Imagine that I am going to do a practical demonstration which will involve folding paper. I wish to fold a sheet several times. It will get smaller after each fold, so I will start with a big sheet.....How big a sheet?.....In order to be precise ~ and modern ~ I 'go metric' and start with a sheet that measures one metre by one metre. I will be able to fold that lots of times.
When I fold it once ~ into half ~ into halves ~ you choose ~ (either is acceptable, I am told) ~ I still have the same amount of paper. As I look at the folded result it has taken on a different shape. I started with a square shape. I now have an oblong ~ rectangle ~ shape. If I fold that again ~ short edge to short edge ~ I then have a square. The square now measures 50cmx50cm since there are 100cm in one metre.
I could set this up as my standard paper size system. Maybe I call it the Square System ~ or the 'S' system for short. It annoys me that there are two or more different shapes, and I wonder if there is another system that preserves the shape. It turns out that there is. Someone else got there ages ago ~ a hundred years or more ago. The system has been set out in great detail and recognised internationally as the 'A' system. What is more there are 'B' and 'C' systems also.
The 'B' system has the same idea, but starts with a different size of paper. The International Standards Organization ~ ISO ~ chose to start the 'B' series with a very large sheet measuring one metre wide and 1.414 metres tall. The 1.414m sounds odd, but there is an interesting explanation which lends a tiny bit of credibility to my 'S' system. Otherwise the 'S' system, using a square as the beginning shape, is a non-starter [117511].
To go back to where I was at the top of the page. If I start with a large sheet of paper ~ 1000mmx1414mm ~ and fold it over and over ~ then the various half sizes make up the 'B' series of papers. To get all the sizes you have to halve the largest number each time. Pictorially you get this.
It is a good idea to count the number of folds. With one fold we call the new size B1. With two folds it becomes B2 ~ then B3 ~ B4 ~ B5, and so on. Becasue this is a thought experiment we caould go on indefinitely. If we are really folding the larger sheet there is a limit to the number of folds that is possible.
It helps to remember that of the two numbers ~ 1000 ~ 1414 ~ only the largest one is halved each time. This gives B1=1000x707 B3= 500x353.5
The International Standards Organization publishes a list of defining properties. This saves a lot of time expaining the differences between foolscap, imperial, pints, gallons, pecks, chains and Nebuchadnezzars. ISO216 lists a set of paper sizes. All countries recognize them. Most nations have seen the advantages of adopting them, although traditional measure will continue to be used for a long time ~ especially in some specialist areas. Artists still buy paper in Imperial sizes
The 'A' and 'B' series of paper sizes have a huge advantage of any other system. In theory it has to do with mathematics and ratios. In practice the continuity of shape with change of size is very important ~ and is the only system that offers that advantage. Hence the World-wide addoption.
I next start thinking about the rectangle shape that I made. Every fold causes the shape to change, but every other one is either square or rectangle. All the squares have a diffierent size, but are the same shape. All the rectangles have the same shape, but are different sizes. If I fold one of the rectangles long edge to long edge then I get yet another ~ different ~ shape of rectangle. It becomes a much thinner shape. That is why I wrote short-edge to short edge.
Going back to one of the squares I decide to trim a thin rectangle away from one side of it. You could try this. Start with any square piece of paper. Trim a slice off the side. The task I have set myself is to try to make a ...but that's a thought for another time and page...