In the illustration a way of expressing the difference in volume of 2 different sized 3-dimensional cubes is given:
The volume of a black cube of size b is diminished by the volume of a red cube with size a.
3d drawing of the volumetric difference of two cubes expressed as the sum of the volumes of 4 bodies
This can be expressed as the sum of the volumes of 4 bodies: 3.(b-a).a.b + (b-a)^3 , which is the sum of the 3 equal sized volumes of the red, green and blue bodies and the yellow cube where two corners of the black and red cube have the same coordinates.
I would like to know if there is an analogue way of expressing the difference of the 'volumes' of two hypercubes in 4d and would like to know if this can be done also in higher dimensions.