The tetrahedron and its Regge conjugate

This week I have been reading the PhD thesis ‘Single and collective dynamics of discretized geometries’ by Dimitri Marinelli. In this post I’ll look at a small portion about Regge calculus, the  tetrahedron and its Regge conjugate.

Regge Calculus is a dynamical theory of space-time introduced in 1961 by Regge as a discrete approximation for the Einstein theory of gravity. The basic idea is to replace a smooth space-time with a collection of simplices. The collective dynamics of these geometric objects is driven by the Regge action and the dynamical variables are their edge lengths – which play the role of the metric tensor of General Relativity. Simplices are the n-dimensional generalization of triangles and tetrahedra. Regge Calculus inspired and is at the base of almost all the present discretized models for a quantum theory of gravity for at least two reasons:

  • It is a discretized model, so it represents a possible atomistic system typical of quantum systems
  • There is a deep connection between the Regge action, the asymptotic of the 6j symbol and a path integral formulation of gravity.

Let’s see  how the Regge transformation acts on a tetrahedral shape. The formulas


and the association between 6j symbol and an Euclidean tetrahedron tell us that any Regge transformation acts on four edges of a tetrahedron keeping a pair of opposite edges unchanged. The Regge-transformed tetrahedra is called `conjugate’.

Using the Ponzano-Regge formula for the 6j,

reggeequ2.18we can immediately say that the volume of a tetrahedron and that of a Regge transformed one must coincide.



The volume of a tetrahedron is also invariant under the Regge transformation of four consecutive edges.

The volume of a tetrahedron, being a function of six parameters, can be expressed in several ways. For the tetrahedron below:

tetrahedron with dihedral angleThe ‘orientated’ volume reads, 


where AABC and AACD are respectively the areas of the triangles ABC and ACD, lAC is the length of the common edge and β is the dihedral angle between these two faces.

The importance of the Regge symmetry is that it constrains the shape dynamics of a single tetrahedron,  it relates different tetrahedra equating their quantum representations and it is the key tool to understand the classical motion of a four-bar linkage mechanical systems and its link to the the quantum dynamics of tetrahedra.

This thesis also contains a section on the Askey scheme which I’ll be following up in future posts:

askey scheme






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