## Geometry & Topology

- Geom. Topol.
- Volume 3, Number 1 (1999), 21-66.

### Classical 6j-symbols and the tetrahedron

#### Abstract

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 $SU\left(2\right)$. This abstract association is traditionally used simply to express the symmetry of the $6j$–symbol, which is a purely algebraic object; however, it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the $6j$–symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spin-off is that a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra.

#### Article information

**Source**

Geom. Topol., Volume 3, Number 1 (1999), 21-66.

**Dates**

Received: 9 January 1999

Accepted: 9 March 1999

First available in Project Euclid: 21 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.gt/1513883138

**Digital Object Identifier**

doi:10.2140/gt.1999.3.21

**Mathematical Reviews number (MathSciNet)**

MR1673272

**Zentralblatt MATH identifier**

0918.22014

**Subjects**

Primary: 22E99: None of the above, but in this section

Secondary: 81R05: Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70] 51M20: Polyhedra and polytopes; regular figures, division of spaces [See also 51F15]

**Keywords**

$6j$–symbol asymptotics tetrahedron Ponzano–Regge formula geometric quantization scissors congruence

#### Citation

Roberts, Justin. Classical 6j-symbols and the tetrahedron. Geom. Topol. 3 (1999), no. 1, 21--66. doi:10.2140/gt.1999.3.21. https://projecteuclid.org/euclid.gt/1513883138