In geometry
A pentagram, or five-pointed polygram, is the first proper star polygon constructed from the diagonals of a regular pentagon as self-intersecting edges that are proportioned in golden ratio, 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
  . Its internal geometry appears prominently in Penrose tilings, and is a facet inside Kepler-Poinsot star polyhedra and Schläfli–Hess star polychora, represented by its Schläfli symbol {5/2}. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges. Generally, star polytopes that are regular only exist in dimensions 2 ⩽  
  
    
      
        n
      
    
    {\displaystyle n}
   < 5. In graph theory, all graphs with 4 or fewer vertices are planar, however, there is a graph with 5 vertices that is not: K5, the complete graph with 5 vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram. By Kuratowskis theorem, a finite graph is planar iff it does not contain a subgraph that is a subdivision of K5, or the complete bipartite utility graph K3,3. A similar graph is the Petersen graph, which is strongly connected and also nonplanar.
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.