Why are certain lattices compatible with only certain point groups and not all point groups?

Question

I will limit this discussion to 2 dimensions for ease of intuition. My understanding of the 17 crystallographic plane groups is that these 17 groups represent all the possible symmetry groups of any 2d crystal structure we may think of (crystal structure=lattice + motif). Thus, if we think of an arbitrary periodic 2d crystal structure, its symmetry group will necessarily correspond to one of the 17 crystallographic plane groups. My problem is that I can think of numerous 2d crystal structures whose plane group is not one of the 17 crystallographic plane groups. Take for example the one shown below

In this example, we have combined a square lattice with a motif that has a 3 fold rotational symmetry. The resulting structure does not have any 3-fold rotational symmetry axis's nor does it have any of the characteristic 4-fold rotational symmetries present in a square lattice. But it still is a periodic repeating structure. It still has the same translational symmetry as the square lattice. It even has horizontal mirror lines. So its overall symmetry group is the combination of a mirror line with a square lattice. Yet if we look at all 17 of the crystallographic plane groups below, we find that a single horizontal mirror plane may only be combined with a rectangular lattice (plane group pm in the image below).

Apparently, if we have a structure with only one mirror plane, the underlying lattice has to be a rectangular lattice. But this is absurd. I have a clear counterexample shown in first image of this post. This counterexample should then be an 18'th crystallographic plane group. But its not for some reason. Why?

I understand that when we create the 17 plane groups by combining point groups with bravais lattices, we only combine lattices with point groups when the lattice has at least as much symmetry as the point group. But why do we limit ourselves to only these combinations? Why do we not combine say a threefold rotation axis with a square lattice as shown above? Does nature somehow discriminate against these combinations? I guess my issues boil down to the following

1. What is the plane group of the structure shown above in fig 11.29 (if it has one)?
2. If it doesn't have a plane group, why do we not include its symmetry group in the list of all 2d plane groups?
3. Does nature ever allow crystals to exist if they do not have one of the 230 space groups or 17 plane groups?

Answer 1

What is the plane group of the structure shown above in fig 11.29 (if it has one)?

It is pm. In this plane group, you are free to choose the lattice parameters $a$ and $b$ (they can be different or the same), and the angle has to be 90 degrees. You can prove that the lattice is rectangular by combined translation and mirror operations.

If it doesn't have a plane group, why do we not include its symmetry group in the list of all 2d plane groups?

It does. Whenever you think you found a new plane group, you can probably shift the origin or take linear combinations of "your" cell axes to get to one that already exists.

Does nature ever allow crystals to exist if they do not have one of the 230 space groups or 17 plane groups?

It has more to do with math and geometry. If you define a crystal in the way it is defined, and assume space is Euclidean, you can enumerate the space and plane groups. You could do that slightly differently, but the way it is done, any crystal can be assigned to one of the plane groups or space groups in the international crystallographic tables.

In this example, we have combined a square lattice with a motif that has a 3 fold rotational symmetry. The resulting structure does not have any 3-fold rotational symmetry axis's nor does it have any of the characteristic 4-fold rotational symmetries present in a square lattice. But it still is a periodic repeating structure. It still has the same translational symmetry as the square lattice.

In the absence of four-fold symmetry, we don't call it a square lattice. It is a coincidence that the length of $a$ and $b$ are equal. If you look closely at your crystal, the packing in the vertical direction is tighter than in the horizontal direction, and it would be possible to push the motifs closer in the horizontal direction. This would then match your expectation that the lengths of $a$ and $b$ are different.

If your crystal had 4-fold symmetry, the packing in the vertical and horizontal direction would be equivalent. If for some reason the 4-fold symmetry breaks down (maybe instead of having four identical proteins coming together, I have pairs of two distinct proteins), the necessity for equal lengths would break down, too. So one comes with the other.

Apparently, if we have a structure with only one mirror plane, the underlying lattice has to be a rectangular lattice.

It does not have to be. A square lattice has $a = b$. A rectangular lattice may have $a = b$ or not, because a square is a special case of a rectangle.