I Miller indices for hexagonal crystal systems

[Mind----Blown]

Hi everyone, to find the draw the direction for a given miller index say, [1234] we first convert this miller index consisting of 4 indices into one containing 3 indices.
To do so, we have a set of formulae prescribed in almost every book. Sadly I haven't been able to come across a single book the gives the derivation of those formulae!
I thought that i could use vector-component method to get the results but that gives totally weird formulae not even close to the ones i see in my textbooks. (have a look at the attached image)

So, can anyone suggest me a textbook, a link or anything that can help me understand the derivation? I'm not finding the enthusiasm for rote-memorising the formulae if i don't know where they come from...

Thanks!

 

[M Quack]

The 4 Miller indices often used with hexagonal crystals are just a convenience. The 3rd index can be dropped. 4 indices are used to make planes with the same symmetry also "look alike" in the 4-notation.
The physical reason is that the a- and b- axes form a 120 degree angle. There is a third axis within the basal plane that has the exact same symmetry. You get this axis, let's call it "d", by rotating the b-axis by another 120 degrees. So instead of using a and be as basis vectors, you could just as well use b and d, or d and a. Absolutely nothing would change, as the 120 degree rotation is the defining feature of a hexagonal crystal.
The redundant 3rd Miller index corresponds to the reciprocal space "d*" axis. The 4th index is along the c-axis.

 

[Mind----Blown]

The 4 Miller indices often used with hexagonal crystals are just a convenience. The 3rd index can be dropped. 4 indices are used to make planes with the same symmetry also "look alike" in the 4-notation.
The physical reason is that the a- and b- axes form a 120 degree angle. There is a third axis within the basal plane that has the exact same symmetry. You get this axis, let's call it "d", by rotating the b-axis by another 120 degrees. So instead of using a and be as basis vectors, you could just as well use b and d, or d and a. Absolutely nothing would change, as the 120 degree rotation is the defining feature of a hexagonal crystal.
The redundant 3rd Miller index corresponds to the reciprocal space "d*" axis. The 4th index is along the c-axis.

yes, i get that but still, how do we arrive at those formulae?
i tried using vectors and their components method but it doesn't work

 

[DrDu]

The German wikipedia contains a simple derivation:
https://de.wikipedia.org/wiki/Millersche_Indizes