Chapter 15-1 

Comparison of Agnesi a^3/(a^2+x^2) to Einstein's E=mc^2

You must be able to distinguish between what is true and what is real.

                                                                                          Albert Einstein

When I started trying to understand the behavior of the Agnesi curve my focus was how a curve that is mathematically correct can behave the way it does.  I looked for faults in the formula; but it is so simple it just couldn’t be wrong.  Looking at the curve in two dimensions allows a person to see the contradictions.  I believe Guido Grande and Maria Agnesi saw these contradictions and both spend a considerable amount trying to understand them.  Unfortunately the calculations required to move the curve to three dimensions were difficult and visualization of the results would be almost impossible.

When I moved the curve to three dimensions and began to rotate and observe it, I noticed that as strange as it was it seemed to look like things we see in nature.  It appeared that if you were to change the symbols somehow it might model Einstein’s equation of e=mc^2, but attempts to do so didn’t work.  The picture I had in my mind was “a” was a volume or 3d space (a^3) and related to “mc^2” in Einstein’s formula.  The problem is you would only have one point using Einstein’s formula, not a curve.  The beauty of the Agnesi curve is you are storing “x” in the space and this accumulation of “x’s” I picture as energy (remember the springs from chapter 2).  Einstein’s and Agnesi’s formulas appeared unrelated.

One day I was in the book store and found Einstein’s 1912 Manuscript on the Theory of Special Relativity. 

Figure 16-1

On the cover of the book (Figure 16-1) was the following formula:

Figure 16-2

There it was, all of the pieces to combine and compare the two formulas.

If you look at the denominator in Einstein’s equation (the square root of 1 - v^2/c^2) does two things; it returns the ratio between the two and removes the units.

Figure 16-3

Figure 16-3 shows what is being calculated in the denominator of Einstein’s equation.  If you remember the Agnesi formula from chapter 2, we were calculating basically the same thing as the denominator of Einstein’s formula.  The big difference is the Agnesi formula is calculated from the base of the circle, where Einstein is working from the center.  It is working from the base that gives us the weirdness in Agnesi’s formula.

Figure 16-4

In the Agnesi equation we are calculating ‘y’ as shown in Figure 16-4.

What if we combine the two formulas?  To do this we need to use the same symbols; figure 16-5 shows ‘a’ being changed to ‘c’, or the speed of light and ‘x’ the distance traveled to ‘v’ the velocity.  This gets at the heart of Einstein’s equation; that distance and time are basically the same thing.

Figure 16-5

Figure 16-6

The right side of Figure 16-6 shows removing the units from the equation to end up with a ratio. 

The left side of figure 16-7 shows the two equations combined.

Figure 16-7

The right side of figure 16-7 shows the equation algebraically simplified.

Most of the work we've done has been calculating mass or    m = e / (c^2 + v^2) .

If you let 'e' equal 'c^3' in the formula you can see the formulas are the same except for the "+ v^2".

Figure 16-7 helps explain Einstein's E=mc^2 formula.  The speed of light defines a starting (empty) space shown as the light blue cube on the right.  The triangles define how velocity changes what happens in space; as you increase velocity you are pushing space into a smaller area converting it to mass.  To say it differently, the energy required to increase velocity causes an increase in mass, the volume is squashed into a smaller space.  If you were to stop pushing (increasing velocity) you would get the force contained in the mass.  If you released the force you would end up with the starting space. 

Figure 16-8

I think the problem with this is there is no locking mechanism to keep the energy contained. The beauty is it combines space, time, energy and mass into one formula.  Time is a sleeper, it enters into the formula through velocity because velocity is stated as distance moved in a given time.

As I said earlier the biggest difference in the Einstein and Agnesi formula's is where the center point is located as shown in figures 16-3 and 16-4.  When I think about Einstein's formula it works like a bank where you can borrow from below the centerline and this creates the force to try and center the system.  The nice part is the borrowing can be used to explain gravity and other phenomenon but with it is a whole other world that needs to be considered.

The Agnesi formula doesn't need a non-existent world to borrow from, it is its own bank.  It has a mechanism to store and retrieve energy and the ability to lock it up when not needed.  It does however require flow, the addition of 'x' or 'v'.  For every pi unit you get one more unit of mass that is locked or bound together.

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