A flat surface is like a halfway point between a Chess board and Manhattan
(the equations only work on my blog)
Distance on a flat plane is measured by the formula below where Δx and Δy represent the differences between the x values and y values of two points that has a distance being measured:
$$\sqrt{ \lvert \Delta x \rvert^2 + \lvert \Delta y \rvert^2 }$$
If one were to put a point down on the Cartesian plane and highlight all of the points that are 1 unit away from said point, the point would have a circle drawn around the point:
The distance formula is the type of measurement that exists in virtually every context under the Sun. However, this type of measurement doesn't work if you can only move vertically and horizontally:
(image courtesy of Wikipedia)
The diagonal path in green is the shortest path that can be taken between the two points in black. However, if diagonal paths are forbidden, then the best one can do is to go 6 vertical steps and 6 horizontal steps with each of the remaining three paths in red, blue, and yellow representing a shortest path to the two points.
This type of measurement where diagonal movement is disallowed and only orthogonal movement is counted is called the Manhattan distance. The Manhattan cannot exist in a regular Euclidean plane because reality does not have set grid lines unless you find yourself in Manhattan.
If one were to draw a point on a Cartesian plane and the x axis and y axis are taken as the grids of the Manhattan distance, the set of all point that are one unit away form a rhombus.
The points of the rhombus can be arrived at by taking a certain number of steps in the orthogonal directions to reach each of the points. This distance can be given an equation that matches that of the flat plane where Δx and Δy again represent the differences between the x values and y values with the only difference being that the differences are raised to the power of 1 instead of 2 and the sum is in a radical of 1 instead of 2.
$$\sqrt[1]{ \lvert \Delta x \rvert^1 + \lvert \Delta y \rvert^1 } = \Delta x + \Delta y$$
This formula above is identical to the distance formula for the normal flat plane with the only difference being that the 2s of the equations are replaced with 1s. Both formulas can be seen as generalizations of the formula below, where p equals 1 in the Manhattan distance, and p equals 2 for normal distances:
$$\sqrt[p]{ \lvert \Delta x \rvert^p + \lvert \Delta y \rvert^p }$$
This equation means that any value p will form a measurement that measures a distance like a weird looking circle described by the equation x^p+y^p=1.
This is a special case of a superellipse where the semimajor axis and semiminor axis are equal and both are equal to 1 unit.
If p were 4, then the measurement would be based on the squircle:
As p approaches ∞, the unit shape looks more and more like a square:
(image courtesy of Wikipedia)
The limit of this measurement is thus treated as if the defining shape was a square:
In this case, a diagonal movement is treated the same as a movement in the orthogonal directions which is the same type of measure used in chess (this type of measurement is named after Pafnuty Lvovich Chebyshev):
(image courtesy of Wikipedia)
A diagonal movement in the Manhattan distance is 2 units, root 2 on the Euclidean plane, and 1 unit on a chess board. In a way the Euclidean distance is in the middle of a chess board and Manhattan (logarithmically).
