Tuesday, 29 May 2018

Kites and Darts and Dragons (Oh MY!)

The geometric description of a quadrilateral with two pairs of congruent adjacent sides (some definitions require the two pairs to be distinct, so that a rhombus is NOT a kite) is drawn from the name of the flying toy which it resembles (although in many cultures the flying kites are not at all shaped like the ones in the U.S.). The toy itself probably drew its name from the bird commonly called a kite, or kyte, in England. The old English form of the word, cyte, is probably from an early German name for an Owl. The OED dates the word kete for the bird as early as 725 and the first reference for its use in reference to a toy was in 1664. John Conway has stated that the use of "kite" for the mathematical object is only a few decades old. He writes, "a few decades ago most authors would say something like ...a "kite-shaped" quadrilateral .... leaving those quotation marks in to show that they felt a bit uneasy about using this informal term in a technical sense. The toy "kite" was in term named after the bird, and about a century ago, people would have put explicit quote marks around this word when using it for the toy!" Given the early dates for its first use for a toy in the OED, I have some reservations about this last part.

It is not clear whether it was the shape of the bird, or its flight behavior that was responsible for its name being given to the toy. A non-convex kite is often called a dart, which I credit to Roger Penrose  who used the name in a proof on a non-periodic tilling of the plane.

Proclus calls this shape the four-sided triangle and speaks of it as a geometric paradox. In Ancient writings it is also called barb-like and hollow-angled.

In a recent discussion group, John Conway responded to the question "Is there a name (other than "kite") for a quadrilateral that looks like a kite --with no parallel sides, but with two pairs of equal sides?" His response was to describe a suggestion for a new word, Strombus. Here is the message in Mr Conway's own words.
I was trying to coin an acceptable word for this for a long time, without success until after being prompted by some considerable discussion on the net about a year ago, I eventually came up with "Strombus", which is derived from the Greek word for a spinning top.
I think it's the best of the terms that were suggested. It's interesting that the word "rhombus" is ultimately derived from the same source, a fact that lends the new term some respectability.
John Conway 

More recently, (2018) I occasioned to hear a Spanish speaking mathematician, Ignacio Larrosa @ilarrosac, describe the shape as a "comet", and when I inquired he added, "In Spanish, 'cometa' is used, and google translated comet this time. For the concave is sometimes used 'saeta', a somewhat poetic synonym of 'flecha' = arrow." On inquiring of other of my Twitter Mathematicians, I found that, in Germany they call both shapes a Dragon, (Olaf Doschke @OlafDoschk) but they call the one a "konk-Drachenv." which abbreviates konkaves Drachenviereck (concave dragon). He even included a tree diagram of the terms. Do note the one on the top right, which we would call an anti-parallelogram (if the two opposite sides are the same length) or if two of the sides are parallel, then a cross-trapezoid (trapezium in UK).

Vincent Pantaloni, who has co-authored a book on "Geometry Snacks" says he has never seen either the convex or concave shape defined in a textbook in France.  He did suggest though, that some might call it a "cerf-volant", which he says is French for the toy kite.

I also have an English textbook that uses trapezion (note the n ending) for the shape we more commonly call a kite. In A Junior Geometry by Noel S. Lydon published in 1903 the definition on page 55 states trapezion is a four-sided figure having two pairs of adjacent equal sides. It goes on to show the method of construction.

During the summer of 2002 while visiting a Chinese exhibit, Land of the Dragon, at the Dennos museum in Traverse City, Michigan, I happened upon the following historical tidbit about kites. I have paraphrased from notes scribbled at the scene.

In 206 BC, General Han Hsin sought to take the castle of an enemy. He flew a kite over the center of the castle, and measured the amount of string that was used. Then he had his scholars calculate the distance to the center of the castle on ground level, and dug a tunnel under the walls of the castle to the described distance and emerged to take the fortress by complete surprise. This little military anecdote is the first historical mention of a kite.
The name for "kite" in Chinese is fengzheng. Feng relates to the wind, and I had originally translated the Kanji for zheng to mean 'to oppose, to resist'. From this I had interpreted a literal meaning something like "wind fighter". Actually, zheng in fengzheng is either 1) a kind of harpsichord, or 2) a kite. A fengzheng then literally means a kite, from the aeolian (wind) harps sometimes attached to the string. [Thanks to Jeffrey Hayden for his corrections in July 2005.

The museum display suggested that the kite made its way out of China during the Tang dynasty (around 700 AD) and spread through Asia and Europe. From the site of "http://www.kiteman.co.uk" I found:
"Its thought that kites were first introduced into Japan by Buddhist missionaries who travelled from China in the Nara period (649-794 AD) and were mainly used in religious and thanks giving ceremonies.

A Japanese dictionary dated 981 AD was the first to record the Japanese word for kite and used the characters for "Kami Tobi" meaning paper hawk - which suggests, to me, that the first kites in Japan were likely bird shaped.  "The Japanese absorbed much of the Chinese culture but they developed their own distinctive kite designs and traditions. They were used from earliest times for practical purposes such as in the construction of many shrines and temples in Japan where large kites were used to lift tiles and other materials up to workers on the roof tops." (Kiteman)

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