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Steam engine: James Watt, working as an instrument maker for the University of Glasgow, was given a scale model Newcomen steam engine to put in working order. Watt recognized the problem as being related to the square–cube law, in that the surface to volume ratio of the model's cylinder was greater than that of the much larger commercial engines, leading to excessive heat loss.

Root vectors. The cuboctahedron's 12 vertices can represent the root vectors of the simple Lie group A 3. With the addition of 6 vertices of the octahedron, these vertices represent the 18 root vectors of the simple Lie group B 3. Metric properties. The area A and the volume V of the cuboctahedron of edge length a are:

If perfect cube ends in 8, the cube root of it must end in 2. If perfect cube ends in 9, the cube root of it must end in 9. Note that every digit corresponds to itself except for 2, 3, 7 and 8, which are just subtracted from ten to obtain the corresponding digit.

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as √ 3 or 3 1/2. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. The square root of 3 is an irrational number.

The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached.

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x 2 + 1 = 0.Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication.