|The Gecko -- is the most intriguing of Dragons. Colourful but bereft of eyelids, lacking claws, yet so provocative. Aristotle insisted that the gecko can run up and down a tree in any way, even with the head downwards. Its physics -- how the gecko climbs walls and even across ceilings -- is beyond Aristotle and Newton -- and involves quantum physics. [ Not that such advanced physics is required to dissect and analyse my geckos.] When challenged by a gecko many try to bar this creature -- but the more creative task is to on adjust the gecko so as to transform it to something more familiar. The genus was properly first described in problem-solving terms by Imre Lakatos, hence the alternate nomenclaure Lakatosian Monster.||
Dragons are formidable problems, challenging, and yet solvable using little or even no algebra or calculus.
But their solutions requires one to use or develop genuine physical insight,
beyond that acquired in childhood.
For more on dragons click here
Eastern Water Dragon, in Sydney Harbour bush.
In his classic work,
Proofs and Refutationas, Imre Lakatos explored how mathematics develops in practice.
He choose as his model for study the evolution of proofs published in
the mathematics literature, both in papers and texts,
of the Euler Theorem, that relates
V + F - E = 2
This formula is very easily checked for the five platonic solids
and it works perfectly for familiar convex polyhedra.
Latatos showed how the discovery of various counter-examples -- the statement of the theorem and the detailed "proofs" evolved over time.
Lakatos delineated two major procedures for dealing with monsters:
Most striking of the monsters that Lakatos discussed were
[ Note the use of specific colours for each visible face of the monstrous dodecahedrons ]
The original articles and the 1976 book by Lakatos, Proofs and Refutations are not freely available online. However this writer recommends for academic readers two online publications:
and An Examination of Counterexamples in Proofs and Refutations
Jesse Lambe Formal Proofs and Refutations, PhD thesis, Stanford, 2009
Now it must be said that while Lakatos was solely concerned with the development of proofs in pure mathematics. here his terminology has been applied to the description of problem categories in qualitative physics.
In A Dragon Hunter's Box a Number of the Dragons are in fact Lakatosian Monsters. To appreciate
2πrTsin(θ) = π r2 ρg H
where ρ is the fluid density, and g is the force of gravity.
Hence, introducing a constant K, the rise in the tube is computed as
H = K/r
That is, the height rise due to surface tension is inversely proportional to the radius of the capillary.
This effect can be demonstrated easily, so that the ideas implied in Inducia Capillaria are part of qualitative physics -- relating to the question formulated somesuch:
One sees that the narrower the tube -- the higher the water roses. Does that mean that for a thin enough tube water would spray out of the top of the tube? Or would water just dribble over the top of the tube?
Inducia Capillaria I
This graphic is reproduced from
Harvey A. Cohen, A Dragon Hunter's Box, Hanging Lake Books, Warrandyte, Victoria, Australia, 1974.
(C) Harvey A. Cohen, 1974, 1975, 2002 and its reproduction other than for private purposes requires written permission.
This dragon is a Lakatosian Monster.
The standard methods of dealing with such monsters are
Moving mouse over the graphic may prove helpful.
The Guide For The Perplexed a four page folio within the Dragon Hunter's Box, provides
a guide to the Snaring -- and Taming -- of Dragons.