Rotations

Put your visualisation skills to the test by seeing which of these molecules can be rotated onto each other.

What groups of transformations map a regular pentagon to itself?

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the minute hand and hour hand had swopped places. What time did the train leave London and how long did the journey take?

Make a footprint pattern using only reflections.

Does changing the order of transformations always/sometimes/never produce the same transformation?

What is the relationship between these first two shapes? Which shape relates to the third one in the same way? Can you explain why?

This problem provides training in visualisation and representation of 3D shapes. You will need to imagine rotating cubes, squashing cubes and even superimposing cubes!