Sunday, January 31, 2010


JSH: Parameterizing conics and math politics

One of the privileges I had recently was re-deriving a conic parameterizations result:

Given x^2 - Dy^2 = 1, in rationals:

y = 2t/(D - t^2) and x = (D + t^2)/(D - t^2)

You get ellipses D < 0 and hyperbolas with D > 0, with the circle at D=-1, which is the one result that you can easily see in standard literature:

See: eqns. 16 & 17

And it was a rare pleasure as usually I have unique results which are new discoveries, and then there is a lot of arguing, people lying about my research, and all kinds of unpleasant things which are about human competition at its worst, so it's fun for me to walk in the footsteps of prior discoverers by figuring out something they already knew, and that parameterization from the record was known to Fermat, so back to the 17th century, so it's been known for hundreds of years.

But here's where things get very fun, as of course, mathematicians call x^2 - Dy^2 = 1, Pell's Equation.

And if you look at the Wikipedia article on it:'s_equation

There is no mention of the rational parameterization because to mathematicians x^2 - Dy^2 = 1 is a Diophantine equation, so they behave as if the rational parameterization does not exist, which is hilarious.

Now I've brought this subject up before, but also I noticed some other simple things about Pell's Equation which I didn't see mentioned, and something interesting happened!!!

The Wikipedia page was updated.

(But NOT the page at MathWorld:

Notice the simple solutions left out which were added at the Wikipedia page. Those are what I noticed.)

Yup, the other things I noticed which had not been there before WERE ADDED, but NOT the rational parameterization because math people consider the equation to be a Diophantine equation so they WILL NOT tolerate its mention as anything else.

To them it is a Diophantine equation and nothing else matters now because they've DEFINED it you see.

Now I like ranting on this subject as it feels good to be right, and I love these impressively stupid things that are at a world level.

And it's interesting to wonder if the people who adjusted to what I said before will bite the bullet and eventually give in and mention the rational parameterization. Or will they fight it!!!

Academics are often idiots. It's that simple.

You are provably often fools. Fools. Stupid on a level that defies belief.

Of course I'll take an opportunity to rip on the academic community like this one. I need it.

It helps me feel better.

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