# Is defined as a zero square

Variation: eccentric 9-square grid

Half a level can be addressed by defining the center-bottom square in a block of nine (green in the picture) as a zero square, and adding 8 more squares (white in the picture) to the left, top and right in each subsequent step. The 9-way system (digits 0-8) is ideal for addressing. You can also address the other half of the levels (in the picture below the red line) by mirroring the above nine blocks down and adding a minus sign to their sector numbers.

Square 0 can also be placed in a corner, e.g. at the bottom left. But then the 4-square grid described below is probably preferable because it is simpler.

A similar one is possible, but the squares then become increasingly flatter rectangles. The system of 6 (digits 0-5) is ideal for addressing.

Variation: 4-square grid

You can do a similar thing 4-square coordinate system design: Square 0 is now in a corner, 3 other squares with numbers 1-3 are around it in a quarter circle (picture below, left).
In the next higher size level, 3 more such blocks of four are added (picture below, middle. For reasons of space, the large green 0 is missing in the large square at the bottom left, in the other large squares the numbering of the small squares is missing.)
For clarification, all small squares are shown with their local numbers in the picture at the bottom right. The blue square has the total address 00 = 0, the yellow 22, the green 32. A quarter level (1 quadrant) can be addressed step by step, it represents all positive numbers (if the addresses are viewed as numbers in the system of four).
You can address the whole level if you address all 4 quadrants individually as above. Each quadrant has a sign to distinguish it: + or - or + i or - i. The whole plane or the whole coordinate system then represents the complex numbers, the most powerful number system: The numbering of the small squares in each quadrant is rotated by 90 degrees compared to the previous one: Then 0 is always inside (towards the intersection of axes), and the following applies, looking outwards from 0: 1 = left, 2 = middle, 3 = right. But one could also number all blocks of four small squares exactly the same. The 4-way system (digits 0-3) is always optimal for addressing.

Variation: 4-triangle grid

Here again a symmetrical solution is possible, i.e. triangle 0 is in the middle (of a total of 4 triangles, green in the picture). In the next stage, 3 more triangles (green numbers) are placed around the overall triangle. In the next stage then again 3 (in the picture without numbers). With each step change, the overall triangle folds over vertically (its tip is alternately above or below). The numbering of the new triangles can therefore be done in 2 ways: either in such a way that the following always applies: 1 = left, 2 = center, 3 = right (as in the picture). Or in such a way that the direction of rotation of the numbers is retained. The 4-way system (digits 0-3) is always optimal for addressing.
Equilateral triangles seem most useful, but this coordinate system works with isosceles triangles as well.

This coordinate system already looks quite exotic. However, areas of application would also be conceivable for this, e.g. in the modeling of any shape using triangles (which are rounded before being displayed on the screen), which is the standard process nowadays.

Variation: rotating square grid

The zero element is a square. Around it 4 isosceles right triangles (with sector numbers 1-4) are placed so that there is another square (green area in the picture). This can be continued indefinitely (the next 4 triangles are white in the picture, sector 0 of this white square is the large green square).
The process can also be interpreted as laying a square rotated by 90 degrees around the previous one, with side length = diagonal of the old square). The square 0 of the respective level is rotated by 90 degrees for successive levels. You can let the sector numbers 1-4 rotate completely (in 4 steps, i.e. by 360 degrees) or return to the old numbering after every 2 steps.
The 5-way system (digits 0-4) is ideal for addressing.

This division of the squares can also be continued inwards, i.e. the smallest squares can be further subdivided in this way (the resulting sectors have numbers with decimal places). The triangles have to be divided according to another method, e.g. the 4-triangle grid. You now have a mixed grid with 2 shapes and 2 division factors: 5 for squares, 4 for triangles.

Variation: 7-circle grid

A grid coordinate system can also be based on circles: 6 additional circles are placed around a center circle (all blue in the picture). This "bundle" is then viewed as a new circle around which 6 more circles of the same size are placed (red in the picture). These all contain the same subdivision into small circles as the center circle (only drawn in this one). In contrast to the above coordinate grids, there are gaps between the individual basic elements (here circles) of the same addressing level. (If you put fewer or more than 6 circles around the center circle, this effect would be even stronger.) But it is possible to make all circles a little larger so that they overlap and the position of each point can be specified as precisely as required.
Compare the similar iris coordinates

The orbits of planets and their moons would be conceivable as an application area, but these are somewhat more irregular. The picture is reminiscent of a flower or a cluster of cells - perhaps this coordinate system is biologically applicable: there, it is not the absolute position that is decisive, but the relative e.g. the position of a cell in a cell cluster.

The 7 system (digits 0-6) is ideal for addressing.

More variations

Many other variations are possible, there are almost no limits to creativity. Shapes that meet 2 conditions are predestined:

- Adding a fixed number of the same forms results in the same form again
- These shapes completely fill the space (tiling)

However, exceptions are also possible: The exception to rule 1 is e.g. the rotating square grid, to rule 2 the 7-circle grid.
Grids that vary periodically in terms of shape and / or number are also possible.

Spatial grid

The above grid coordinate systems of the plane are also possible in a space with 3 or more dimensions. This was already described in the parent article for the 9-square grid, which in 3-dimensional space becomes a cube with 3 * 3 * 3 = 27 subordinate cubes.

In general, the number of subsectors per sector usually increases. Then another number system is optimal for sector numbering, except when the new number of subsectors is equal to a power of the old one.