Sum of Two Cubes: Physical Models

cults3d

####x^3 + y^3 = (x+y)(x^2 - xy+y^2) Physical Model for the Sum of Two Cubes This set of 3D blocks demonstrate, physically, the well-known algebraic fact that x^3 + y^3 = (x+y)(x^2-xy+y^2). While it is possible to show that the equation is true in a physical sense, it is a bit of a stretch toward physical modeling. The algebraic manipulation might be more aesthetically appealing. The big ideas covered are: (1) If the area of a polygon is (x^2-xy+y^2) square units, then, when extended (or extruded) for (x+y) units in the 3rd dimension, the resulting solid has a volume of (x+y)(x^2-xy+y^2); (2) the four solids x^2(x-y), xy(x-y), xy^2, and y^3 can be arranged into two cubes: x^3 and y^3, as shown in the pictures. Note that the term * (-xy)* is represented by the missing piece at the bottom. A box is also included for help with the arrangement of the blocks for the (x^2-xy+y^2) base. The height of the box is (x + y). Have fun!