Whereas the hexagonal Golomb graph and the regular hexagonal tiling generate chromatic numbers of 4 and 7, respectively, a chromatic coloring of 5 can be attained under a more complicated graph where multiple four-coloring Moser spindles are linked so that no monochromatic triples exist in any coloring of the overall graph, as that would generate an equilateral arrangement that tends toward a purely hexagonal structure. The plane contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.Five points are needed to determine a conic section, in the same way that two points are needed to determine a line.
All Mathieu groups are subgroups of 
  
    
      
        
          
            M
          
          
            24
          
        
      
    
    {\displaystyle \mathrm {M} _{24}}
  , which under the Witt design 
  
    
      
        
          
            W
          
          
            24
          
        
      
    
    {\displaystyle \mathrm {W} _{24}}
   of Steiner system S(5, 8, 24) emerges a construction of the extended binary Golay code 
  
    
      
        
          
            B
          
          
            24
          
        
      
    
    {\displaystyle \mathrm {B} _{24}}
   that has 
  
    
      
        
          
            M
          
          
            24
          
        
      
    
    {\displaystyle \mathrm {M} _{24}}
   as its automorphism group.