The Area of the Medial Parallelogram of a Tetrahedron by David N. Yetter


This  is a nice paper which  finds a simple formula for the area of the medial parallelogram of a tetrahedron in terms of the lengths of the six edges. This is interesting to me because in  simplicial models for quantum gravity, the formula is needed to deal the problem of length operators.

The paper finds that given a pair of non-incident edges in a tetrahedron, the medial parallelogram determined by the pair is the parallelogram whose vertices are the mid-points of the remaining four edges.

The area of the medial parallelogram determined by the edges of lengths d and e in the tetrahedron is



Numerical work with sage math 1

What is going on here? Well what I want to try and do initially is generate random discrete spaces, triangulate them and calculate their properties using simplex complexes based on tetrahedra.

As starting point I am using  Sage  to generate a pointset of 20 random points which is then triangulated and  plotted in 3d.

triangulate random points 1

There are several research routes I can consider  following on from this:

Classical 6j–symbols and the tetrahedron by Justin Roberts

I have been reading this paper over the weekend together with some other papers on spin networks (which I’ll post about later). The goal of this paper is to prove and explain the classical 6J symbol by using geometric quantization. A classical 6j –symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). It has a deep geometric significance -Ponzano and Regge, expanding on work of Wigner, gave an asymptotic formula relating the value of the 6j – symbol, when the dimensions of the representations are large, to the volume of an Euclidean tetrahedron whose edge lengths are these dimensions.


Quantum tetrahedron and its classical limit by Daniel R. Terno

A great paper which clearly explains the geometry of both the classical and quantum tetrahedron. Its aim was to show that classical information that is retrieved from a quantum tetrahedron is intrinsically fuzzy – in fact for a single tetrahedron it was shown that  the  uncertainty in the  dihedral angles is inversely proportional the surface area. The paper states that a  quantum tetrahedron is a useful toy model to investigate the classical limit of LQG and  in provides boundary states in studies of the graviton propagator. 

Review of  classical and quantum tetrahedra

Six numbers that determine the shape of a tetrahedron, for example,  the six edges of a tetrahedron, or four facial areas and two independent dihedral angles between them. There are number of useful relations between areas, angles and volume in this paper.

If the outward normals to the faces are  labelled as Ji , then lengths are twice the triangular areas,

Ji = 2Ai

Because it is  a closed surface a tetrahedron satisfies the closure condition

Ji = 0

Angles between the triangular faces – the  inner dihedral angles  are related to the outer dihedral angles θij by ,

Jij Ji · Jj = JiJj cos θij

The volume squared of the tetrahedron can be expressed in terms of the area vectors,

V2 = -(1/36 ) J1 · J2 × J3


In the quantized problem the four normals J1, J2, J3, J4 are identified with the generators of SU(2). So that,

J^2 |ji,mi>   = j(j+1) |ji,mi>  and Jz |ji,mi> = mi|ji,mi>

Asymptotically the Casimir Operator

J= sqrt[ J (J+1)] ~ J + 1/2

The Quantum Tetrahedron in 3 and 4 Dimensions by John C. Baez

John Baez is one of my favourite physicists. He has done a lot of work on quantum gravity especially with regard to loops and knot theory. He used to maintain the ‘ This weeks finds in Mathematical Physics’ webpage.  In this paper Baez states that recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the ‘quantum tetrahedron’.  By starting with a classical phase space whose points correspond to geometries of the tetrahedron in R3, he uses geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labelled by the areas of the faces of the tetrahedron together with another quantum number, such as  the area of one of the parallelograms formed by midpoints of the tetrahedron’s edges.


Parallelogram formed by midpoints of the tetrahedron’s edges

By repeating the procedure for the tetrahedron in R4, he also  obtains a Hilbert space with a basis labelled solely by the areas of the tetrahedron’s faces.