This picture shows the Binary Tetrahedral Group
embedded as a subgroup of the matrix group SU(2).
SU(2) is geometrically a 3-dimensional sphere of
radius one; here we see a conformal projection of that
sphere into 3-space. The point 1 is at the intersection of the three black circles of longitude. The six points +/- i,j,k lie at the intersections of these circles with the yellow equatorial sphere. Eight other group points lie inside the sphere, at a distance pi/3 from 1. They are a,b,c,d (colored violet) and -a^2,-b^2,-c^2,-d^2. Six of the nine points outside the equatorial sphere are visible here. They appear larger because they are closer to the projection point (near -1). The image of the point -1 itself has been suppressed. It would occupy the whole background. This image was produced using GeomView modules developed at the Geometry Center by Summer Institute student Rebecca Frankel. |

The *Tetrahedral Group* is the group of orientation-
preserving symmetries
of an equilateral tetrahedron.
If the vertices of the tetrahedron are labeled A,B,C,D, each
of the symmetries may be represented as a permutation of
these four symbols. Each even permutation corresponds to
an orientation-preserving symmetry. The twelve are

- the identity
- eight rotations about axes going through a vertex and
the opposite face. There are four vertices; each rotation can be
by 120 degrees or 240 degrees:

(ABC), (ACB), (ABD), (ADB), (ACD), (ADC), (BCD), (BDC), - three 180-degree rotations about axes joining
the midpoints of two opposite edges:

(AB)(CD), (AC)(BD), (AD)(BC).

The Special Unitary group SU(2) consists of 2 by 2 complex matrices of the form

/ \ | (x + iy) (z + iw) | |(-z + iw) (x - iy) | \ /with determinant x^2 + y^2 + z^2 + w^2 = 1. There is a group homomorphism SU(2) --> SO(3) which can be described in terms of x,y,z,w but which can be more geometrically defined using

/1 0\ /i 0\ /0 1\ /0 i\ \0 1/ \0 -i/ \-1 0/ \i 0/respectively. The rules for multiplying quaternions follow from the rules for matrix multiplication; equivalently: i^2 = j^2 = k^2 = -1, ij = -ji = k, jk = -kj = i, ki = -ik = j, extended linearly. The condition x^2 + y^2 + z^2 + w^2 = 1 defines a

/ \ | (x - iy) (-z - iw) | | (z - iw) (x + iy) | \ /so the multiplicative inverse of the unit quaternion x + iy + jz + kw is

x - iy - jz - kw.

The vector (a,b,c) in R^3 can be identified with
the *pure imaginary* quaternion ai + bj + ck;
then each quaternion q defines a linear map
L(q) : R^3 --> R^3 by L(q)(ai + bj + ck) =
q x (ai + bj + ck) x q^(-1), using quaternionic
multiplication. It is straightforward to check
that L(q) is also pure imaginary; the fact that
the determinant of a product is the product of the
determinants then guarantees that L(q) is a
length-preserving, and therefore orthogonal,
transformation; the question of orientation
can be answered by noting that L(1) is the
identity and that SU(2) is connected. Also,
the definition yields immediately that L is
a homomorphism from SU(2) to SO(3):
L(pq) = L(p)L(q) for any
two unit quaternions p and q. Finally,
two unit quaternions p and q give the same SO(3)
element if and only if p = +/- q; the condition
is equivalent to the three equations L(q^(-1)p)i = i,
L(q^(-1)p)j = j, L(q^(-1)p)k = k;
writing q^(-1)p = x + iy + jz + kw
in these three equations
yields x^2 = 1, y = z = w = 0, so q^(-1)p = +/- 1,
i.e. p = +/- q.

For obvious reasons, then, the map L is called the
double-covering homomorphism from SU(2) to SO(3).
Since the tetrahedral group is a 12-element
subgroup of SO(3), the SU(2) matrices which map
to elements of the tetrahedral group will form a
24-element subgroup of SU(2). This is the
*Binary Tetrahedral Group*.

Click here for an explicit
description of the elements of the Binary Tetrahedral
Group in terms of quaternions.

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