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.
On the other hand, the square of four 42, equivalently the fourth power of two 24, is sixteen; the only number that has 
  
    
      
        
          a
          
            b
          
        
      
    
    {\displaystyle a^{b}}
   = 
  
    
      
        
          b
          
            a
          
        
      
    
    {\displaystyle b^{a}}
   as a form of factorization.