Interpolate & Extrapolate

Interpolate (pronounced in-tur-puh-leyt)

(1) To introduce (something additional or extraneous) between other things or parts; interject; interpose; intercalate; to make additions, interruptions, or insertions.

(2) In mathematics, to estimate (a value of a function) between the values already known or determined.

(3) To alter a text by the insertion of new matter (with a long history of being applied especially if done deceptively or without authorization but technically a neutral term and can be used either way).

(4) To insert (additional or spurious material) in this manner.

1605–1615: From the Latin interpolātus, past participle of interpolātus & interpolāre (to make new, refurbish, touch up; to give a new appearance to), the construct being inter- (between, among, together) + -polā- (verb stem (akin to polīre (to smooth or polish) + -tus (the past participle suffix) from polare, from the primitive Indo-European root pel- (to thrust, strike, drive), the connecting notion being "to full cloth".  The sense evolved in Latin from the neutral "refurbish" to the slightly more loaded "alter appearance of" to the actually accusative "falsify” (especially or specifically by adding new material".  By the early fifteenth century Middle English had gained interpolen in a similar sense and by the 1650s also interpolator, from the Late Latin interpolator (one who corrupts or spoils), agent noun from past participle stem of Latin interpolāre.  The noun interpolation (that which is interpolated) dates from the 1670s and appears to have evolved both from the seventeenth century French interpolation and directly from the Latin interpolationem (nominative interpolatio) from the past participle stem of interpolāre.  Interpolate, interpolated & interpolating are verbs, interpolater (or interpolator) & interpolation are nouns, interpolable, interpolatory, interpolative are adjectives and interpolatively is an adverb.

Extrapolate (pronounced ik-strap-uh-leyt)

(1) To infer (an unknown) from something that is known; an evidence-based conjecture.

(2) In statistics, to estimate (the value of a variable) outside the tabulated or observed range.

(3) In mathematics, to estimate (a function that is known over a range of values of its independent variable) to values outside the known range.

(4) To perform extrapolation.

1830s: The construct was extra- + -polate (extracted and borrowed from interpolate).  The verb extrapolate in the sense of “make an approximate calculation by inferring unknown values from trends in the known data" became popular among astronomers, statisticians, economists & mathematicians after appearing in an 1862 Harvard Observatory account of Comet Donati (Donati's Comet (C/1858 L1 & 1858 VI)) in 1858).  In contemporary accounts, it was said to have been a word used since the 1830s by English mathematician and astronomer Sir George Airy (1801-1892).  Extrapolation (an approximate calculation made by inferring unknown values from trends in the known data) dates from 1867 and was the noun of action from extrapolate by analogy with the long-established interpolation although the original sense was "an inserting of intermediate terms in a mathematical series", the transferred sense of "drawing of a conclusion about the future based on present tendencies" adopted since 1889.  Extrapolate, extrapolated & extrapolating are verbs, extrapolater (or extrapolator) & extrapolation are nous, extrapolable, extrapolatory, extrapolative are adjectives and extrapolatively is an adverb.

Extrapolation and Interpolation

The common root of the words is the Latin verb (polīre) meaning “to polish” which in this context means “adding finish” to a data-set by adding what’s missing but the prefix is most useful in distinguishing between the two, inter- meaning “between” or “among,” and extra-, “outside” or “beyond”.  The two words look similar and at first glance it’d be not unreasonable to assume they might be antonyms but, although related in use and tangled in history, they are used in different ways and, one highly nuanced and the other sometimes applied correctly but inducing the drawing of erroneous or at least misleading conclusions.  Interpolation refers to inserting something between other things, while extrapolation is the act of drawing conclusions about something unknown based on what is known.  In mathematics, the meanings are uncontroversial in that interpolation is the process of determining an unknown value within a sequence based on other points in that set, while extrapolation is the process of determining an unknown value outside of a set based on the existing data (often expressed as a “curve”).  Interpolation is a commonly used tool of mathematicians, statisticians and others in the data-based sciences where it’s necessary to determine a function’s value based on the value of other points, an unknown value within the sequence is determined based on what else is in the sequence.

Interpolation, used beyond mathematics can be a loaded word because it’s the act of introducing something (additional or extraneous) between other parts, usually in text or musical notation and thus the technical equivalent of “insert” or (sometimes) “interject or interpose”.  Interpolation can thus be a merely neutral description but because of the history of the word (in Latin it evolved from the neutral "refurbish" to the slightly more loaded "alter appearance of" to the actually accusative "falsify” (especially or specifically by adding new material"), can imply that what has been inserted is spurious, false, misleading or done with some other nefarious purpose.  It’s thus a word which needs to be used with caution lest implications be drawn where no inference was intended.

A big word with lots of syllables, interpolate may be unfamiliar to many and that’s maybe why sometimes it’s been used apparently in an attempt to impart some sense of gravitas or perhaps disguise what’s really happening.  In pop music, sampling, the interpolation of other people’s music into one’s own is now probably a sub-genre and it’s well understood although, despite the involvement of courts and copyright lawyers, the distinctions between sampling, interpretation and actual appropriation although well-trimmed, remain frayed at the margins and all three can be interpolated.  One derided as a form of plagiarism, sampling seems to have gained respectability, at least among those who practice the art, the critical legal device apparently being to sample by using a fragment from a previously recorded song, but re-recording rather than directly copying the original.  The origin of the practice appears to be as the work-around for when the copyright holder refuses to license the original for sampling purposes.  Use in this way, only a publisher’s permission is required although in some common-law jurisdictions, the original can be subject to a compulsory licensing regime.

Extrapolation is related to deduction, an act of drawing a conclusion about something unknown based on what is known so the verb extrapolate is often used synonymously with infer and deduce.  However, in mathematics, while the act of interpolation involves a closed data set with defined low and high values, extrapolation involves estimating the value of a variable or function outside an observed range so it can be necessary to understand the context (social, economic etc) of the numbers being used in the exercise.  A Roll-Royce dealership which has a good month and sells ten cars should probably not from that data-set extrapolate that in the year ahead they will sell 120; other factors need to be considered beyond the simple math.

Xanax (Alprazolam), a fast-acting benzodiazepine.  It is marketed as anti-anxiety medication.

Lindsay Lohan released the track Xanax in 2019.  With a contribution from Finnish pop star Alma (Alma-Sofia Miettinen; b 1996), the accompanying music video was said to be “a compilation of vignettes of life”, Xanax reported as being inspired by Ms Lohan’s “personal life, including an ex-boyfriend and toxic friends”.  Structurally, Xanax was quoted as being based around "an interpolation of" Better Off Alone, by Dutch Eurodance-pop collective Alice Deejay, slowed to a Xanax-appropriate tempo.

Lindsay Lohan risked going straight to Hell by creating a promotional meme featuring Pope Francis (b 1936; Roman Catholic Pope since 2013).  Cryptically captioned Blessed Be The Fruit, it included an image of the art-work used for her debut album Speak (2004).  Given the problems he's expected to manage, solve or conceal (depending on the circumstances), most would forgive the pope if he popped the odd Xanax.

The original photograph (top left) was taken in 2013 during a mass conducted in the Catedral Basílica do Santuário Nacional de Nossa Senhora Aparecida (Cathedral Basilica of the National Shrine of Our Lady Aparecida) in Aparecida, Brazil.  His Holiness was at the time administering communion.  It has since proved a popular photograph for meme-makers interpolating optical discs.

Xanax by Lindsay Lohan

I don't like the parties in LA, I go home

In a bad mood, pass out, wake up alone

Just to do it all over again, oh

Looking for you

Only one reason I came here

Too many people, I can't hear

Damn, I got here at ten

Now it's 4 AM

I can't be in this club

It's too crowded and I'm fucked

Ain't nobody here for love

Ain't nobody care about us

I got social anxiety, but you're like Xanax to me, yeah

Social anxiety, when you kiss me, I can't breathe

No, I can't be in this club

It's too crowded and I'm fucked

Ain't nobody here for love

Ain't nobody care 'bout us

 

I got social anxiety, but you're like Xanax to me, yeah

Social anxiety, when you kiss me, I can't breathe, yeah

 

But you're like Xanax to me

When you kiss me, I can't breathe

 

I try to stay away from you, but you get me high

Only person in this town that I like

Guess I can take one more trip for the night

Just for the night

 

Only one reason I came here

Too many people, I can't hear

Damn, I got here at ten

Now it's 4 AM

 

I can't be in this club

It's too crowded and I'm fucked

Ain't nobody here for love

Ain't nobody care about us

I got social anxiety, but you're like Xanax to me, yeah

Social anxiety, when you kiss me, I can't breathe

No, I can't be in this club

It's too crowded and I'm fucked

Ain't nobody here for love

Ain't nobody care 'bout us

 

I got social anxiety, but you're like Xanax to me, yeah

Social anxiety, when you kiss me, I can't breathe, yeah

 

But you're like Xanax to me

When you kiss me, I can't breathe

 

But you're like Xanax to me

When you kiss me, I can't breathe

Xanax lyrics Universal © Music Publishing Group

Epitaph

Epitaph (pronounced ep-i-taf or ep-i-tahf)

(1) A commemorative inscription on a tomb or mortuary monument about the person buried at that site.

(2) A brief poem or written passage composed in commemoration of a dead person.

(3) A final judgment on a person or thing.

(4) To commemorate in or with an epitaph.

(5) To write or speak after the manner of an epitaph. 

1350–1400: From the Middle English epitaphe (inscription on a tomb or monument), from the Old French epitafe, from the twelfth century Old French epitaphe, from the Latin epitaphium (funeral oration, eulogy), from the Ancient Greek epitáphion (over or at a tomb; a funeral oration), (noun use of neuter of ἐπιτάφιος (epitáphios) ((words) spoken on the occasion of a funeral), the construct being epi- (From the Ancient Greek ἐπί (epí) (at, over; on top of; in addition to (in a special use in chemistry, it denotes an epimeric form))) + τάφος (táph(os)) (tomb) + -ion (the noun-adjectival suffix).  Táphos (tomb, burial, funeral) was related to taphē (interment) & thaptō (to bury) of uncertain origin.  It has long been thought derived (like the Armenian damban (tomb)) from the primitive Indo-European root dhembh- (to dig, bury) but recent scholarship has cast doubts and some etymologists suggest both the Armenian and Greek could be borrowings.  There were equivalent words in the Old English and regional variations were many; the one which survived longest was byrgelsleoð.

The companion words, which differ not only in nuance but in convention of use, include eulogy (an oration about the dead, delivered usually at a funeral or memorial service), obituary (something in written form published soon after death which provides a potted biography and epigraph (a quote engraved on a tombstone, variously plaintive, humorous or barbed).  Not quite the same but very to the point is the Latin hic jacet (literally “here lies”).  Epitaph is a noun or verb (used with object), epitaphic, epitaphial, epitaphed & epitaphless are adjectives, epitaphically is an adverb and epitaphist is a noun.  The noun plural is epitaphs.

,

Jonathan Swift's marble memorial, St Patrick's Cathedral, Dublin.

One of the most celebrated epitaphs in English was saeva indignatio (literally “savage indignation”) which appeared on the tomb of the delightfully wicked Anglo-Irish satirist & poet Jonathan Swift (1667-1745), expressing a resigned contempt at human folly.  Swift is probably best remembered for Gulliver's Travels (1726) but it was A Modest Proposal (1729) which defined the genre of satire and work in this vein is often still labeled "Swiftian".  Swift started his political life as a Whig but ended it a Tory, becoming an Anglican cleric who was appointed Dean of St Patrick's Cathedral, Dublin.

Swift not only wrote his own epitaph but left instructions also for the stonemason and the authorities of Saint Patrick’s Cathedral, the memorial to be rendered in black marble, mounted seven feet from the ground, the large letters to be deeply cut and strongly gilded.  His specifications were followed but the stridency of Swift's Latin displeased a few who, finding it harsh or inelegant, didn't always reproduce it with complete fidelity.  The translation into modern English is Here is laid the body of Jonathan Swift.....where savage indignation can no longer tear his heart. Depart, wayfarer, and imitate if you can a man who to his utmost strenuously championed liberty.  Fellow Irish poet William Butler Yeats (1865–1939) rendered it as the punchier Swift has sailed into his rest; savage indignation there cannot lacerate his breast.  Imitate him if you dare, world-besotted traveller; he served human liberty.

Epitaph (1990) by Charles Mingus (CBS–466631 2).

Charles Mingus (1922–1979) was an American double bassist, pianist, composer and bandleader and one of the seminal figures in jazz.  Although lauded for the way his bands would interpolate passages of collective improvisation into performance pieces, he was influential also in his structured compositions, some of which were, by the standards of the genre, unusually long.  None however matched his Epitaph, comprising over four-thousand measures (a grouping of beats, which indicates the meter of a particular piece of music) and demanding more than two hours to perform, ranking with epic-length pieces such as Wynton Marsalis’s (b 1961) Blood On The Fields (1997) and Carla Bley’s (b 1936) Escalator Over The Hill (1968-1971); only Wadada Leo Smith’s (b 1941) sprawling Ten Freedom Summers (2012), unfolding over five hours, runs longer.

It’s not clear how long Mingus worked on Epitaph and its gestation may have absorbed as long as Ten Freedom Summers (thirty-four years in the making) because fragments of Epitaph were performed as early as 1962 although whether it was then envisaged as what it became is unknown.  It was only after his death, while Mingus’s work was being catalogued, that the whole of Epitaph was assembled and the score compiled.  This enabled the piece to be performed in 1989 by a thirty-piece orchestra, conducted by Gunther Schuller (1925-2015) and produced by Mingus's widow, Sue Graham Mingus (b circa 1933).  It has since had a number of performances, several in 2007, and the complete score has been published.

Lindsay Lohan reading the epitaphs, graveyard scene in I know who killed me (2007).

Epitaph, full of melodies, is rewarding and not entirely unfamiliar because Mingus over the years included several snatches in live recordings and concerts preformed with smaller bands, playfully sampling the music of a few others in sections although that’s not typical of Epitaph, a work all have noted for its originality.  A two-hour suite for thirty-one musicians is not necessarily unwieldy but Epitaph is complicated and really demands a band both familiar with each-other and well-rehearsed.  It’s not the sort of piece suited to an ensemble, however virtuosic, assembled for a one-off performance and the definitive performance which one day will be released will likely have been carefully edited and polished from any number of studio sessions.  Technically, it’s challenging for a conductor, there are shifts between melodic strains which sometimes are sudden and sometimes overlap, parts apparently unresolved skid to a stop, tempos pick-up at various paces and there’s an underlying cross-talking between extreme-register instruments; doubtlessly it's no less difficult for the musicians, two pianists, two bassists, a drummer and two percussionists needing peacefully to co-exist although, this is Mingus and that means creative tension is lives between the notes.  Even once détente was established however, there's still the piece itself to conquer, not all of it in the familiar language of jazz for there are vertiginous jumps in register, fast phrases slurring effortlessly to the languid and the jar sometimes of the polytonality of which American composers of the twentieth century were so fond.  Critics and other aficionados of the art were enchanted but it’s suspected there were those who dipped in and out of their CD and listened just to the bits they liked.

Algorithm

Algorithm (pronounced al-guh-rith-um)

(1) A set of rules for solving a problem in a finite number of steps.

(2) In computing, a finite set of unambiguous instructions performed in a prescribed sequence to achieve a goal, especially a mathematical rule or procedure used to compute a desired result.

(3) In mathematics and formal logic, a recursive procedure whereby an infinite sequence of terms can be generated.

1690s: From the Middle English algorisme & augrym, from the Anglo-Norman algorisme & augrimfrom, from the French algorithme, re-fashioned (under mistaken connection with Greek αριθμός (arithmos) (number)) from the Old French algorisme (the Arabic numeral system) from the Medieval Latin algorismus, a (not untypical) mangled transliteration of the Arabic ⁧الخَوَارِزْمِيّ⁩ (al-ḵawārizmiyy), the nisba (the part of an Arabic name consisting a derivational adjective) of the ninth century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī and a toponymic name meaning “person from Chorasmia” (native of Khwarazm (modern Khiva in Uzbekistan)).  It was Muḥammad ibn Mūsā al-Khwārizmī works which introduced to the West some sophisticated mathematics (including algebra). The earlier form in Middle English was the thirteenth century algorism from the Old French and in English, it was first used in about 1230 and then by the English poet Geoffrey Chaucer (circa 1344-1400) in 1391.  English adopted the French term, but it wasn't until the late nineteenth century that algorithm began to assume its modern sense.  Before that, by 1799, the adjective algorithmic (the construct being algorithm + -ic) was in use and the first use in reference to symbolic rules or language dates from 1881.  The suffix -ic was from the Middle English -ik, from the Old French -ique, from the Latin -icus, from the primitive Indo-European -kos & -ḱos, formed with the i-stem suffix -i- and the adjectival suffix -kos & -ḱos.  The form existed also in the Ancient Greek as -ικός (-ikós), in Sanskrit as -इक (-ika) and the Old Church Slavonic as -ъкъ (-ŭkŭ); A doublet of -y.  In European languages, adding -kos to noun stems carried the meaning "characteristic of, like, typical, pertaining to" while on adjectival stems it acted emphatically; in English it's always been used to form adjectives from nouns with the meaning “of or pertaining to”.  A precise technical use exists in physical chemistry where it's used to denote certain chemical compounds in which a specified chemical element has a higher oxidation number than in the equivalent compound whose name ends in the suffix -ous; (eg sulphuric acid (H₂SO₄) has more oxygen atoms per molecule than sulphurous acid (H₂SO₃).  The noun algorism, from the Old French algorisme was an early alternative form of algorithm; algorismic was a related form.  The meaning broadened to any method of computation and from the mid twentieth century became especially associated with computer programming to the point where, in general use, this link is often thought exclusive.  The spelling algorism has been obsolete since the 1920s.  Algorithm, algorithmist, algorithmizability, algorithmocracy, algorithmization & algorithmics are nouns, algorithmize is a verb, algorithmic & algorithmizable are adjectives and algorithmically is an adverb; the noun plural is algorithms.

Babylonian and later algorithms

An early Babylonian algorithm in clay.

Although there is evidence multiplication algorithms existed in Egypt (circa 1700-2000 BC), a handful of Babylonian clay tablets dating from circa 1800-1600 BC are the oldest yet found and thus the world's first known algorithm.  The calculations described on the tablets are not solutions to specific individual problems but a collection of general procedures for solving whole classes of problems.  Translators consider them best understood as an early form of instruction manual.  When translated, one tablet was found to include the still familiar “This is the procedure”, a phrase the essence of every algorithm.  There must have been many such tablets but there's a low survival rate of stuff from 40 centuries ago not regarded as valuable.

So associated with computer code has the word "algorithm" become that it's likely a goodly number of those hearing it assume this was its origin and any instance of use happens in software.  The use in this context, while frequent, is not exclusive but the general perception might be it's just that.  It remains technically correct that almost any set of procedural instructions can be dubbed an algorithm but given the pattern of use from the mid-twentieth century, to do so would likely mislead or confuse confuse many who might assume they were being asked to write the source code for software.  Of course, the sudden arrival of mass-market generative AI (artificial intelligence) has meant anyone can, in conversational (though hopefully unambiguous) text, ask their tame AI bot to produce an algorithm in the syntax of the desired coding language.  That is passing an algorithm (using the structures of one language) to a machine which interprets the text and converts it to language in another structure, something programmers have for decades been doing for their clients.

A much-distributed general purpose algorithm (really more of a flow-chart) which seems so universal it can be used by mechanics, programmers, lawyers, physicians, plumbers, carpet layers, concreting contractors and just about anyone whose profession is object or task-oriented.   

The AI bots have proved especially adept at such tasks.  While a question such as: "What were the immediate implications for Spain of the formation of the Holy Alliance?" produces varied results from generative AI which seem to range from the workmanlike to the inventive, when asked to produce computer code the results seem usually to be in accord with a literal interpretation of the request.  That shouldn't be unexpected; a discussion of early nineteenth century politics in the Iberian Peninsular is by its nature going to to be discursive while the response to a request for code to locate instances of split infinitives in a text file is likely to vary little between AI models.  Computer languages of course impose a structure where syntax needs exactly to conform to defined parameters (even the most basic of the breed such as that PC/MS-DOS used for batch files was intolerant of a single missing or mis-placed character) whereas something like the instructions to make a cup of tea (which is an algorithm even if not commonly thought of as one) greatly can vary in form even though the steps and end results can be the same.

An example of a "how to make a cup of tea" algorithm.  This is written for a human and thus contains many assumptions of knowledge; one written for a humanoid robot would be much longer and include steps such as "turn cold tap clockwise" and "open refrigerator door".

The so-called “rise of the algorithm” is something that has attracted much comment since social media gained critical mass; prior to that algorithms had been used increasingly in all sorts of places but it was the particular intimacy social media engenders which meant awareness increased and perceptions changed.  The new popularity of the word encouraged the coining of derived forms, some of which were originally (at least to some degree) humorous but beneath the jocularity, many discovered the odd truth.  An algorithmocracy describes a “rule by algorithms”, a critique in political science which discusses the implications of political decisions are being made by algorithms, something which in theory would make representative and responsible government not so much obsolete as unnecessary.  Elements of this have been identified in the machinery of government such as the “Robodebt” scandal in Australia in which one or more algorithms were used to raise and pursue what were alleged to be debts incurred by recipients of government transfer payments.  Despite those in charge of the scheme and relevant cabinet ministers being informed the algorithm was flawed and there had been suicides among those wrongly accused, the politicians did nothing to intervene until forced by various legal actions.  While defending Robodebt, the politicians found it very handy essentially to disavow connection with the processes which were attributed to the algorithm.

The feeds generated by Instagram, Facebook, X (formerly known as Twitter) and such are also sometimes described as algorithmocracies in that it’s the algorithm which determines what content is directed to which user.  Activists have raised concerns about the way the social media algorithms operate, creating “feedback loops” whereby feeds become increasingly narrow and one-sided in focus, acting only to reinforce opinions rather than inform.  In fairness, that wasn’t the purpose of the design which was simply to keep the user engaged, thereby allowing the platform to harvest more the product (the user’s attention) they sell to consumers (the advertisers).  Everything else is an unintended consequence and an industry joke was the word “algorithm” was used by tech company CEOs when they didn’t wish to admit the truth.  A general awareness of that now exists but filter bubbles won’t be going away but what it did produce were the words algorithmophobe (someone unhappy or resentful about the impact of algorithms in their life) and algorithmophile (which technically should mean “a devotee or admirer of algorithms” but is usually applied in the sense of “someone indifferent to or uninterested in the operations of algorithms”, the latter represented by the great mass of consumers digitally bludgeoned into a state of acquiescent insensibility.

Some of the products are fighting back: The Algorithm: How AI Decides Who Gets Hired, Monitored, Promoted, and Fired and Why We Need to Fight Back Now (2024) by  by Hilke Schellmann, pp 336, Hachette Books (ISBN-13: 978-1805260981).

Among nerds, there are also fine distinctions.  There are subalgorithms (sub-algorithm seems not a thing) which is a (potentially stand-alone) algorithm within a larger one, a concept familiar in many programming languages as a “sub-routine” although distinct from a remote procedure call (RPC) which is a subroutine being executed in a different address space.  The polyalgorithm (again hyphens just not cool) is a set of two or more algorithms (or subalgorithms) with instructions for choosing which in some way integrated.  A very nerdy dispute does exist within mathematics and computer science around whether an algorithm, at the definitional level, really does need to be restricted to a finite number of steps.  The argument can eventually extend to the very possibility of infinity (or types of infinity according to some) so it really is the preserve of nerds.  In real-world application, a program is an algorithm only if (even eventually), it stops; it need not have a middle but must have a beginning and an end.

There is also the mysterious pseudoalgorithm, something les suspicious than it may first appear.  Pseudoalgorithms exist usually for didactic purposes and will usually interpolate (sometime large) fragments of a real algorithm bit it may be in a syntax which is not specific to a particular (or any) programming language, the purpose being illustrative and explanatory.  Intended to be read by humans rather than a machine, all a pseudoalgorithm has to achieve is clarity in imparting information, the algorithmic component there only to illustrate something conceptual rather than be literally executable.  The pseudoalgorithm model is common in universities and textbooks and can be simplified because millions of years of evolution mean humans can do their own error correction on the fly.

Of the algorithmic

The Netflix algorithm in action: Lindsay Lohan (with body-double) during filming of Irish Wish (2024).  The car is a Triumph TR4 (1961-1967), one of the early versions with a live rear axle, a detail probably of no significance in the plot-line.

The adjective algorithmic has also emerged as an encapsulated criticism, applied to everything from restaurant menus, coffee shop décor, choices of typefaces and background music.  An entire ecosystem (Instagram et al) has been suggested as the reason for this multi-culture standardization in which a certain “look, sound or feel” becomes “commoditised by acclamation” as the “standard model” of whatever is being discussed.  That critique has by some been dismissed as something reflective of the exclusivity of the pattern of consumption by those who form theories about what seem not very important matters; it’s just they only go to the best coffee shops in the nicest parts of town.  In popular culture though the effect of the algorithmic is widespread, entrenched and well-understood and already the AI bots are using algorithms to write music will be popular, needing (for now) only human performers.  Some algorithms have become well-known such as the “Netflix algorithm” which presumably doesn’t exist as a conventional algorithm might but is understood as the sets of conventions, plotlines, casts and themes which producers know will have the greatest appeal to the platform.  The idea is nothing new; for decades hopeful authors who sent manuscripts to Mills & Boon would receive one of the more gentle rejection slips, telling them their work was very good but “not a Mills & Boon book”.  To help, the letter would include a brochure which was essentially a “how to write a Mills & Boon book” guide and it included a summary of the acceptable plot lines of which there were at one point reputedly some two dozen.  The “Netflix algorithm” was referenced when Falling for Christmas, the first fruits of Lindsay Lohan’s three film deal with the platform was released in 2022.  It was an example of followed a blending of several genres (redemption, Christmas movie, happy ending etc) and the upcoming second film (Irish Wish)  is of the “…always a bridesmaid, never a bride — unless, of course, your best friend gets engaged to the love of your life, you make a spontaneous wish for true love, and then magically wake up as the bride-to-be.” school; plenty of familiar elements there so it’ll be interesting to see if the algorithm was well-tuned.

Math of the elliptic curve: the Cox–Zucker machine can help.

Some algorithms have become famous and others can be said even to have attained a degree of infamy, notably those used by the search engines, social media platforms and such, the Google and TikTok algorithms much debated by those concerned by their consequences.  There is though an algorithm remembered as a footnote in the history of linguistic oddities and that is the Cox–Zucker machine, published in 1979 by Dr David Cox (b 1948) and Dr Steven Zucker (1949–2019).  The Cox–Zucker machine (which may be called the CZM in polite company) is used in arithmetic geometry and provides a solution to one of the many arcane questions which only those in the field understand but the title of the paper in which it first appeared (Intersection numbers of sections of elliptic surfaces) gives something of a hint.  Apparently it wasn’t formerly dubbed the Cox–Zucker machine until 1984 but, impressed by the phonetic possibilities, the pair had been planning joint publication of something as long ago as 1970 and undergraduate humor can’t be blamed because they met as graduate students at Princeton University.  The convention in academic publishing is for authors’ surnames to appear in alphabetical order and the temptation proved irresistible.

Clerestory

Clerestory (pronounced kleer-stawr-ee or kleer-stohr-ee)

(1) In architecture, a portion of an interior rising above adjacent rooftops, fitted with windows admitting daylight.

(2) In church architecture, a row of windows in the upper part of the wall, dividing the nave from the aisle and set above the aisle roof (associated particularly with the nave, transept and choir of a church or cathedral.

(3) In transportation vehicles (usually busses, railroad cars and occasionally cars & vans), a raised construction, typically appended to the roof structure and fitted with (1) windows to admit light or enhance vision or (2) slits for ventilation (or a combination of the two).

1375–1425: From the late Middle English, the construct being clere (clear (in the sense of “light” or “lighted”)) + story (from storey (a level of a building).  The word is obviously analyzed as “a story (upper level) with light from windows”.  Storey was from Middle English stori & storie, from the Anglo-Latin historia (picture), from the Latin, from the Ancient Greek ἱστορία (historía) (learning through research, narration of what is learned), from ἱστορέω (historéō) (to learn through research, to inquire), from ἵστωρ (hístōr) (the one who knows, the expert, the judge).  In the Anglo-Latin, historia was a term from architecture (in this case “interior decorating” in the modern sense) describing a picture decorating a building or that part of a building so decorated.  The less common alternative spellings are clearstory & clerstory.  Clerestory is a noun and clerestoried is an adjective; the noun plural is clerestories.

From here was picked up the transferred sense of “floor; level”.  The later use in church architecture of “an upper story of a church, perforated by windows” is thought simply to be a reference to the light coming through the windows and there is nothing to support the speculation the origin was related to a narrative (story) told by a series of stained glass windows, illuminated by sunlight.  Historians have concluded the purpose of the design was entirely functional; a way of maximizing the light in the interior space.  The related architectural design is the triforium.  The noun triforium (triforia or triforiums in the plural) (from the Medieval Latin triforium, the construct being tria (three) + for (opening) + -ium) describes the gallery of arches above the side-aisle vaulting in a church’s nave.  The –ium suffix (used most often to form adjectives) was applied as (1) a nominal suffix (2) a substantivisation of its neuter forms and (3) as an adjectival suffix.  It was associated with the formation of abstract nouns, sometimes denoting offices and groups, a linguistic practice which has long fallen from fashion.  In the New Latin, it was the standard suffix appended when forming names for chemical elements.

Clerestories which once shone: Grand Central Terminal (the official abbreviation is GCT although the popular form is "Grand Central Station" (often clipped to "Grand Central")), Midtown Manhattan, New York City, 1929 (left) and the same (now dimmed) location in a scene from the Lindsay Lohan film Just my Luck (2006) (right).  Because of more recent development in the surrounding space, the sunlight no longer enters the void through the clerestoried windows is such an eye-catching way.  In modern skyscrapers, light-shafts or atriums can extend hundreds of feet.

Tourist boat on a canal cruise, Amsterdam, the Netherlands.

Boats with clerestory windows are commonly seen on waterways like the canals of Amsterdam and are valued by tour-guides because they make excursions possible in (almost) all-weathers.  Those designing passenger busses and train carriages were also early adopters of clerestory windows, initially because they were a source of “free” light but as packaged tourism developed into “sightseeing”, tour operators recognized the potential and commissioned versions optimized for outward visibility, essentially a form of “value-adding” which made even the dreary business of bus travel from one place to the next more of a “sightseeing” experience, something probably most valued by those afforded a greater vista to observe in mountainous regions.

Greyhound Scenicruiser in original livery.

A variation of the idea was used for long-distance busses in North America, the best known of which remains the Greyhound Scenicruiser (PD-4501), featuring a raised upper deck with a clerestory windscreen.  Although the view through that was enjoyed by many passengers, the design was less about giving folk a view and more a way to maximize revenue within the length restrictions imposed by many US states.  What the upper deck did was allow a increase in passenger numbers because the space their luggage would absorb in a conventional (single layer) design could be re-allocated to people, their suitcases (and in some cases also freight, another revenue stream) relegated to the chassis level which also improved weight distribution and thus stability.

A predecessor: 1930 Lancia Omicron.

Famous as it became, the Scenicruiser, 1001 of which were built by General Motors (GM) between 1954-1956, was to cause many problems for both manufacturer and operator, the first of which caused by the decision to use twin-diesel engines to provide the necessary power for the new, heavy platform.  The big gas (petrol) units available certainly would have provided that but their fuel consumption would not only have made their operation ruinously expensive (both the fuel burn and the time lost by needing frequently to re-fill) and the volume of gas which would have to be carried would have both added to weight and reduced freight capacity.  Bigger GM diesel units weren’t produced in the early 1950s so the twin engines were a rational choice and the advantages were real, tests confirming that even when fully loaded, the coupled power-train would be sufficient for hill climbing while on the plains, the Scenicruiser happily would cruise using just a single engine.  However, as many discovered (on land, sea and in the air), running two engines coupled together is fraught with difficulties and these never went away, the busses eventually adopting one of GM’s new generation of big-displacement diesels as part of the major re-building of the fleet in 1961, a programme which also (mostly) rectified some structural issues which had been recognized.  Despite all that, by 1975 when Greyhound retired the model, the company still had hundreds in daily service and many of those auctioned off were subsequently used by other operators and in private hands, some are still running, often as motor homes or (mostly) static commercial displays or museum exhibits.

1951 Pegaso Z-403 (left) and 1949 Brill Continental (right).

GM’s Scenicruiser was influential and clerestory windscreens soon proliferated on North American roads although the idea wasn’t new.  The Spanish manufacturer Pegaso (a creation of Generalissimo Francisco Franco’s (1892-1975; Caudillo of Spain 1939-1975) industrial policy) between 1951-1957 produced the Z-403 (1951-1957) which used the same design and before even that a bus with an almost identical profile had been sold in the US by the JG Brill Company, albeit with a conspicuous lack of success.  As early as 1930 the Italian concern Lancia offered the Omicron bus with a 2½ half deck arrangement with a clerestoried upper windscreen.  The Omicron’s third deck was configured usually as a first-class compartment but at least three which operated in Italy were advertised as “smoking rooms”, the implication presumably that the rest of the passenger compartment was smoke-free.  History doesn't record if the bus operators were any more successful in enforcing smoking bans than the usual Italian experience.

1968 Oldsmobile Vista Cruiser.

One quirky offering which picked up the Scenicruiser’s clerestoried windscreen was the Oldsmobile Vista Cruiser (1964-1977), a station wagon which, until the release of the third series in 1973 featured one at the leading edge of a raised roof section, the glass ending midway over the backseat, the shape meaning it functioned also as a “skylight”.  Above the rear side windows were matching clerestories which might sound a strange thing to add above a luggage compartment but during those years, a “third seat” was a popular option to install in the space, transforming the things into eight or nine seat vehicles; families were bigger then.  The Vista Cruiser sold well and Buick later adopted the idea for some of its range until the concept was abandoned in 1972 because of concerns about upcoming safety regulations.  Such rules were however never imposed and both manufacturers revived the idea in the 1990s for their (frankly ugly) station wagons and when the Buick was discontinued in 1996, it was the last full-sized station wagon to be made in the US, the once popular market segment cannibalized to the point of un-viability by the mini-van (people-mover) and the sports utility vehicle (SUV).

The arrangement of a series of windows in a high-mounted row, borrowed from architecture, became familiar on train carriages and buses, especially those which plied scenic routes.  Usually these were added to the coachwork of buses built on existing full-sized commercial chassis which could seat 40-60 passengers but in the 1950s, there emerged the niche of the smaller group tour, either curated to suit a narrower market or created ad-hoc by hotels or operators; smaller vehicles were required and these offered the additional advantage of being able sometimes to go where big buses could not.  In places like the Alps, where those on the trip liked to look up as well as out, rows of clerestoried windows were desirable.

1959 Volkswagen Microbus Deluxe (23 Window Samba). 

The best known of these vehicles was the Volkswagen “Samba”, a variation of the Microbus, one of the range of more than a dozen a models built on the platform of the Type 2, introduced in 1950 after a chance sighting by a European distributer of a VW Beetle (Type 1) chassis which had been converted by the factory into a general-purpose utility vehicle.  The company accepted the suggestion a market for such a thing existed and in its original, air-cooled, rear-engined configuration, it remained in production well into the twenty-first century.  Between 1951-1966, the Microbus was available in a “Deluxe” version which featured both a folding fabric sunroof and rows of rows of clerestoried windows which followed the curve of the sides of the roof.  Available in 21 & 23 window versions, these are now highly collectable and such is the attraction there’s something of a cottage industry in converting Microbuses to the clerestoried specification but it’s difficult exactly to emulate the originals, the best of which can command several times the price of a fake (a perfectly restored genuine Samba in 2017 selling at auction in the US for US$302,000).  Such was the susceptibility to rust, the survival rate wasn’t high and many led a hard life when new, popular with the tour guides who would conduct bus-loads of visitors on (slow) tours of the Alps, the sunroof & clerestory windows ideal for gazing at the peaks.  To add to the mood, a dashboard-mounted valve radio was available as an option, something still for many a novelty in the early 1950s.  The Microbus Deluxe is rarely referred to as such, being almost universally called the “Samba” and the origin of that in uncertain.  One theory is it’s a borrowing from the Brazilian dance and musical genre that is associated with things lively, colorful, and celebratory, the link being that as well as the sunroof and windows, the Deluxe had more luxurious interior appointments, came usually in bright two-tone paint (other Type 2s were usually more drably finished) and featured lashings of external chrome.  It’s an attractive story but some prefer something more Germanic: Samba as the acronym for the business-like phrase Sonnendach-Ausführung mit besonderem Armaturenbrett (sunroof version with special dashboard).  However it happened, Samba was in colloquial use by at least 1952 and became semi official in 1954 when the distributers in the Netherlands added the word to their brochures.  Production ended in July 1967 after almost 100,000 had been built.

1966 Volkswagen Samba (21 Window, Left) and 1962 Volkswagen Samba (23 Window, Right).  The 23 Window van is a conversion of a Microbus and experts (of which there seem to be many, such is the following these things have gained) say it's close to impossible exactly to replicate a factory original.  In theory, the approach would be to take the parts with serial numbers (tags, engine, gearbox etc) from a real Samba which has rusted into oblivion (something not uncommon) and interpolate these into the sound body of a Microbus with as close a build date as possible.  Even then, such are the detail differences that an exact replication would be a challenge.  Because the Sambas received the same running changes and updates as the rest of the Microbus range, there was much variation in the details of the specification over the years but the primary distinction is between the “21” & “23” window vans, the difference accounted for by the latter’s pair of side-corner windows to the left & right of the rear top gate opening.  In 1964, when the rear doors were widened, the curved windows in the roof were eliminated because there would no longer be sufficient metal in the coachwork to guarantee structural integrity.