Encoding with prime factors series

Aesthetic Computing chapter

Patterns and forms constructed of multitudes of similar, yet unique, objects always have been hallmarks of my work. Usually, when working with multitudes of objects, the exact number involved was arbitrary and the variation found in the individual objects often was based on random numbers.

In the Encoding with prime factors series, the numbers of objects involved are almost always particular and significant. The objects themselves have unique configurations, but rather than being based on random values, each object encodes a specific positive integer — based on the prime factorization of the integer. As that factorization is unique for every integer, each of the objects is unique.

The Works: Prime numbers section contains the entire series, along with other works incorporating prime numbers. The Glossary below defines some terms used in connection with this series.

This work is explained in depth in a chapter I wrote for Aesthetic Computing (MIT Press, 2006).

Glossary

“The beginning of wisdom is to call things by their rights names.” — Chinese Proverb

An artist’s lexicon

I do not create images that are meant to be purely representational, in the sense that I am not creating works meant to convey an interpretation of a particular physical object. At the same time, I do not think of the work as purely abstract. The images I create contain objects with a physical level of detail and realism. I am not bothered that they are often initially mistaken for photographs. This is part of the very purposeful ambiguity in my work. The idea that these things looks as if they could exist — that they simultaneously resemble any number of forms, objects or materials in a physically realistic manner — makes more powerful the moment of realization that they are entirely imagined and artificial.

Early on, I made the decision not to give verbal titles to my work. While I do not wish to give a particular verbal label to individual images, when thinking of the ideas I am exploring and discussing the themes in my work, it is very useful to maintain a consistent vocabulary. Internally, this vocabulary most often serves as a textual shorthand, allowing me to record ideas of greater complexity in my sketches than I would be able to draw in the same amount of time. The idea of visually encoding objects with representations of numbers based of the prime factorizations of those numbers is the most complex theme to date and has resulted in the largest of these vocabularies.

Some of the terms are generalizations or partial definitions from the world of mathematics, others are the words and phrases I have come to use in my sketches to describe specific ideas, objects and patterns. What follows are the definitions for the terms as I use them.

A prime number is a positive integer greater than one that is wholly divisible (no remainder) only by itself and one. Positive integers other than one which are not prime are called composite numbers. The factors of a positive integer are the integers that evenly divide it. The fundamental theorem of arithmetic states that every positive integer greater than one can be expressed uniquely as a product of prime numbers, apart from the rearrangement of factors. This unique group is known as the set of prime factors of the particular number.

Source integers are the positive integers — those integers greater than or equal to one. This is the numerical domain of this body of work.

An encoded object is the embodiment or manifestation of a single source integer. The encoded object is constructed of one or more elements — individual visual indications (form, material, color, etc.) representing some part of the encoding of a particular source integer, the way that the dots and dashes make up visual representations of the letters and numbers of Morse Code.

The combination of forms and materials and the application of those forms and materials in a specific way to visually encode the prime factorization of a particular positive integer is an encoding scheme.

I use the term identity to refer to the multiplicative identity of one in the series of positive integers and to refer to a single, specific element in each encoded object. The identity in mathematics is the number one and is so called because multiplying any number by one results in the original number. (This is a simplified version of the mathematical concepts of identity and multiplicative identity.) In that sense and in this work, the number one also is a factor of any number. I take advantage of that fact by including a specific element which not only encodes the multiplicative identity of one but also provides a visual starting point from which the object can be decoded. The identity element in a sense marks the beginning of the sentence that is an encoded number.

By using the identity and always using it as the starting point for interpretation, the encoded object may be oriented in three-dimensional space without constraint. Encoded objects may be constructed from left to right, right to left, clockwise, counterclockwise, up, down or sideways — in whatever direction best suits the theme and composition of the encoding scheme and the final image.

The reading spine of an encoded object starts at the identity element and continues through the encoded elements in the order in which they are encoded and interpreted. In some encoding schemes, the reading spine is visible, in others it is implied.

Each positive integer greater than one can be factored into a unique sequence of prime factors. Each of the factors of the given integer is a used factor. Those prime numbers which are not factors of the given number and which fall between one and the largest prime factor for the given integer are considered skipped factors. The idea of a skipped factor is distinct from the mathematical concept of a prime gap, defined as the number of positive integers between two consecutive prime numbers (e.g., the prime gap between 23 and 29 would be 5). Prime numbers larger than the largest factor of a given integer are ignored. Because of this, an encoded object will always end with a used factor element.

Indicator elements are those which show that either a single factor in the sequence of prime numbers is used (to the first power) or a single factor is skipped.

The purpose of a bracket element varies with whether the bracket is showing a used factor (with an exponent greater than one) or a series of unused factors in the sequence of prime numbers. Brackets for used factors enclose the exponent of the factor. The exponent is recursively encoded following the same rules as the overall encoding. Brackets for skipped factors enclose the encoding of the number of factors in the sequence of prime numbers which are to be passed over in the interpretation of the visual encoding. (It will be illustrated later that the nesting of brackets can cause ambiguity if the opening and closing brackets are not visually differentiated.)

External resources

Two excellent sources for definitions of mathematical terms are Wolfram MathWorld and The Prime Pages’ Prime Glossary.