Let \xi=-\frac32-\frac12\sqrt{1+4\phi}\approx -2.86676039917 be the smallest (most negative) zero of the polynomial P=x^2+3x+\phi^{-2}, where \phi is the
golden ratio. Equivalently, \xi = -1-\sqrt{\phi + \sqrt{\phi + \sqrt{\phi + \cdots}}}\, = -1-\beta where \beta \approx 1.86676039917 () is a root of \beta^2-\beta-\phi=0. Let the point p be given by :p= \begin{pmatrix} \phi^{-1}\xi+\phi^{-3} \\ \xi \\ \phi^{-2}\xi+\phi^{-2} \end{pmatrix} . Let the matrix M be given by :M= \begin{pmatrix} 1/2 & -\phi/2 & 1/(2\phi) \\ \phi/2 & 1/(2\phi) & -1/2 \\ 1/(2\phi) & 1/2 & \phi/2 \end{pmatrix} . M is the rotation around the axis (1, 0, \phi) by an angle of 2\pi/5, counterclockwise. Let the linear transformations T_0, \ldots, T_{11} be the transformations which send a point (x, y, z) to the
even permutations of (\pm x, \pm y, \pm z) with an even number of minus signs. The transformations T_i constitute the group of rotational symmetries of a
regular tetrahedron. The transformations T_i M^j (i = 0,\ldots, 11, j = 0,\ldots, 4) constitute the group of rotational symmetries of a
regular icosahedron. Then the 60 points T_i M^j p are the vertices of a small snub icosicosidodecahedron. The edge length equals -2\xi, the circumradius equals \sqrt{-4\xi-\phi^{-2}}, and the midradius equals \sqrt{-\xi}. For a small snub icosicosidodecahedron whose edge length is 1, the circumradius is :R = \frac12\sqrt{\frac{\xi-1}{\xi}} \approx 0.5806948001339209 Its midradius is :r = \frac12\sqrt{\frac{-1}{\xi}} \approx 0.2953073837589815 The other zero of P plays a similar role in the description of the
small snub icosicosidodecahedron. == See also ==