"From the time Alexander the Great cut the Gordian knot, knots and their many shapes and forms have pervaded many facets of our lives. Magicians, artists, and philosophers alike have been intrigued by such knots as the trefoil knot, which has no beginning or end."
This is the first page of a long section, "Knots in the Mysteries of Life." You may be wondering, what happened to page 235, since today, August 23rd, is the 235th day of the year? Here's the explanation:
Lesson 0.6 of Michael Serra's Discovering Geometry is called "Knot Designs." And there you go -- a natural reading would put today's Lesson 0.6 one day away from the knot section in Pappas -- so tantalizingly close to lining up.
And so I couldn't resist starting the Pappas knot section today. We'll go back to page 235 tomorrow, before returning to knots with page 237 on Friday.
The Pappas section on knots focuses on a branch of mathematics called "knot theory." She tells us that "knot theory is a very recent field of topology." I've written about topology on the blog before, and in fact Pappas herself discusses topology in her book (see my May 10th post). Two figures are topologically equivalent if one can be bent, stretched, tied, or untied to form the other.
During this entire Pappas science chapter, I've been writing about the science class I taught -- and thinking about how I should have taught these lessons in my own class. Of course, knot theory isn't a suitable topic for middle school science.
Then again, Pappas points out that knot theory is related to the sciences. She tells us that 19th century physicist Lord Kelvin -- of temperature fame -- once believed that atoms were knots in the ether:
"Although his theory was not true, the mathematical study of knots is a very current topic today."
...and may ultimately have a link to physics after all. But Pappas will get to that later in this section.
Meanwhile, let's see what Serra has to say about knots. This is Lesson 0.6 in my old Second Edition, while it's Lesson 0.5 in the modern editions, as those editions omit my 0.5.
Serra begins:
"Knots have played very important roles in cultures all over the world. Before the Chinese use ideograms, they recorded events by using a system of knots."
The Pappas section and Serra lesson have much in common. Both books depict a Celtic knot. As Serra explains, the ancient Celts carved various knot designs in stone.
But Serra has a more inclusive definition of knot than Pappas. He writes:
"Knot designs are geometric designs that appear to weave in or interlace like a knot."
By Serra's definition, the Olympic rings form a knot. The Pappas definition is more in line with the mathematical definition of knot, but we will "not" get to that definition until Friday's post.
Today's worksheet is based on Serra's definition, so there is much emphasis on rings. Of the three questions I selected, one of them has the students draw a knot using a compass (so the shapes will end up being rings). The other two are puzzles involving interlocking rings.
One of the puzzle questions asks students to sketch five rings linked together such that all five can be separated by cutting open one ring. This is easy -- just link four rings to a center ring. The other question is a classic -- the Borromean rings are three rings such that all three are linked, and yet no two of them are linked.
Here is a link to a solution to the Borromean ring puzzle:
http://im-possible.info/english/articles/borromeo/index.html
According to the link above, the Borromean rings are physically impossible -- unless the rings deviate slightly from perfect circularity. Hence the link labels this as an "impossible figure" not unlike the op art from Monday's lesson. Of course, we can draw op art, including perfectly circular Borromean rings, on paper with a compass.
Actually, I found a few interesting links involving Borromean rings. Here is Evelyn Lamb, who writes a math column for Scientific American once or twice per month ("Roots of Unity"):
https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-borromean-rings/
Recall that knots and science are related. Here is a Borromean ring consisting of three atoms:
https://www.livescience.com/9776-strange-physical-theory-proved-40-years.html
And here's a link to Borromean onion rings, courtesy one of my favorite mathematicians, Vi Hart:
https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/thanksgiving-math/v/borromean-onion-rings
Pappas introduce knots by writing about Alexander the Great and the Gordian knot. Actually, Serra mentions the Gordian knot as well, even though I didn't include this question on my worksheet.
The recreational math website Cut the Knot is actually named after the Gordian knot. Here is a link to Alexander (Bogomolny) the Great, who explains why he chose that name:
https://www.cut-the-knot.org/logo.shtml
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