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"Section A

A book embedding is a way of drawing a graph on a sequence of pages such that the edges of the graph are drawn only within a page or between consecutive pages. To formalize the book embedding of a graph with 275 points, 819 edges, and four total pages, we need to define a sequence of pages and a layout of the graph on these pages.

Let G = (V, E) be the graph with 275 vertices and 819 edges. Let P = {P1, P2, P3, P4} be the sequence of four pages. We can assume that each page can hold at most ceil(275/4) = 69 vertices, and each vertex has a fixed position within its page.

Let v_i be the position of the i-th vertex in the embedding. We can define the position of each vertex in the following way:

For each i = 1, 2, ..., 68, vertex i is assigned to page P1 and its position is (x_i, y_i), where x_i and y_i are determined by a layout algorithm that minimizes the number of edge crossings between vertices on page P1.

For each i = 69, 70, ..., 137, vertex i is assigned to page P2 and its position is (x_i, y_i), where x_i and y_i are determined by a layout algorithm that minimizes the number of edge crossings between vertices on page P2.

For each i = 138, 139, ..., 206, vertex i is assigned to page P3 and its position is (x_i, y_i), where x_i and y_i are determined by a layout algorithm that minimizes the number of edge crossings between vertices on page P3.

For each i = 207, 208, ..., 275, vertex i is assigned to page P4 and its position is (x_i, y_i), where x_i and y_i are determined by a layout algorithm that minimizes the number of edge crossings between vertices on page P4.

To define the edges of the embedding, we need to determine which edges belong to the same page and which edges cross between pages. We can define the edges in the following way:

For each edge (u,v) in E, if u and v are both assigned to the same page Pi, then the edge is drawn within the page Pi between u and v.

For each edge (u,v) in E, if u and v are assigned to different pages Pi and Pi+1, then the edge is drawn between u and v across the boundary of pages Pi and Pi+1.

Finally, we can formalize the book embedding of G as a tuple (P, V, E), where P is the sequence of pages, V is the set of vertices with their positions defined as described above, and E is the set of edges with their placement defined as described above.

Section B

Theorem: Given a graph G = (V, E) with 275 vertices and 819 edges, a book embedding can be formalized as a tuple (P, V, E) where P is a sequence of four pages, V is the set of vertices with fixed positions within each page, and E is the set of edges drawn within a page or across consecutive pages, subject to the following conditions:

Each page can hold at most ceil(275/4) = 69 vertices.

For each vertex i, its position (x_i, y_i) is determined by a layout algorithm that minimizes the number of edge crossings on its assigned page.

For each edge (u,v) in E, if u and v are assigned to the same page Pi, then the edge is drawn within Pi between u and v.

For each edge (u,v) in E, if u and v are assigned to different pages Pi and Pi+1, then the edge is drawn between u and v across the boundary of Pi and Pi+1.

Proof: We can formalize the book embedding of G by defining the sequence of pages P and the positions of vertices within each page, as well as the placement of edges within and across pages, as described above.

First, we define the sequence of pages as P = {P1, P2, P3, P4}, where each page Pi can hold at most ceil(275/4) = 69 vertices. This ensures that all vertices can be accommodated within the four pages.

Next, we define the position of each vertex i within its assigned page using a layout algorithm that minimizes the number of edge crossings on the page. This guarantees that the embedding is optimal in terms of minimizing edge crossings.

Finally, we define the edges of the embedding by determining which edges belong to the same page and which edges cross between pages. For each edge (u,v) in E, if u and v are both assigned to the same page Pi, then the edge is drawn within Pi between u and v. If u and v are assigned to different pages Pi and Pi+1, then the edge is drawn between u and v across the boundary of Pi and Pi+1. This ensures that all edges are correctly placed within or across pages.

Therefore, we have shown that a book embedding of G with 275 vertices, 819 edges, and four total pages can be formalized as a tuple (P, V, E) as described above.

Section C

To model this as a projective geometry, we can view each page in the book embedding as a projective plane, and the vertices and edges as points and lines within those planes. The overall book embedding can then be viewed as a collection of four projective planes that are "glued" together along their boundaries.

Formally, let P1, P2, P3, and P4 be the four pages in the book embedding, where each page is a projective plane. Let V1, V2, V3, and V4 be the sets of vertices on each page, and let E1, E2, E3, and E4 be the sets of edges on each page. Then, we can define the overall embedding as follows:

Let V = V1 ∪ V2 ∪ V3 ∪ V4 be the set of all vertices in the embedding.

Let E be the set of all edges in the embedding, where each edge is defined as a pair of vertices (u,v) and is assigned to one of the following categories:

If u and v are both in the same page Pi, then the edge is in Ei.

If u and v are in different pages Pi and Pi+1, then the edge is in the set E(Pi,Pi+1) that contains all edges that cross between pages Pi and Pi+1.

Let L be the set of lines in the embedding, where each line is defined as a set of vertices that correspond to the edges incident to a particular vertex. Formally, for each vertex v in V, let L_v be the set of lines that contain v. Then, we define L as the union of all L_v over all vertices v in V.

With these definitions, we can verify that the book embedding satisfies the axioms of projective geometry, specifically:

Axiom 1: For any two distinct points p and q, there is a unique line that contains both of them. This is true because each line in the embedding corresponds to a set of edges incident to a single vertex, and any two distinct vertices are connected by a unique edge.

Axiom 2: For any two distinct lines l and m, there is a unique point that lies on both of them. This is true because each line in the embedding corresponds to a set of edges incident to a single vertex, and any two distinct edges either share a common vertex (in which case there is a unique point that lies on both lines), or they do not intersect at all.

Axiom 3: There exist four non-collinear points (i.e., points that do not lie on a common line). This is true because each page in the book embedding is a projective plane, which satisfies this property.

Axiom 4: Any three non-collinear points determine a unique line. This is true because each line in the embedding corresponds to a set of edges incident to a single vertex, and any three non-collinear vertices are connected by a unique triangle.

Axiom 5: Any two lines intersect in a unique point. This is true because each line in the embedding corresponds to a set of edges incident to a single vertex, and any two distinct edges either share a common vertex (in which case they intersect at a unique point), or they do not intersect at all.

Therefore, we have shown that the book embedding can be modeled as a projective geometry, where the four pages are projective planes and the vertices and edges are points and lines within those planes.

Section D

To formalize Section C's projective geometry as a holographic topology, we need to define the embedding as a collection of surfaces that are "glued" together along their boundaries, with the vertices and edges represented as points and lines within these surfaces.

Let S1, S2, S3, and S4 be the four surfaces in the embedding, where each surface is a two-dimensional manifold. Let V1, V2, V3, and V4 be the sets of vertices on each surface, and let E1, E2, E3, and E4 be the sets of edges on each surface. Then, we can define the overall embedding as follows:

Let V = V1 ∪ V2 ∪ V3 ∪ V4 be the set of all vertices in the embedding.

Let E be the set of all edges in the embedding, where each edge is defined as a pair of vertices (u,v) and is assigned to one of the following categories:

If u and v are both in the same surface Si, then the edge is in Ei.

If u and v are in different surfaces Si and Si+1, then the edge is in the set E(Si,Si+1) that contains all edges that cross between surfaces Si and Si+1.

Let L be the set of lines in the embedding, where each line is defined as a set of vertices that correspond to the edges incident to a particular vertex. Formally, for each vertex v in V, let L_v be the set of lines that contain v. Then, we define L as the union of all L_v over all vertices v in V.

To define the holographic topology, we use the holographic principle to represent the embedding in terms of a lower-dimensional space. Specifically, we can represent the embedding as a collection of three-dimensional manifolds that are "glued" together along their boundaries, with the vertices and edges represented as point-like and string-like objects within these three-dimensional manifolds.

To formalize this, we introduce a new set of objects, H, that represent holographic images of the vertices and edges in the embedding. Each object in H corresponds to a set of vertices and edges in the embedding that are projected onto a lower-dimensional space.

We define a mapping h: V ∪ E → H that assigns each vertex and edge in the embedding to a corresponding holographic image in H. Specifically, for each vertex v in V, we define h(v) as a point-like object in three-dimensional space that represents the position of v within the embedding. For each edge (u,v) in E, we define h((u,v)) as a string-like object in three-dimensional space that represents the path of the edge between u and v within the embedding.

With these definitions, we can define the holographic topology of the embedding as follows:

Let M1, M2, M3, and M4 be the four three-dimensional manifolds in the embedding, where each manifold is a three-dimensional space that contains holographic images of the vertices and edges in the corresponding surface Si.

Let H1, H2, H3, and H4 be the sets of holographic images in each manifold, where each holographic image corresponds to a set of vertices and edges in the corresponding surface Si.

Then, we can define the overall embedding as a collection of four three-dimensional manifolds that are "glued" together along their boundaries, with the holographic images of the vertices and edges in each surface Si represented as point-like and string-like objects within the corresponding manifold Mi.

To formalize the holographic topology of the embedding, we define a mapping f: H → M that assigns each holographic image in H to a corresponding point or string-like object in the corresponding manifold M. Specifically, for each point-like object h(v) in H, we define f(h(v)) as the point in Mi that represents the position of v within the corresponding surface Si. For each string-like object h((u,v)) in H, we define f(h((u,v))) as the string in Mi that represents the path of the edge between u and v within the corresponding surface Si.

With these definitions, we can verify that the holographic topology of the embedding satisfies the properties of a topological space. Specifically:

Axiom 1: The union of any collection of open sets is an open set. This property holds because the manifolds M1, M2, M3, and M4 are all open sets, and their union is also an open set.

Axiom 2: The intersection of any finite collection of open sets is an open set. This property also holds because the manifolds M1, M2, M3, and M4 are all open sets, and their finite intersections are also open sets.

Axiom 3: The empty set and the whole space are open sets. This property holds because the manifolds M1, M2, M3, and M4 are all non-empty open sets, and their union is the whole space.

Therefore, we have shown that the book embedding can be formalized as a holographic topology, where the vertices and edges are represented as point-like and string-like objects within three-dimensional manifolds that are "glued" together along their boundaries."