orthogonality: one construct, many systems

In linear algebra, two vectors are orthogonal when their inner product is zero (geometrically, at right angles), and an orthogonal set spans its space along mutually independent directions, so each component can be read off without interference from the others. Schmidt's 1907 Zur Theorie der linearen und nichtlinearen Integralgleichungen gave the orthogonalisation procedure, now called Gram–Schmidt, for turning any independent set into an orthogonal one. The referent is independence-as-perpendicularity among directions in a vector or function space, the abstract idea that reappears, unchanged, in experimental design and in software architecture.[1]

Unit

The inner product between two directions, zero when independent

Recovery means

Not applicable

Measured by

Checked by a zero inner product between directions, or an identity Gram matrix for a whole set

In experimental design, a design is orthogonal when its factors are arranged so that the effect of each can be estimated independently of the others (the columns of the design matrix are mutually orthogonal) so that treatments do not confound and the analysis separates cleanly. Fisher's The design of experiments built this independence into factorial and balanced block designs. It is the same perpendicular-directions construct as in linear algebra, now realised in how data are deliberately collected rather than in an abstract space.[2]

Unit

The off-diagonal cross-products of the design, driven to zero

Recovery means

Not applicable

Measured by

Verified when factor contrasts are uncorrelated across the design (an orthogonal design or array)

In software engineering, orthogonality is the design principle of giving components non-overlapping responsibilities so that changing one has no unintended effect on the others, keeping systems easier to understand, test, and modify. Hunt and Thomas's The pragmatic programmer made the term a working maxim for practitioners. It is the identical independence-and-non-interference construct as orthogonal vectors or orthogonal designs, transposed to the structure of a codebase.[3]

Unit

The degree of coupling between components, driven toward independence

Recovery means

Not applicable

Measured by

Judged by how localised the effect of a change is: low coupling and clear separation of concerns