Dixon elliptic functions

<p>In mathematics, the Dixon elliptic functions sm and cm are two <a href="/facts/Elliptic_function/mF03oI04">elliptic functions</a> (<a href="/facts/Doubly_periodic_function/r8R5RRAc">doubly periodic</a> <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic functions</a> on the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>) that map from each <a href="/facts/Regular_hexagon/DPzuCzWE">regular hexagon</a> in a <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a> to the whole complex plane. Because these functions satisfy the identity 
  
    
      
        
          cm
          
            3
          
        
        ⁡
        z
        +
        
          sm
          
            3
          
        
        ⁡
        z
        =
        1
      
    
    {\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1}
  
, as <a href="/facts/Function_of_a_real_variable/bbNG3DnN">real functions</a> they parametrize the cubic <a href="/facts/Fermat_curve/31LFyJ5p">Fermat curve</a> 
  
    
      
        
          x
          
            3
          
        
        +
        
          y
          
            3
          
        
        =
        1
      
    
    {\displaystyle x^{3}+y^{3}=1}
  
, just as the <a href="/facts/Trigonometric_function/czej7ur0">trigonometric functions</a> sine and cosine parametrize the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> 
  
    
      
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        =
        1
      
    
    {\displaystyle x^{2}+y^{2}=1}
  
.
</p><p>They were named sm and cm by <a href="/facts/Alfred_Cardew_Dixon/vkhTloGh">Alfred Dixon</a> in 1890, by analogy to the trigonometric functions sine and cosine and the <a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic functions</a> sn and cn; <a href="/facts/G%25C3%25B6ran_Dillner/VF6XTPoN">Göran Dillner</a> described them earlier in 1873.
</p>

Dixon elliptic functions open-in-new

Dixon elliptic functions