page_title: Bhargava cube
section_title: The three forms
hierarchical_section_title: Bhargava cube / Quadratic forms associated with the Bhargava cube / The three forms
caption_attribution_description: English: Bhargava cube showing the pair of opposite faces corresponding to M3 and N3.
context_page_description: In mathematics, in number theory, a Bhargava cube is a configuration consisting of eight integers placed at the eight corners of a cube. This configuration was extensively used by Manjul Bhargava, a Canadian-American Fields Medal winning mathematician, to study the composition laws of binary quadratic forms and other such forms. To each pair of opposite faces of a Bhargava cube one can associate an integer binary quadratic form thus getting three binary quadratic forms corresponding to the three pairs of opposite faces of the Bhargava cube. These three quadratic forms all have the same discriminant and Manjul Bhargava proved that their composition in the sense of Gauss is the identity element in the associated group of equivalence classes of primitive binary quadratic forms. Using this property as the starting point for a theory of composition of binary quadratic forms Manjul Bhargava went on to define fourteen different composition laws using a cube.
context_section_description: Let the cube be formed by the integers a, b, c, d, e, f, g, h. The pairs of matrices associated with opposite edges are denoted by (M₁, N₁), (M₂, N₂), and (M₃, N₃). The first rows of M₁, M₂ and M₃ are respectively [a b], [a c] and [a e]. The opposite edges in the same face are the second rows. The corresponding edges in the opposite faces form the rows of the matrices N₁, N₂, N₃ (see figure).
  The quadratic form associated with the faces defined by the matrices 
   (see figure) is
  The discriminant of a quadratic form Q₁ is
  The quadratic form associated with the faces defined by the matrices 
   (see figure) is
  The discriminant of a quadratic form  Q₂ is
  The quadratic form associated with the faces defined by the matrices 
   (see figure) is
  The discriminant of a quadratic form  Q₃ is
  Manjul Bhargava's surprising discovery can be summarised thus:
  If a cube A gives rise to three primitive binary quadratic forms Q₁, Q₂, Q₃,  then Q₁, Q₂, Q₃ have the same discriminant, and the product of these three forms is the identity in the group defined by Gauss composition. Conversely, if Q₁, Q₂, Q₃ are any three primitive binary quadratic forms of the same discriminant whose product is the identity under Gauss composition, then there exists a cube A yielding Q₁, Q₂, Q₃.
caption: Bhargava cube showing the pair of opposite faces  M3 and N3.
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