The Miller indices- searching for a proof

[antonni]

Hi all, first let me post this (from "Elementary Crystallography An Introduction to the Fundamental Geometrical Features of Crystals" by Buerger):
https://www.physicsforums.com/attachments/1-png.82644/
Can someone please explain me the proof of Lemma 1? I just can not see it with the "mini proof" provided. Maybe a rigorous mathematical proof?

Thank you,

Anton

 

[antonni]

don't know really how to reword it...but others can try

 

[dipstik]

i tried for quite a while to derive the results using a*(x-x_0)+b*(y-y_0)+c*(z-z_0)=0 equations, but got B*C+B^2*C/A+C^2*B/A planes to reach A*B*C along the x axis...

there may be a more elegant approach considering the number of latice points within the quadrant (from origin to rational plane) where each point is where a new parallel plane would originate. but then you would have to remove all the degenerate planes

here is another place that states the same and does a rough sketch: https://books.google.com/books?id=SHzeQ49ZlH4C&pg=PA12&lpg=PA12&dq=ABC+planes+miller&source=bl&ots=QLFbVXZoyf&sig=RJbIjR6Nd4Gm_AJx0QClHio1GFU&hl=en&sa=X&ei=pDFmVcKoCsXUsAWiuID4DA&ved=0CDkQ6AEwAw#v=onepage&q=ABC planes miller&f=false

 

[antonni]

Thanks for the reply...yes, just one of those things everyone takes for granted and think its basic knowledge, but not straight forward at all

ill try it again