$3\mathrm{.}21.$ Suppose that a base cuboid has three dimensions, A$,$ B$,$ C$,$ with the following number of cells:

$|$A$|=1,000,000,|$B$|=100,$ and $|$C$|=1000.$ Suppose that each dimension is evenly partitioned into

$10$ portions for chunking.

$($a$)$ Assuming each dimension has only one level, draw the complete lattice of the cube.

$($b$)$ If each cube cell stores one measure with four bytes, what is the total size of the computed

cube if the cube is dense?

$($c$)$ State the order for computing the chunks in the cube that requires the least amount of space

and compute the total amount of main memory space required for computing the $2-$D

planes