What you think you think you know you know
Newly minted voices, bottomless holes, and other representations and equivalences of binary forms
“A.I. Eeee! Ohhhh! You!!!!”
“The machine covers the ground behind it in thick black syrup”
“This seems to be working well. What could go wrong?”
The question we must ask ourselves is: which conditions should be placed on our forms and on Σ such that the resulting form F is indeed a binary quadratic form? Also, how can we guarantee that such a Σ in fact exists? Finally, from our previous section, we have some intuition that the composition of two forms is not unique: how can we avoid this problem?
Firstly, in our discussion of Legendre’s theory of composition, we have proved the existence of a transformation Σ giving rise to a new form. For example, our expression for Y in ?? can be written in terms of x, y, z,w as above, and thus gives us a transformation by setting q = 0, q′ = 2a, q′′ = °æ2a′ and q′′′′ = b °æ b′. Similarly, the expression for X, given in [x], can also be rewritten in this way.
Let’s look at an example.
Example 3.1. Let f(x, y) = 2x2 + 3y2 and g(z,w) = z2 + 6w2. We define Σ by
X = xz − 3yw, Y = 2xw + yz,
and thus obtain
(2x2 + 3y2)(z2 + 6w2) = 2(xz)2 + 12(xw)2 = 3(yz)2 + 18(yw)2 = 2X2 + 3Y 2.
For f(x, y) = 5x2 +3xy +6y2 and g(z,w) = 2z2 +3zw+15w2 of discriminant Δ = −111, define Σ by X = xz − 3yw, Y = 5xw + 2yz + 3yw,
and thus obtain
(5x2 + 3xy + 6y2)(2z2 + 3zw + 15w2) = 10X2 + 3XY + 3Y 2.
This seems to be working well. What could go wrong?
—Zola Mooney, “Composition of Binary Quadratic Forms”, Level-4 Honours project, School of Mathematics and Statistics, University of Glasgow, Friday 1st March, 2024
“Jim Pulk was said to have slayed the dragon by baking a poisoned pie.”
Within the flat lands of the South Downs, there are several springs which rise to the surface, creating pools that are thought to be bottomless. These are known as ‘Knucker Holes’. It was believed that the South Downs were filled with holes like these and that they went down to the other side of the world.
A famous legend tells of a dragon called the Knucker who lived in a pool near the Sussex village of Lyminster. The water of the pool was said to have healing properties and was once used to cure illnesses.
Different legends have surrounded this pool and its famous dragon, the Knucker. One story tells us of a farmer's boy from Lyminster called Jim Pulk, (however some say the man was called Jim Puttock from Wick, a nearby village). Jim Pulk was said to have slayed the dragon by baking a poisoned pie, which he placed on a cart, only for the dragon to eat the entire cart and the horses who towed it as well. Once the dragon was dead, Jim Pulk cut off the head of this large water beast and allegedly took it to the pub, the Six Bells Inn to celebrate. However, Pulk poisoned himself after killing the dragon (some say he didn’t wash his hands), and died while at the Six Bells. Jim Pulk is allegedly buried in Lyminster churchyard.
Sussex has several other folkloric stories of Dragon’s in places such as St. Leonard’s Forest, where the 6th century French hermit St. Leonard battled with a Dragon in the ancient woodland. Or on Bignor Hill, which was believed to be the home of a Dragon (similar to Chrome Hill, also known as ‘The Dragon’s Back’ in Derbyshire).
— Otis Jordan, “Sussex: Seven Scenes of the South Downs”, Ceremonial County Series Vol.I - West Sussex | Cheshire, Folklore Tapes, March 2, 2024
— Otis Jordan, “Sussex: Seven Scenes of the South Downs”, Ceremonial County Series Vol.I - West Sussex | Cheshire, Folklore Tapes, March 2, 2024
“I am confused about the rules at first, but quickly get engrossed in the game.”
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