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.
There are also a total of five Coxeter groups that generate non-prismatic Euclidian honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets, as with the order-5 5-cell honeycomb and the order-5 120-cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10.