When the desired rotation matrix has been selected, it is assigned by
HYPERCUBE
, to a special matrix, rote
. The
assignment can be made conveniently by employing a double loop, the inner
loop for the sequence of numbers across a row of the matrix and the outer
loop for the sequence of rows.
The calculation of the coordinates of the rotated hypercube is done by
"multiplying" the matrix cube
by the selected rotation matrix
rote
. The product of the two matrixes is stored temporarily in
a third matrix called temp
, and it gives the coordinates of
the rotated hypercube. That product is found according to the rules of
matrix multiplication, a standard operation on matrixes of numbers. It
results from an orderly orgy of multiplications embedded in three nested
loops.
Temp
, like vert
and cube
, is a
16-by-4 matrix of 64 numbers: it gives four coordinates for each of the 16
vertexes of the rotated hypercube. Each number temp(i,j
) in
the matrix is designated by a pair of indexes i
and
j
. For example, temp(13,3)
is the third
coordinate of the 13th vertex of the rotated hypercube. Its value is the
sum of four products of numbers drawn in a precisely defined way from the
matrix cube
and from the matrix rote
. The
innermost loop in the program can therefore be indexed by the letter
k
, which runs from I to 4, and that loop returns the value of
one entry of temp
.
The k
loop is then placed inside the intermediate loop that
has index j
, The j
loop computes all four
coordinates of the ith vertex of the rotated hypercube according to
the procedure I have just outlined; in other words, it fills in all four
entries in the ith row of temp
. Finally the
j
loop is placed inside the outermost loop, which has the
index i
. The i
loop calculates all 16 rows of
temp
, and when it is completed, temp
gives the
coordinates of all the vertexes of the rotated hypercube, resplendent in
its new position. In order to display it on the computer monitor one more
double loop is needed that replaces the old position coordinates of the
hypercube in cube
with the newly calculated coordinates from
temp
.
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area. |