So[0,1)^1 is a line interval, [0,1)^2 is a unit square and [0,1)^3 is the unit cube, and [0,1]^d is a d-dimensional cube.
2.Only one boundary can be included
It includes 0 but not 1 because it can only the context is usually that practitioners want a region where one edge will wrap to the opposite edge.
Thus they treat [0,1)^2 as if it is actually a 2-dimensional torus.
thus the the 2 boundaries acutally map to the same point, so you can only include one of them.
In our case as we are using x %1 = fractional part of x, the fractional part could be 0, if x=3.0, but it could never be exactly 1.
So[0,1)^1 is a line interval, [0,1)^2 is a unit square and [0,1)^3 is the unit cube, and [0,1]^d is a d-dimensional cube.
2.Only one boundary can be included
It includes 0 but not 1 because it can only the context is usually that practitioners want a region where one edge will wrap to the opposite edge. Thus they treat [0,1)^2 as if it is actually a 2-dimensional torus.
thus the the 2 boundaries acutally map to the same point, so you can only include one of them. In our case as we are using x %1 = fractional part of x, the fractional part could be 0, if x=3.0, but it could never be exactly 1.
3) the mapping from the circle to the surface of the sphere is described here https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_p...
the entire top edge of the square maps to the north pole, and the entire bottom edge maps to the south pole.
4.) t0 is the first point, t3 is the 4-th point.
Hope that helps!