Quaternions and spatial rotation around the "z" axis, then flip it 180 degrees around the "x" axis. Take the other book, flip it 180° around "x" axis first, and 90° clockwise around "z" later. The two books do not end up parallel. This shows that, in general, the composition of two different rotations around two distinct spatial axes will not commute. The vector cross product, used to define the axis–angle representation, does confer an orientation ("handedness") to space: in a three-dimensional vector space, the three vectors in the equation will always form a right-handed set (or a left-handed set, depending on how the cross
Gauss–Bonnet theorem and hyperbolic trigonometry, the area of a triangle is proportional to the amount by which its interior angles fail to add up to 180°, or equivalently by the (inverse) amount by which its exterior angles fail to add up to 360°. The area of a spherical triangle is proportional to its excess, by Girard's theorem – the amount by which its interior angles add up to more than 180°, which is equal to the amount by which its exterior angles add up to less than 360°. The area of a hyperbolic triangle, conversely is proportional to its "defect," as established