Multidimensional Space - Microsoft SQL Server 2008 Analysis Services - page 15

In Figure 2.1, we have only one sales transaction represented by a point in the data space.
If we represent every sales transaction of the product by a point on the multidimensional
space, those points, taken together, constitute a “fact space” or “fact data.”

It goes without saying that actual sales are much less than the number of sales possible if
we were to sell each of our products to all our customers each month of the year. That's
the dream of every manager, of course, but in reality it doesn't happen.

The total number of possible points creates a theoretical space. The size of the theoretical
space is defined mathematically by multiplying the size of one dimension by the product
of the sizes of the other two. In a case where you have a large number of dimensions,
our theoretical space can became huge; but no matter how large the space gets, it
remains limited because each dimension is distinct and is limited by the distinct number
of its members.

The following list defines some more of the common terms we use in describing a multidi-
mensional space:

. A tuple is a coordinate in multidimensional space.

. A slice is a section of multidimensional space that can be defined by a tuple.

Each point of a geometric space is defined by a set of coordinates, in a three-dimensional
space: x , y , and z . Just as a geometric space is defined by a set of coordinates, multidimen-
sional space is also defined by a set of coordinates. This set is called a tuple .

For example, one point of the space shown in Figure 2.1 is defined by the tuple ([Club 2%
Milk], [Edward Melomed], [March]).

An element on one or more dimensions in a tuple could be replaced with an asterisk (*)
indicating a wildcard. In our terminology, that is a way to specify not a single member but
all the members of this dimension. By specifying an asterisk in the tuple, we turn the
tuple from a single point into a subspace (actually, a normal subspace). This sort of normal
subspace is called a slice .

You might think of an example of a slice for the sales of all the products in January to all
customers as written (*, *, [January]). But for simplicity, the wildcards in the definitions of
slice are not written; in our case, it would be simply ([January]). Figure 2.2 shows the slice
that contains the sales that occurred during January.

You can think of many other slices, such as the sales of all the products to a specific
customer ([Edward Melomed]), the sales of one product to all customers ([Club 2%
Milk]), and so on.