Thursday 10 November 2022

#33 Lune, Lens and related terms.....from old Math Terms notes

Lune, Lens and related terms

The word lune is from Luna, the Roman Goddess of the Moon. The more ancient Indo-European root is leuk, which relates to light. Lunatic is from the same root, perhaps because the ancients believed it was the effects of the moon that accounted for the crazy behavior.
Lune is used in mathematics to describe two different ideas. [see figure below] The most common is the area on a sphere between two semicircles with endpoints at the same poles. Imagine two lines of longitude running from the North Pole to the South pole. The surface of the Earth between the two lines is called a lune. There is also a two dimensional shape called a lune. The crescent area formed between two excentric circles that share a common chord is also called a lune. This shape plays an important part in the mathematics of the ancient Greeks. The definition I have used is from "The Mechanic's Assistant: A Thorough Practical Treatise on Mensuration" by D. M. Knappen, published in 1849. The Wolfram Mathworld web site, which is usually one of the top sources for math topics defines the planer lune this way, "A lune is a plane figure bounded by two circular arcs of unequal radii, i.e., a crescent." These two definitions do NOT seem to be the same to me. Imagine the crescent formed by two excentric circles of the same radius. I wonder why this must be excluded from the Wolfram definition. Under either construction, there are two crescent regions formed. Students will not see many planer lunes in their math experience, but the ones they do seem almost all to be constructed so that the smaller radius circle has its center on the common chord of the two circles. I do not think this is a property of lunes in genral. To be sure, I asked the memebers of the Philomathes Yahoo discussion group, and Colin McLarty responded with, "Neither arc needs to have its center on the common chord, and Hippocrates did a case where neither one does. Since he was interested in squaring lunes, and he was so far as we know pretty much limited to Euclidean means, he necessarily dealt with only very special cases. It is unlikely he ever worried about a precise general definition of "lune" -- esp. whether or not the case of equal radii would qualify to be called a lune." [see a suggested definition from David W. Cantrell farther below]

A spherical lune is sometimes also called a digon, since it is a two-sided polygon (and less appropriately called a bigon since that mixes both Latin and Greek roots... and leads too often to the quip, "let bigons be bigons".)

The plane lune was studied in depth by the Greek mathematician Hippocrates of Chios (around 400 BC) in an attempt to find a way to find a square with the same area as a given circle using only the classic tools of compass and straightedge. He discovered that a lune formed by the chord which is the hypotenuse of a central right angle has an area equal to the area of the right triangle. This lune is still called the Lune of Hippocrates. Later he found two other plane lunes that could also be "squared". A nice site on the lunes which can be squared is here. The problem of "Squaring the Circle" was one of the three great unsolved problems of Greek mathematics. Much later it was proven using algebraic tools that the task was impossible, but the Greeks discovered lots of important mathematical ideas in their search to prove the impossible.

Hippocrates of Chios, the mathematician, is often confused with Hippocrates of Cos, who is considered to be the father of medicine, and for whom the Hippocratic Oath is named. This should not be two unexpected since they both lived at the same time, and came from relatively close islands off the coast of modern day turkey. This link shows a map of the area. There is an arrow pointing out the island of Cos, just northwest of Rhodes. North of Cos just south of Lesbos is the island of Chios. Between them is the Island of Samos, which was the birthplace of Pythagorus.

Two similar looking objects that may be included here are "lens" and "gibbous". The area of intersection between two circles is often called a lens, because it resembles a biconvex lens. The word lens actually is drawn from the word lentil, for its symmilarity to the lentil seed. Gibbous is derived from the Latin root for "hunchback" and can be applied to any object that is convex or rounded, but is most commonly used to describe the phases of the moon between a half-moon and a full-moon.

There is, or once was, a special name for the intersection between two identical circles and is called the Vesci Pescies. Here is a description from "Geometry in Art" by Hilton Andrade De Mello. "The 'Vesica Piscis' is an important symbol based on the circle. It is generated by the intersection of two identical circles, as shown in Figure 9.2 with one circle centered at point A and the other centered at point B. The area common to both circles, shown in blue, is called 'Vesica Piscis' because it resembles a vesicle, i.e., a receptacle. As seen later in this chapter, the “Piscis”, which is Latin for 'fish', is associated with the symbol adopted by early Christianity." Images below are from the e-text of the book.


David W. Cantrell suggested what I think are very clear definitions of lune and lens in a discussion on another list, and graciously sent me a copy: ”Given two circular disks, A and B, having a nonempty intersection and such that neither is entirely contained in the other, three regions are formed. (Think of a Venn diagram.) One of the regions, A intersect B, is convex; I suggest that "lens" be used to name that kind of shape. Neither of the other two regions, A-B and B-A, is convex; I suggest that "lune" be used to name that kind of shape. “

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