Cubic curves

The reason I am interested in cubic curves is that they may be the simplest mathematical representations of the twisting action.

In “Prometheus and Chronos” I tried to build a conceptual model of particles based on the hypothesis of intrinsic flow in curved primordial threads. The curvature is the result of a symmetry breaking that I refer to as the “twisting action.”

As a result of the twisting action a multitude of curved forms of the fundamental thread are possible. Let us consider the simplest form shown in the figure below. Note the 2 semi-circles and the flow direction as indicated by the arrows. The primordial flow shown by the arrows is intrinsic.

Why is there a “twisting action” in the first place? This question cannot be answered in the context of physics. Mystery will remain but the progress will be measured by the economy of our theory. The simplest and the most efficient theory with the smallest number of free parameters yielding to calculations that explain and predict the observations  will be accepted as the most worthy.

My approach to particle physics (curvatures in primordial threads) is not too far off from the string theory approach. Conceptually, the “primordial thread” is somewhere between a “field” and a “string/filament.” I am sure we will converge at some point.

Let’s get back to the cubic curves. The shape of the “twisting action” shown in the figure above can be modeled mathematically.

The cubic curve $y^3 - y=a (x^3-x)$ where a is a constant may be the simplest mathematical representation of the “twisting action.”

Note that in the picture above the graph of $y^3 - y = 1.2 (x^3-x)$ passes through 9 points (note the blue dots which are placed on a 3×3 grid). The coordinates of these blue dots (9 points) are: (-1,0), (0,0), (1,0), (-1,1), (0,1), (1,1), (-1,-1), (0,-1), (1,-1)

When a=1 the 2 lobes of the curve $y^3 - y=a (x^3-x)$ touch each other.

When $a=10$ the shape changes into

There are infinite number of cubic curves. Even the $y^3 - y=a (x^3-x)$ type curves are infinite in number.

Weierstrass equation

The cubic curve $y^2 = x^3 + ax + b$  known as the Weierstrass equation has special significance. Here’s an example curve where $a=1$ and $b=1$

Another example of Weierstrass equation

The following paper is an excellent introduction to the Weierstrass equation. It reminds us that cubic curves are related to elliptic curves.

https://people.maths.ox.ac.uk/szendroi/cubic.pdf

The last section of  this paper shows that the string theorists are equally interested in the cubic curves.

I should mention that the cubic curves are very important in cryptography as well.

In the “Determinantal representations of cubic curves” A. Buckley and T. Košir mention that:

“Elliptic curves have profound influence in mathematics. Since ancient times they turn up in the most astonishing places, joining together algebra and geometry. Recently they have become popular in number theory (cryptography of elliptic curves), optimization (semidefinite programming SDP) and also in theoretical physics (mirror symmetry of elliptic curves). The abundance of results is due to the following two classical facts for smooth plane cubics:”

• It can be brought by a change of coordinates into the Weierstrass canonical form, or equivalently the Hesse canonical form.
• It can be equipped by a group law (induced by the Jacobian group variety).

There are many resources on the web. Here’s a small selection

Wolfram MathWorld: Cubic curve

http://mathworld.wolfram.com/EllipticCurve.html

http://hyperelliptic.org/EFD/g1p/index.html

http://graphics.cs.cmu.edu/nsp/course/15-462/Spring04/slides/10-curves.pdf

https://www.math.hmc.edu/~ursula/teaching/math189/finalpapers/elaine.pdf

http://en.wikipedia.org/wiki/List_of_curves

http://www.math.ucla.edu/~baker/149.1.02w/handouts/dd_splines.pdf

http://www.engr.sjsu.edu/ragarwal/ME165/ME165_Lecture_Notes_files/Chapter_4_CURVES.pdf

http://blog.wolframalpha.com/2011/08/03/generating-polar-and-parametric-plots-in-wolframalpha/