Taxicab distance

Taxicab distance, also known as rectilinear distance, city block distance, Manhattan distance, and other names, is a metric in taxicab geometry for measuring distance, as an alternative to Euclidean distance (straight-line, or "as the crow flies"). Minecraft uses taxicab distance as an efficient range measurement in several elements of the game.

Calculation
In Minecraft, the distance between two points in three-dimensional space can be measured in taxicab or Euclidean distance, depending on the situation.

Given two points in 3-D space $$(x_1, y_1, z_1)$$ and $$(x_2, y_2, z_2)$$, the straight-line distance (Euclidean distance) between them is:
 * Euclidean distance
 * $$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}$$

The square root operation, if it needs to be done many times during a game tick, can be processor intensive, particularly on mobile devices that may have slow processors. Too many processor-intensive operations can introduce lag in the game.

The taxicab distance is a simplified calculation that eliminates the square root by simply calculating the absolute value (positive value) of the difference between each coordinate value:
 * Taxicab distance
 * $$d = |x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2|$$

A consequence of this calculation is that a taxicab "circle" drawn with a constant taxicab radius appears as a diamond, in which the taxicab radius equals the Euclidean radius only in the four cardinal directions (north, south, east, west), and the radius is shorter in all other directions. Similarly, in three dimensions, a taxicab "sphere" appears as an octahedron.

Taxicab
Minecraft makes use of taxicab distance in various situations:
 * Light from light sources such as torches, campfires, lanterns, glowstone, etc. illuminate surfaces according to taxicab distance.
 * Torches also also melt snow layers within 2 blocks and ice within 3 blocks taxicab distance.
 * Heat blocks melt snow layers and ice within 2 blocks taxicab distance, but do not emit light.
 * Liquid (water or lava) flowing horizontally on a flat surface flows from the source in taxicab distance. For example, if flowing water is forced into a 1-block-wide stream and turns corners, it travels 7 blocks. If it is allowed to flow freely on a surface, it also travels 7 blocks taxicab distance.
 * In Bedrock Edition, Simulation distance is the distance from the player at which blocks get tick updates (such as lightning strikes), and beyond which mobs despawn; this also uses taxicab distance.
 * A sponge absorbs water up to 7 blocks away in taxicab distance (an octahedral volume centered on the sponge).
 * Phantoms in Java Edition spawn 20–34 blocks above the player, and off to the side by a taxicab distance of up to 10 blocks.
 * If bone meal is used on a grass block, then grass, tall grass, or flowers form on random grass blocks up to 7 taxicab blocks away. Also, using bone meal on sea pickles that are on coral blocks creates more sea pickles on nearby coral blocks out to a horizontal taxicab distance of 2.
 * A block of copper is oxidized according to a taxicab distance search for unwaxed copper blocks within 4 blocks of the block being considered for oxidation.
 * Shulkers can move along only one of the three grid axes at a time and on integer steps, meaning that they move according to taxicab geometry.
 * Some redstone components produce block updates up to two blocks away by taxicab distance, including up and down.

Euclidean
Situations in which Euclidean distance is used:
 * Mob despawning range in Java Edition.
 * Disks in rivers and lakes generate as blocks within a Euclidean distance from a central block.
 * Effects of explosions.
 * Lightning rods redirect lightning within a Euclidean spherical volume centered on the rod, having a radius of 128 blocks in Java Edition and 64 blocks in Bedrock Edition.
 * The command.
 * Nether portal searches for destination portals.
 * Splash potions affect mobs inside a 4-block radius Euclidean sphere.
 * Target selectors used in commands use Euclidean distances.
 * A zombie siege is started based on Euclidean distance between a player and the village center.
 * Finally, a splash text displays the misspelled word "Euclidian!"