polykite-tiling-explorer.py

"""
Polykite Tiling Explorer

Author: OpenAI GPT-4

Description:
This script explores the tiling properties of polykites, specifically those with 8 components.
It generates all possible polykites with 8 components, canonicalizes them to avoid duplicate
analysis, and then computes their isohedral numbers to identify interesting candidates for further
mathematical exploration.

The script is based on the following functions:

1. generate_all_polykites: Generates all possible polykites with a given number of components.
2. canonicalize: Computes the canonical representation of a polykite to avoid duplicate analysis.
3. isohedral_number: Computes the isohedral number of a polykite by attempting to tile the plane using backtracking.
4. get_coordinates: Extracts the coordinates of the vertices of a polykite.
5. tile_plane: Checks if a given tiling of a polykite covers the entire plane within a specified bounding box.
6. backtrack: Backtracking function used within the isohedral_number function to explore all possible tilings of a polykite.
7. get_transformations: Generates a list of possible transformations to apply to a polykite.
8. apply_transformation: Applies a given transformation (rotation, reflection, and translation) to a polykite.
9. can_place: Checks if a transformed polykite can be placed at a specific position without overlapping other polykites.
10. main: Serves as the integration point for all the other functions.

Please note that this implementation is a high-level representation and might require adjustments
based on your specific needs, libraries, and representation of polykites.
"""

import networkx as nx

def generate_all_polykites(n_kites):
    def recursive_generate(current_polykite, remaining_kites):
        if remaining_kites == 0:
            yield current_polykite
        else:
            for edge in current_polykite.edges():
                new_polykite = add_kite_to_edge(current_polykite, edge)
                if new_polykite and not has_self_intersection(new_polykite):
                    yield from recursive_generate(new_polykite, remaining_kites - 1)

    def add_kite_to_edge(polykite, edge):
        new_polykite = polykite.copy()
        node1, node2 = edge
        neighbors1 = [n for n in polykite.neighbors(node1) if n != node2]
        neighbors2 = [n for n in polykite.neighbors(node2) if n != node1]

        if not neighbors1 or not neighbors2:
            return None

        new_node = max(new_polykite.nodes()) + 1
        new_polykite.add_node(new_node)

        new_polykite.add_edge(new_node, node1)
        new_polykite.add_edge(new_node, node2)

        for neighbor in neighbors1:
            if neighbor not in neighbors2:
                new_polykite.add_edge(new_node, neighbor)
                break

        return new_polykite

    def has_self_intersection(polykite):
        # Check if any nodes have more than two neighbors
        for node in polykite.nodes():
            if len(list(polykite.neighbors(node))) > 2:
                return True
        return False

    # Start with a single kite as the base polykite
    initial_polykite = create_kite()

    return list(recursive_generate(initial_polykite, n_kites - 1))

def create_kite():
    kite = nx.Graph()
    kite.add_nodes_from(range(4))
    kite.add_edges_from([(0, 1), (1, 2), (2, 3), (3, 0), (0, 2)])
    return kite


def canonicalize(polykite):
    def get_coordinates(polykite):
        coordinates = []
        for node, data in polykite.nodes(data=True):
            coordinates.append(data['pos'])
        return coordinates

    def normalize_coordinates(coords):
        min_x, min_y = min(coords, key=lambda p: (p[0], p[1]))
        return [(x - min_x, y - min_y) for x, y in coords]

    def rotate_90(coords):
        return [(y, -x) for x, y in coords]

    def reflect(coords):
        return [(-x, y) for x, y in coords]

    coordinates = get_coordinates(polykite)
    normalized_coordinates = normalize_coordinates(coordinates)

    canonical_form = normalized_coordinates

    for _ in range(3):
        normalized_coordinates = rotate_90(normalized_coordinates)
        canonical_form = min(canonical_form, normalized_coordinates)
        reflected_coordinates = reflect(normalized_coordinates)
        canonical_form = min(canonical_form, reflected_coordinates)

    return canonical_form


def isohedral_number(polykite):
    def tile_plane(polykite, current_tiling):
        # Define a bounding box for the area of the plane we want to check for tiling
        bounding_box = (-10, -10, 10, 10)

        # Initialize an empty plane grid
        plane = [[False for _ in range(bounding_box[2] - bounding_box[0] + 1)] for _ in range(bounding_box[3] - bounding_box[1] + 1)]

        # Fill the grid with the current tiling
        for tiled_polykite in current_tiling:
            for node, data in tiled_polykite.nodes(data=True):
                x, y = data['pos']
                if bounding_box[0] <= x <= bounding_box[2] and bounding_box[1] <= y <= bounding_box[3]:
                    plane[y - bounding_box[1]][x - bounding_box[0]] = True

        # Check if the entire grid is covered
        for row in plane:
            for cell in row:
                if not cell:
                    return False

        return True

    def backtrack(polykite, current_tiling):
        if tile_plane(polykite, current_tiling):
            return True

        # Try to place a new copy of the polykite in the plane
        for transformation in get_transformations(polykite):
            new_polykite = apply_transformation(polykite, transformation)
            if can_place(new_polykite, current_tiling):
                current_tiling.append(new_polykite)
                if backtrack(polykite, current_tiling):
                    return True
                current_tiling.pop()

        return False

    def get_transformations(polykite):
        # Generate possible translations, rotations, and reflections of the polykite
        transformations = []
        for dx in range(-10, 11):
            for dy in range(-10, 11):
                for angle in [0, 90, 180, 270]:
                    for reflection in [False, True]:
                        transformations.append((dx, dy, angle, reflection))
        return transformations

    def apply_transformation(polykite, transformation):
        dx, dy, angle, reflection = transformation
        transformed_polykite = polykite.copy()

        # Apply translation
        for node in transformed_polykite.nodes():
            x, y = transformed_polykite.nodes[node]['pos']
            transformed_polykite.nodes[node]['pos'] = (x + dx, y + dy)

        # Apply rotation
        for node in transformed_polykite.nodes():
            x, y = transformed_polykite.nodes[node]['pos']
            if angle == 90:
                transformed_polykite.nodes[node]['pos'] = (-y, x)
            elif angle == 180:
                transformed_polykite.nodes[node]['pos'] = (-x, -y)
            elif angle == 270:
                transformed_polykite.nodes[node]['pos'] = (y, -x)

        # Apply reflection if needed
        if reflection:
            for node in transformed_polykite.nodes():
                x, y = transformed_polykite.nodes[node]['pos']
                transformed_polykite.nodes[node]['pos'] = (-x, y)

        return transformed_polykite

    def can_place(polykite, current_tiling):
        # Check if the polykite can be placed in the current tiling without overlaps or gaps
        for existing_polykite in current_tiling:
            for node1, data1 in polykite.nodes(data=True):
                for node2, data2 in existing_polykite.nodes(data=True):
                    if data1['pos'] == data2['pos']:
                        return False
        return True

    current_tiling = [polykite]
    isohedral_count = 0

    while backtrack(polykite, current_tiling):
        isohedral_count += 1

    return isohedral_count


def main():
    polykites = generate_all_polykites(8)
    seen_polykites = set()
    for polykite in polykites:
        canonical_polykite = canonicalize(polykite)
        if canonical_polykite not in seen_polykites:
            seen_polykites.add(canonical_polykite)
            iso_number = isohedral_number(polykite)
            if iso_number >= 64:
                print(f"Found polykite with isohedral number >= 64: {canonical_polykite}")
                # Further analyze the polykite with other mathematical techniques

if __name__ == "__main__":
    main()