Final answer:
Spherical rectangles cannot exist on the unit sphere S(0,1) because the geometry on a curved surface like a sphere does not allow for right angles to exist at each vertex of a quadrilateral, unlike on a flat plane.
Step-by-step explanation:
The question asks whether spherical rectangles exist on the unit sphere S(0,1) with all four angles measuring 90 degrees. By definition, a spherical rectangle would be a quadrilateral on the sphere having four right angles. However, this is not possible on a sphere. Unlike on a flat surface, where a grid system can have perfect right angles, such as the streets and avenues in cities, on the curved surface of a sphere, the concept of a rectangle does not hold in the same way. When dealing with coordinates on a sphere, creating a grid system involves defining circles that serve a similar function to rectangular grids on flat surfaces. These are lines of latitude and longitude, which can intersect at right angles at certain points, but the shapes formed are not rectangles as the sides are not straight lines but curves.
The concept of using grid systems on a sphere helps in understanding that the geometry on curved surfaces differs from plane geometry, wherein quadrilaterals with four right angles can easily exist. It is also pertinent to note that shapes on a sphere, such as a 'spherical rectangle,' do not follow the same rules as those in plane geometry and thus we cannot have a quadrilateral with four right angles on a sphere. Instead, we have spherical shapes that consist of curved lines, making it impossible to form right angles at each vertex of a quadrilateral.