On isometric sequences of colored spaces Mitsugu Hirasaka (jont work with Masashi Shinohara) Pusan National University Combinatorics Seminar at Shanghai Jiao Tong University December 24, 2017. 1 / 58
Introduction 2 / 58
Introduction I like to use a white board. 3 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. 4 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. 5 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. 6 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). 7 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). We can put another point forming a regular triangle. 8 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). We can put another point forming a regular triangle. That’s all. 9 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). We can put another point forming a regular triangle. That’s all. Though I am conservative, let me add one new relation, say β to the existing distance, say α . 10 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). We can put another point forming a regular triangle. That’s all. Though I am conservative, let me add one new relation, say β to the existing distance, say α . The distances from the fourth points to the former three points are βββ , ββα or βαα . 11 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). We can put another point forming a regular triangle. That’s all. Though I am conservative, let me add one new relation, say β to the existing distance, say α . The distances from the fourth points to the former three points are βββ , ββα or βαα . We can draw the following pictures. 12 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). We can put another point forming a regular triangle. That’s all. Though I am conservative, let me add one new relation, say β to the existing distance, say α . The distances from the fourth points to the former three points are βββ , ββα or βαα . We can draw the following pictures. Moreover, we can draw other kinds of four points without regular triangles. 13 / 58
Introduction I like to use a white board. Because I feel something created from the vacant white space. Let me put one point in this Euclidean plane. Let me make one of his friends. Since I am conservative, I want to avoid to make new relations (distances). We can put another point forming a regular triangle. That’s all. Though I am conservative, let me add one new relation, say β to the existing distance, say α . The distances from the fourth points to the former three points are βββ , ββα or βαα . We can draw the following pictures. Moreover, we can draw other kinds of four points without regular triangles. Can we add the fifth point to these pictures in which only α and β appear? 14 / 58
Distances and Triangles 15 / 58
Distances and Triangles Do you remember LLL, LAL and ALA? 16 / 58
Distances and Triangles Do you remember LLL, LAL and ALA? These are properties for two triangles to be congruent. 17 / 58
Distances and Triangles Do you remember LLL, LAL and ALA? These are properties for two triangles to be congruent. How many congruence classes of triangles are there? 18 / 58
Distances and Triangles Do you remember LLL, LAL and ALA? These are properties for two triangles to be congruent. How many congruence classes of triangles are there? For X ⊆ R n we set A ( X ) := { d ( x , y ) | x , y ∈ X , x � = y } , and 19 / 58
Distances and Triangles Do you remember LLL, LAL and ALA? These are properties for two triangles to be congruent. How many congruence classes of triangles are there? For X ⊆ R n we set A ( X ) := { d ( x , y ) | x , y ∈ X , x � = y } , and we denote by T ( X ) the set of congruence classes of triangles in X . 20 / 58
Distances and Triangles Do you remember LLL, LAL and ALA? These are properties for two triangles to be congruent. How many congruence classes of triangles are there? For X ⊆ R n we set A ( X ) := { d ( x , y ) | x , y ∈ X , x � = y } , and we denote by T ( X ) the set of congruence classes of triangles in X . We observe that | A ( X ) | ≤ | T ( X ) | if | X | ≥ 5. 21 / 58
Distances and Triangles Do you remember LLL, LAL and ALA? These are properties for two triangles to be congruent. How many congruence classes of triangles are there? For X ⊆ R n we set A ( X ) := { d ( x , y ) | x , y ∈ X , x � = y } , and we denote by T ( X ) the set of congruence classes of triangles in X . We observe that | A ( X ) | ≤ | T ( X ) | if | X | ≥ 5. Here we aim to prove it and show what happens if the equality holds. 22 / 58
Examples 23 / 58
Examples (i) It is easy to find X ⊆ R n with | A ( X ) | = � | X | � if n is enough large, 2 24 / 58
Examples (i) It is easy to find X ⊆ R n with | A ( X ) | = � | X | � if n is enough large, 2 � | X | � so that | T ( X ) | = . 3 25 / 58
Examples (i) It is easy to find X ⊆ R n with | A ( X ) | = � | X | � if n is enough large, 2 � | X | � so that | T ( X ) | = . 3 (ii) If X ⊆ R 3 forms an octahedron, then | A ( X ) | = 2 and | T ( X ) | = 2. 26 / 58
Examples (i) It is easy to find X ⊆ R n with | A ( X ) | = � | X | � if n is enough large, 2 � | X | � so that | T ( X ) | = . 3 (ii) If X ⊆ R 3 forms an octahedron, then | A ( X ) | = 2 and | T ( X ) | = 2. (iii) If X ⊆ R 2 forms a regular hexagon, then | A ( X ) | = 3 and | T ( X ) | = 3. 27 / 58
Examples (i) It is easy to find X ⊆ R n with | A ( X ) | = � | X | � if n is enough large, 2 � | X | � so that | T ( X ) | = . 3 (ii) If X ⊆ R 3 forms an octahedron, then | A ( X ) | = 2 and | T ( X ) | = 2. (iii) If X ⊆ R 2 forms a regular hexagon, then | A ( X ) | = 3 and | T ( X ) | = 3. (iv) If X ⊆ R n is a regular simplex, then | A ( X ) | = | T ( X ) | = 1. 28 / 58
Generalize X ⊆ R n to colored spaces 29 / 58
Generalize X ⊆ R n to colored spaces For a set X and positive integer k 30 / 58
Generalize X ⊆ R n to colored spaces For a set X and positive integer k � X � we denote by the set of k -subsets of X . k 31 / 58
Generalize X ⊆ R n to colored spaces For a set X and positive integer k � X � we denote by the set of k -subsets of X . k A pair ( X , r ) is called a colored space 32 / 58
Generalize X ⊆ R n to colored spaces For a set X and positive integer k � X � we denote by the set of k -subsets of X . k A pair ( X , r ) is called a colored space � X � if r is a function whose domain is . 2 33 / 58
Generalize X ⊆ R n to colored spaces For a set X and positive integer k � X � we denote by the set of k -subsets of X . k A pair ( X , r ) is called a colored space � X � if r is a function whose domain is . 2 For Y , Z ⊆ X we say that Y is isometric to Z 34 / 58
Generalize X ⊆ R n to colored spaces For a set X and positive integer k � X � we denote by the set of k -subsets of X . k A pair ( X , r ) is called a colored space � X � if r is a function whose domain is . 2 For Y , Z ⊆ X we say that Y is isometric to Z if there exists a bijection f : Y → Z such that 35 / 58
Generalize X ⊆ R n to colored spaces For a set X and positive integer k � X � we denote by the set of k -subsets of X . k A pair ( X , r ) is called a colored space � X � if r is a function whose domain is . 2 For Y , Z ⊆ X we say that Y is isometric to Z if there exists a bijection f : Y → Z such that � Y � r ( U ) = r ( f ( U )) for each U ∈ . 2 36 / 58
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