1. What does it mean for a symmetry of R to be direct or opposite? [3pt]...

 

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1. What does it mean for a symmetry of R³ to be direct or opposite? [3pt] 2. Explain the connection between permutations and the symmetries of the dodecahedron. [3pt] 3. What is the number of symmetries (the order of the symmetry group) of each of the Platonic solids. [3pt] 4. Describe all isometries of R³ that give the symmetries of this object (describing an isometry includes information like the axis and angle of a rotation, the plane that a reflection is in, etc.); [4pt] 5. Every rotation in 3 dimensions can described by a unit quaternion. How unique is this description; if multiple quaternions give the same rotation, what is the relationship between them? [2pt] 1. What does it mean for a symmetry of R³ to be direct or opposite? [3pt] 2. Explain the connection between permutations and the symmetries of the dodecahedron. [3pt] 3. What is the number of symmetries (the order of the symmetry group) of each of the Platonic solids. [3pt] 4. Describe all isometries of R³ that give the symmetries of this object (describing an isometry includes information like the axis and angle of a rotation, the plane that a reflection is in, etc.); [4pt] 5. Every rotation in 3 dimensions can described by a unit quaternion. How unique is this description; if multiple quaternions give the same rotation, what is the relationship between them? [2pt]

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1 When a symmetry of R is direct it means that the action of the symmetry on a point in the space will result in a point in the same direction When a ... View the full answer