Eastern Water Dragon
Eastern Water Dragon
Found lurking near Middle Harbour, Sydney, April 2015




The Gecko -- is the most intriguing of Dragons. Colourful but bereft of eyelids, lacking claws, yet so provocative - with behaviour that is so at odds with common sense. Aristotle reported 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 -- was way beyond Aristotle and Newton -- and involves quantum physics. [ Not that any such advanced physics is required to dissect and analyse my geckos.] When challenged by a gecko many try to bar this creature -- "This just can't be" as a reason that it shouldn't be even thought about -- but the real task is to manipulate (adjust) the gecko so as to transform it to a recognisable creature. The genus, and the methods of dealing with individual specimens, was properly first described in problem-solving terms by Imre Lakatos, hence the alternate nomenclature Lakatosian Monster, and the operations names monster barring and monster adjustment. . The genus is described here.
Lid of THE Box Dragons in Qualitative Physics and Mathematics
Dragons are formidable problems. challenging, and yet solvable using little or even no algebra or calculus. A collection of Dragons, focussing on mechanics, was published long ago, in the out-of-print collection:

Harvey A. Cohen, A Dragon Hunterís Box

where the Dragons were magnificently illustrated by Ms Jeni Rawson, who wrote out the text by hand, and embellished, sometimes with adapted borrowing from classic illustrative artists such as Doré.

The Dragon Hunter's Box included material interpretting Dragon Hunting -- forays at Dragons -- in terms of ideas developed by Thomas Kuhn and Imre Lakatos to explain the evolution of physics and mathematics. An important analysis of Dragons in relation to Piagetian Conservation puzzles and to Marvin Minsky's Frames theory of knowledge was published as Harvey A. Cohen, The Art of Snaring Dragons, MIT Artificial Intelligence Laboratory Memo 338, (1975) downloadable as PDF Click for PDF download which was the first, and defining paper, in the field of qualitative physics (as a domain of study within cognition and instruction).

The lid of the Dragon Hunter's Box(to left) lists demons, imps and monsters as included; a web page elucidating Labatosian Monsters within the collection is here
One Dragon from the original compilation is YoYo, or strictly speaking, YOYO in the Upper Case, and yoyo in the lower case:

The YoYo problem -- a yoyo sitting vertically on a flat surface -- has two cases:
YOYO UPPER CASE the cord, wrapped around the YoYo core, comes out on the top-side of the yoyo.  and is steadily pulled to the right.  
In yoyo, lower case, the YoYo cord, wrapped around the YoYo core, comes out on the bottom-side of the yoyo, and is steadily pulled to the right.
In both cases, the question is, does the YoYo roll to the right or to the left?

Which way does the YoYo go ? Left or right? In both cases?



ZsaZsa says:
Things always go in the direction they are pulled. In both cases
the cord is pulled to the right,
so in both cases the YoYo rolls to the right. TseTse says:
In YOYO (UPPER CASE) the cord turns the YoYo clockwise about its center,
so the YoYo rolls to the right.
In yoyo (lower case) the cord turns the YoYo anticlockwise about its center,
so the YoYo rolls to the left.   Mouse ZzaZza and TseTse

Which way does the YoYo go ? Left or right? In both cases?

Do you need help? Even if you don't, try moving the mouse over the frabjous, beasties, ZsaZsa, TseTse, and Garbo web versions of the creatures of the Dragon Hunter's Box. But whose suggestions should one listen to, or are they both right or both wrong ?


ZsaZsa says -- Darlings --
In both cases the cord is pulled to the right.  In UPPER CASE cord is drawn off the top of the YoYo to compensate, and so YOYO rolls right.  In lower case, the yoyo rolls to the right to compensate for the end of the cord going in that direction
The "Hint" page for YoYo from A Dragon Hunter's Box can be viewed by Clicking Here or in PDF by clicking on the token Click for PDF download

Where does Piaget fit in ?


A sentient being has to have a capability at qualitative physics and mathematics in order to interact with the world about us. The development of our knowledge of the world tends to go by stages, where at each stage a simpler scheme is replaced by a more sophisticated. This is also true with regard to the development of Science, as stressed by Thomas Kuhn, and in the development of mathematical theory as discovered by Imre Lakatos.

Children's ability to use quantity and number likewise undergoes an amazing transformation at about the age of five, as evidenced by the following:

Note that the top diagram, with one egg in each cup, is what the child initially views. Then in front of the child the eggs are removed the cup, and spread out as shown in the lower diagram.
In this game for children, invented by the Swiss psychologist Piaget, a simple question is asked, to which both the younger and older children give a confident answer. This is an instance of a mathematical Dragon. Just what is going in children's head both before and after this stage is reached is discussed in Heuristic Growth and Intellectual Development.

So are Dragons the same as Piagetian Spatial and Conservation Puzzles?


Webmaster apologies for the mess below -- this page is being substantially revised and reorganised.


Links to the Dragons now on-line are included in the subset of Dragons::
The Milko Bottle Problem aka the dragon Milko
The Ilya Revel
Inducia Capillaria I a Lakatosian Monster
Inducia Capillaria II a Mercuric Lakatosian Monster
The Coin Problem
The Flying Omnibus: How and Why Aeroplanes Fly
The BumbleBee Paradox: Why It Can't Fly But Does
The Satellite Paradox: Negative Mass in Low Earth Orbit
Executive Toy -- A Challenge in Metacognition
Cyber -- a 3D Perceptual Dragon
Ayelet's Challenge -- a mathematical puzzle
The Ant on the Box Problem - a new mathematical Dragon
Sir Arthur Eddington's Cricket Puzzle -- For Cricket Fans only
The Gecko -- is the most intriguing of Dragons. Colourful but bereft of eyelids, lacking claws, yet so provocative. Aristotle reported 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 -- was way beyond Aristotle and Newton -- and involves quantum physics. [ Not that any such advanced physics is required to dissect and analyse my geckos.] When challenged by a gecko many try to bar this creature -- find a reason that it shouldn't be handled at all -- but the real task is to manipulate (adjust) the gecko so as to transform it a recognisable -- and thus solvable -- puzzle. The genus, and the methods of dealing with individual specimens, was properly first described in problem-solving terms by Imre Lakatos, hence the alternate nomenclature Lakatosian Monster, and the operations names monster barring and monster adjustment. . The genus is described here.


Just what is a Dragon?
A number of examples are given here
The canonical analysis of Dragons in relation to Piagetian Conservation puzzles and to Marvin Minsky's Frames theory of knowledge was published as Harvey A. Cohen, The Art of Snaring Dragons, MIT Artificial Intelligence Laboratory Memo 338, (1975). The link is to a text-searchable version of the 1975 revised version.
Colour phograph of the Lipson LEGO model of \"Belvedere\" There is an interesting correspondence between Dragons and the category of visual illusions -- the Impossible Figures -- that form the basis of Escher's famous lithographs.

The Escher lithographs stimulate. In Waterfall for instance, one traces the flow of water down, down, down, until the water flows into its starting location. Simply impossible.

Recently LEGO models based on the Escher Lithographs, Balcony, Belvedere, Waterfall, Ascending and Descending. and Relativity were produced by Andrew Lipson. Photographs taken of these models match the Escher lithographs on which they are based. Yet these are real constructible models of the impossible.
Note that the 2007 version of Internet Explorer (unlike earlier versions) does not support Java required to view Lipson's animations.
Or are Lipson's models Geckos ? [ aka Lakatosian Monsters ], a special category of Dragon discussed here ?
And lastly, here is a mathematical riddle, called the Case of the Missing Dollar