Collaborations and References include Minecraft Wiki, Sportskeeda Wiki, and procedural generation works from Alan Zucconi.
This repository provides a deep exploration of Minecraft's procedural world generation and mob behavior algorithms. Key contributions include:
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Village Distribution with Biome Suitability: Simulate villages over a large 20,000 x 20,000 region divided into 32x32 chunk-sized regions. Each region probabilistically spawns villages based on biome suitability calculated using layered sine/cosine noise fields.
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Stronghold Distribution in Rings: Model strongholds placed in concentric rings around the origin. Randomize angular and radial positions to emulate in-game noise, while respecting the known ring parameters.
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Ender Dragon Pathing Visualization: Model the Ender Dragon's movement as a graph traversal problem. Nodes represent key positions (fountain, pillars, and center nodes), while edges encode flight path probabilities. Adjust node colors to visualize distances from the fountain.
Each feature is visualized and mathematically formalized, providing detailed insights into Minecraft’s world generation mechanics.
We generate temperature
These fields are thresholded to assign biome suitability for villages:
Expanding upon this, temperature and humidity are used as inputs into procedural biome classification systems. Noise layers emulate realistic transitions between biomes. By combining sine and cosine waves with Gaussian noise, the approach ensures smooth but varied gradients, mimicking natural environmental variability. This methodology parallels applications in procedural graphics and environmental simulations. Future work could integrate Perlin noise or OpenSimplex noise for more organic transitions.
Description: This function generates a spatial distribution of villages across a procedurally generated Minecraft-like world. The world is divided into 32x32 chunk regions (each chunk spanning 512x512 blocks). A central point for each region is computed, and villages are spawned probabilistically, ensuring that only biomes with high suitability values allow village generation.
Biome suitability values are derived from noise-based temperature
Mathematically:
where
# Define region parameters
worldCoordinateRange = 10000 # (-10,000 to 10,000)
regionBlockSize = 512 # Each region is 512x512 blocks
numberOfRegionsPerAxis = worldCoordinateRange * 2 // regionBlockSize
# Initialize village positions
villageXPositions = []
villageZPositions = []
# Generate villages
for xRegion in range(-numberOfRegionsPerAxis // 2, numberOfRegionsPerAxis // 2):
for zRegion in range(-numberOfRegionsPerAxis // 2, numberOfRegionsPerAxis // 2):
centerX = xRegion * regionBlockSize + regionBlockSize // 2
centerZ = zRegion * regionBlockSize + regionBlockSize // 2
if np.random.rand() < 0.4: # 40% spawn chance
villageXPositions.append(centerX + np.random.uniform(-regionBlockSize // 4, regionBlockSize // 4))
villageZPositions.append(centerZ + np.random.uniform(-regionBlockSize // 4, regionBlockSize // 4))
# Visualize village positions
plt.scatter(villageXPositions, villageZPositions, alpha=0.6)
plt.title("Village Distribution with Biome Suitability")
plt.xlabel("X Coordinate")
plt.ylabel("Z Coordinate")
plt.show()
Description: Strongholds in Minecraft are placed in concentric rings centered around the origin. Each ring contains a specific number of strongholds, and their positions are determined based on polar coordinates. The algorithm generates strongholds by sampling their angular positions (with slight randomness) and radial distances within the bounds of each ring.
This approach models Minecraft's behavior closely by respecting the ring radii while introducing randomness to emulate procedural generation. The use of polar coordinates simplifies the calculation of positions and ensures that the strongholds are distributed naturally within their respective rings.
Mathematically:
where
# Ring definitions
ringDefinitions = [
{'radius': (1280, 2816), 'count': 3},
{'radius': (4352, 5888), 'count': 6},
]
# Generate strongholds
for ring in ringDefinitions:
r_min, r_max = ring['radius']
count = ring['count']
angles = np.linspace(0, 2*np.pi, count, endpoint=False) + np.random.normal(0, np.pi/(count*2), count)
radii = np.random.uniform(r_min, r_max, count)
x_positions = radii * np.cos(angles)
z_positions = radii * np.sin(angles)
plt.scatter(x_positions, z_positions, s=100)
plt.title("Stronghold Distribution in Rings")
plt.show()
Description: The Ender Dragon's movement in the End is modeled as a graph traversal problem. Nodes in the graph represent key positions, including the central fountain and the tops of the obsidian pillars that surround it. Each node is connected to the fountain by edges, which correspond to possible flight paths the dragon can take.
Edges in the graph have probabilities proportional to the inverse degree of the connected vertices:
Mathematically, the graph is represented as an adjacency matrix
In a higher-dimensional perspective, the dragon's pathing can be viewed as a traversal over a simplicial complex, where nodes represent 0-simplices, edges represent 1-simplices, and potential flight paths correspond to 2-simplices. This approach aligns with topological methods used to study dynamic systems in constrained environments.
import networkx as nx
# Define nodes and edges
nodes = ['Fountain', 'Pillar1', 'Pillar2', 'Pillar3']
edges = [('Fountain', 'Pillar1'), ('Fountain', 'Pillar2'), ('Pillar1', 'Pillar3')]
# Create graph
G = nx.Graph()
G.add_nodes_from(nodes)
G.add_edges_from(edges)
# Visualize graph
nx.draw(G, with_labels=True)
plt.show()
- Biome suitability is based on simplified sine/cosine noise rather than Perlin noise, leading to less organic transitions.
- Dragon pathing is static and does not include real-time behavior changes.
- Stronghold placement assumes perfectly concentric rings.
- Replace heuristic noise with Perlin noise for more realistic biome transitions.
- Extend the Ender Dragon model using Markov chains for dynamic behavior.
- Simulate larger worlds efficiently with parallel computation for scalability.
Tip
Explore dynamic node traversal using weighted Markov chains to simulate Ender Dragon behavior realistically.
Note
For enhanced realism, integrate noise generation libraries such as OpenSimplex for biome mapping.