Tag Archives: Spin network

Ising Spin Network States for Loop Quantum Gravity by Feller and Livine

This week I have been studying a great paper by Feller and Livine on Ising Spin Network States for Loop Quantum Gravity. In the context of loop quantum gravity, quantum states of geometry are defined as
spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. In this paper the authors develop the condensed matter point of view on extracting the physical and geometrical information out of spin network states: they
introduce  Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. They construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints which entirely characterize the states. By studying their phase diagram distance can be reconstructed from the correlations in the various phases.

The line of research pursued in this paper is at the interface between condensed matter and quantum information and quantum gravity on the other: the aim is to understand how the distance can be recovered from correlation and entanglement between sub-systems of the quantum gravity state. The core of the investigation is the correlations and entanglement entropy on spin network states. Correlations and especially entropy are of special importance
for the understanding of black holes dynamics. Understanding the microscopic origin of black holes entropy is one the major test of any attempt to quantify gravity and entanglement between the horizon and its environment degrees of freedom appears crucial

A spin network state is defined on a graph, dressed
with spins on the edge and intertwiners at the vertices.
A spin on an edge e is a half-integer je 2 N=2 giving an
irreducible representation of SU(2) while an intertwiner
at a vertex v is an invariant tensor, or singlet state, between
the representations living on the edges attached
to that vertex. Spins and intertwiners respectively carry
the basic quanta of area and volume. The authors build the spin
network states based on three clear simplifications:

  1.  Use a fixed graph, discarding graph superposition and graph changing dynamics and  work with a fixed regular
  2. Freeze all the spins on all the graph edges. Fix to smallest possible value, ½, which correspond to the most basic excitation of geometry in loop quantum gravity, thus representing a quantum geometry directly at the Planck scale.
  3. Restricted  to 4-valent vertices, which represent the basic quanta of volume in loop quantum gravity, dual to quantum tetrahedra.

These simplifications provide us with the perfect setting to map spin network states, describing the Planck scale quantum geometry, to qubit-based condensed matter models. Such models have been extensively studied in statistical physics and much is known on their phase diagrams and correlation functions, and we hope to be able to import these results to the context of loop quantum gravity. One of the most useful model is the Ising model whose relevance goes from modeling binary mixture to the magnetism of matter. We thus naturally propose to construct and investigate Ising spin
network states.

The paper reviews the definition of spin network and analyzes the structure of 4-valent intertwiners between spins ½ leading to the
effective two-state systems used to define the Ising spin network states. Different equivalent definitions are given in terms of the high and low temperature expansions of the Ising model. The loop representation of the spin network is then obtained and studied as well as the associated density which gives information about
parallel transport in the classical limit. Section III introduces

It then  introduces a set of local Hamiltonian constraints for which
the Ising state is a unique solution and elaborates on their usefulness for understanding the coarse-graining of
spin network  and the dynamic of loop quantum gravity.

After this, it discusses the phase diagram of the Ising
states and their continuum limit as well as the distance
from correlation point of view.


Ising Spin Network State

Spin network basis states  define the basic excitations of the quantum geometry and they are provided with a natural interpretation in terms of discrete geometry with the spins giving the quanta of area and the intertwiners giving the quanta of volume.


These 4-valent vertices will be organized along a regular lattice. The 3d diamond lattice and the 2d square lattice are considered . Looking initially at the 2d square the square lattice: in this setting, the space of 4-valent interwiners between four spins ½ is two dimensional – it can be decomposed into spin 0 and spin 1 states by combining the spins by pairs, as


Different such decompositions exist and are shown as  a graphical representation below. There are three such decompositions, depending on which spins are paired together, the  s, t and u channels.


The spin 0 and 1 states in the s channel  can be explicitly written in terms of the up and down states of the four spins:


Those two states form a basis of the intertwiner space.  There transformation matrices between this basis and the two other channels:


Let’s look at the  intertwiner basis  defined in terms of the square volume operator U of loop quantum gravity. Since the spins, and the area quanta, are fixed, the only freedom left in the spin network states are the volume quanta defined by the intertwiners. This will provide the geometrical interpretation of our spin network states as
excitations of volumes located at each lattice node. For a 4-valent vertex, this operator is defined as:



where J are the spin operators acting on the i link.

The volume itself can then obtained by taking the square root of the absolute value of U . Geometrically, 4-valent intertwiners are interpreted as representing quantum tetrahedron, which becomes the building block of the quantum geometry in loop quantum gravity  and spinfoam models. U takes the following form in the s channel basis:


The smallest  possible value of a chunk of space is the square volume ±√3/4 in Plank units.

The two oriented volume states of û,  | u ↑,↓〉 , can be considered as
the two levels of an effective qubit. Let’s now define a pure spin network state which maps its quantum fluctuations on the thermal fluctuations of a given classical statistical model such as the Ising model by


This state represents a particular configuration of the spin network and the full state is a quantum superposition of them all. Defined as such, the state is unnormalized but its norm is easily computed using the Ising partition function ZIsing:


The intertwiner states living at each vertex are now entangled and carry non-trivial correlations. More precisely this state exhibits Ising correlations between two vertex i, j:


Those correlations are between two volume operators at different vertex which are in fact components of the 2- point function of the gravitational field. So understanding how those correlations can behave in a non-trivial way is a first step toward understanding the behavior of the full 2-point gravity correlations and for instance recover the inverse square law of the propagator.

The generalization to 3d is straightforward. Keeping the requirement that the lattice be 4-valent the natural
regular lattice is the diamond lattice



Using the usual geometrical interpretation of loop quantum gravity, this lattice can be seen as dual to a triangulation of the 3d space in terms of tetrahedra dual to each vertex. This can be seen as an extension of the more used cubic lattice better suited to loop quantum gravity. The Ising spin network state and the whole
set of results which followed are then identical :

  • The wave function


  • The Hamiltonian constraints and their algebra  are the same.


In 3d, the Ising model also exhibits a phase transition

Information about the 2-points correlation functionssuch as long distance behavior at the phase transition or near it can be obtained using methods of quantum field theory. In d dimensions we have


where K(r) are modified Bessel functions and ξ is the
correlation length. For the three-dimensional case, we
have the simple and exact expression



In this paper, the authors have introduced a class of spin network
states for loop quantum gravity on 4-valent graph. Such 4-valent graph allows for a natural geometrical interpretation in terms of quantum tetrahedra glued together into a 3d triangulation of space, but it also allows them to be map the degrees of freedom of those states to effective qubits. Then we can define spin network states corresponding to known statistical spin models, such as the Ising model, so that the correlations living on the spin network are exactly the same as those models.
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Deformations of Polyhedra and Polygons by the Unitary Group by Livine

This week whilst preparing some calculations I have been looking at the paper ‘Deformations of Polyhedra and Polygons by the Unitary Group’. In the paper the author inspired by loop quantum gravity, the spinorial formalism and the structures of twisted geometry discusses the phase space of polyhedra in three dimensions and its quantization, which serve as basic building of the kinematical states of discrete geometry.

They show that the Grassmannian space U(N)/(U(N − 2) × SU(2)) is the space of framed convex polyhedra with N faces up to 3d rotations. The framing consists in the additional information of a U(1) phase per face. This provides an extension of the Kapovich-Milson phase space  for polyhedra with fixed number of faces and fixed areas for each face – see the post:

Polyhedra in loop quantum gravity

They describe the Grassmannian as the symplectic quotient C2N//SU(2), which provides canonical complex variables for the Poisson bracket. This construction allows a natural U(N) action on the space of polyhedra, which has two main features. First, U(N) transformations act non-trivially on polyhedra and change the area and shape of each individual face. Second, this action is cyclic: it allows us to go between any two polyhedra with fixed total area  – sum of the areas of the faces.

On quantization, the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners, which is interpreted as the space of quantum polyhedra. By performing a canonical quantization from the complex variables of C2N//SU(2) all the classical features are automatically exported to the quantum level. Each face carries now a irreducible representation of SU(2) – a half-integer spin j, which defines the area of the face. Intertwiners are then SU(2)-invariant states in the tensor product of these irreducible representations. These intertwiners are the basic
building block of the spin network states of quantum geometry in loop quantum gravity.

The U(N) action on the space of intertwiners changes the spins of the faces and each Hilbert space for fixed total area defines an irreducible representation of the unitary group U(N). The U(N) action is cyclic and allows us to generate the whole Hilbert space from the action of U(N) transformation on the highest weight
vector. This construction provides coherent intertwiner states peaked on classical polyhedra.

At the classical level, we can use the U(N) structure of the space of polyhedra to compute the averages of polynomial observables over the ensemble of polyhedra and  to use the Itzykson-Zuber formula from matrix models  as a generating functional for these averages. It computes the integral over U(N) of the exponential of the matrix elements of a unitary matrix tensor its complex conjugate.

At the quantum level the character formula, giving the trace of unitary transformations either over the standard basis or the coherent intertwiner basis, provides an extension of the Itzykson-Zuber formula. It allows us in principle to generate the expectation values of all polynomial observables and so their spectrum.

This paper defines and describe the phase space of framed polyhedra, its parameterization in terms of spinor variables and the action of U(N) transformations. Then it shows  how to compute the averages and correlations of polynomial observables using group integrals over U(N) and the Itzykson-Zuber integral as a generating function. It discusses the quantum case, with the Hilbert space
of SU(2) intertwiners, coherent states and the character formula.

The paper also investigates polygons in two dimensions and shows that the unitary group is replaced by the orthogonal group and that
the Grassmannian Ø(N)/(Ø(N −2)×SO(2)) defines the phase space for framed polygons. It then discusses the issue of gluing such polygons together into a consistent 2d cellular decomposition, as a toy model for the gluing of framed polyhedra into 3d discrete manifolds. These constructions are relevant to quantum gravity in 2+1 and 3+1 dimensions, especially to discrete approaches based on a description of the geometry using glued polygons and polyhedra such as loop quantum gravity  and dynamical

The paper’s goal is to clarify how to parametrize the set of polygons or polyhedra and their deformations, and to introduce mathematical tools to compute the average and correlations of observables over the ensemble of polygons or polyhedra at the classical level and then the spectrum and expectation values of geometrical operators on
the space of quantum polygons or polyhedra at the quantum level.

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Numerical work using mathematica 26: The Gauge invariant area operator in the flux formulation of LQG

This week I been following up the posts:

The starting point is to take the action of the area operator in the spin representation, and to consider the following normalized trace of the area operator:

gauge invariant

If this give a well-defined operator on either the kinematical Hilbert space or on the Hilbert space of fully gauge-invariant wave functions, and also if we took the limit Λfix →∞ it is possible to  read off the spectrum from the representation. There, the oscillatory behaviour of the sine function is suppressed by a factor of 1/dwhich leads to a discrete spectrum for sufficiently small spins j.



Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.1 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.




Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.3 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.




Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.05, 0.1 and 0.3 and j=1…100

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Polyhedra in spacetime from null vectors by Neiman

This week I have been studying a nice paper about Polyhedra in spacetime.

The paper considers convex spacelike polyhedra oriented in Minkowski space. These are classical analogues of spinfoam intertwiners. There is  a parametrization of these shapes using null face normals. This construction is dimension-independent and in 3+1d, it provides the spacetime picture behind the property of the loop quantum gravity intertwiner space in spinor form that the closure constraint is always satisfied after some  SL(2,C) rotation.These  variables can be incorporated in a 4-simplex action that reproduces the large-spin behaviour of the Barrett–Crane vertex amplitude.

In loop quantum gravity and in spinfoam models, convex polyhedra are fundamental objects. Specifically, the intertwiners between rotation-group representations that feature in these theories can be viewed as the quantum versions of convex polyhedra. This makes the parametrization of such shapes a subject of interest for  LQG.
In kinematical LQG, one deals with the SU(2) intertwiners, which correspond to 3d polyhedra in a local 3d Euclidean frame. These polyhedra are naturally parametrized in terms of area-normal vectors: each face i is associated with a vector xi, such that its norm
equals the face area Ai, and its direction is orthogonal to the face. The area normals must satisfy a ‘closure constraint’:


Minkowski’s reconstruction theorem guarantees a one-to-one correspondence between space-spanning sets of vectors xi that satisfy (1) and convex polyhedra with a spatial orientation. In
LQG, the vectors xi correspond to the SU(2) fluxes. The closure condition  then encodes the Gauss constraint, which also generates spatial rotations of the polyhedron.

In the EPRL/FK spinfoam, the SU(2) intertwiners get lifted into SL(2,C) and are acted on by SL(2,C) ,Lorentz, rotations. Geometrically, this endows the polyhedra with an orientation in the local 3+1d Minkowski frame of a spinfoam vertex. The polyhedron’s
orientation is now correlated with those of the other polyhedra surrounding the vertex, so that together they define a generalized 4-polytope. In analogy with the spatial case, a polyhedron with spacetime orientation can be parametrized by a set of area-normal
simple bivectors Bi. In addition to closure, these bivectors must also satisfy a cross-simplicity


In this paper, the author presents a different parametrization of convex spacelike polyhedra with spacetime orientation. Instead of bivectors Bi, they associate null vectors i to the polyhedron’s
faces. This parametrization does not require any constraints between the variables on different faces. It is unusual in that both the area and the full orientation of each face are functions of the data on all the faces. This construction, like the area-vector and
area-bivector constructions above, is dimension-independent. So we can parametrize d-dimensional convex spacelike polytopes with (d − 1)-dimensional faces, oriented in a (d + 1)-dimensional Minkowski spacetime.  These variables can be to construct an action principle for a Lorentzian 4-simplex. The action principle reproduces the large spin behaviour of the Barrett–Crane spinfoam vertex. In particular, it recovers the Regge action for the classical simplicial gravity, up to a possible sign and the existence of additional,degenerate solutions.

In d = 2, 3 spatial dimensions, the parametrization is  contained in the spinor-based description of the LQG intertwiners. There, the face normals are constructed as squares of spinors. It was observed that the closure constraint in these variables can always be satisfied by acting on the spinors with an SL(2,C) boost.  The simple spacetime picture presented in this paper is new. Hopefully, it will contribute to the geometric interpretation of the modern spinor and twistor variables in LQG.

The parametrization
Consider a set of N null vectors liμ in the (d + 1)-dimensional Minkowski space Rd,1, where i = 1, 2, . . . ,N and d ≥2.  Assume the following conditions on the null vectors liμ.

  • The liμ span the Minkowski space and  N ≥  d + 1.
  • The  liμ  are either all future-pointing or all past-pointing.

The central observation in this paper is that such sets of null vectors are in one-to-one correspondence with convex d-dimensional spacelike polytopes oriented in Rd,1.

Constructing the polytrope

Consider a set {liμ} ,take the sum of the liμ normalized
to unit length:


The unit vector nμ is timelike, with the same time orientation as the liμ. Now take nμ to be the unit normal to the spacelike polytope. To construct the polytope in the spacelike hyperplane ∑ orthogonal to nμ define the projections of the null vectors liμinto this hyperplane:


The spacelike vectors siμ  automatically sum up to zero. Also, since the liμ span the spacetime, the siμ must span the hyperplane ∑ . By the Minkowski reconstruction theorem, it follows that the siμ are the (d − 1)-area normals of a unique convex d-dimensional polytope in . In this way, the null vectors li define a d-polytope oriented in spacetime.

Basic features of the parametrization.

The vectors  are liμ  associated to the polytope’s (d −1)-dimensional faces and are null normals to these faces. The orientation of a spacelike (d − 1)-plane in Rd,1 is in one-to-one correspondence with the directions of its two null normals. So each liμ carries partial information about the orientation of the ith face. The second null normal to the face is a function of all the liμ. It can be expressed as:


where  nμ is given by


Similarly, the area Ai of each face is a function of the
null normals liμ to all the faces:


The total area of the faces has the simple expression:


A (d+1)-simplex action

To construct a (d + 1)-simplex action that reproduces in the d = 3 case the large-spin behaviour of the Barrett–Crane spinfoam vertex.

At the level of degree-of-freedom counting, the shape of a (d +1)-simplex is determined by the (d + 1)(d + 2)/2 areas Aab of its (d − 1)-faces. These areas are directly analogous to the spins that appear in the Barrett–Crane spinfoam. Let us fix a set of values for Aab and consider the action:


Then restrict to the variations where:


The stationary points of the action  have the following properties. For each a, the vectors  labμ define a d-simplex with unit normal naμ
and (d − 1)-face areas Aab.


A (d − 1)-face in a (d + 1)-simplex, shared by two d-simplices a and b. The diagram depicts the 1+1d plane orthogonal to the face. The dashed lines are the two null rays in this normal plane.


The d-simplices automatically agree on the areas of their shared (d −1)-faces. The two d-simplices agree not only on the area of their shared (d − 1)-face, but also on the orientation of its (d − 1)-plane in spacetime. In other words, they agree on the face’s area-normal bivector:


The area bivectors defined  automatically satisfy closure and cross-simplicity:


We conclude that the stationary points are in one-to-one correspondence with the bivector geometries of the Barrett-Crane model with an action of the form:



Numerical work with sagemath 25: The Wigner D matrix

This week I have been studying the paper ‘A New Realisation of Quantum Geometry’. I’ll review the paper in the next post and follow up that with an analysis of the area operator in  the SU(2) case.

What I’m posting at the moment is some exploratory work looking at the behaviour of the Wigner D matrix elements and U(1)area operator using python and sagemath.

Below I look at the general behaviour of the Wigner D matrix elements:



Below  Here I  look at the behaviour of the spectrum of the area operator with U(1).



Below I look at the how the area varies with μ for the values 0.1, 0.3, 0.5:



Group field theories generating polyhedral complexes by Thürigen

This week I have been studying recent developments in  Group Field Theories. Group field theories are a generalization of matrix models which provide both a second quantized reformulation of loop quantum gravity as well as generating functions for spin foam models. Other posts looking at this include:

While states in canonical loop quantum gravity are based on graphs with vertices of arbitrary valence, group field theories have been defined so far in a simplicial setting such that states have support only on graphs of fixed valency. This has led to the question whether group field theory can indeed cover the whole state space of loop quantum gravity.

The paper discusses  the combinatorial structure of the complexes generated by the group field theory partition function. These new group field theories strengthen the links between the various quantum gravity approaches and  might also prove useful in the investigation of renormalizability.

The combinatorial structure of group field theory
The common notion of GFT is that of a quantum field theory on group manifolds with a particular kind of non-local interaction vertices. A group field is a function of a Lie group G and the GFT is defined by a partition function


the action is of the form:


The evaluation of expectation values of quantum observables O[φ], leads to a series of Gaussian integrals evaluated
through Wick contraction which are catalogued by Feynman diagrams Γ,


where sym(Γ) are the combinatorial factors related to the automorphism group of the Feynman diagram Γ:

The specific non-locality of each vertex is captured by a boundary graph. In the interaction term in each group field term  can be represented by a graph consisting of a k-valent vertex connected to k univalent vertices. One may further understand the graph as
the boundary  of a two-dimensional complex a with a single internal vertex v. Such a one-vertex two-complex a is called a spin foam atom.


The GFT Feynman diagrams in the perturbative sum have the structure of two complexes because Wick contractions effect bondings of such atoms along patches. The combinatorial
structure of a term in the perturbative sum is then a collection of spin foam atoms, one for each vertex kernel, quotiented by a set of bonding maps, one for each Wick contraction. Because of this construction such a two-complex will be called a spin foam molecule.


The crucial idea to create arbitrary boundary graphs in a more efficient way is to distinguish between virtual and real edges and obtain arbitrary graphs from regular ones by contraction of the
virtual edges.


In terms of these contractions, any spin foam molecule can be obtained from a molecule constructed from labelled regular graphs.


This paper has aimed to  generalization of GFT to be compatible with LQG. It has clarified the combinatorial structure underlying the amplitudes of perturbative GFT using the notion of spin foam atoms and molecules and discussed their possible spacetime interpretation.

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The geometry of the tetrahedron and asymptotics of the 6j-symbol

This week I have been studying a useful PhD thesis ‘Asymptotics of quantum spin networks‘ by van der Veen. In this post I’ll look a the section dealing with the geometry of the tetrahedron.

Spin networks and their evaluations

In the special case of the tetrahedron graph the author presents a complete solution to the rationality property of the generating series of all evaluations with a fixed underlying graph using the combinatorics of the chromatic evaluation of a spin network, and a complete study of the asymptotics of the  6j-symbols in all cases: Euclidean, Plane or Minkowskian ,using the theory of Borel transform.


 The three realizations of the dual metric tetrahedron depending on the sign of the Cayley-Menger determinant det(C). Also shown are the two exponential growth factors Λ ± of the 6j-symbol.

The author also computes the asymptotic expansions for all possible colorings,  including the degenerate and non-physical cases. They find that  quantities in the asymptotic expansion can be expressed as geometric properties of the dual tetrahedron graph interpreted as a metric polyhedron.

letting ( Γ,γ ) denote the tetrahedral spin network colored by an admissible coloring . Consider the planar dual graph which is also a tetrahedron with the same labelling on the edges. Regarding the labels of as edge lengths one may ask whether ( Γ,γ ) can be realized as a metric Euclidean tetrahedron with these edge lengths. The admissibility of ( Γ,γ ) implies that all faces of  the dual tetrahedron satisfy the triangle inequality. A well-known theorem of metric geometry implies that ( Γ,γ ) can be realized in exactly one of three flat geometries

(a) Euclidean 3-dimensional space R³
(b) Minkowskian space R²’¹
(c) Plane Euclidean R².

Which of the above applies  is decided by the sign of the Cayley-Menger determinant det(C) which is a degree six polynomial in the six edge labels.

Its assumed that ( Γ,γ ) is nondegenerate in the sense that all faces of are two dimensional. In the degenerate case, the evaluation of 6-symbol is a ratio of factorials, whose asymptotics are easily obtained from Stirling’s formula.

The geometry of the tetrahedron

The asymptotics of the 6j-symbols are related to the geometry of the planar dual tetrahedron. The 6j-symbol is a tetrahedral spin network ( Γ,γ ) admissibly labeled as shown below:


with  γ= (a, b, c, d, e, f). Its dual tetrahedron  ( Γ,γ ) is also labelled by  γ. The tetrahedron and its planar dual, together with an ordering of the vertices and a colouring of the edges of the dual is depicted below. When a more systematic notation for the edge labels is needed  they will be denoted them by dij it follows that

(a, b, c, d, e, f) = (d12, d23, d14, d34, d13, d24)


We can interpret the labels of the dual tetrahedron as edge lengths in a suitable flat geometry. A condition that allows one to realize ( Γ,γ )  in a flat metric space such that the edge lengths equal the edge labels. Labeling the vertices of ( Γ,γ ). We can formulate such a condition in terms of the Cayley-Menger determinant. This is
a homogeneous polynomial of degree 3 in the six variables a2, . . . , f2. A  definition of the determinant is:

Given numbers dij we can define the  Cayley-Menger matrix by

Cij = 1−d²ij/2 for i, j ≥1


Cij = sgn(i−j) for when i = 0 or j = 0.

In terms of the coloring = (a, b, c, d, e, f) of a tetrahedron, we have:


The sign of the Cayley-Menger determinant determines in what space the tetrahedron can be realized such that the edge labels equal the edge lengths

  • If det(C) > 0 then the tetrahedron is realized in Euclidean space R³.
  •  If det(C) = 0 then the tetrahedron is realized in the Euclidean plane R².
  • If det(C) < 0 then the tetrahedron is realized in Minkowski space R²’¹.

In each case the volume of the tetrahedron is given by


The 6 dimensional space of non-degenerate tetrahedra consists of regions of Minkowskian and regions of Euclidean tetrahedra. It turns out to be a cone that is made up from one connected component of three dimensional Euclidean tetrahedra and two connected components of Minkowskian tetrahedra. The three dimensional Euclidean and Minkowskian tetrahedra are separated by Plane tetrahedra. The Plane tetrahedra also form two connected components, representatives of which are depicted below:


The tetrahedra in the Plane component that look like a triangle with an interior point are called triangular and the Plane tetrahedra from that other component that look like a quadrangle together with its diagonals are called quadrangular.  The same names are used for the corresponding Minkowskian components.

An integer representative of the triangular Plane tetrahedra is not easy to find as the smallest example is (37, 37, 13, 13, 24, 30).

Let’s look at the dihedral angles of a tetrahedron realized in either of the three above spaces. The cosine and sine of these angles can be expressed in terms of certain minors of the Cayley-Menger
matrix. Define the adjugate matrix ad(C) whose ij entry is (−1)i+j times the determinant of the matrix obtained from C by deleting the i-th row and the j-th column. Define Ciijj to be the matrix obtained from C by deleting both the i-th row and column and the j-th row and column.

The Law of Sines and the Law of Cosines are well-known formulae for a triangle in the Euclidean plane. let’s look at the Law of Sines and the Law of Cosines for a tetrahedron in all three flat geometries.

If we let θkl be the exterior dihedral angle at the opposite edge ij. The following formula is valid for all non-degenerate tetrahedra:


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