real world columns: sequences

The most common structure of things and information is the linear sequence. A collection of items arranged in one dimension with each item adjacent to two others. In England there is that wonderful physical manifestation of such a sequence; the queue. When more than one person is waiting to buy bread, catch a taxi or whatever, they will inevitably form an orderly line and take strict turns. But what are the abstract qualities of a sequence?

If we break down a sequence into its abstract essentials we have three key manifestations:

1) A loop, basically the idea of things arranged in one dimension with items having nearest neighbors. There is no beginning, no end and no direction.

2) A loop with one special item. By flagging one particular item in a loop we can identify a starting point. But there are still two possible directions to go in. A loop with one special item is not useful and has no physical manifestation that I can think of.

3) A loop with two special items. Now things start to get interesting and useful. We actually need two special unique items in order to be able to define a direction. Imagine item A and item B are uniquely identified and are next to each other. Using them we can specify the two different directions A to B and onwards and B to A and onwards.

This idea of direction only coming into being when we have a loop with two uniquely defined items is a bit strange. However, it becomes clearer when we think of a chain (a sequence that is not a loop). By splitting a loop and opening it out into a chain we basically identify two items in it; the item at one end and the item at the other end. Imagining the situation as a chain is a simpler conceptual handle on the same underlying situation as a loop with two unique items. Of course, once direction is possible like this we can include that as part of the specification.

When it comes to applying abstract sequences to the physical world there are all sorts of strange problems. There is the idea of ‘next’ and ‘previous’. There is also the whole science of the labeling of sequences. In the real world we are continually encountering and navigating sequences. From page numbers to floors in a tall building, from carriages in a train to seats in the theater.

Usually we are only aware of the power and ease of navigating sequences when something goes wrong. Take road junctions for example; in the UK, the road junctions on freeways (motorways) are numbered consecutively. This helps with error checking, if you are looking for exit 5 and you see signs for exit 6 then you know that you have gone too far and missed it. By some bizarre glitch of planning, England’s M37 freeway does not actually have a junction 6, the numbering goes straight from 5 to 7, completely disrupting the use of the numbers as an error checking device for those that do not know about this glitch. In Holland freeway junctions are known not by numbers but by the area they are in, which is fine in that it gives you contextual information, but it does then lose the sequential information that helps you to spot the errors of missing a junction.

As well as the obvious differences in numbering floors in different countries (I’d be willing to jump from a first floor window in America but not in England) there are strange examples of designed breaks in sequences. The mathematics tower in UMIST in Manchester (UK) used the letters of the alphabet to label the floors (planners and architects tried some weird ideas out in the sixties) someone anticipated problems with letters that looked like numbers such as I and O, so they took all the vowels out of the alphabet (and didn’t anticipate any problems with that!). The result is considerable confusion and head scratching for those using the elevators and especially the stairs. On the same theme there is the story (it must be fictitious, surely) of a tall building where floor 13 was left out because no one ever wanted an apartment there due to the unlucky connotations. This didn’t cause any big problems until someone tried to do a bungee jump off the top and based the length of the elastic on the number of floors, they overlooked the missing floor and as a result the elastic was twenty feet too long, with tragic consequences.

But back to queuing for bread; in Italy one summer we found ourselves in a very crowded bakers with absolutely no queue whatsoever. After getting to the front and standing there for what seemed like a week we worked out that the order of serving depended on who responded when the baker said; ‘next’. The problem was that we weren’t sure exactly which of his many words meant ‘next’. Eventually the locals at the front realized our predicament and after a hasty discussion and gesturing we were served. In that situation there was a sequence, but it had no physical manifestation. It was an abstract sequence, even more abstract than my opening analysis. It was a sequence distributed through the minds of many customers and servers and augmented with factors such as age, respect, attention, and favor with the baker. A situation which I would never want to try and represent in a software model!