Orthogonal

Orthogonal (pronounced awr-thog-uh-nl)

(1) Of, pertaining to or involving right angles; perpendicular

(2) In mathematics (sometimes as orthographic), pertaining to or involving right angles or perpendiculars.

(3) In mathematics, of a system of real functions defined so that the integral of the product of any two different functions is zero; of a system of complex functions defined so that the integral of the product of a function times the complex conjugate of any other function equals zero.

(4) In mathematics, of two vectors having an inner product equal to zero.

(5) In mathematics, of a linear transformation defined so that the length of a vector under the transformation equals the length of the original vector.

(6) In mathematics (and applied fields such as engineering or statistics), of a square matrix defined so that its product with its transpose results in the identity matrix.

(7) In crystallography, referable to a rectangular set of axes.

(8) Figuratively, something having no bearing on the matter at hand; independent of or irrelevant to another thing or each other.

(9) In art, (1) the descriptor of the lines of perspective which can be mapped onto an image pointing to the vanishing point & (2) in the literature of art criticism a technical term which refers to work consisting exclusively of horizontal or vertical line and thus angles which, if they exist, are right angles.

1565–1575: From (the now obsolete) orthogonium (right triangle), from the French orthogonal, from either the Late Latin orthogōnium & orthogōnālis, from the Latin orthogōnius (right-angled).or directly from the Greek orthogṓnion (neuter) (right-angled), the construct being ortho- + -gōn(ion) + -al.  Ortho- (straight, correct; proper), was from the Ancient Greek ὀρθός (orthós), from the Proto-Hellenic ortwós, from the primitive Indo-European hr̥dwós, from herd- (upright) and was cognate with the Latin arduus and the Sanskrit ऊर्ध्व (ūrdhvá).  The –gon element was from the Ancient Greek γωνία (gōnía) (corner, angle), from the primitive Indo-European ǵónu (knee).  The -al suffix was from the Middle English -al, from the Latin adjectival suffix -ālis, or the French, Middle French and Old French –el & -al.  It was use to denote the sense "of or pertaining to", an adjectival suffix appended (most often to nouns) originally most frequently to words of Latin origin, but since used variously and also was used to form nouns, especially of verbal action.  The alternative form in English remains -ual (-all being obsolete).  Orthogonal is a noun & adjective, orthogonality & orthogonalization are nouns and orthogonally & orthonormal are adverbs; the noun plural is orthogonals.

As adjectives, orthogonal & orthographic are synonymously and the choice is dictated by preference, habit or the rhythm of the text although, to avoid confusion, they probably shouldn’t both be used in the same document.  The Ancient Greek ὀρθογώνιον (orthogṓnion) and the Classical Latin orthogonium originally denoted a rectangle and it was in this sense the words were used by the early mathematicians although the use was later extended to mean a right triangle.  By the twelfth century (especially among engineers and architects) the post-Classical Latin orthogonalis came to mean a right angle or something related to a right angle.  In the modern era, in science and mathematics, derived forms have been coined as required (biorthogonal, psuedo-orthogonal et al) and there’s also the mysterious semiorthogonal which would seem oxymoronic given orthogonal is a description of a mathematically defined absolute.  In figurative use, orthogonal is used to suggest something is unrelated or irrelevant to whatever is being discussed but because it’s so rarely used outside of mathematics, engineering, architecture or art criticism, it’s probably a term to avoid though it may be worth a point or two in Philosophy 101.

Lindsay Lohan in an unusual cage cutout top, the lines assuming or relaxing from the orthogonal as the body moves (maybe an instance of "a shifting semiorthogonal").  The combo was a Black Pash shirt with & Alaia skirt, The World's First Fabulous Fund Fair in aid of The Naked Heart Foundation, The Roundhouse, London, February 2015.   The t-strap sandals were by Miu Miu) but an an opportunity was missed by not adding a sympathetic clutch purse.

Richard Nixon, the Franklins & the Orthogonians

When Richard Nixon (1913-1994; US president 1969-1974) attended Whittier College in California (1930-1934), it was still formally affiliated with the Religious Society of Friends (the Quakers), a link it would maintain until the post-war years.  However, in many ways it was little different to wholly secular institutions in the US (except that one hour a day in chapel was mandatory), including the fraternities and sororities, the still almost exclusively single-sex, student-run societies which were established on many a basis and have evolved variously although a number of the male-only ones often attract attention related to their epic levels of alcohol consumption.  As was the case with many colleges until recent decades, many of those which Richard Nixon found when he arrived had been formed as literary societies and while there were four sororities (for women), there was but one fraternity.

That was the Franklin Society which in 1921 had been the first fraternity founded at Whittier College, beginning as a literary society that based itself on the "virtues" espoused by US founding father and polymath (and confessed Freemason) Benjamin Franklin (1706-1790).  Nixon, then unpolished and obviously a shop-keeper’s son, thought them snobby and elitist, an opinion either formed or reinforced when the Franklins rebuffed attempts to join and readily he accepted the suggestion be assume the presidency of the fraternity formed by other students resentful at being rejected by the Franklins.  This was the Orthogonians, the name (based on the now standard translation “right angles”) meaning “the straight shooters”, their motto an earthy “beans, brains, brawn and bowels”.  The Orthogonians did not enjoy the black tie lifestyle of the Franklins and weren’t invited to the best parties but several of Nixon’s biographers have traced from his fraternity experience many of the characteristics which would remain identifiable throughout this political career.  He learned that in life there are few stars but many supporting players and the man who can align himself with their interests you can gain their loyalty, something of real practical value in systems where everyone has one vote; it was the origin of his idea that elections and contests of ideas can be won by being appealing to the “silent majority”, something which would emerge as a political strategy during his presidency.  It taught him also that being hated was no obstacle to political success as long as one was hated by the right people, something he proved by beating the Franklin’s nominee for the position of student body president.  To this day, the Frankin’s website still boasts: “Among those who have been denied membership to this exclusive society is former president Richard Nixon.”  The Orthogoian Society still exists.

The master and the apprentice: Donald Trump and Richard Nixon, Houston, Texas, 1989.  It’s said the subject of Iran was raised in their discussions.

Some biographers have made much of the Franklin-Orthogonian contrast in the making of Nixon the younger.  In Nixonland (2008), Rick Perlstein’s (b 1969) thesis was that between 1965-1972, Nixon crafted a national conflict by exploiting the the mutual fear and hatred between the country’s elite Franklins and the “ordinary people”, the Orthogonians.  Some criticized the approach but Nixonland was a vivid approach to the era in which the divisions in the US became more exposed than they had been for a century and which was a prelude to the cross-cutting cleavages which have for decades characterized the country’s politics.  Nixon didn’t invent the politics of resentment but he expressed them in a language more easily understood by even the unsophisticated and more offensive than ever to those he labeled “the elites”.  In their own ways, to their radically different constituencies, Bernie Sanders (b 1941; senior US senator (Independent, Vermont) since 2007) and Donald Trump (b 1946; US president 2017-2021) are both inheritors of the Nixon legacy, two Franklins telling the Orthogonians they’re here to help.