Friday 29 March 2024

Rhombus from Math Terms Notes

 The rhombus is a quadrilateral with four sides of congruent length(and whether you include squares, or don't, is a matter of taste.) It is sometimes called a rhomb, and sometimes a diamond and sometimes, if you are French especially, a lozenge. In classical Latin, a rhombus was a diamond shaped instrument that was whirled on a string to make a whirring sound, aha, but why. 

If you have ever been to a rodeo, you have probably been impressed with the agility and courage of the rodeo clowns as they distract the bull after the rider departs from the bull. You may even think it was a sport created by cowboys of the old American west. The truth however, is that playing tag with a bull may date back to the ancient Greek civilizations around 2000 BC. Archaeologists working on Minoan ruins found pots with illustrations that seemed to show that taunting a bull was a popular pastime for young males of that culture. The picture below is from a fresco found when the castle of King Minos was excavated.

It seems that sometimes, however, the bull was bored by the whole routine. It is hard to be macho if the bull is doing his "Ferdinand" routine and smelling the daisies so, to prod the animal into more ferocious activity, the young men began twirling an object on a string around their heads that made a roaring noise.

You may have seen other objects used to make a sound in a similar manner. Hopi Indians use something like that in their dances and you may have seen your science teacher twirl a length of plastic tube to make various "roaring" sounds. Such objects today are called bull roarers. I always thought it was because they were presumed to sound like a bull. Now I am less sure. The ancient Minoan object that twisted as it twirled and made the roaring sound was called a rhomb. The root began to be used in words that suggested rotation or twisting motions, such as spinning tops, but none of the others seem to have made it into modern language. The use that did prevail was for shapes that looked like the four sided object that they swung on the end of the string. This is how we came to call the equilateral quadrilateral a rhombus... and that's no bull.

Euclid uses the word rombos and in his translation Heath says it is apparently drawn from the Greek word rembw, to turn round and round. He also points out that Archimedes used the term solid rhombus for two right circular cones sharing a common base. Euclid extended the idea in using rhomboid to name the shape we more commonly call a parallelogram. Since the definition of rhomboid (romboeides) comes before the definition of parallel lines, Euclid defines the rhomboid as (in Heath's words)," that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled."

The term rhomb is often used for the same shape, and many people, particularly young students, refer to a rhombus as a diamond. Some use the term only for non-square rhombi (or rhombuses). It is especially interesting to work with early elementary students who will identify a shape as a square when the sides are horizontal and vertical, and then call it a diamond when the shape is rotated 45 degrees, even while they watch. The word diamond seems to be a mutation of the word adamant. A person is adamant if they are firm or unyielding in their attitude or position. The origin seems to be from the common root anti and the Greek word deme which meant to force or break (as in training an animal) which is the root of our present word domesticate. Together the two roots meant unbreakable, and the word was commonly used for hard metals and gems, and extremely difficult people.

The French word lozenge is also used for the non-square rhombus by some people, although I have never seen the term in a current math book. The word comes from the Gaulish word lausa for "flat stone". 
Lozenge was used by Robert Recorde in 1551 in Pathway to Knowledge: "Defin., The thyrd kind is called losenges or diamondes whose sides bee all equall, but it hath neuer a square corner" *J Miller

As with many geometric terms, there are two common definitions that are still in use for a rhombus. Some think of geometric names with an inclusive approach, and they usually define rhombus as it is defined in The Concise Oxford English Dictionary, "rhombus n. ( pl. rhombuses or rhombi / -b / ) Geometry a quadrilateral whose sides all have the same length." Notice that in this definition, a square would be a rhombus also. Others, who want definitions to describe how things are different from each other will define it the way it is defined in The Oxford American Dictionary of Current English, "rhombus n. a parallelogram with oblique angles and equal sides." Note that the Oblique angles rules out the case of a square.

The word Rhomboid which means rhom-like was commonly used in the 19th century for a parallelogram which was neither a rectangle nor a rhombus. Today it is more often used for a solid figure with six faces in which each face is a parallelogram and opposite faces in pairs lie in parallel planes. Some crystals are formed in 3D rhomboids. It is also sometimes called a rhombic prism. The term shows up frequently in science terminology referring to both its two and three dimensional meaning.  
My thanks to Mary O'Keeffe for the suggestions to explore the origins of this word.

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