Editing Taxicab distance

[[File:Taxicab vs euclidean.png|thumb|right|400px|Taxicab distance is the distance a taxicab would travel on streets connecting square city blocks.]]

'''Taxicab distance''', also known as '''rectilinear distance''', '''city block distance''', '''Manhattan distance''', and other names, is a metric in [[wikipedia:Taxicab geometry|taxicab geometry]] for measuring distance, as an alternative to [[wikipedia:Euclidean distance|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 ==
The distance between two points in three-dimensional space can be measured either by Euclidean distance or taxicab distance, depending on whether geometric accuracy or calculation efficiency is desired.

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

This is geometrically the most accurate calculation. The square root operation, if it needs to be done many times during a [[Tick|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.

;Taxicab distance
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:
:<math>d = |x_1 - x_2| + |y_1 - y_2| + |z_1 - z_2|</math>

This is a less accurate but more efficient calculation suitable for efficiently processing numerous or rapidly-repeating calculations in a short time. 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 [[wikipedia:Octahedron|octahedron]].

== Usage in ''Minecraft'' ==
[[File:MinecraftTaxicab.png|thumb|right|Taxicab distance as represented by ''Minecraft'' blocks rather than city streets.]]
=== Taxicab ===
''Minecraft'' makes use of taxicab distance in various situations:
* [[Light]] from light sources such as [[torch]]es, [[campfire]]s, [[lantern]]s, [[glowstone]], etc. illuminate surfaces according to taxicab distance.
* [[Torch]]es also also melt snow layers within 2 blocks and ice within 3 blocks taxicab distance. 
* [[Heat block]]s{{only|education}} 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.<!-- Simulation distance is also a video setting in Java Edition. Do not add Java edition simulation distance here without verifying that it also uses taxicab distance. -->
* A [[sponge]] absorbs water up to 7 blocks away in taxicab distance (an octahedral volume centered on the sponge).
* [[Phantom]]s 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.
* [[Shulker]]s 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 [[Redstone mechanics#Redstone block updates|block updates]] up to two blocks away by taxicab distance, including up and down.

<gallery>
File:Water spread v1-13.png|Horizontal spread of water.
File:Lava spread v1-13.png|Horizontal spread of lava.
File:LightEmittingBlocks.png|Taxicab illumination from various light-emitting blocks.
File:Lightning distribution BE.png|The effect of simulation distance (the diamond area) on [[lightning]] strikes around a [[lightning rod]].
</gallery>

=== Euclidean ===
Situations in which Euclidean distance is used:
* Mob [[Spawn#Despawning|despawning range]] in ''Java Edition''.
* [[Disk]]s in rivers and lakes generate as blocks within a Euclidean distance from a central block.
* Effects of [[explosion]]s.
* [[Lightning rod]]s 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|locate}} command.
* [[Nether portal#Portal search|Nether portal]] searches for destination portals.
* [[Splash potion#Using|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!"