There are 12 geometrically distinct tetrahedra with vertices in the binary tetrahedral group. Conformal images of three of them are shown here. The first is the simplest: the generalized octant (hexadecant?) AAAA with volume 1/16 of the total volume. Here it is positioned with its vertices at 1,i,j,-k. The second is the largest, ADDD, with volume almost 1/4 the total volume. Here it is shown with vertices i,j,k,a^2. The third is the smallest, BBEE, shown here with vertices i,j,d,-a^2; its volume is barely more than 1/100 the total volume. These images were produced using GeomView modules developed by Geometry Center Summer Institute student Rebecca Frankel of MIT. Larger versions of these images, with more detailed explanations, can be reached through links below.

**NOTATION:** Edge
lengths and dihedral angles are encoded as before:

p = pi/2,
g = pi/3, b = v = 2*pi/3,

with in addition

h = pi/4, i = 3*pi/4.

In the table below each row corresponds
to a single 3-simplex type.
(Note that the set of faces
determines the simplex up to a mirror-symmetry). In the first column each
simplex is assigned a 1-letter code. In the second column the
simplexes are listed lexicographically for convenience in reference.
In the
third column the list of faces is
rewritten in a more topological
order

/\ /1 \ /----\ / \ 3/ \ /_2_\/_4_\

In the fourth column the six edge lengths of those faces are recorded, in the order 01,02,03,12,13,23 (referring to the vertices in the following copy of the diagram)

0 /\ / \ 1----2 / \ / \ / \/ \ 0-----3----0

In the fifth column the
six dihedrals are recorded, in the same order,
except that the first one is the
dihedral *opposite* 01, etc. These dihedrals
were calculated as in Vinberg's paper.
The sixth column records the relative frequency of each
simplex: (1/24) times the number each occurred in an
enumeration of all simplexes of the form 1,x,y,z.
The seventh column records the volume relative to
the total volume of SU(2). See
Calculation of Volumes for the explanation of how
these volumes were computed.

Highlighted symbols link to conformal images of the tetrahedra in SU(2), prepared with GeomView modules developed by Rebecca Frankel.

Code Lexic. Topol. Edges Dihedrals Rel.Freq. Volume K AAAA AAAA pppppp pppppp 2 1/16 O ABBB BBAB gggppp hhhbbb 8 1/64 N ABCC BCAC ggbppp hhibgg 24 5/192 M ACCD DCAC bbgppp iihbgg 24 11/192 L ADDD DDAD bbbppp iiibbb 8 15/64 Q BBCC CBBC* gbppgg gbppgg 24 1/48 V BBEE BEBE gggpgg ggpbgg 12 1/96 S BDFF DFBF bbbpgg bbpbgg 24 7/96 P CCDD CDCD* bpbgpb bpbgpb 24 5/48 U CCEF ECCF* ggbgpb gpbggb 24 1/32 T CCFF FCCF* bbggpb bpgggb 24 5/96 R DDFF DFDF bbgpbb bbpbbb 12 17/96 BBBB BBBB ggppgg (singular**) 3 0 BCCD DCBC bbppgg (singular**) 12 0 CCCC CCCC bpggpb (singular**) 6 0 DDDD DDDD bbppbb (singular***) 3 1/2

* occur in non-congruent mirror-image pairs.

** these three have volume zero, corresponding to the decompositions B+B=B+B, D=B+2*C, C+C=C+C respectively;

*** this degenerate tetrahedron has all dihedrals equal to pi, and encloses one-half the volume of the 3-sphere.

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Tony Phillips

tony@math.sunysb.edu

March 26 1997