This chapter begins the process of making a 3D interpretation of the plaid model. The idea is to group together certain of the light points and think of them as instances of 1-dimensional worldlines rather than as a succession of points. It fixes an even rational parameter p/q and uses the notation from Section 1.2, i.e., ω = p + q. Section 4.2 explains a different notion of adjacency for the ω × ω blocks, dividing up the plaid model. Section 4.3 says what it means for two horizontal light points in remotely adjacent blocks to be different instances of the same particle. Section 4.4 does the same for the vertical particles. Section 4.5 shows a few pictures of spacetime diagrams and discuss their symmetries. Section 4.6 proves a technical lemma, the Bad Tile Lemma, which is very similar in spirit to Theorem 1.4.
{"title":"Particles and Spacetime Diagrams","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.8","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.8","url":null,"abstract":"This chapter begins the process of making a 3D interpretation of the plaid model. The idea is to group together certain of the light points and think of them as instances of 1-dimensional worldlines rather than as a succession of points. It fixes an even rational parameter p/q and uses the notation from Section 1.2, i.e., ω = p + q. Section 4.2 explains a different notion of adjacency for the ω × ω blocks, dividing up the plaid model. Section 4.3 says what it means for two horizontal light points in remotely adjacent blocks to be different instances of the same particle. Section 4.4 does the same for the vertical particles. Section 4.5 shows a few pictures of spacetime diagrams and discuss their symmetries. Section 4.6 proves a technical lemma, the Bad Tile Lemma, which is very similar in spirit to Theorem 1.4.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"6 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":"126412273","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 Horizontal Lemma. The proof follows a similar outline as in Chapter 10. The chapter is organized as follows. Section 11.2 uses symmetry to reduce the Horizontal Lemma to a simpler statement. Section 11.3 modifies the construction as in the vertical case, reducing the Horizontal Lemma to the simpler Lemma 11.4. Section 11.4 proves two easy technical lemmas. Section 11.5 proves the Λ' version of Lemma 11.4. Section 11.6 keeps track of the signs and prove Lemma 11.4 as stated.
{"title":"The Horizontal Lemma","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.15","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.15","url":null,"abstract":"This chapter contains the statement and proof of the Horizontal Lemma. The proof follows a similar outline as in Chapter 10. The chapter is organized as follows. Section 11.2 uses symmetry to reduce the Horizontal Lemma to a simpler statement. Section 11.3 modifies the construction as in the vertical case, reducing the Horizontal Lemma to the simpler Lemma 11.4. Section 11.4 proves two easy technical lemmas. Section 11.5 proves the Λ' version of Lemma 11.4. Section 11.6 keeps track of the signs and prove Lemma 11.4 as stated.","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":"124159757","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 defines the plaid model. The input to the plaid model is a rational A = p/q ɛ (0, 1) with pq even. The output is the union PLA of plaid polygons. The chapter is organized as follows. Section 1.2 defines some auxiliary quantities associated to the parameter. Section 1.3 defines six families of parallel lines. The first two families are the horizontal and vertical lines comprising the boundary of the integer square grid, and are called grid lines. The remaining four families are called slanting lines. Section 1.4 explains the assignment of masses and capacities to the lines. Section 1.5 defines the concept of a light point. The light points are certain intersections between slanting lines and grid lines. Section 1.6 assigns unit vectors to the light particles. Finally, Section 1.7 puts everything together and gives the definition of PLA.
{"title":"Definition of the Plaid Model","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.5","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.5","url":null,"abstract":"This chapter defines the plaid model. The input to the plaid model is a rational A = p/q ɛ (0, 1) with pq even. The output is the union PLA of plaid polygons. The chapter is organized as follows. Section 1.2 defines some auxiliary quantities associated to the parameter. Section 1.3 defines six families of parallel lines. The first two families are the horizontal and vertical lines comprising the boundary of the integer square grid, and are called grid lines. The remaining four families are called slanting lines. Section 1.4 explains the assignment of masses and capacities to the lines. Section 1.5 defines the concept of a light point. The light points are certain intersections between slanting lines and grid lines. Section 1.6 assigns unit vectors to the light particles. Finally, Section 1.7 puts everything together and gives the definition of PLA.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"1 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":"122560843","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 proves the Quasi-Isomorphism Theorem modulo two technical lemmas, which will be dealt with in the next two chapters. Section 18.2 introduces the affine transformation TA from the Quasi-Isomorphism Theorem. Section 18.3 defines the graph grid GA = TA(Z2) and states the Grid Geometry Lemma, a result about the basic geometric properties of GA. Section 18.4 introduces the set Z* that appears in the Renormalization Theorem and states the main result about it, the Intertwining Lemma. Section 18.5 explains how the Orbit Equivalence Theorem sets up a canonical bijection between the nontrivial orbits of the plaid PET and the orbits of the graph PET. Section 18.6 reinterprets the orbit correspondence in terms of the plaid polygons and the arithmetic graph polygons. Everything is then put together to complete the proof of the Quasi-Isomorphism Theorem. Section 18.7 deduces the Projection Theorem (Theorem 0.2) from the Quasi-Isomorphism Theorem.
{"title":"The Quasi-Isomorphism Theorem","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.22","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.22","url":null,"abstract":"This chapter proves the Quasi-Isomorphism Theorem modulo two technical lemmas, which will be dealt with in the next two chapters. Section 18.2 introduces the affine transformation TA from the Quasi-Isomorphism Theorem. Section 18.3 defines the graph grid GA = TA(Z2) and states the Grid Geometry Lemma, a result about the basic geometric properties of GA. Section 18.4 introduces the set Z* that appears in the Renormalization Theorem and states the main result about it, the Intertwining Lemma. Section 18.5 explains how the Orbit Equivalence Theorem sets up a canonical bijection between the nontrivial orbits of the plaid PET and the orbits of the graph PET. Section 18.6 reinterprets the orbit correspondence in terms of the plaid polygons and the arithmetic graph polygons. Everything is then put together to complete the proof of the Quasi-Isomorphism Theorem. Section 18.7 deduces the Projection Theorem (Theorem 0.2) from the Quasi-Isomorphism Theorem.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"41 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":"132059594","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: 10.7551/mitpress/3877.003.0008
S. Ullman
This book uses the methodology of artificial intelligence to investigate the phenomena of visual motion perception: how the visual system constructs descriptions of the environment in terms of objects, their three-dimensional shape, and their motion through space, on the basis of the changing image that reaches the eye. The author has analyzed the computations performed in the course of visual motion analysis. Workable schemes able to perform certain tasks performed by the visual system have been constructed and used as vehicles for investigating the problems faced by the visual system and its methods for solving them.Two major problems are treated: first, the correspondence problem, which concerns the identification of image elements that represent the same object at different times, thereby maintaining the perceptual identity of the object in motion or in change. The second problem is the three-dimensional interpretation of the changing image once a correspondence has been established.The author's computational approach to visual theory makes the work unique, and it should be of interest to psychologists working in visual perception and readers interested in cognitive studies in general, as well as computer scientists interested in machine vision, theoretical neurophysiologists, and philosophers of science.
{"title":"Three-Dimensional Interpretation","authors":"S. Ullman","doi":"10.7551/mitpress/3877.003.0008","DOIUrl":"https://doi.org/10.7551/mitpress/3877.003.0008","url":null,"abstract":"This book uses the methodology of artificial intelligence to investigate the phenomena of visual motion perception: how the visual system constructs descriptions of the environment in terms of objects, their three-dimensional shape, and their motion through space, on the basis of the changing image that reaches the eye. The author has analyzed the computations performed in the course of visual motion analysis. Workable schemes able to perform certain tasks performed by the visual system have been constructed and used as vehicles for investigating the problems faced by the visual system and its methods for solving them.Two major problems are treated: first, the correspondence problem, which concerns the identification of image elements that represent the same object at different times, thereby maintaining the perceptual identity of the object in motion or in change. The second problem is the three-dimensional interpretation of the changing image once a correspondence has been established.The author's computational approach to visual theory makes the work unique, and it should be of interest to psychologists working in visual perception and readers interested in cognitive studies in general, as well as computer scientists interested in machine vision, theoretical neurophysiologists, and philosophers of science.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"3 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":"115329230","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 is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ denote the second iterate of the outer billiards map defined on R 2 − P. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the X-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.
{"title":"Pinwheels and Quarter Turns","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.18","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.18","url":null,"abstract":"This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ denote the second iterate of the outer billiards map defined on R\u0000 2 − P. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the X-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"14 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":"128595990","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: 10.23943/princeton/9780691181387.003.0013
R. Schwartz
This chapter puts together the ingredients from the last three chapters—the Segment Lemma, the Horizontal Lemma, and the Vertical Lemma—and proves Theorem 8.2. The three technical lemmas do not mention the partition of the space X at all, but they do give a lot of control over how the nature of the particles tracked by points in the plaid grid Π influences the image of such grid points under the classifying map Φ. What remains is to compare the three results above to the partition and determine whether everything matches. The remainder of the chapter is organized as follows. Section 12.2 carries out the program to show that these containers are each a union of two prisms. Section 12.3 discusses some extra symmetry of the partition. Section 12.5 compares the prism containers in the vertical case to the partition of X and deduces that Theorem 8.2 is true for the vertical unit integer segments. Section 12.4 compares the prism containers in the vertical case to the partition of X and deduces that Theorem 8.2 is true for the horizontal unit integer segments. The two results together complete the proof.
{"title":"Proof of the Main Result","authors":"R. Schwartz","doi":"10.23943/princeton/9780691181387.003.0013","DOIUrl":"https://doi.org/10.23943/princeton/9780691181387.003.0013","url":null,"abstract":"This chapter puts together the ingredients from the last three chapters—the Segment Lemma, the Horizontal Lemma, and the Vertical Lemma—and proves Theorem 8.2. The three technical lemmas do not mention the partition of the space X at all, but they do give a lot of control over how the nature of the particles tracked by points in the plaid grid Π influences the image of such grid points under the classifying map Φ. What remains is to compare the three results above to the partition and determine whether everything matches. The remainder of the chapter is organized as follows. Section 12.2 carries out the program to show that these containers are each a union of two prisms. Section 12.3 discusses some extra symmetry of the partition. Section 12.5 compares the prism containers in the vertical case to the partition of X and deduces that Theorem 8.2 is true for the vertical unit integer segments. Section 12.4 compares the prism containers in the vertical case to the partition of X and deduces that Theorem 8.2 is true for the horizontal unit integer segments. The two results together complete the proof.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"21 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":"134206024","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 aims to prove Theorem 0.4, the Graph Master Picture Theorem. Theorem 0.4 is proven in two different ways, the first proof is discussed here; it deduces Theorem 0.4 from Theorem 13.2, which is a restatement of [S1, Master Picture Theorem] with minor cosmetic changes. The chapter is organized as follows. Section 13.2 discusses the special outer billiards orbits on kites. Section 13.3 defines the arithmetic graph, which is an arithmetical way of encoding the behavior of a certain first return map of the special orbits. Section 13.4 states Theorem 13.2, a slightly modified and simplified version of [S1, Master Picture Theorem]. Section 13.5 deduces Theorem 0.4 from Theorem 13.2 and one extra piece of information. Finally, Section 13.6 lists the polytopes comprising the partition associated to Theorems 13.2 and 0.4.
{"title":"Graph Master Picture Theorem","authors":"R. Schwartz","doi":"10.2307/j.ctv5rf6tz.17","DOIUrl":"https://doi.org/10.2307/j.ctv5rf6tz.17","url":null,"abstract":"This chapter aims to prove Theorem 0.4, the Graph Master Picture Theorem. Theorem 0.4 is proven in two different ways, the first proof is discussed here; it deduces Theorem 0.4 from Theorem 13.2, which is a restatement of [S1, Master Picture Theorem] with minor cosmetic changes. The chapter is organized as follows. Section 13.2 discusses the special outer billiards orbits on kites. Section 13.3 defines the arithmetic graph, which is an arithmetical way of encoding the behavior of a certain first return map of the special orbits. Section 13.4 states Theorem 13.2, a slightly modified and simplified version of [S1, Master Picture Theorem]. Section 13.5 deduces Theorem 0.4 from Theorem 13.2 and one extra piece of information. Finally, Section 13.6 lists the polytopes comprising the partition associated to Theorems 13.2 and 0.4.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"29 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":"115588179","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: 10.23943/princeton/9780691181387.003.0006
R. Schwartz
This chapter explains the 3-dimensional interpretation of the plaid model. Section 5.2 stacks the blocks on top of each other in such a way that remotely adjacent blocks appear actually adjacent to each other in the stack. Section 5.3 shows how to modify the spacetime diagrams constructed in Section 4.3 and 4.4 so that they are unions of embedded loops, much like the plaid polygons. This modification is called pixilation. Section 5.4 shows that the plaid model construction and the pixilation processes are compatible with each other. Section 5.5 uses the compatibility of all the constructions to create polyhedral surfaces which simultaneously interpolate between the plaid polygons and the pixelated spacetime diagrams. These surfaces are viewed as spacetime diagrams for the plaid polygons; they are called spacetime plaid surfaces. Finally, Section 5.6 indulges in some discussion and speculation.
{"title":"Three-Dimensional Interpretation","authors":"R. Schwartz","doi":"10.23943/princeton/9780691181387.003.0006","DOIUrl":"https://doi.org/10.23943/princeton/9780691181387.003.0006","url":null,"abstract":"This chapter explains the 3-dimensional interpretation of the plaid model. Section 5.2 stacks the blocks on top of each other in such a way that remotely adjacent blocks appear actually adjacent to each other in the stack. Section 5.3 shows how to modify the spacetime diagrams constructed in Section 4.3 and 4.4 so that they are unions of embedded loops, much like the plaid polygons. This modification is called pixilation. Section 5.4 shows that the plaid model construction and the pixilation processes are compatible with each other. Section 5.5 uses the compatibility of all the constructions to create polyhedral surfaces which simultaneously interpolate between the plaid polygons and the pixelated spacetime diagrams. These surfaces are viewed as spacetime diagrams for the plaid polygons; they are called spacetime plaid surfaces. Finally, Section 5.6 indulges in some discussion and speculation.","PeriodicalId":205299,"journal":{"name":"The Plaid Model","volume":"26 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":"115739792","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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