There are also two infinite families of duoprisms and antiprismatic prisms in the fourth dimension. Four-dimensional differential manifolds have some unique properties. There is only one differential structure on 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
   except when 
  
    
      
        n
      
    
    {\displaystyle n}
   = 
  
    
      
        4
      
    
    {\displaystyle 4}
  , in which case there are uncountably many. The smallest non-cyclic group has four elements; it is the Klein four-group.
Five-dimensional and every higher dimension: three regular convex 
  
    
      
        n
      
    
    {\displaystyle n}
  -polytopes, all within the infinite family of regular 
  
    
      
        n
      
    
    {\displaystyle n}
  -simplexes, 
  
    
      
        n
      
    
    {\displaystyle n}
  -hypercubes, and 
  
    
      
        n
      
    
    {\displaystyle n}
  -orthoplexes.The fourth dimension is also the highest dimension where regular self-intersecting figures exist:

Two-dimensional: infinitaly many regular star polygons. Three-dimensional: four regular star polyhedra, the regular Kepler-Poinsot star polyhedra. Four-dimensional: ten regular star polychora, the Schläfli–Hess star polychora. They contain cells of Kepler-Poinsot polyhedra alongside regular tetrahedra, icosahedra and dodecahedra.