Linearity

In common usage, linearity refers to a mathematical relationship or function that can be graphically represented as a straight line, as in two quantities that are directly proportional to each other, such as voltage and current in an RLC circuit, or the mass and weight of an object.

Example

A crude but simple example of this concept can be observed in the volume control of an audio amplifier. While our ears may (roughly) perceive a relatively even gradation of volume as the control goes from 1 to 10, the electrical power consumed in the speaker is rising geometrically with each numerical increment. The "loudness" is proportional to the volume number (a linear relationship), while the wattage is doubling with every unit increase (a non-linear, exponential relationship).

In mathematics

In mathematics, a linear map or linear function f(x) is a function that satisfies the following two properties:

The homogeneity and additivity properties together are called the superposition principle. It can be shown that additivity implies homogeneity in all cases where α is rational; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers. If f is assumed to be continuous as well, then this can be extended to show homogeneity for any real number α, using the fact that rationals form a dense subset of the reals.

Writing system

A writing system is any conventional method of visually representing verbal communication. While both writing and speech are useful in conveying messages, writing differs in also being a reliable form of information storage and transfer. The processes of encoding and decoding writing systems involve shared understanding between writers and readers of the meaning behind the sets of characters that make up a script. Writing is usually recorded onto a durable medium, such as paper or electronic storage, although non-durable methods may also be used, such as writing on a computer display, in sand, or by skywriting.

The general attributes of writing systems can be placed into broad categories such as alphabets, syllabaries, or logographies. Any particular system can have attributes of more than one category. In the alphabetic category, there is a standard set of letters (basic written symbols or graphemes) of consonants and vowels that encode based on the general principle that the letters (or letter pair/groups) represent speech sounds. In a syllabary, each symbol correlates to a syllable or mora. In a logography, each character represents a word, morpheme, or other semantic units. Other categories include abjads, which differ from alphabets in that vowels are not indicated, and abugidas or alphasyllabaries, with each character representing a consonant–vowel pairing. Alphabets typically use a set of 20-to-35 symbols to fully express a language, whereas syllabaries can have 80-to-100, and logographies can have several hundreds of symbols.

Nonlinear gameplay

A video game with nonlinear gameplay presents players with challenges that can be completed in a number of different sequences. Each player sees only some of the challenges possible, and the same challenges may be played in a different order. Conversely, a video game with linear gameplay will confront a player with a fixed sequence of challenges: every player sees every challenge and sees them in the same order.

A nonlinear game will allow greater player freedom than a linear game. For example, a nonlinear game may permit multiple sequences to finish the game, a choice between paths to victory, or optional side-quests and subplots. Some games feature both linear and nonlinear elements, and some games offer a sandbox mode that allows players to explore an open world game environment independently from the game's main objectives, if any objectives are provided at all.

A game that is significantly nonlinear is sometimes described as being open-ended or a sandbox, though that term is used incorrectly in those cases, and is characterized by there being no "right way" of playing the game. Whether intentional or not, a common consequence of open-ended gameplay is emergent gameplay.