Quadrays and Areaby Kirby Urner 

If a and b have the same length d and the angle between them is 90 degrees, then our parallelogram is a square and its area is defined as d x d or "d squared". The game of quadrays includes xyz area among its library of methods but defaults to a different conceptualization of the process for deriving area from two vectors. 


The area varies with the angle between two fixedlength vectors according to either method, as the resulting parallelogram or triangle changes shape. We will need a method to determine the angle between any two quadrays. One way to get this angle is to expand or reduce the lengths of both a and b to unity by applying a scalar of 1/distance. This will result in two vectors a' and b' with the same angle between them as between the original a and b, but both with length one: 
a' = a * 1/D(a) D(a') = 1 b' = b * 1/D(b) D(b') = 1 

The sin( ) function accepts an angle and outputs the corresponding ratio of oppositeSide/hypotenuse, while its inverse, arcsin( ), accepts this ratio as input and outputs the angle's measure. In other words: sin(angle/2) = D(a'  b')/2 angle = 2 * arcsin[D(a'  b')/2] 
One other consideration is that our native D( ) function returns
its length in radians, meaning in units equaling the radius of our
densepacked ivm spheres e.g. 

The function D(vector), developed in An Introduction to Quadrays, returns a vector's length or distance. Putting all of these facts together, we have enough information to design the required area methods: 

The default area is equal to xyzArea multiplied by a constant, which is to be expected, since both vary linearly in proportion to edges a, b and the sine of the included angle. xyzArea = D(a)/2 * D(b)/2 * sin(angle) Area = D(a)/2 * D(b)/2 * sin(angle) * Root(4/3) However, we should remember another important difference: xyzArea refers to the area of a parallelogram, whereas the default method refers to the area of a triangle with half the area of the corresponding parallelogram. If we halve the xyzArea, then both numbers refer to the same geometric shape and we have the basis for a conversion constant. Area = K * xyzArea/2 K = 2 * Area/xyzArea K = 2 * Root(4/3)According to our default method, the concentric hierarchy icosahedron has an area of 20, the octahedron an area of 8, and the tetrahedron an area of 4. 


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