A cube is one of the Platonic solid shapes, named after the ancient Greek philosopher Plato. A cube consists of eight corners, twelve edges, and six faces. Its proper name is a "hexahedron." Of course, it's also a familiar object in our popular culture, being a handy shape for stackable boxes, the most common shape for a pair of dice used in gaming, and a fascinating puzzle in the Rubik's cube.
The formulas for finding various metrics of a cube are well-established:
• Face = Edge ^ 2
• Volume = Edge ^ 3
• Surface Area = Face * 6
• Face Diagonal = Edge * √2
• Cube Diagonal = Edge * √3
These equations are common sense to us now, but we arrived at them by years of puzzling them out over generations.
We owe our cube study to ancient Greek mathematician Pythagoras (500 B.C.), for coming up with the Pythagorean theorem, which shows that the diagonal (called the hypotenuse) side of any right triangle (a triangle with one corner being exactly 90 degrees) as the square root of the sums of the squares of the other two sides.
This projects into the shape of the square, which can be taken as the combination of two right triangles, and from there into three dimensions for the inside diagonal of a cube.
Finding these was a challenge, because it turns out the square root of two is an irrational number, just like the number Pi. This means that the number can be calculated to an infinite number of decimal places without repeating and without terminating. The square root of two was the first number proven to be irrational in mathematical proofs, but in fact the square roots of most natural integers are all irrational except for the few perfect square numbers - 4, 9, 16, 25, 36, and so on.
Cubed numbers also play a role in the most famous mathematical paradox in history, known as Fermat's last theorem. French mathematician Pierre de Fermat (17th century) proposed that no known values existed for a, b, and c, such that (a^3 + b^3) = c^3. This is not true of square numbers, as the Pythagorean theorem finds integer sets such as 3, 4, and 5; 3^2 = 9, 4^2 = 16, 5^2 = 25, and 9 + 16 = 25, making this a typical first example in finding right triangles. Fermat wondered if the same could apply to cubic numbers, but, in notes found after his death, he claimed to have found a proof that (a^3 + b^3) = c^3 was impossible. Alas, he did not leave the proof itself.
Over three centuries of frustration went by as mathematicians tried to prove the theorem before English Oxford professor Andrew Wiles published a formal confirmation of the hypothesis in 1995 - at last laying the matter to rest.
