mg.metric geometry - Tetrahedra with prescribed face angles

Tuesday, 11 May 2010

mg.metric geometry - Tetrahedra with prescribed face angles

I am not sure if the 3-dimensional problem formulated in the question is the proper analogue of the 2-dimensional one for triangles - essentially because of the appearance of Dehn invariants and such.
At least the following modification of the question can be answered using the results of Dupont and Sah:

Given a combination of side lengths and dihedral angles $sum l_iotimes frac{theta_i}{2pi}inmathbb{R}otimes(mathbb{R}/mathbb{Z})$ , is there a euclidean polytope having this element as Dehn invariant?

The answer is given by an exact sequence which you can find in Section 4 of J.L. Dupont and C.-H. Sah: Homology of euclidean groups of motions made discrete and euclidean scissors congruences. Acta Math. 164 (1990), 1--27:

$0to mathcal{P}(mathbb{R}^3)/mathcal{Z}_2(mathbb{R}^3)stackrel{D}{longrightarrow} mathbb{R}otimes(mathbb{R}/mathbb{Z}) stackrel{J}{longrightarrow} H_1(SO(3),mathbb{R}^3)to 0$
In this sequence,

$mathcal{P}(mathbb{R}^3)$ are scissors congruence classes of polytopes in

$mathbb{R}^3$ ,

$mathcal{Z}_2(mathbb{R}^3)$ are the scissors congruence classes of prisms,

$D$ is the Dehn invariant and

$J(lotimes frac{theta}{2pi})= frac{1}{2}lfrac{dcostheta}{sintheta}$ using the identification

$H_1(SO(3),mathbb{R}^3)congOmega^1_{mathbb{R}}$ with absolute Kähler differentials.

So, if you are given the six dihedral angles for the tetrahedron, it is at least in principle possible to figure out if there are six side lengths which give a realizable Dehn invariant. Unfortunately the theorem does not tell you if the Dehn invariant will be realizable by a tetrahedron - the theorem generally does not tell you how to construct the polytope realizing the Dehn invariant...

Anyway, there are analoguous exact sequences for hyperbolic and spherical scissors congruence classes. For hyperbolic scissors congruences you get in particular

$mathcal{P}(mathbb{H}^3)stackrel{D}{longrightarrow}mathbb{R}otimes(mathbb{R}/mathbb{Z})to H_2(SL_2mathbb{C},mathbb{Z})^-to 0$
where

$mathcal{P}(mathbb{H}^3)$ is the group of scissors congruence classes in hyperbolic 3-space, and

$H_2(SL_2mathbb{C},mathbb{Z})^-$ is the

$-1$ -eigenspace of complex conjugation on

$H_2(SL_2mathbb{C},mathbb{Z})$ . For spherical scissors congruence the

$+1$ -eigenspace appears. This can be found in papers of Dupont, Sah, or the book "Scissors congruences, group homology and characteristic classes" by J.L. Dupont. The map

$mathbb{R}otimes(mathbb{R}/mathbb{Z})to H_2(SL_2mathbb{C},mathbb{Z})^-$ can be identified with the reduction

$S^2mathbb{C}^timesto K_2(mathbb{C})$ from a symmetric square of the units of

$mathbb{C}$ to

$K_2$ , though that may not necessarily be considered explicit. At least, this tells you that there is a precise obstruction to realizing a linear combination of side lengths and dihedral angles as the Dehn invariant of some hyperbolic or spherical polytope. It is probably further significant work to produce precise conditions for realizability by tetrahedra.

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Tuesday, 11 May 2010