# Whats up with the Bravais Lattices?

Hello everyone.

We're doing 'The Solid State' in our school where I first went through the concept of Unit Cells Lattices. Can someone help me through with -

1) Why are there only 14 such lattices?

2) How can one give a rigorous proof about the non existence of unit cells of 'End-centered cubic lattice' and 'face centered tetragonal lattice'.

Thank you!

Note by Priyansh Sangule
6 years ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

2) Even if you put particles on edge centers, it will again form same type of but smaller simple cubic unit cells. And same is the case of face centered tetragonal lattice.

- 6 years ago

Indeed. This is what my teacher explained me. BTW what about the other cases? Do all of them have the same reasoning?

- 6 years ago

May be they exist but , their properties may be similar to one of those 14. And this can be reason for having only 14 bravais lattice.

- 6 years ago

1 is a bit difficult to show.

For 2, the main idea is that if you did place lattice points at those places, you'd get smaller unit cells as repeating patterns.

To begin with, I suggest you look up the Wallpaper Symmetry Groups first.

(by the way, I do not have the full answer to your question)

Hey, thanks for your views bro! :D

- 6 years ago

The answer to the first could be that only 14 bravais lattice are seen practically yet.

- 6 years ago

Nope, they are mathematical structures - not empirically derived structures in chemistry.

You both are somewhat right. They do are mathematical structures but we exclude the ones which are practically not possible.

- 6 years ago

Which ones are they?

Like, Edge Centric and other cases...

- 6 years ago

As far as my understanding goes, there are no edge centric bravais lattices because inserting lattice points at the midpoint of every edge it instead transform into another kind of Bravais Lattice

Your understanding is, well, alright as always! $\ddot\smile$

- 6 years ago

Yes. I think this is more appropriate. Thanks!

- 6 years ago

See, both of the things you have asked are based on observations so there is actually no answer. The reason agnishom gave is what I don't know.

- 6 years ago