OpenAI 模型推翻离散几何 80 年核心猜想——单位距离问题的新突破

import numpy as np
import matplotlib.pyplot as plt
from itertools import combinations

def count_unit_distances(points, tolerance=1e-9):
    """计算点集中距离恰好为 1 的点对数量"""
    count = 0
    unit_pairs = []
    for i, j in combinations(range(len(points)), 2):
        dist = np.linalg.norm(points[i] - points[j])
        if abs(dist - 1.0) < tolerance:
            count += 1
            unit_pairs.append((i, j))
    return count, unit_pairs

def generate_grid_with_unit_edges(n_side):
    """经典构造：正方形网格，边长为 1，每条边就是一个单位距离"""
    points = []
    for i in range(n_side):
        for j in range(n_side):
            points.append(np.array([float(i), float(j)]))
    return np.array(points)

def generate_hexagonal_lattice(n_rings):
    """六角格构造：相邻点距离为 1，密度更高"""
    points = []
    for ring in range(n_rings + 1):
        for k in range(6 * ring):
            angle = k * np.pi / (3 * ring) if ring > 0 else 0
            if ring == 0:
                points.append(np.array([0.0, 0.0]))
            else:
                # 六角格的精确坐标
                sector = k // ring
                pos_in_sector = k % ring
                base_angle = sector * np.pi / 3
                r = ring
                x = r * np.cos(base_angle) + pos_in_sector * np.cos(base_angle + np.pi / 3)
                y = r * np.sin(base_angle) + pos_in_sector * np.sin(base_angle + np.pi / 3)
                points.append(np.array([x, y]))
    return np.array(points)

def visualize(points, unit_pairs, title="Unit Distance Configuration"):
    """可视化点集和单位距离连线"""
    fig, ax = plt.subplots(figsize=(8, 8))
    ax.scatter(points[:, 0], points[:, 1], s=30, zorder=3, color='black')
    for i, j in unit_pairs:
        ax.plot([points[i][0], points[j][0]],
                [points[i][1], points[j][1]],
                color='red', linewidth=0.8, alpha=0.6)
    ax.set_aspect('equal')
    ax.set_title(f"{title}\nn={len(points)}, unit pairs={len(unit_pairs)}")
    plt.tight_layout()
    plt.savefig("unit_distance.png", dpi=150)
    plt.show()

# --- 实验 1：正方形网格 ---
grid = generate_grid_with_unit_edges(5)  # 5×5 网格，25 个点
count, pairs = count_unit_distances(grid)
visualize(grid, pairs, "Square Grid (side=1)")

# --- 实验 2：六角格 ---
hex_pts = generate_hexagonal_lattice(3)  # 3 层六角格
count2, pairs2 = count_unit_distances(hex_pts)
visualize(hex_pts, pairs2, "Hexagonal Lattice")

# --- 打印统计 ---
for label, pts, c, p in [("Grid", grid, count, pairs),
                          ("Hex", hex_pts, count2, pairs2)]:
    n = len(pts)
    ratio = c / n
    print(f"{label}: n={n}, U(n)={c}, U(n)/n={ratio:.2f}")