Patterns (e.g., such as images, waveforms such as sounds, electromagnetic waves, or other signals, etc.) are synthesized, modulated and/or analyzed through the use of a computer programmed with a novel mathematical formula. The formula acts as a linear operator and can be used to create a variety of shapes, waveforms, and other representations. The formula greatly enhances ability in computer operations and provides a great savings in computer memory and a substantial increase in computing power.