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In the book Chemistry: The Central Science, they introduced the following 5 types of two-dimensional lattices:enter image description here

The book said that the blue square represents the unit cell, the black dots are lattice points, and the vector a and b are lattice vectors. They don't say what the solid black lines represent. I feel as though they are redundant.

According to the illustration, hexagonal lattice is just a special case of rhombic lattice where γ = 120°. So why does a rhombic lattice have 2 types of unit cells (the blue and the green) whereas a hexagonal lattice only has 1 (and why do the solid black lines for these two lattices are different?). The same question also applies to rectangular lattice. Isn't it just a special case of oblique lattice where γ = 90°?

Why does it make sense to group lattices into 5 types, and what distinguishes them?

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  • $\begingroup$ Remember the geometry classes for kids? Why is there such thing as a square, isn't that just a special case of rectangle? Indeed it is. Then again, isn't a rectangle just a special case of parallelogram? Indeed it is. Same thing here. $\endgroup$
    – Ivan Neretin
    May 23, 2021 at 16:28
  • $\begingroup$ Different but related in Math SE: Are planar symmetry groups and wallpaper groups the same things? $\endgroup$
    – uhoh
    May 24, 2021 at 3:53
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    $\begingroup$ While there may be exactly 5 types of two-dimensional lattices in a book, that's just a custom, made by people who didn't think quasicrystals could exist. $\endgroup$
    – Mithoron
    May 24, 2021 at 18:46

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First, about the black lines. I think they are helpful for the hexagonal and the rhombic lattice. For the hexagonal system, it shows you the hexagons. One hexagon contains exactly the same space as one conventional unit cell. You just have move (translate by a lattice edge) certain areas as shown below.

enter image description here

For the rhombic lattice, it is nice to have the black lines when you are looking at the green cell. Without it, you would not understand why you would call it rhombic.

So why does a rhombic lattice have 2 types of unit cells (the blue and the green) whereas a hexagonal lattice only has 1 (and why do the solid black lines for these two lattices are different?).

The green cell is a centered cell, which makes the math easier because it has right angles. On the other hand, you suddenly have a translation by $½a + ½b$ on top of the lattice translations that you have the keep track of. In fact, the rhombic type is usually described as a centered rectangular lattice, e.g. https://en.wikipedia.org/wiki/Bravais_lattice#In_2_dimensions.

The same question also applies to rectangular lattice. Isn't it just a special case of oblique lattice where γ = 90°?

Compared to the oblique lattice, all the other lattices are special cases. They aren't special cases because of some rare coincident, but because of the presence of point group symmetries, i.e. mirror planes and twofold, threefold, fourfold and sixfold rotations. A mirror plane will always allow you to choose a unit cell with a right angle. A fourfold axis will always allow you to choose a unit cell that is square, and a threefold or sixfold axis will always allow you to choose a unit cell that is of the hexagonal type. As you can see, the conventional unit cell is not a hexagon but a rhombus with a 60 degree angle. This makes the math easier.

I don't know why the illustrator chose to offset the black lines and the unit cell by half a unit cell length in both $a$ and $b$ for all lattice types (except the hexagonal one). Maybe it is for better visibility. In general, any point in the unit cell will have symmetry mates in other unit cells related by lattice translations.

If you want to see some examples of lattices that contain objects, check out this link: https://www.iucr.org/education/pamphlets/21/supplement

enter image description here

Above is the illustration for a lattice that contains a hexagonal axis. In this case, you have six identical objects in the unit cell, related by point group symmetry. They do no fit directly into the conventional unit cell, but it has the same volume as six objects (you can try to draw the black lines showing hexogons in your mind). The object is called the asymmetric unit, and it can have various shapes as long as it maps onto the unit cell filling the space without overlap.

The animation below highlights the object in different orientations generated by the hexagonal symmetry.

enter image description here

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    $\begingroup$ +1 very nice answer! There's also a discussion of how to think about the "centered" case (in 2 and 3 dimensions) in this answer. $\endgroup$
    – uhoh
    May 24, 2021 at 3:53
  • $\begingroup$ +1 Just for the animation. How do you do that? $\endgroup$
    – user10186832
    Jul 7, 2021 at 14:50
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    $\begingroup$ @user10186832 Thanks, I traced the bird in powerpoint, made it transparent and overlayed it on the different positions in six different slides. From there, you could export slides as images and use a service like ezgif to make a gif. Instead, I used tools I have on my computer already (not-free Camtasia to record the screen, get rid of unwanted material, and turn it into an animated gif). $\endgroup$
    – Karsten ♦
    Jul 7, 2021 at 14:59
  • $\begingroup$ Here is the powerpoint $\endgroup$
    – Karsten ♦
    Jul 7, 2021 at 15:11
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For the square, rectangular, and hexagonal lattices, the black lines show Wigner-Seitz primitive unit cells. They contain all points closer to a given lattice point than to any other lattice point. Every lattice type has an infinite number of primitive unit cells that each contain just one lattice point. The diagrams show some of those that may be helpful for recognizing the symmetries. The green cell shown for the rhombic lattice contains 2 points and is non-primitive. The different lattice types are distinguished by the symmetries they possess that are in addition to their lattice translation symmetries. For example, the rectangular lattice contains a 2-fold rotation symmetry and 2 reflection lines, while the square lattice contains a 4-fold rotation and 4 reflection lines.

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    $\begingroup$ No, the black lines don't show Wigner-Seitz cells. For oblique lattice these cells aren't parallelograms: they have six sides. See this animation. $\endgroup$
    – Ruslan
    May 24, 2021 at 7:58
  • $\begingroup$ Thanks for pointing out this error. I have edited my answer. $\endgroup$
    – 10ppb
    May 24, 2021 at 11:34

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