# Trigonal lattice with each angle equal to 120degree

• PrinceOfDarkness
In summary, when the angles between the three sides of a trigonal reach 120 degrees, it forms a structure similar to a football. Regular pentagons do not satisfy translational symmetry, but it is possible that a pentagon with different side lengths could form a lattice. However, no such pentagon has been found yet.
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

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

## 1. What is a trigonal lattice with each angle equal to 120 degrees?

A trigonal lattice with each angle equal to 120 degrees is a type of crystal lattice structure in which the unit cell is composed of three identical axes, with each axis forming an angle of 120 degrees with the other two axes.

## 2. How is a trigonal lattice with each angle equal to 120 degrees formed?

A trigonal lattice with each angle equal to 120 degrees is formed when atoms or molecules are arranged in a repeating pattern in three dimensions, resulting in a lattice structure with three-fold rotational symmetry and equal angles between the axes.

## 3. What are the properties of a trigonal lattice with each angle equal to 120 degrees?

Some properties of a trigonal lattice with each angle equal to 120 degrees include high symmetry, strong bonding between atoms or molecules, and the ability to transmit light in specific directions due to its anisotropic nature.

## 4. What are some examples of materials with a trigonal lattice with each angle equal to 120 degrees?

Examples of materials with a trigonal lattice with each angle equal to 120 degrees include quartz, calcite, and boron nitride. These materials are commonly used in electronics, optics, and structural applications due to their unique properties.

## 5. How does the structure of a trigonal lattice with each angle equal to 120 degrees affect its physical and chemical properties?

The structure of a trigonal lattice with each angle equal to 120 degrees greatly influences its physical and chemical properties. The strong bonding between atoms or molecules and the high symmetry of the lattice result in unique properties such as high strength, optical anisotropy, and piezoelectricity.

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