the spacetime diagrams in terms of patterns of oriented lines. Section 6.2 assigns directions to all the particle lines. This is done using a key feature of the oriented plaid model which has already been established: the directions of all instances of a particle are the same. The Curve Turning Theorem is also stated at the end of this section. Section 6.3 proves a technical result about the spacing of the particle lines in spacetime diagrams. This result will help with the proof of the Curve Turning Theorem. Section 6.4 and 6.5 prove the Curve Turning Theorem, respectively, in the vertical and the horizontal case. Section 6.6 gives two applications of the Curve Turning Theorem.
{"title":"Pixellation and Curve Turning","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.10","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.10","url":null,"abstract":"This chapter revisits the idea of pixelated spacetime diagrams considered in Chapter 5. It proves a technical result, the Curve Turning Theorem, which gives a way to understand the spacetime diagrams in terms of patterns of oriented lines. Section 6.2 assigns directions to all the particle lines. This is done using a key feature of the oriented plaid model which has already been established: the directions of all instances of a particle are the same. The Curve Turning Theorem is also stated at the end of this section. Section 6.3 proves a technical result about the spacing of the particle lines in spacetime diagrams. This result will help with the proof of the Curve Turning Theorem. Section 6.4 and 6.5 prove the Curve Turning Theorem, respectively, in the vertical and the horizontal case. Section 6.6 gives two applications of the Curve Turning Theorem.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"221 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130641733","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 is the first of four chapters giving a self-contained proof of Theorem 0.7. Section 23.2 describes a sequence of even rationals {pn/qn} that converges to A. Section 23.3 states the two main technical results, the Box Theorem and the Copy Theorem. Section 23.4 shows how to choose a sequence {cn}. Section 23.5 states three auxiliary results about arc copying in the plaid model. Section 23.6 deduces the Box Theorem from one of these auxiliary lemmas. Section 23.7 deduces the Copy Theorem from the auxiliary lemmas and some elementary number theory. Thus, after this chapter ends, the only remaining task is to prove the auxiliary copy lemmas and prove a few lemmas in elementary number theory.
{"title":"Infinite Orbits Revisited","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.27","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.27","url":null,"abstract":"This is the first of four chapters giving a self-contained proof of Theorem 0.7. Section 23.2 describes a sequence of even rationals {pn/qn} that converges to A. Section 23.3 states the two main technical results, the Box Theorem and the Copy Theorem. Section 23.4 shows how to choose a sequence {cn}. Section 23.5 states three auxiliary results about arc copying in the plaid model. Section 23.6 deduces the Box Theorem from one of these auxiliary lemmas. Section 23.7 deduces the Copy Theorem from the auxiliary lemmas and some elementary number theory. Thus, after this chapter ends, the only remaining task is to prove the auxiliary copy lemmas and prove a few lemmas in elementary number theory.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"300 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114581203","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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