ABSTRACT
Suppose, however, that we require the quadrature formula to be symmetric under rotation. That is, whenever the node (x1, y1) is in (74.2), then the nodes ( -x,, y,), (x,, -y,), and ( -x,, -y,) should also be in (74.2). This constraint adds the following restrictions to (74.3): w1 = w2 = w3 = w4, x1 = x2 = -x3 = -x4, and f/1 = -y2 = -y3 = f/4· With these constraints, the equations in (74.3) become:
318 VI Numerical Methods: Techniques
(74.4)
[2) The groups under which the nodes are mapped into themselves are the reflection groups of polyhedra. In three dimensions there only exist three such groups:
(A) the extended tetrahedral group of order 24 (the orbits of this group can have 24, 12, 6, 4, or 1 point(s));
(B) the extended octahedral group of order 48 (the orbits of this group can have 48, 24, 12, 8, 6, or 1 point(s));
(C) the extended icosahedral group of order 120 (the orbits of this group can have 120, 60, 30, 20, 12, or 1 point(s)).
