Trapezoid

Trapezoid (pronounced trap-uh-zoid)

(1) In British English, a quadrilateral plane figure having no parallel sides.

(2) In US English, a quadrilateral plane figure having two parallel and two non-parallel sides.

(3) In anatomy, a small bone in the wrist that is situated near the base of the index finger and that articulates with the second metacarpal, trapezium, capitate, and scaphoid bones.

1706: From the New Latin trapezoīdēs, from the Late Greek trapezoeidḗs (trapezium-like in shape), coined by the mathematician Euclid (4^th-3^rd century BC) from trapeze (literally “table”) + -oeides (shaped).  Originally it described a quadrilateral figure a quadrilateral figure having only two sides parallel, the purists insisting that describes a trapezium.  The terms trapezium and trapezoid have swapped meanings in North America compared with the rest of the world.  Trapezoid & trapezium are nouns, trapezoidal is an adjective; the noun plural is trapezoids or trapeziums depending on one's definition.

The trapezoid’s different definitions

Of all the squabbles in the world, one of the most improbably enduring concerns the definition of what constitutes a trapezoid.  Disagreements between mathematicians are of course not unusual and those involving the efficacy of this or that proof of some arcane have lasted sometimes centuries but differences over the definition of something basic are rare.  They do however happen.  It was once the case that in the UK the value of a billion was held to be a million millions, on the logical basis that a million was a thousand thousands.  The Americans decided that was silly because a number expressing a million millions was (at the time) of use only to cosmologists and argued, no less logically that to set a billion at a thousand millions would make it genuinely useful.  In the end, thankfully, the US view prevailed.

An irregular quadrilateral in the UK, a trapezium in the US (left); a trapezium in the UK, a trapezoid in the US (right).

But it seems strange there could be differences over what makes a trapezoid, a flat 2D shape with four straight sides.  Classically, it had one pair of parallel sides (usually the top and bottom); the parallel called the bases, the non-parallel the legs (the distance from one base to another is known as the altitude).  The faction which holds a trapezoid has only one pair of parallel sides (meaning they can never be parallelograms) adheres to what’s known as the exclusive definition.  The others (now in the majority) believe trapezoids have at least one pair of parallel sides, so they can be a special type of parallelogram and this is known as the inclusive definition which encompasses the “taxonomy of quadrilaterals” (an ordered group category of quadrilaterals (four-sided shapes)).  In the US and Canada a quadrilateral shape with at least one pair of parallel sides is known as a trapezoid. This is what is called a trapezium outside those countries but fortunately, mathematicians everywhere agree there are three main types of trapezoids:

(1) The right trapezoid (a shape with a pair of right angles).
(2) Isosceles trapezoid (a shape in which the non-parallel sides have the same length).
(3) Scalene trapezoid (a shape where all four sides are of unequal length).

Peugeot 504 with the original “trapezoid” headlamp lens (left), the four-lamp arrangement used in the US, Australia (after 1973) and some export markets (centre) and the single light adopted for the “poverty spec” & later utility models (right).

The Peugeot 504 in one form or another was in production for almost half a century, lasting from 1968 until 2006 although the mainstream (non-utility) models were discontinued in France in 1983.  A machine of extraordinary virtue which was refined enough for European roads yet sufficiently robust successfully to endure the harsh conditions often found in the former colonies of the old French Empire, it also gained an admiring audience in rural Australia where distances were vast, roads often rough and mechanics sometimes hard to find.  Among cars using a conventional suspension (no hydraulics, air bellows or other exotica), only Jaguar’s equally remarkable XJ6 (another debutante from 1968) could match its ride quality.  The 504’s headlights were often described as trapezoid but that was never true (on either side of the Atlantic) because not only were no two sides parallel, the outer was actually curved.  At most they could be could be thought trapezoidish or trapezoidesque but some were not even that.  As a means to assist in producing low-cost variants, some were made with a single, circular lamp instead cost-saving measure while versions with four (smaller) units were offered in some markets.  In the US this was to satisfy laws introduced as an industry-protection measure while in Australia it was to assist in meeting the local content rules which offered taxation advantages for the assembly plants although the change was popular with rural buyers used to frequently broken lights on their unsealed roads, the round units cheap and available at every petrol (gas) station or country store.

A woman in Chador contemplates placing a bid on Mr Ahmadinejad's then 34 year old Peugeot 504, Abadan, Iran, February 2011.

In November 2010, it was announced President Mahmoud Ahmadinejad (b 1956; president of the Islamic Republic or Iran 2005-2013) would be auctioning his 1977 Peugeot 504 to raise money for a charity dedicated to providing affordable housing for low-income families.  With low mileage and said to be in immaculate condition, there was much interest and what proved to be the most expensive 504 ever made was sold in March 2011 to the highest bidder for US$2.5 million.  The 504's engine and suspension were used in later models of the Paykan, the long running (1966-2015) Iranian version of the Rootes Group's Arrow (the best known version of which was the Hillman Hunter (1967-1979)).  The transplant came about because in 1978 Peugeot-Talbot took control of Chrysler's European operations, the US corporation having in 1967 absorbed Rootes Group.

A metallic gray Peugeot 504.

Another infamous Peugeot 504 owner was Archbishop Paul Marcinkus (1922–2006; president of the Institute for the Works of Religion (the "Vatican Bank") 1971-1989) who had a metallic gray one with a “lovely leather interior”.  The archbishop was gay which was nothing unusual among the bishops and cardinals in the Vatican (indeed it’s probably unusual not to be gay in such circles) but even in that colourful milieu his sexual appetite was considered “in the upper range” and his fondness for Swiss Guards was both well known and the subject of some mirth among the Curia; often the archbishop would "lend" his Peugeot to his favourite Swiss Guards to use on their expeditions for this and that.  Highlights of the archbishop’s life included (1) being for a time ensconced behind the Vatican walls and protected by diplomatic immunity to ensure he’d not have to face interrogation from the various authorities interested in matters related to certain transactions at the Vatican Bank, (2) being associated with Italian banker Roberto Calvi (1920–1982), chairman of the bank Banco Ambrosiano (which collapsed in 1982) and known as Banchiere di Dio (God's Banker); found hanged from the scaffolding under London's Blackfriars Bridge) and (3) being accused of complicity in the murder of Pope John Paul I (1912–1978; pope August-September 1978).  There’s no compelling evidence the 33 day pontificate of JPI ended with his murder and nor is there anything but “a bit of circumstantial” suggest Marcinkus may have been involved.  Most historians concluded JPI died from natural causes and the marvellous conspiracy theory hinges on the suggestion the pope was planning to institute reforms in the Vatican Bank which had been linked to financial corruption involving the Banco Ambrosiano and the Propaganda Due (P2) Masonic Lodge.

Archbishop Marcinkus (far left) & Paul VI (1897-1978; pope 1963-1978, centre left) meet US Baptist preacher & civil rights activist Reverend Dr Martin Luther King Jr (1929–1968, centre right) & US Baptist preacher & civil rights activist Reverend Ralph Abernathy Sr (1926–1990, far right), the Vatican, 18 November 1964.  The US delegation was on the Rome leg of a European tour.

Even by the standards of the cult, the P2 Masonic Lodge was secretive which was understandable given how deeply it was involved in political corruption, financial irregularities and organized crime and finally it was banned in 1981 after the extent of its criminality became just too much for elements within the Italian state to continue the protection for years provided.  Founded in 1945, P2 originally was just another Masonic lodge under the Grand Orient of Italy but it came under the control of businessman and fascist Licio Gelli (1919–2015) who became Venerable Master and transformed it into a kind of shadow state.  In the manner Masons have practiced for centuries, P2 infiltrated institutions and recruited influential figures including politicians (including then media figure Silvio Berlusconi (1936-2023; prime minister of Italy 1994-1995, 2001-2006 & 2008-2011), military officers, judges, journalists and businessmen, most with some degree of fascist sympathy.  Essentially, P2’s agenda was a kind of “MIGA” (Make Italy Great Again) program and in the way these things are done, their plan for a latter day Il Risorgimento (Resurgence, the nineteenth century movement which culminated in the unification of Italy (1961)) their project, although presented as a “Plan for Democratic Rebirth” was actually a plot to undermine democracy and take control of the government.  P2 deeply was implicated in the scandals swirling around the Vatican Bank and the collapse of Banco Ambrosiano in 1982 and when finally investigated, almost a thousand officials and others were found to have ties to the cult; the prime minister, Arnaldo Forlani (1925–2023; Prime Minister of Italy 1980-1981), was compelled to resign although at the time of his death he was both the oldest living and longest-lived Italian prime minister so there was that.  As the scandal unfolded, Venerable Master Licio Gelli was expelled from the Masons (presumably because he’d committed the unforgivable sin of “being caught”) and arrested, triggering years of court cases, escapes from custody, hiding in other countries and pleading not guilty.  In the Italian way, despite receiving long sentences he spent very little time in prison and by 2003 seemed content P2 had in a way succeeded because the “democratic rebirth plan” was “being implemented by Silvio Berlusconi”.

Peugeot 504 Coupé: The early headlamp treatment (1969-1974, left) and the later (1974-1983, centre).  The 2018 Peugeot E-Legend concept car (right).

The rather lovely 504 coupé & cabriolet (1969-1983) both eschewed the trapezoidesque, the early versions using four lamps sometimes described a “quartic” but, being rectangles with rounded ends, they’re actually closer to what mathematicians call a “stadium”.  Later models used essentially the same internals but were mounted behind a single lens in a shape called a “rounded rectangle”.  Although not quintessentially “French” like the classic Citroëns (DS, GS, SM, CX), the 504 coupé & cabriolet were elegant and capable and it a shame the French industry had produced nothing to match them in recent decades.  Peugeot’s designers however seem aware of the appeal and in 2018 displayed their E-Legend concept car, an attractive take on the 504 coupé’s lines.  Nothing like it ever reached production and the range remained dreary and predictable.

Variations of the trapezoid are widely used shapes for handbags.  Noted handbag fan Lindsay Lohan (in brunette, blonde and red) demonstrates some of the interpretations.