There are lots of patterns to be found in the picture from my previous post!
This is a really remarkable tiling. Here are a few special properties I know of:

First of all, I hope you realized that the pattern can be extended infinitely to cover the whole plane with a hexagonal tiling. One way to convince yourself of this is to look at the horizontal strips of hexagons labelled . Each row of hexagons will just have repeating forever; and each row is a copy of the row below it, offset by two and a half hexagons.

Of course, there are seven different colors used (also indicated by the numbers ).

Instead of seeing it as made up of a bunch of repeating strips, you can also see it as made of a bunch of repeating parallelograms:
Each parallelogram has four zeros at its corners, and contains one of every other color/number in its interior. (There’s actually nothing special about zero; you can make parallelograms using any number you choose for the corners.)

Every hexagon touches six other hexagons, exactly one with each other color/number. For example, a hexagon labelled 1 will touch six hexagons labelled 0, 2, 3, 4, 5, and 6. (Can you see how to prove this? Think about how the whole tiling is built out of copies of the strip .) This explains what’s so special about having seven colors!

If you pick any color and look at all the hexagons of that color, they are always arranged in the same pattern, at the vertices of an equilateral triangular grid. For example, below I have arbitrarily chosen to highlight all the number 3 hexagons:
This means that if you pick any two colors/numbers, there is always some translation that will move all the hexagons of one color to the places that used to be occupied by the hexagons of the other color.
Here’s what it looks like if we draw all these triangular grids overlaid on top of each other (with brighter colors just for fun). I think it’s remarkable that seven equilateral triangular grids fit together so precisely!

In fact, since each hexagon touches one of each other color, for any two colors we can move all the hexagons of one color to those of the other by translating just one “hexagon unit”—that is, each hexagon will move to a hexagon next to it. For example, if we want to move all the number 3 hexagons so they match up with where the number 5 hexagons used to be, just move everything one hexagon down and to the right.
I’ll reproduce the original image here so you can refer to it while thinking about this and the following properties:

Each of the six directions we can translate corresponds to adding by a different number mod seven. For example, it’s easy to see that moving to the left corresponds to adding 1 (adding 1 to 6 wraps back around to 0 because we take the remainder when dividing by 7). In other words, pick any hexagon and look at its number; the hexagon to its left will have a number which is 1 bigger (mod 7). Moving down and to the right is adding 2: for example, starting at the 0 in the middle and following the line of hexagons down and to the right, we find , where each number is two more than the previous (mod 7). Down and to the left is adding 3; and so on. I’ll let you work out the other three directions.

As pointed out in a comment by Will, rotating corresponds to multiplication mod 7! For example, rotating 60 degrees counterclockwise around 0 corresponds to multiplying by 3 (mod 7). In counterclockwise order, the six numbers we find around 0 are . We can check that , and , , , and so on. Rotating the other direction is multiplying by 5. Rotating by 120 degrees is multiplying by 2 or 4; rotating by 180 degrees is multiplying by 6.

Pick a hexagon; say it contains the number . We already know that none of the hexagons it touches contain . But in fact, none of the hexagons those hexagons touch contain either (except for the original hexagon we chose). This is because each hexagon touches exactly one copy of every other color/number. So, in other words, each hexagon is contained in two layers of hexagons, none of which share the same color with the central hexagon. For example, if we pick a zero hexagon and go two layers out from there, we can see that the zero in the middle is the only zero (and there are exactly 3 copies of each other number):
Let me know in the comments if you see any other patterns that I missed!
I want to add to the comment on rotating correspond to multiplication mod 7 .
Rotating around 0 clockwise and anticlockwise does give multiplication by 3 and 5 respectively, but multiplication by the other numbers form slightly different patterns.
1) Multiplication by 2 gives us two distinct “triangles”. We have and . So the numbers $1, 2, 4$ are related by multiplication by 2. Notice that these three numbers always sits in a triangle formation, which going anticlockwise from 1 is .
Similarly, we have . This gives us the triangle which going anticlockwise gives us multiplication by 2.
By the what’s been mentioned before, the triangles and tessellate the picture (if we ignore the 0)
2) Multiplication by 4 gives the same triangles as multiplication by 2, but this time we rotate clockwise! (This is because so multiplication by 4 is the same as dividing by 2. This was already seen with multiplication by 3 and 5, as .
3) Multiplication by 6 (or 1) seems less interesting at first sight, as we can only get pairs of numbers. But if we look at the three pairs of numbers we see that they fit naturally into a triangular shape, either pointing down or pointing up depending of your choice (going clockwise from 1, either the triangle or the triangle $latex {1,6,4,3,2,5} ). Again, ignoring the zero, the either one of the triangles tessellate the picture.
Yes, great insights!
Following on from your point #3: if you quotient the plane by the group of translations that maps any one of those parallelograms on to another one – or, equivalently, cut out just one parallelogram and identify each edge with the opposing one – then this is one of many ways to construct a torus on which is drawn a map with 7 regions all touching each other.
But this particular way of doing it also equips the resulting torus with a geometry in which the 7 regions form a tiling of the torus with _regular_ hexagons, which I’ve always thought makes it the cutest of all such constructions.
Oh yes, good point! I was aware of being able to have a map on a torus with 7 regions all touching each other, but I never realized before you could do it with regular hexagons! So this shows that some toroidal graphs have chromatic number at least 7. (I guess it turns out 7 is also maximal, though I don’t know off the top of my head how to prove that.)
This 7coloring of the hexagonal tessellation of the plane is closely related to a 7coloring of the cubic tessellation of space: start by arbitrarily assigning one cube color 0. The color of the cube to the west of any given cube is determined by adding 1 mod 7 to the color of the given cube (and the color of the cube to the east by subtracting 1 mod 7.) Similarly the color of the cube to the north of a given cube is determined by adding 2 mod 7 to the color of the given cube, the color of the cube to the south by subtracting 2 mod 7, the color the cube below by adding 3 mod 7, and the color of the cube above by subtracting 3 mod 7. This produces a map on the 3torus in which the six cubes sharing a face with a given cube are colored differently from the given cube.
Making planar slices of this map gives 7colorings of various planar maps. Unless I’m mistaken, you can’t get your hexagonal coloring directly in this way, but you can get a 7coloring of the trihexagonal tiling (also called the kagome lattice) that can be thought of as an elaboration of your map. (I don’t have time to draw it, but the coloring I have in mind is as follows: color the six hexagons sharing a vertex with a given hexagon according to your scheme. To color the triangles, note that there are three neighboring hexagons and three secondnearest hexagons. Color the triangle using the color that is not used in these six hexagons.)
Like this? http://ozark.hendrix.edu/~yorgey/hosted/Kagome.png
A late addition – I was pondering hexagonal tiling and this tesselation and ended up doodling this: https://yomikoma.dreamwidth.org/38411.html
Very nice!
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