There is a deep connection between fundamental symmetry groups pertaining to the field of Algebra in Mathematics, and the natural occurrence of symmetry in the physical realm. Some of the most fundamental symmetry groups, such as the Platonic Solids Symmetry Groups, are associated with the Platonic solids.

The Golden Ratio

\(\varphi = (1+\sqrt{5})/2 = 2\cos(\pi/5) \approx 1.618,\)

and its inverse

\(\omega= 1/\varphi = \varphi-1=(\sqrt{5}-1)/2=2\cos(2\pi/5) \approx 0.618,\)

are the only non-integer irrational algebraic numbers that appear in the character table and matrix representations of the largest and more complex of these symmetry groups, the binary icosahedral group \(I^*\).

The Icosahedral group \( I\) is isomorphic to \( A_5\), where \( A_5\) is the alternating group of even permutations of five objects. Starting from \( I\), one may define the B.I.G \( I^*\) as the preimage of \( I\) under the \( 2:1\) covering homomorphism \(Spin(3) \rightarrow SO(3)\) of the special orthogonal group \( SO(3)\) by the Spin Group \( Spin(3)\).

It follows that the binary icosahedral group is discrete subgroup of the Spin Group \( Spin(3)\) of order \( 120\).

Let

be the algebra of Quaternions, where \( \{1,i,j,k\}\) denotes its’ usual basis. \( \mathbb H\) is a four-dimensional associative non-commutative normed division algebra.

Then the binary icosahedral group is a discrete subgroup of the unit quaternions

\( Sp(1) = \{q \in \mathbb H \ | \ |q|=1 \}\),

under the isomorphism \( Spin(3) \cong Sp(1)\), where \( Sp(1) = U(1,\mathbb H)\) stands for the Symplectic Group

Let \( \chi^l_{I^*}(\alpha)\), \( l,\alpha \in \{1,\cdots,9\}\), denote the \( 9×9\) character matrix of the Binary Icosahedral Group \( I^*\). It is given by

\( \chi^l_{I^*}(\alpha) =\left(\begin{array}{*{9}{r}} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 2 & -2 & 0 & 1 & -1 & \varphi & -\omega & -\varphi & \omega\\ 3 & 3 & -1 & 0 & 0 & \varphi & -\omega & \varphi & -\omega\\ 4 & -4 & 0 & -1 & 1 & 1 & 1 & -1 & -1\\ 5 & 5 & 1 & -1 & -1 & 0 & 0 & 0 & 0\\ 6 & -6 & 0 & 0 & 0 & -1 & -1 & 1 & 1\\ 4 & 4 & 0 & 1 & 1 & -1 & -1 & -1 & -1\\ 3 & 3 & -1 & 0 & 0 & -\omega & \varphi & -\omega & \varphi\\ 2 & -2 & 0 & 1 & -1 & -\omega & \varphi & \omega & -\varphi\\ \end{array}\right) \)

The quaternionic representation of the \( 120\) elements of the Binary Icosahedral Group \( I^*\) is decomposed into the following four subsets: \( I^* = I_1^* \bigcup I_2^* \bigcup I_3^* \bigcup I_4^*\) , with

\( \left\{ \begin{align} I_1^* & = \left\{\pm k^{2n/5}\right\}_{n=1,\cdots,5},\\ I_2^* & = \left\{\pm j \cdot k^{2n/5}\right\}_{n=1,\cdots,5},\\ I_3^* & = \left\{\pm \gamma \left(k^{2n/5} \cdot (i+\omega k) \cdot k^{2m/5}\right) \right\}_{n,m=1,\cdots,5},\\ I_4^* & = \left\{\pm \gamma \left(k^{2n/5} \cdot (i+\omega k) \cdot j \cdot k^{2m/5}\right) \right\}_{n,m=1,\cdots,5}, \end{align} \right. \)

where \( \omega = 1/\varphi = (\sqrt{5}-1)/2\) is the inverse of the Golden Ratio \( \varphi\), and \( \gamma = 1/\sqrt{1+\omega^2}\). The four sets have cardinality \( |I_1^*| = |I_2^*| = 10\) elements, and \( |I_3^*| = |I_4^*| = 50\) elements.

One of the most sought after model of Cosmology today is called the Poincaré Dodecahedral Space (P.D.S.). It is a homogeneous ‘space-form’ (a spherical 3-manifold) given by the quotient space of the 3-Sphere with the Binary Icosahedral Group (B.I.G.), referred to as \( 2I\) (John Conway) or \( I^*\).

The P.D.S. is defined as the quotient space:

\(X = S^3/I^*\),

where \( S^3\) is the 3-Sphere, a geometric object of dimension \( 3\) that forms the boundary of a ball in four dimensions. The 3-Sphere identified with unit quaternions \( Sp(1)\) is also isomorphic to the special unitary matrices of degree \( 2\): \( Sp(1) \cong SU(2)\). As a manifold \( S^3\) is diffeomorphic to \( SU(2)\), whereby \( SU(2)\) is a compact, connected Lie group.

In the definition of this model for the universe, the Golden Ratio together with its inverse \( \omega = 1/\varphi\) are the key non-integer values of the Binary Icosahedral Group \( I^*\).

Poincaré Dodecahedral Sphere, Jeffrey Weeks

References:

• J.R. Weeks, Curved Spaces.
• J.R. Weeks, ‘The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds’, Second edition, 2001.
• J.R. Weeks, ‘The PDS and the Mystery of the Missing Fluctuations‘, Not. AMS, 51 n° 6, pp. 610-619, June/July (2004).
• P. R. Girard, ‘The quaternion group and modern physics‘, Eur. J. Phys., 5, 25-32 (1984).
• J-P. Luminet, ‘A cosmic hall of mirrors‘, Physics World, pp. 3-8, September 2005.
• J. Conway and D. A. Smith, ‘On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry‘, A K Peters, Ltd., 2003.
• K. Tapp, ‘Matrix Groups for Undergraduates‘, Student Mathematical Library, Volume 29, 2005.
• Y. Kosmann-Schwarzbach, ‘Groupes et Symétries, Groupes finis, groupes et algèbres de Lie, représentations‘, Les Editions de l’Ecole Polytechnique, 2° Edition, 2010.
• Joseph A. Wolf, Spaces of Constant Curvature: Sixth Edition, University of California, Berkeley, CA

Work in progress, to appear in ArXiv.org, as “A Review of Quaternions and the Binary Icosahedral Group”.