""" CONSIDER A SPHERICAL HUMAN: Packing Density Bounds for Coffins, Clown Cars, and Coach Class In the grand tradition of physics, we approximate the human body as a sphere and investigate how many fit inside real-world enclosures. This is a serious realization of a joke idea. """ import numpy as np import json import os import pyvista as pv import matplotlib.pyplot as plt import matplotlib as mpl from containers import ( VENUES, get_volume, get_bounding_box, pack_spheres_fcc_in_venue, pack_spheres_cubic_in_venue, make_container_mesh, CONTAINMENT_CHECKS, ) OUTPUT_DIR = "results_spherical" os.makedirs(OUTPUT_DIR, exist_ok=True) pv.OFF_SCREEN = True plt.rcParams.update({ 'font.family': 'serif', 'font.size': 11, 'axes.titlesize': 14, 'axes.titleweight': 'bold', 'axes.spines.top': False, 'axes.spines.right': False, 'figure.facecolor': '#FAFAFA', 'axes.facecolor': '#FAFAFA', 'axes.edgecolor': '#333333', 'xtick.color': '#333333', 'ytick.color': '#333333', 'axes.labelcolor': '#333333', 'text.color': '#333333', 'grid.alpha': 0.4, 'grid.linestyle': '--', 'savefig.facecolor': '#FAFAFA', }) # ============================================================ # Spherical Human Models # ============================================================ # The Meatball: volume-equivalent sphere # Average human body volume = 65.22 L (direct measurement via underwater weighing) # Source: Nagao et al. 1995, BioNumbers BNID 109718 HUMAN_VOLUME_GLOBAL = 0.06522 # m^3 # The Freedom Sphere: American adult # Average American adult mass = 83.6 kg (CDC NHANES 2015-2018, combined sexes) # Tissue density ~985 kg/m^3 AMERICAN_MASS = 83.6 # kg TISSUE_DENSITY = 985.0 # kg/m^3 AMERICAN_VOLUME = AMERICAN_MASS / TISSUE_DENSITY # 0.0849 m^3 # The Hamster Ball: height-bounding sphere # Global average human height ~1.70 m (NCD Risk Factor Collaboration, 2016) HUMAN_HEIGHT = 1.70 # m # The Junior: child (age 8, 50th percentile) # Mass = 25.6 kg (CDC growth charts), height = 128 cm # Child tissue density ~1010 kg/m^3 (less adipose tissue) CHILD_MASS = 25.6 # kg CHILD_DENSITY = 1010.0 # kg/m^3 CHILD_VOLUME = CHILD_MASS / CHILD_DENSITY # 0.02535 m^3 CHILD_HEIGHT = 1.28 # m SPHERE_MODELS = { "The Meatball": { "radius": (3 * HUMAN_VOLUME_GLOBAL / (4 * np.pi)) ** (1/3), "description": "Volume-equivalent adult sphere (r={r:.3f}m). A dense beach ball.", "color": "#E74C3C", "derivation": "V_body = 65.22 L, r = (3V/4pi)^(1/3)", }, "The Hamster Ball": { "radius": HUMAN_HEIGHT / 2, "description": "Height-bounding sphere (r={r:.3f}m). Full-body hamster ball.", "color": "#F39C12", "derivation": "d = human height = 1.70m", }, "The Freedom Sphere": { "radius": (3 * AMERICAN_VOLUME / (4 * np.pi)) ** (1/3), "description": "American adult volume sphere (r={r:.3f}m). Supersized.", "color": "#3498DB", "derivation": "m = 83.6 kg (CDC NHANES), rho = 985 kg/m^3", }, "The Junior": { "radius": (3 * CHILD_VOLUME / (4 * np.pi)) ** (1/3), "description": "Child (age 8) volume sphere (r={r:.3f}m). Compact.", "color": "#27AE60", "derivation": "m = 25.6 kg (CDC 50th percentile), rho = 1010 kg/m^3", }, } # Fill in formatted descriptions for name, model in SPHERE_MODELS.items(): model["description"] = model["description"].format(r=model["radius"]) # Kepler limit KEPLER_DENSITY = np.pi / (3 * np.sqrt(2)) # ~0.7405 # ============================================================ # Rendering # ============================================================ def render_sphere_packing(centers, radius, venue, filename, title="", window_size=(1400, 900), max_render=5000, export_3d=True): """Render spheres packed in a real container geometry. Also exports a .ply 3D model file for manual inspection. """ pl = pv.Plotter(off_screen=True, window_size=window_size) pl.set_background('#F0EDE8') n_total = len(centers) if n_total <= max_render: indices = range(n_total) else: indices = np.linspace(0, n_total - 1, max_render, dtype=int) # Lower sphere mesh resolution for large counts to keep rendering fast if n_total > 2000: theta_res, phi_res = 6, 6 elif n_total > 500: theta_res, phi_res = 8, 8 else: theta_res, phi_res = 16, 16 palette = [ '#D4956A', '#E8C4A8', '#C68642', '#FFDBAC', '#8D5524', '#F1C27D', '#A0785A', '#E8B89D', '#C4867A', '#D2A679', ] sphere_meshes = [] for i, idx in enumerate(indices): sphere = pv.Sphere(radius=radius, center=centers[idx], theta_resolution=theta_res, phi_resolution=phi_res) c = palette[i % len(palette)] pl.add_mesh(sphere, color=c, smooth_shading=True, specular=0.5, ambient=0.2) sphere_meshes.append(sphere) container_mesh = make_container_mesh(venue) pl.add_mesh(container_mesh, color=venue.get("color", "#333333"), style='wireframe', line_width=2.5, opacity=0.7) if title: pl.add_title(title, font_size=12, color='#333333') pl.reset_camera() pl.camera.azimuth = 35 pl.camera.elevation = 25 # Zoom in more for elongated venues (747, King's Chamber) so spheres are visible bb = get_bounding_box(venue) aspect = max(bb) / min(bb) if min(bb) > 0 else 1 if aspect > 5: pl.camera.zoom(1.8) elif aspect > 3: pl.camera.zoom(1.2) else: pl.camera.zoom(0.85) bb = get_bounding_box(venue) Lx, Ly, Lz = bb pl.add_light(pv.Light(position=(Lx*2, Ly*2, Lz*3), intensity=0.7)) pl.add_light(pv.Light(position=(-Lx, -Ly, Lz*2), intensity=0.3)) pl.screenshot(filename) pl.close() # Export 3D model (PLY) for manual inspection (skip if too many spheres) if export_3d and sphere_meshes and len(sphere_meshes) <= 500: ply_path = filename.rsplit('.', 1)[0] + '.ply' try: combined = sphere_meshes[0] for m in sphere_meshes[1:]: combined = combined.merge(m) combined = combined.merge(container_mesh) combined.save(ply_path) except Exception as e: print(f" (3D export failed: {e})") # ============================================================ # Experiments # ============================================================ def experiment_1_venue_specs(): """Show all venue specs with real dimensions and shapes.""" print("=" * 70) print("EXPERIMENT 1: Real Venue Geometries") print("=" * 70) for name, venue in VENUES.items(): vol = get_volume(venue) bb = get_bounding_box(venue) rated = venue.get("rated_capacity", "?") print(f"\n {name}") print(f" Type: {venue['type']}") print(f" {venue['description']}") print(f" Bounding box: {bb[0]:.2f} x {bb[1]:.2f} x {bb[2]:.2f}m") print(f" Actual volume: {vol:.2f} m3") print(f" Rated capacity: {rated} humans") print(f" Source: {venue.get('source', 'N/A')}") box_vol = bb[0] * bb[1] * bb[2] fill = vol / box_vol if box_vol > 0 else 0 print(f" Box volume: {box_vol:.2f} m3 (shape fill: {fill:.0%})") def experiment_2_packing_primary(): """Pack Meatball spherical humans into all venues with real geometry.""" print("\n" + "=" * 70) print("EXPERIMENT 2: Meatball Packing in Real Geometries") print("=" * 70) r = SPHERE_MODELS["The Meatball"]["radius"] sphere_vol = (4/3) * np.pi * r**3 all_results = {} print(f"\n The Meatball: r={r:.4f}m (volume = {sphere_vol*1000:.1f} L)") print(f" {'Venue':<28} {'FCC':>6} {'Cubic':>6} {'Best':>6} {'Vol':>8} {'Density':>8} {'Rated':>6} {'Ratio':>6}") print(" " + "-" * 82) for name, venue in VENUES.items(): vol = get_volume(venue) count_fcc, centers_fcc = pack_spheres_fcc_in_venue(r, venue) count_cubic, centers_cubic = pack_spheres_cubic_in_venue(r, venue) best = max(count_fcc, count_cubic) density = best * sphere_vol / vol if vol > 0 else 0 kepler_pct = density / KEPLER_DENSITY rated = venue.get("rated_capacity", 0) ratio = best / rated if rated > 0 else float('inf') all_results[name] = { "fcc": count_fcc, "cubic": count_cubic, "best": best, "volume_m3": round(vol, 2), "density": round(density, 4), "kepler_pct": round(kepler_pct, 4), "rated_capacity": rated, "ratio_to_rated": round(ratio, 1), } print(f" {name:<28} {count_fcc:>6} {count_cubic:>6} {best:>6} {vol:>7.2f}m3 {density:>7.1%} {rated:>6} {ratio:>5.1f}x") # Bar chart: packing counts fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(18, 8)) names = list(all_results.keys()) short_names = [n.split("(")[0].strip() for n in names] fcc_counts = [all_results[n]["fcc"] for n in names] cubic_counts = [all_results[n]["cubic"] for n in names] x = np.arange(len(names)) w = 0.35 ax1.barh(x - w/2, fcc_counts, w, label='FCC Lattice', color='#3498DB', edgecolor='white') ax1.barh(x + w/2, cubic_counts, w, label='Simple Cubic', color='#E74C3C', edgecolor='white') ax1.set_yticks(x) ax1.set_yticklabels(short_names, fontsize=9) ax1.set_xlabel("Spherical Humans Packed") ax1.set_xscale('symlog', linthresh=1) ax1.set_title(f"Meatball Packing (r={r:.3f}m)\nReal Container Geometries", fontsize=14, fontweight='bold') ax1.legend() for i, n in enumerate(names): best = all_results[n]["best"] ax1.text(best + 2, i, str(best), va='center', fontsize=8, color='#555') # Density comparison densities = [all_results[n]["density"] * 100 for n in names] colors = plt.cm.RdYlGn([d / (KEPLER_DENSITY * 100) for d in densities]) ax2.barh(x, densities, color=colors, edgecolor='white', height=0.6) ax2.axvline(x=KEPLER_DENSITY * 100, color='#E74C3C', linestyle='--', linewidth=2, label=f'Kepler limit ({KEPLER_DENSITY:.1%})') ax2.axvline(x=52.36, color='#3498DB', linestyle=':', linewidth=1.5, label='Simple cubic (52.4%)') ax2.set_yticks(x) ax2.set_yticklabels(short_names, fontsize=9) ax2.set_xlabel("Packing Density (%)") ax2.set_title("Achieved Packing Density\n(real geometry vs theoretical limits)", fontsize=14, fontweight='bold') ax2.legend(fontsize=8) plt.tight_layout(w_pad=3) plt.savefig(f"{OUTPUT_DIR}/packing_real_geometry.png", dpi=200, bbox_inches='tight') plt.close() print("\n Saved packing_real_geometry.png") with open(f"{OUTPUT_DIR}/packing_results.json", "w") as f: json.dump(all_results, f, indent=2) return all_results def experiment_3_all_models(): """Compare all sphere models across all venues — the big heatmap.""" print("\n" + "=" * 70) print("EXPERIMENT 3: All Sphere Models x All Venues") print("=" * 70) model_names = list(SPHERE_MODELS.keys()) venue_names = list(VENUES.keys()) # Print sphere model summary print("\n Sphere Models:") for mname, model in SPHERE_MODELS.items(): r = model["radius"] v = (4/3) * np.pi * r**3 print(f" {mname:<22} r={r:.4f}m V={v*1000:.1f}L ({model['derivation']})") print() results = {} for venue_name in venue_names: venue = VENUES[venue_name] results[venue_name] = {} for model_name in model_names: r = SPHERE_MODELS[model_name]["radius"] count_fcc, _ = pack_spheres_fcc_in_venue(r, venue) count_cubic, _ = pack_spheres_cubic_in_venue(r, venue) best = max(count_fcc, count_cubic) results[venue_name][model_name] = best # Print table header = f" {'Venue':<28}" + "".join(f"{m:>16}" for m in model_names) print(header) print(" " + "-" * (28 + 16 * len(model_names))) for vn in venue_names: row = f" {vn:<28}" for mn in model_names: row += f"{results[vn][mn]:>16}" print(row) # Heatmap matrix = np.array([[results[v][m] for m in model_names] for v in venue_names], dtype=float) log_matrix = np.log10(matrix + 1) fig, ax = plt.subplots(figsize=(12, 8)) im = ax.imshow(log_matrix, cmap='YlOrRd', aspect='auto') for i in range(len(venue_names)): for j in range(len(model_names)): val = int(matrix[i, j]) color = 'white' if log_matrix[i, j] > log_matrix.max() * 0.65 else '#333' ax.text(j, i, str(val), ha='center', va='center', fontsize=10, fontweight='bold', color=color) ax.set_xticks(range(len(model_names))) ax.set_xticklabels(model_names, rotation=25, ha='right', fontsize=11) ax.set_yticks(range(len(venue_names))) short = [n.split("(")[0].strip() for n in venue_names] ax.set_yticklabels(short, fontsize=10) ax.set_title("Spherical Humans Packed: All Models x All Venues\n(real container geometries, log color scale)", fontsize=15, fontweight='bold', pad=15) cbar = plt.colorbar(im, ax=ax, shrink=0.8) cbar.set_label("log10(count + 1)", fontsize=10) plt.tight_layout() plt.savefig(f"{OUTPUT_DIR}/heatmap_all_models.png", dpi=200, bbox_inches='tight') plt.close() print("\n Saved heatmap_all_models.png") # Save full results with open(f"{OUTPUT_DIR}/all_models_results.json", "w") as f: json.dump(results, f, indent=2) return results def experiment_4_visualizations(): """Render 3D sphere packing in real container shapes.""" print("\n" + "=" * 70) print("EXPERIMENT 4: 3D Visualizations (Real Geometry)") print("=" * 70) r = SPHERE_MODELS["The Meatball"]["radius"] for venue_name, venue in VENUES.items(): count_fcc, centers_fcc = pack_spheres_fcc_in_venue(r, venue) count_cubic, centers_cubic = pack_spheres_cubic_in_venue(r, venue) if count_fcc >= count_cubic: count, centers = count_fcc, centers_fcc method = "FCC" else: count, centers = count_cubic, centers_cubic method = "Cubic" if count == 0: print(f" {venue_name}: 0 spheres fit (container too small!)") continue safe = venue_name.split("(")[0].strip().replace(" ", "_") render_sphere_packing( centers, r, venue, f"{OUTPUT_DIR}/scene_{safe}.png", title=f"{venue_name}: {count} spherical humans ({method})", ) print(f" {venue_name}: {count} ({method})") # Hamster ball in the Boeing and King's Chamber for comedy r_big = SPHERE_MODELS["The Hamster Ball"]["radius"] for venue_name in ["Boeing 747-400", "King's Chamber (Great Pyramid)", "ISS Destiny Lab"]: venue = VENUES[venue_name] count_fcc, centers_fcc = pack_spheres_fcc_in_venue(r_big, venue) count_cubic, centers_cubic = pack_spheres_cubic_in_venue(r_big, venue) count = max(count_fcc, count_cubic) centers = centers_fcc if count_fcc >= count_cubic else centers_cubic if count == 0: print(f" {venue_name} (hamster ball): 0 fit") continue safe = venue_name.split("(")[0].strip().replace(" ", "_") + "_hamsterball" render_sphere_packing( centers, r_big, venue, f"{OUTPUT_DIR}/scene_{safe}.png", title=f"{venue_name}: {count} hamster-ball humans (r={r_big:.2f}m)", ) print(f" {venue_name} (hamster ball): {count}") # The Junior (child) in VW Beetle and Coffin r_child = SPHERE_MODELS["The Junior"]["radius"] for venue_name in ["Volkswagen Beetle (Classic)", "Standard Coffin"]: venue = VENUES[venue_name] count_fcc, centers_fcc = pack_spheres_fcc_in_venue(r_child, venue) count_cubic, centers_cubic = pack_spheres_cubic_in_venue(r_child, venue) count = max(count_fcc, count_cubic) centers = centers_fcc if count_fcc >= count_cubic else centers_cubic if count == 0: print(f" {venue_name} (The Junior): 0 fit") continue safe = venue_name.split("(")[0].strip().replace(" ", "_") + "_funsize" render_sphere_packing( centers, r_child, venue, f"{OUTPUT_DIR}/scene_{safe}.png", title=f"{venue_name}: {count} The Junior spherical children (r={r_child:.3f}m)", ) print(f" {venue_name} (The Junior): {count}") def experiment_5_reality_check(): """Compare spherical packing predictions to rated/world-record capacities.""" print("\n" + "=" * 70) print("EXPERIMENT 5: Reality Check") print("(Spherical humans vs actual humans — who wins?)") print("=" * 70) comparisons = [] for name, venue in VENUES.items(): rated = venue.get("rated_capacity", 0) world_record = venue.get("world_record", None) row = {"venue": name, "rated": rated, "world_record": world_record} for model_name in SPHERE_MODELS: r = SPHERE_MODELS[model_name]["radius"] count_fcc, _ = pack_spheres_fcc_in_venue(r, venue) count_cubic, _ = pack_spheres_cubic_in_venue(r, venue) best = max(count_fcc, count_cubic) row[model_name] = best comparisons.append(row) # Print table print(f"\n {'Venue':<28} {'Rated':>6} {'WR':>6} {'Meatball':>10} {'Hamster':>10} {'Freedom':>10} {'Child':>10}") print(" " + "-" * 90) for row in comparisons: wr = str(row["world_record"]) if row["world_record"] else "-" print(f" {row['venue']:<28} {row['rated']:>6} {wr:>6} " f"{row['The Meatball']:>10} {row['The Hamster Ball']:>10} " f"{row['The Freedom Sphere']:>10} {row['The Junior']:>10}") # Chart: Meatball vs Rated fig, ax = plt.subplots(figsize=(14, 8)) venue_labels = [r["venue"].split("(")[0].strip() for r in comparisons] rated_vals = [r["rated"] for r in comparisons] meatball_vals = [r["The Meatball"] for r in comparisons] child_vals = [r["The Junior"] for r in comparisons] x = np.arange(len(venue_labels)) w = 0.25 ax.bar(x - w, rated_vals, w, label='Rated Capacity', color='#2ECC71', edgecolor='white') ax.bar(x, meatball_vals, w, label='Meatball (adult)', color='#E74C3C', edgecolor='white') ax.bar(x + w, child_vals, w, label='The Junior (child)', color='#3498DB', edgecolor='white') # Mark world records for i, row in enumerate(comparisons): if row["world_record"]: ax.plot(i - w, row["world_record"], 'k*', markersize=15, zorder=5) ax.annotate(f'WR: {row["world_record"]}', (i - w, row["world_record"]), textcoords="offset points", xytext=(10, 10), fontsize=8, fontweight='bold') ax.set_xticks(x) ax.set_xticklabels(venue_labels, rotation=35, ha='right', fontsize=9) ax.set_ylabel("Humans Packed") ax.set_yscale('symlog', linthresh=1) ax.set_title("Reality Check: Spherical vs Actual Humans\n" "(rated capacity, world records, and spherical predictions)", fontsize=14, fontweight='bold') ax.legend(fontsize=9) ax.grid(True, alpha=0.3, axis='y') plt.tight_layout() plt.savefig(f"{OUTPUT_DIR}/reality_check.png", dpi=200, bbox_inches='tight') plt.close() print("\n Saved reality_check.png") # Child Advantage Factor chart fig, ax = plt.subplots(figsize=(12, 7)) caf = [c / m if m > 0 else float('inf') for c, m in zip(child_vals, meatball_vals)] caf_plot = [v if v != float('inf') else 0 for v in caf] # can't plot inf colors = ['#333333' if v == float('inf') else '#E74C3C' if v > 2.5 else '#F39C12' if v > 1.5 else '#3498DB' for v in caf] bars = ax.barh(venue_labels, caf_plot, color=colors, edgecolor='white', height=0.6) ax.axvline(x=1, color='#333', linewidth=1.5, linestyle='-') ax.set_xlabel("Child Advantage Factor (The Junior count / Meatball count)") ax.set_title("The Child Advantage Factor (CAF)\n" "How many more spherical children fit than adults?", fontsize=14, fontweight='bold') for bar, val in zip(bars, caf): if val == float('inf'): ax.text(0.15, bar.get_y() + bar.get_height()/2, '\u221e (5 children, 0 adults)', va='center', fontsize=9, fontweight='bold', color='white') elif val > 0: ax.text(bar.get_width() + 0.05, bar.get_y() + bar.get_height()/2, f'{val:.1f}x', va='center', fontsize=9) plt.tight_layout() plt.savefig(f"{OUTPUT_DIR}/child_advantage_factor.png", dpi=200, bbox_inches='tight') plt.close() print(" Saved child_advantage_factor.png") # Save results with open(f"{OUTPUT_DIR}/reality_check.json", "w") as f: json.dump(comparisons, f, indent=2, default=str) return comparisons # ============================================================ # MAIN # ============================================================ if __name__ == "__main__": print("=" * 70) print(" CONSIDER A SPHERICAL HUMAN") print(" Packing Density Bounds for Coffins, Clown Cars, and Coach Class") print("=" * 70) print() # Print sphere models print("Sphere Models:") for name, model in SPHERE_MODELS.items(): r = model["radius"] v = (4/3) * np.pi * r**3 d = 2 * r print(f" {name:<22} r = {r:.4f}m d = {d:.3f}m V = {v*1000:.1f}L") print() experiment_1_venue_specs() results = experiment_2_packing_primary() all_models = experiment_3_all_models() experiment_4_visualizations() reality = experiment_5_reality_check() print("\n" + "=" * 70) print("All experiments complete.") print(f"Results saved to ./{OUTPUT_DIR}/") print("=" * 70)