There exists two different families of Magic Meshes. They are called Type 2 and Type 3. Type 2 is the family that Magic Squares belong to, and Type 3 is the family that Magic Hexagons belong to.
The properties of the Type 2 family are similar to those of Magic Squares, ie.. all rows are identical in length, and add to the same value. You can add a constant value to every cell and the result is still completely magic. Therefore, every Type 2 Magic Mesh that I have generated uses the numbers 0 to k, k being 1 less than the total number of cells in the figure. The following is an example of a 5 sided Type 2 Magic mesh with 4 cells per side:
It uses the numbers 0 to 19 once and only once, and each row of yellow cells adds to 38. Every row is 4 cells in length.
A Type 2, 5 Sided Order 5 follows.
Every row is 5 cells in length, and it uses the numbers 0 thru 30 once and only once. Each row adds to 67. Unlike a regular Magic Square, 67 is NOT the only Magic value that this 5 sided, 5 cells per side Type 2 can add to. It can, using the same numbers 0 thur 30, add to from 67 up to 83. as seen below:
There exists an uncountable total number of Type 2 Magic Meshes, starting with the 960 three sided, 4 cells per side arrangement using the numbers 0 thru 11 that adds to 22.
Type 3, like Type 2, are a series of n-sided figures nested from outside to inside, with a certain number of cells arranged per figure such that they can be "connected" to form rows that add to the same value. Type 2's cells decrease by 2 as you go from outside to inside, ie... the external row of the previous Type2 5 sided 5 cells per side, consists of 3 figures from outside to inside haveing 5 cps, 3 cps, and the central 1 cps. If a Type 2 had an even number of cells per external side, ie.. 4, it would only consist off 2 geometric figures, a 4 cps and a 2 cps. Type 3 do not decrease by 2, but instead, by 1. All Type 3 Magic meshes have a central single cell, and increase inward to outward by 1. The rows are not identical in length, and so, you cannot transform a type 3 by adding a single constant value to each cell. Type 2 figures can have no diagonals, diagonals, or "diaganoids", that is, a line of cells thaty start on a vertice, pass thru the central cell, and end in the middle of the side opposite the originating vertice. Type 3 can have Diagonals, and "Half Diags", ie... a line of cells that start from a vertice and terminate in the central cell. The following is a 5 sided type 3, with 4 cells per side, using the numbers -45 to -15 once and only once, to add to -144: