From Cubes to Hex: Deriving SC/BCC/FCC/HCP as AOL Projections

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Projected Coordinates (u, v)


        
Basis and Mapping

        
    
          
  • (111) plane basis:
            e₁ = (1, −1, 0)/√2,
            e₂ = (1, 1, −2)/√6
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  • Projection: [u,v] = [r·e₁, r·e₂]
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  • Hex axial (q,r) from (u,v); s = −(q + r)
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  • HCP supercell uses (a, c/a); projections land on the same AOL hex plane.
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How to Cite This Article


    

      

        

          
APA

          
        

        
Allen, J. J. S. (2025). From Cubes to Hex: Deriving SC/BCC/FCC/HCP as AOL Projections. Pattern Field Theory. https://www.patternfieldtheory.com/articles/crystal-systems-radii-packing/cubes-to-hex.php

      


      

        

          
MLA

          
        

        
Allen, James Johan Sebastian. "From Cubes to Hex: Deriving SC/BCC/FCC/HCP as AOL Projections." Pattern Field Theory, 2025, https://www.patternfieldtheory.com/articles/crystal-systems-radii-packing/cubes-to-hex.php.

      


      

        

          
Chicago

          
        

        
Allen, James Johan Sebastian. "From Cubes to Hex: Deriving SC/BCC/FCC/HCP as AOL Projections." Pattern Field Theory. October 9, 2025. https://www.patternfieldtheory.com/articles/crystal-systems-radii-packing/cubes-to-hex.php.

      


      

        

          
BibTeX

          
        

        
@article{allen2025pft,
  author  = {James Johan Sebastian Allen},
  title   = {From Cubes to Hex: Deriving SC/BCC/FCC/HCP as AOL Projections},
  journal = {Pattern Field Theory},
  year    = {2025},
  url     = {https://www.patternfieldtheory.com/articles/crystal-systems-radii-packing/cubes-to-hex.php}
}