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

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) 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 gas station or country store.  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.

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.

The trapezoid is a widely used shape for handbags.  Noted handbag fan Lindsay Lohan demonstrates some of the variations.