Symmetries of the Hypercube and the Hyperdiamond Using Quaternions
F. van der Valk (TU Delft - Electrical Engineering, Mathematics and Computer Science)
P.M. Visser – Mentor (TU Delft - Mathematical Physics)
Jeroen Spandaw – Mentor (TU Delft - Analysis)
D.C. Gijswijt – Graduation committee member (TU Delft - Discrete Mathematics and Optimization)
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Abstract
Quaternions prove to be a useful tool when determining rotation groups of regualr polytopes and other object in three and four dimensions. In this thesis it will be studied how the unit quaternions relate to the special orthogonal groups in three and four dimension. Thereafter, this theory is applied to the normal subgroups of the binary octahedral groups as well as the cube and its dual the octahedron. The normal subgroups of the binary octahedral group form 4-dimensional objects, for example the 4-orthoplex or 16-cell and the hyperdiamond or 24-cell.
Symmetries that keep the origin in place, consist of orientation preserving and orientation
flipping isometries, called rotations and reflections respectively. The special orthogonal group in n dimensions, containing all real n × n orthogonal matrices with positive determinant, is precisely the group containing all rotations in n dimensions. The map Φ : H_1→SO(3) defined by q → Φ_q, where Φ_q (p) = qpq^−1 for p∈H is a rotation of ImH ≅ R^3, is a two-to-one surjective homomorphism. The map Ψ : H_1 × H_1 →SO(4), defined by q →Ψ_(q_1,q_2), where
Ψ_(q_1,q_2) (p) = q_1pq_2^−1 is a rotation of H ≅ R^4, is a two-to-one surjective homomorphism.
The n-cube is the generalization of square and cube in general dimensions. Using Φ the binary octahedral group 2O can be established, which describes all quaternions that describe rotational symmetries of the cube and its dual the octahedron. 2O contains 48 unit quaternions, which is twice as many elements as the number of rotational symmetries of the cube and octahedron, due to the two-to-one relation of Φ.
The binary octahedral group 2O describes an object in four dimensions. Its normal subgroups do as well, the most interesting are Q_8, which describes a 4-orthoplex or 16-cell and has the hypercube or 8-cell as dual, and 2T, which describes a hyperdiamond or 24-cell. The binary rotation groups of these polytopes and the object that 2O itself describes, are studied in particular. These binary rotation groups can all be constructed in the same manner.
Write N ∈ {Q_8,2T,2O}, then 2O_N := {(q_1,q_2) ∈ 2O × 2O : q_1N = q_2N} ⊂ H_1 × H_1 is equal to the entire binary rotation group of N. This means that for the rotation group of N, denote this as O_N ⊂ SO(4), it holds that O_N ≅ 2O_N/{±(1, 1)}, where the isomorphism is given by the restriction of Ψ : H_1 × H_1→SO(4) to 2O_N. For the remaining normal subgroups {1} and {±1}, this construction only gives a subgroup of the entire binary rotation group.