[10 points] Consider the cube in R with its vertices at (+1,1,1). (a) Choose a flag...

 

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[10 points] Consider the cube in R³ with its vertices at (+1,1,1). (a) Choose a flag consisting of a vertex, edge, and face of this polyhedron. Find the planes in the kaleidoscope corresponding to this choice. Give inequalities, in terms of the coordinates (x,y,z), that define the spherical triangle obtained from the fundamental region of the kaleidoscope, on the appropriately centred unit sphere (i.e. the yellow region on the picture of the sphere below). [4,3] *432 (b) Find formulas for the reflections across each of these planes of a point (x,y,z) € R³. (Hint: you can use the formula R₁(x)=x2, where ri is any normal vector to the plane II, but this is probably overkill.) (c) Calling these three reflections R₁, R₂, R3. Find the orders of the rotations R₁ R₂, R₁ R₂, R₂ R3. Describe these rotations geometrically, by their axes and angles of rotation. [10 points] Consider the cube in R³ with its vertices at (+1,1,1). (a) Choose a flag consisting of a vertex, edge, and face of this polyhedron. Find the planes in the kaleidoscope corresponding to this choice. Give inequalities, in terms of the coordinates (x,y,z), that define the spherical triangle obtained from the fundamental region of the kaleidoscope, on the appropriately centred unit sphere (i.e. the yellow region on the picture of the sphere below). [4,3] *432 (b) Find formulas for the reflections across each of these planes of a point (x,y,z) € R³. (Hint: you can use the formula R₁(x)=x2, where ri is any normal vector to the plane II, but this is probably overkill.) (c) Calling these three reflections R₁, R₂, R3. Find the orders of the rotations R₁ R₂, R₁ R₂, R₂ R3. Describe these rotations geometrically, by their axes and angles of rotation.

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The flag we choose consists of the vertex 1 the edge 2 and the face 3 The pla... View the full answer