Flatland and Planiversal Machines

It is amazing how certain things make an lasting, even an indelible, impression. For example, when Edwin Abbott Abbot (sic) wrote Flatland: A Romance of Many Dimensions in 1884, he could not have imagined that his satire taking place in a two-dimensional universe would inspire a feature film in 2007. I couldn't, but here it is, all the same.

Just as unlikely, but equally true, there is another film called Flatland, also inspired by the same book, and released in the same year. Here's the trailer.

You can read the original book at Project Gutenberg here and a lovely digitized copy of the original publication on the Internet Archive here.

I caught the bug at third hand. In the July 1980 edition of Scientific American, Martin Gardner published an article that included the most wondrous two-dimensional machines: lock and key combinations, steam engines, and more. One could look at the diagram and comprehend the entire operation of the machine. Since the wonderful internet has provided me with so many seemingly lost treasures, I went searching for that article and found it. It is called "The Wonders of a Planiverse."

The article was, in fact, the review of a 98-page monograph called  Two-Dimensional Science and Technology. published in 1979 by A.K. Dewdney, a professor of Computer Science at the University of Western Ontario. His home page is here. Gardner's review exercised the same fascination on many others as it had on me: over two-thousand letters and comments reached Dewdney. As a result, he edited A Symposium on Two-Dimensional Science and Technology, which came out in 1981; The Second Symposium on Two-dimensional Science and Technology, in 1986; and a novel, The Planiverse, in 1986.

Let us have a look at a couple of the machines that captured my attention. They are comprised of just a few building blocks that would work as well in a two-dimensional space as a three-dimensional one.

To make a faucet, combine them like this.

To make a steam engine, combine them like this.

As Dewdney writes, and Gardner quotes,

It is amusing to think that the rather exotic design pressures created by the planiversal environment could cause us to think about mechanisms in such a different way that entirely novel solutions to old problems arise. The resulting designs, if steriversally practical, are invariably space-saving.

This is true, but they might not be as easily manufactured. An addendum to Gardner's article gives a letter from John S. Harris, of Brigham Young University's English Department, who points out that one famous engineer did design two-dimensionally.

As I examined Alexander Keewatin Dewdney's planiversal devices in Martin Gardner's article o n science and technology in a two-dimensional universe, I was struck with the similarity of the mechanisms to the lockwork of the Mauser military pistol of 1895. This remarkable automatic pistol (which had many later variants) had no pivot pins or screws in its functional parts. Its entire operation was through sliding cam surfaces and two-dimensional sockets (called hinges by Dewdney). Indeed, the lockwork of a great many firearms, particularly those of the 19th century, follows essentially planiversal principles. For examples see the cutaway drawings in Book of Pistols and Revolvers by W. H . B. Smith. 

Gardner suggests an exhibit of machines cut from cardboard, and that is exactly how the firearms genius John Browning worked. He would sketch the parts of a gun on paper or cardboard, cut out the individual parts with scissors (he often carried a small pair in his vest pocket), and then would say to his brother Ed, "Make me a part like this." Ed would ask, "How thick, John?" John would show a dimension with his thumb and forefinger, and Ed would measure the distance with calipers and make the part. The result is that virtually every part of the 100 or so Browning designs is essentially a two-dimensional shape with an added thickness. 

This planiversality of Browning designs is the reason for the obsolescence of most of them. Dewdney says in his enthusiasm for the planiverse that "such devices are invariably space-saving." They are also expensive to manufacture. The Browning designs had to be manufactured by profiling machines: cam-following vertical milling machines. In cost of manufacture such designs cannot compete with designs that can be produced by automatic screw-cutting lathes, by broaching machines, by stamping, or by investment casting. Thus although the Browning designs have a marvelous aesthetic appeal, and although they function with delightful smoothness, they have nearly all gone out of production. They simply got too expensive to make.

Three-D printers may help to change that.