Turning tables

It is useful to have some operations that turn one proportion table into another.  In this posting, we deal with some operations that can be applied to many different kinds of table, not just proportion tables.

Let's start with the simple operation of swapping (or transposing) two rows, here the second and the third :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(2, 4, 5, 9),(6, 12, 15, 27),(4, 8, 10, 18)] `,

or two columns, here the second and the fourth :

`\qquad\qquad[(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)]  \qquad->\qquad [(2, 9, 5, 4),(4, 18, 10, 8),(6, 27, 15, 12)]`.

Repeatedly transposing various pairs of rows can give every possible reordering (or permutation) of the rows :
 
`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(6, 12, 15, 27),(2, 4, 5, 9),(4, 8, 10, 18)] `,

and similarly for arbitrary column permutations :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(5, 9, 4, 2),(10, 18, 8, 4),(15, 27, 12, 6)] `.

We can combine arbitrary row permutations and arbitrary column permutations to get arbitrary table permutations.  How the rows are reordered is independent of how the columns are reordered.  Combining the most recent row and column reorderings, we get :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(15, 27, 12, 6),(5, 9, 4, 2),(10, 18, 8, 4)] `.

All of the foregoing operations simply rearrange the proportion table to make a proportion table that contains exactly the same information.  When one can make table from another using only rearrangement, the tables are said to be equivalent up to rearrangment.  All of the tables in this posting are equivalent up to permutation---they are in a sense the same `3 xx 4` table, just displayed differently.

Does this account for reliable ways to convert a proportion table to a proportion table?  In much later postings, when we deal with group tables and have more restrictions, the answer will be yes.

Here, however, the answer is no---there are other possibilities.  One such possibility can be made by giving the table a quarter turn, e.g.

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] `.

It should be clear that in the rotated version, rows are proportional to rows and columns to columns, as before, i.e. that the rotated version of the original proportion table is also a proportion table.  We could not have made the rotated version from permutations alone or the original, either, because permutations only ever turn a `3 xx 4` table into a `3 xx 4` table, never into a `4 xx 3` table, as this quarter turn has just done.

Of course, this quarter-turned table can now be permuted to give many `4 xx 3` tables, all of which are equivalent up to permutation.  Indeed, there will be as many distinct `4 xx 3` tables are there are distinct `3 xx 4` tables.

Does this exhaust all the possibilities?  What if one makes a quarter turn in the other direction?  It turns out that the two different ways of quarter-turning the original table lead to two tables that differ by a half turn,

`\qquad\qquad\qquad  [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] `,

and

`\qquad\qquad\qquad  [(9,18,27),(5,10,15),(4,8,12),(2, 4, 6)] `.

Unlike a quarter turn, a half turn can be made just from permutations:  reverse the order of the rows, and reverse the order of the columns.  In this way, we get

`\qquad\qquad\qquad  [(6, 4, 2),(12,8,4),(15,10,5),(27,18,9)] \qquad->\qquad [(9,18,27),(5,10,15),(4,8,12),(2, 4, 6)] `,

and we also get

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad  [(27,15,12,6),(18,10,8,4),(9,5,4,2)] `.

It should also be mentioned here that rotating by a quarter turn clockwise and reversing the order of the columns (or, alternatively, rotating a quarter of a turn anticlockwise and reversing the order of the rows) is often called transposing the table.

Transposing a table is like flipping a rectangular piece of paper while keeping the original top left corner at the top and left, going from portrait orientation to landscape orientation, or vice versa :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad  [(2, 4, 6),(4,8,12),(5,10,15),(9,18,27)] `.
 
Transposing in this sense is a little different from transposition as introduced above.  Transposition swaps a pair of rows, or else a pair of columns. Transposing, on the other hand turns all the columns to rows and all the rows to columns, in particular way.

We shall actually have little use either for quarter turns or for transposing, while we are dealing with proportion tables.  Transposing become important when dealing with matrices, which come much later.