The hobbit, the dragon, and the green knight

by @ulaulaman about #TheHobbit #JRRTolkien #Smaug #mathematics

Gandalf and Bilbo by David Wenzel
The Peter Jackson's Hobbit movie trilogy is arrived to a conclusion, so it could be a good point to write a little, funny curious post about the science and the Tolkien's novel. We start with a paper published last year(1) (2013) in which the researchers find the cause of the triumph of good over evil:
Bilbo Baggins, a hobbit, lives in a hole in the ground but with windows, and when he is first encountered he is smoking his pipe in the sun overlooking his garden (it is worth noting [parenthetically] that smoking is itself associated with skeletal muscle dysfunction6). Dwarves and wizards smoke too, and the production of smoke rings is unfortunately glamourised. The hobbit diet is clearly varied as he is able to offer cake, tea, seed cake, ale, porter, red wine, raspberry jam, mince pies, cheese, pork pie, salad, cold chicken, pickles and apple tart to the dwarves who visit to engage him in the business of burglary. The dwarves also show evidence of a mixed diet and, importantly, although they “like the dark, the dark for dark business”, they do spend much time above ground and have plenty of sun exposure during the initial pony ride in June that begins their trip to the Lonely Mountain.
(...)
Gollum, himself "as dark as darkness" lives in the dark, deep in the Misty Mountains. He does, however, eat fish, although the text describes these only as "blind" and it is not clear whether they are of an oily kind and thus a potential source of vitamin D. He sometimes eats goblins, but they rarely come down to his lake, suggesting that fish play little part in the goblin diet. Interestingly, these occasional trips to catch fish are undertaken at the behest of the Great Goblin, leading one to speculate that his enhanced diet may have helped him to achieve his pre-eminent position within goblin society. In due course, the Great Goblin is replaced by the Son of the Great Goblin. While simple nepotism is a likely explanation, we are unable to exclude an epigenetic process whereby the son’s fitness to rule has been influenced by parental vitamin D exposure.(1)
So the secret is in the diet!
Another great character from The Hobbit is Smaug, the dragon. Its physiology is really peculiar (read also Disco Blog):
We have shown for the first time that [the Organ of Feuerwerk] consists of brown adipose tissue, and that it has an extensive sympathetic innervation; presumably the nerves enable it, as with other deposits of brown adipose tissue, to metabolise rapidly and to create high temperatures, as happened in our experiment. The Ducts of Kwentsch would provide a copious flow of liquid that would both cool the hot expired gas and also form steam, one of the characteristic features of dragons and related species. We found that the skin of the snout contained deposits of asbestos; these were first described by Bestos (1910) but without chemical analysis, and were previously assumed to be hair follicles (Negus, 1958). We speculate that the desert lizard rubs sand into its nasal skin, and possesses the specialized metabolic processes to convert it to asbestos.(2)

Sir Gawain and the Green Knight by John Howe
The most serious part of the post is devoted to the shield of Sir Gawain in the ancient poem Sir Gawain and the Green Knight described as follow by the narrator:
And why that noble prince bare the pentangle I am minded to tell you, though my tale tarry thereby. It is a sign that Solomon set ere-while [some time ago], as betokening truth; for it is a figure with five points and each line overlaps the other, and nowhere hath it beginning or end, so that in English it is called "the endless knot".
The pentacle on Gawain's shield is a geometric shape also known as the pentagram, much loved by the Pythagoreans. The reason is simple: it is linked to the golden section.
The golden ratio is defined as the positive solution of this quadratic equation: \[\frac{1+x}{x} = \varphi\] where $\varphi = 1.618034...$
The pentagram, the star on the shield of Gawain, for instance, is constructed from a regular pentangle. When this has side length 1, then its diagonals, or lines that intersect in the staff, have length $\varphi$2. Also from the pentangle is then possible to construct a pentagram fractal.
On the other hand the pentagram is said, in the poem, Endless Knot and the reason is simple: it is possible to build a knot that has the same shape.
Knot theory has its origins in the 18th century, developed by Vandermonde, but it is studied starting from the 19th century with Gauss.
According to knot theory, a knot is a closed circuit. The first is the circle, the trivial- or unknot, followed by the others. As for the numbers, even in this case there are the prime knots, i.e. those knots which can not be composed using other knots. Each knot, then, measure its complexity based on the number of crossings. For example for 3 and 4 we crossings 1 knot, while for 5 junctions, as in the case of the Gawain's symbol, we have 2 knots.

Alexander cuts the Gordian Knot by Jean-Simon Berthelemy
The Gordian knot was made by Midas, the adopted son of the farmer became King Gordius of Phrygia, in honor of his adoptive father, arrived in the city on the day of his coronation on a horse-drawn cart, as he wanted an ancient prophecy. And just a prophecy, ancient for Alexandrian historians, but in fact posthumous, said that those who would have untied the knot would conquer Asia, something that Alexander had every intention of doing (untie the knot and conquer Asia!). Obviously, the legend saw that he broke it cutting with a sword, betraying the teachings of his teacher Aristotle.
And we close with a knot of our times, intricate but ultimately equivalent to a simple unknot: The Gordian unknot by Wolfgang Haken:
Ah! Merry Christmas!
Further reading:
Symbolism of the Pentangle in Sir Gawain and the Green Knight
Teaching Tolkien's Translations of Medieval Literature: "Sir Gawain and the Green Knight, Sir Orfeo and Pearl"
(1) Hopkinson J.A. (2013). The hobbit — an unexpected deficiency, The Medical Journal of Australia, 199 (11) 805-806. DOI: http://dx.doi.org/10.5694/mja13.10218
(2) Georgy S.T. (2002). The pyrophysiology and sexuality of dragons, Respiratory Physiology , 133 (1-2) 3-10. DOI: http://dx.doi.org/10.1016/s1569-9048(02)00129-5 (pdf)

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