The 5-simplex is the five-dimensional analogue of the 5-cell, or 4-simplex; the fifth iteration of 
  
    
      
        n
      
    
    {\displaystyle n}
  -simplexes in any 
  
    
      
        n
      
    
    {\displaystyle n}
   dimensions.
A Veronese surface in the projective plane 
  
    
      
        
          
            P
          
          
            5
          
        
      
    
    {\displaystyle \mathbb {P} ^{5}}
   of a conic generalizes a linear condition for a point to be contained inside a conic. There are 
  
    
      
        5
      
    
    {\displaystyle 5}
   Platonic solids in three-dimensional space: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The dodecahedron in particular contains pentagonal faces, while the icosahedron, its dual polyhedron, has a vertex figure that is a regular pentagon.