Thinking four dimensionally

Prompted by Alan Davies' maths programme on the telly just now, in which they didn't explain four-dimensional thinking very well IMHO (i.e. a fourth dimension in space, not time), I shall try and explain it better (this is something that has bugged me for ages).

Let's imagine you run a shoe shop, and you keep a record of all the shoes you sell in a year.  For each sale, you record four items of information (four 'dimensions'):
1. What size are the shoes.
2. At what time of the day you sold them.
3. What type of shoes they are (e.g. are the shoes for a baby, a child, a teenager, a woman or a man).
4. The date on which you sold them.

Anybody can imagine a long list of total daily sales figures for all sales of all types of shoe, from 1 January to 31 December. Let's call that one-dimensional for now. (You could argue it's two-dimensional already, as just giving a total number of pairs sold for the whole year would be one-dimensional, but this is my thought experiment, so don't interrupt).

We can also easily imagine a table, with the dates 1 January to 31 December down the side and 'type of shoe' across the top, with a number in each square. Let's call that two-dimensional.

It's not too difficult to imagine how we'd add a third dimension, for example how many pairs you sold on each date of each type in every two-hour period that the shop is open. Instead of a flat table with rows and columns, you'd have a cuboid, with dates 1 January to 31 December along the length; 'type of shoe' across the width; and 'time of day sold' down the side. This divides the cuboid into little cells, each with a number in it: I've done a rough example and filled in a few cells which show that you sold:

- 5 pairs of women's shoes on 4 January between 9 am and 11 am;
- 2 pairs of babies' shoes on 4 January between 1 pm and 3 pm; and
- 4 pairs of babies' shoes on 6 January between 9 am and 11 am.


The number '4', for example, is aligned with 6 January lengthways, with 9 - 11 am vertically and with 'baby' across; the number '4' is aligned in three dimensions.

Once you have pictured the cuboid, it is also easy to imagine that instead of the labels down the side being 'time of day' they would be 'shoe size', and then presenting the same raw data but with different numbers in all the cells, which is still three-dimensional.

But what happens if you try to show all four 'dimensions' (i.e. date; type of shoe; time of day; and shoe size) at the same time? We can imagine writing down these variables each time a pair of shoes is sold, so we know the data is* not complicated, but what does it look like (short of copping out and sub-dividing each day into hours lengthways)?

Well, all you have to do is sub-divide the three-dimensional cuboid space into four dimensions, so that each number can be clearly aligned with date (in one dimension, call it 'lengthways'); with 'type of shoe' (in another dimension, call it 'across'); with time of day (another dimension, call it 'down') and with shoe size (another dimension, call it ...)

* I refuse to treat 'data' as a plural noun, even though it would be the grammatically correct thing to do.