Trigonal lattice with each angle equal to 120degree

[PrinceOfDarkness]

1. What happens when the angles between the three sides of a trigonal reach 120degrees?

I know that at 90degrees, it becomes a simple cubic. At 109degrees, it becomes a body centered cubic. At 60 degrees, it becomes a face centered cubic.
What about 120degrees? I think that perhaps it should form a structure like a football?

2. Can pentagons form a lattice? I know that regular pentagons don’t satisfy translational symmetry. But can any pentagon with different side lengths form a lattice?
I tried making one, but no pentagon satisfies the translation symmetry.



Many thanks.

- Farrukh

 

[OlderDan]

For #1 write two vectors in the x-y plane at 120° separation angle. Then write a general vector in 3-D and take the dot procuct with each of the two in-plane vectors and see where it has to point to make the angles to both of those vectors 120°

For #2, if it did exist you would probably find it here

http://www.mathpuzzle.com/tilepent.html