Mathematics of fax machines

Summary: Fax machines revolutionized the process of sending and receiving documents.

A fax machine enables documents, including illustrations and other graphical elements, to be transmitted over a distance and reproduced by the receiver. The roots of the word “facsimile” are from the Latin words facere, meaning “to make,” and similis, meaning “like.” In the nineteenth century, Alexander Bain developed what some refer to as the first fax machine. His system transmitted information using analog telegraph lines. The sending and receiving equipment was timed using matched pendulums. At the receiving end, an electrically powered stylus recorded messages on a roll of paper. Current from the stylus turned the chemical coating on the paper blue, transcribing the signals’ dots and dashes. Frederick Bakewell demonstrated a chemical fax machine at the 1851 London Exhibition, and the first commercial telefax service began operation in 1865, predating the telephone.

A more modern ancestor is the radio facsimile, developed in 1924, which used radio waves to wirelessly transmit images and is still used in the early twenty-first century to transmit weather information. Modern fax machines scan an input sheet line by line to produce rows of pixels. Algorithms used in fax machines take advantage of the fact that there are white and black pixels in order to compress the data. For example, David Huffman’s variable-length lossless codes and their variations, originally invented in the 1950s, assign binary codes to patterns of pixels using probabilistic methods. The codes are shorter than the strings they replace, reducing overall file size. To optimize compression, symbols with higher probabilities or frequencies of occurrence are assigned shorter codes. The International Telecommunications Union, based in Geneva, Switzerland, makes recommendations for data compression standards. To derive one code called the “Group 3 code,” the organization applied the Huffman algorithm to eight representative samples to assign a code to each run length. Fax machines transmit documents in minutes instead of hours thanks to compression algorithms.

Fax Machine Technology

Modern fax machines utilize the technology of the telephone and the copy machine. Fax machines developed in the 1970s could scan a document and encode and transmit it over telephone lines to another fax machine, which could record and reproduce the document. Fax machines became common in offices as they replaced the need to send paper documents by messenger service or mail, were much quicker than retyping a document for telex, and could send any type of graphical information. Japan played an important role in developing modern fax machines, which used electronic circuits to replace mechanical parts and greatly increased the speed of transmission and reduced the size and price of the machine. Because the Japanese language incorporates many Chinese characters (kanji), the ability to transmit graphical images was particularly useful in that country.

Sending a document by fax requires two fax machines—one to send the document and one to receive it. The sending machine uses a sensor to scan the document, usually line by line, and to convert the pattern of black and white elements on the page into a code (several coding standards exist). The fax machine is not “reading” text—in the sense of converting the letters into meaning—but only recording their shape. For this reason, fax machines are as adept at sending images and diagrams as they are at sending text. The scanned data are compressed in order to reduce the number of bits to be transmitted and thus to speed up the process. The speed of transmission depends in part on how much information, such as text or diagrams, as opposed to blank space is contained on the page being scanned. The receiving fax machine decodes and uncompresses the information and uses it to re-create and to print the sent document. In the 1980s, most fax machines used thermal printing, which required the use of special paper that turns black when exposed to heat. However, in the twenty-first century, most fax machines print on standard white copy paper using either laser or inkjet printing technology.

Internet fax (efax, or online fax) technology has supplemented and, in some cases replaced, the use of traditional fax machines. There are a number of different services offering Internet fax capability, and although they differ in some details (for instance, can the machine receive, send, or both) the principle is the same: they provide a means to transmit facsimile documents to and from computers either as e-mail attachments or through a dedicated phone number or Internet site.

Bibliography

Brain, Marshall. “How Fax Machines Work.” http://communication.howstuffworks.com/fax-machine.htm.

McConnell, Kenneth R., Dennis Bodson, and Stephen Urban. FAX: Facsimile Technology and Systems. 3rd ed. Boston: Artech House, 1999.

Salomon, David. A Guide to Data Compression Methods. New York: Springer, 2002.

Smithsonian Institution. “From Carbons to Computers: The Changing American Office.” http://www.smithsonianeducation.org/scitech/carbons/start.html.