Position Coding

Advertised as “a computer in a pen”, the Fly Pentop computer (ref web site) hit the stores in the fall of 2005. One application of the pen is a four-function calculator that the user draws on a special kind of paper called “Fly paper”. The pen seems to know what numbers have been written and where on the paper the numbers lie, because after tapping on what was drawn (e.g. 5, +, 3, 7, =), the computer announces the result of “42”. The pen can also read handwriting, and there are a number of games printed on “Fly paper” that are based on the pen knowing the location of objects on the paper. How does the pen know where symbols and objects are on the paper? And what does the following quote from a recent article (cite Wired) about this computer mean: “The pen knows where it is within a 1.8-million-square-mile grid”? The Fly Pentop Computer uses paper which has a position code lightly printed on it. A position coding pattern is an array of symbols in which subarrays of a certain fixed size appear at most once. So, each subarray uniquely identifies a location in the larger array, which means there is a bijection of some sort from this set of subarrays to a set of coordinates. The key to “Fly paper” and other examples of position codes is a method to read the subarray and then convert it to coordinates. So, for the calculator application, when the calculator is first drawn, the pen reads a symbol such as “2” through optical character recognition, and also reads the position of the “2” by interpreting the position code on the paper. When the “2” is later tapped, the position code is read again, and the fact that a “2” is there is recovered from the pen’s memory. Position coding makes use of ideas from discrete mathematics and number theory. In this paper, we will describe the underlying mathematics of two position codes, one being the Anoto code that is the basis of “Fly paper”. Then, we will present two new codes, one which uses binary wavelets as part of the bijection.