Ticket to Ride

Let Α be the center of α with the axes of the glide-reflection γ passing through Α. Since (αγ)² is a minimal translation (with the conventions of qn 35(ii)), we consider a fundamental square of which Α, Αγ² and Α (αγ)² are three of the vertices. The transformation α²γ and its conjugates by α, γ and γα give four reflections with axes parallel to the diagonals of the fundamental square. Six glide-reflections are obtained from γ by repeated conjugation by α and γ with axes parallel to the diagonals of the fundamental square and intersecting the square. These axes all contain 4-centres. The transformation αγ is a glide-reflection with axis parallel to the translation (αγ)² and by successive conjugation by the quarter-turns α^–1, (αγ²)^–1, and γ^–2α^–1γ² we obtain four more glide-reflections with axes parallel to the sides of the fundamental square, but not passing through any 4-centre. [+]