Curves

 

It Takes Two

It takes two, baby
It takes two, baby
Make a dream come true
It just takes two

Not Four,  Two.

It is a standard concept that there are four fundamental curves:

1.      a circle,

2.      an ellipse,

3.      a parabola, and

4.      a hyperbola. 

I would suggest that there are really only two curves: an ellipse and a hyperbola. A circle is a special, perfect, case of an ellipse, and a parabola is a special case between a hyperbola and an ellipse.

A curve is defined by its major axis, a, and its minor axis, b.  An ellipse is a curve where the major and minor axes are both less than infinity.  There is no other restriction on a or b.  A circle is the special case of an ellipse where a = b, which is why it is said to be perfect.

A hyperbola is defined by having both axes, a and b equal to infinity.  A parabola is that case where also a ≠b.  But this is between an ellipse and an hyperbola.  Saying that at least one of the axes, a or b, is less than infinity makes it a partial  ellipse.  Saying that at least one of the axes is equal to infinity makes it a partial hyperbola.  A parabola is a mathematical curve where one of the axes is very, very large but NOT infinity.  It looks like it is different than a hyperbola and an ellipse, but actually it is partially both at the same time.

Error

 

O Holy Night

O holy night! The stars are brightly shining,
It is the night of the dear Savior's birth.
Long lay the world in sin and error pining.
Till He appeared and the Spirit felt its worth.

There is still sin and error.

The mathematical statement 5±1 does not mean that there are only two answers: 4 and 6.  It means that any value between 4 and 6 is correct.  The term 5 in this example is the true answer, the mean/median/mode of a normal distribution, and the term 1 is the standard error.  Mathematically it could have also been expressed as 4 ≤ x ≤6, where x is the solution. 

The standard error is the square root of the variance.  A random number is a number with a non-zero variance, which means that it also has a non-zero standard error.  We can try to make that error as small as possible but “To err is Human”.  Scientists can design their experiments such that there is minimal error, but there will always be some error.  Engineers deal with the fact that there is variance and design systems such that the solution considers variance, error, randomness.

The solution of x²-1 as (x-1)*(x+1), x±1, should not be taken to mean that there are only two answers.  It should be taken to mean that any value of x between -1 and 1 is correct.  Otherwise you are effectively assuming that there is NO error. You might as well assume that there is no sin as well!

Absolutely

 

My Back Pages

Yes, my guard stood hard when abstract threats too noble to neglect
Deceived me into thinking I had something to protect
Good and bad, I define these terms quite clear, no doubt, somehow
Ah, but I was so much older then I'm younger than that now

Life is NOT a battle between Good and Bad.

Good and bad are relative terms, just like odd and even; black and red; chaos and order; Republican and Democrat; less filling and tastes great, etc.  An absolute has no relative terms.  It is the unity of all of those relative terms.  That is why, in mathematics, infinity is both odd and even. This is also not a new concept, it is a New Testament concept that goes back almost 2000 years,  Gal 3:28.  The distinction really is between an absolute and no absolute.  Thus the choice is not good or bad. It is good, bad or nothing.  Trying to make it as if the choice ss only between good and bad is trying to deceive.  Be younger than that now.

Relativistic Gravity

 I’m Sorry

You tell me, mistakes
Are part of being young
But that don't right
The wrong that's been done
I'm sorry
(So sorry) So sorry
Please accept my apology

Mistakes are obviously part of being old too!

In a previous blog post I regretted not being able to show my work that the Lortenz Transform which is traditionaly expressed as √(1-(v/c)² should be expressed as 1/(1-ln(½)*ln(1-(v/c)²)).  My work obviously had a mistake and while this appears to be a useful approximation, ...it is wrong.

On a flat surface the hypotenuse/radius of a triangle is r=√(a²+b²).  I had said that on a hyperbolic surface this is r=1/(1+ln(.5)*ln(a²+b²)), which I arrived as a solution and it seems to work, but I did not save my work.  The correct solution, which I did save this time, on a hyperbolic surface is 
r=ln(cosh(a²+b²)*cosh(n*π) ± sinh(a²+b²)*sinh(n*π)).  This makes the solution to the Lorentz Transform, which can be solved from Einstein's Triangle of Energy as ln(cosh(1-v²/c²)±sinh(1-v²/c²)), where cosh(1-v²/c²) is the constant term, location, and sinh(1-v²/c²) is the variance, uncertainty.  Logarithms are not defined for negative numbers, i.e. when the uncertainty is greater than the constant term, so the Lorentz Transfrom is properly in the range between 1 and ln(cosh(1-v²/c²)+sinh(1-v²/c²)).  At a speed of zero, the location dominates and the uncertainty is almost zero.  The uncertainty increases and the location decreases as the speed of the particle approaches the speed of light. 

This means that the formulas originally given in https://dbeagan.blogspot.com/2023/04/on-beyond-einstein.html should instead be :

“If space is not flat, but is hyperbolic then the equations might instead be

Momentum =  m_0* ln(cosh(1-v²/c²)±sinh(1-v²/c²))*v

Force=  m_0* ln(cosh(1-v²/c²)±sinh(1-v²/c²))*v ∂v
         =  m_0* ln(cosh(1-v²/c²)±sinh(1-v²/c²)) *a

Energy = ∫ m_0* ln(cosh(1-v²/c²)±sinh(1-v²/c²))*v ∂v=      
            =   m_0* ln(cosh(1-v²/c²)±sinh(1-v²/c²))*c²

This solution does not create a paradox at v=c , and it is undefined, not imaginary, when v>c. The rest mass, m_{0 ,} is always greater than zero.

The Second Law of Thermodynamics requires that the energy of a system of objects will seek the state of lowest energy and any reduction in the energy of the system will be equal to an increase in the entropy of the system. In curved, hyperbolic, space, two masses will each seek to lower their energy and approach a common center along a geodesic. This change in energy will be accompanied by a change in momentum. This change in momentum could be viewed in flat space as an apparent force, like centrifugal force, and NOT an intrinsic force. The apparent force of gravity is these masses seeking to lower their energy, maximize their entropy, and this is

G*m_01* ln(cosh(1-v₁²/c²)±sinh(1-v₁²/c²))*

                       m_02* ln(cosh(1-v₂²/c²)±sinh(1-v₂²/c²))/  exp(-k*d₁₂);

G=6.67×10^-11

where d₁₂ is the distance between mass 1 and mass 2, m_0x is the rest mass of mass x,  v_x  is the velocity of mass x, and c is the speed of light and k is the average distance. "

This does mean that:

In a curved universe, gravity should be an apparent force and should NOT be combined with the three intrinsic (electromagnetic, weak nuclear, and strong nuclear) forces in a Unified Field Theory.

In a hyperbolic universe, there is a discontinuity at the Big Bang and our universe may be only one sheet in a asymmetrical two-sheeted hyperboloid.

In a hyperbolic universe, there is only one absolute, and that absolute is both random AND deterministic.

If there is one absolute, then there is also only one choice: choosing that absolute, or not choosing that absolute, aka absolute zero. 

In a hyperbolic universe, regressions and statistics using least squares should be redone; the formula for what is called the standard deviation is in fact the formula for error; and the Bessel adjustment, n/(n-1), is not necessary.

In a hyperbolic universe, when there is no error, every moment about the mean should be 0, not just those odd movements where the moment is currently expressed as multiples of i and even movements which are multiples of i² , which are currenly expressed as a real number is - 1.

The universe has a variance of .822,  and thus its standard deviation can never be 0.

Rotations

 

Twist and Shout

Well, shake it up, baby, now
Twist and shout
Come on, come on, come, come on, baby, now
Come on and work it on out

Is life twisted?

The surface, with a coefficient of the imaginary axis=0 and a rotation of that imaginary axis of π, passes through the origin of a two-sheeted hyperboloid where those two sheets intersect at that origin. This surface in complex Cartesian coordinates can be described as -r +0*i.  where r is the real radius and i is the imaginary number, √-1.  But in doing so there is the need for a hyperbolic trigonometric function, the hyperbolic cosine, on that surface to be rotated, twisted, by π.  This is because of the nature of that function, which is undefined, and undefinable, for negative numbers.   In order to not allow negative  numbers, the hyperbolic cosine must be twisted, rotated, by π, in order to pass from one sheet of that two-sheeted hyperboloid to the other sheet through that origin.  It must also be translated at a point which is equal to the variance of that sheet, when the value of that hyperbolic cosine is equal to the variance of that sheet.

This sounds complicated but it is not.   A fluid in a pipe or channel passes from a laminar, calm, domain to a turbulent, white water, domain, when the flow in that pipe, or channel approaches its capacity.  Flow in traffic passes between a congested and an uncongested domain when the volume of traffic on a road is equal to the capacity.  When passing between these domains the fluid flow and traffic volume appear to follow a hyperbolic cosine.  

The variance of the surface is always a multiple of .5²*π²/3, in other words n*.25*π²/3.  This has a minimum value of 0.822467...., when n=1. 

This means that the hyperbolic cosine on the surface which includes our universe has both a mean/median locational position and a non-negative variance.  Since it has both parameters, that makes it a random function.  While that mean/median location can take on many values, if the variance was zero, it would be flat-lined, i.e. dead, not alive. By having a non-zero vaviance and twisting, it is alive and able to shout.

 

Harvard President Resigns

 

Branded

Branded, scorned as the one who ran.
What do you do when you're branded, and you know you're a man.

Wherever you go, for the rest of your life
You must prove, you're a man.

My sympathy for former Harvard President Claudine Gay

It is hard for a Brown alum who had to endure chats from the Harvard attendees at sporting events of “If it’s Brown, flush it down” and “You may have beat us now, but you’ll work for us someday” to defend a Harvard President, but the “Christian” Nationalists who have demanded her resignation and appear to have won, have not been very Christian.  I believe that Christ’s own words were “Judge not, lest ye be judged” Matthew 7:1-3 and “He that is without sin among you, let him cast the first stone" John 8:6-7.  But the criticism was coming mostly from the Two Corinthians crowd, so clearly the Bible means nothing to them, despite what they claim.

Real

 

Compared to What?

I love the lie and lie the love
A-Hangin' on, with push and shove
Possession is the motivation
That is hangin' up the God-damn nation
Looks like we always end up in a rut (everybody now!)
Tryin' to make it real, compared to what? C'mon baby!

Real compared to Imaginary?

Euler’s formula, e^ix=cos(x)+sin(x)*i, can be multiplied by a variable r on each side of the equal sign.  It then becomes r*e^ix=r*cos(x)+r*sin(x)*i.  This is also the statement of a complex number in polar coordinates where r is the radius in polar coordinates and x is the angle of rotation of the imaginary axis. In Cartesian, rectangular, coordinates, this is a+b*i, where a = r*cos(x) and b=r*sin(x).  The angle x, is tan^-1(r*sin(x)/(r*cos(x))) which is simply x  The radius in polar coordinates is r²=r²*cos²(x)+r²*sin²(x).  This is traditionally solved as r=√(a²+b²)=r*√(cos²(x)+sin²(x)) where a = r*cos(x) and b=r*sin(x).  Because of the identity for a circle, cos²(x) + sin²(x)=1 for all x, this means r=r, and thus the square root is the solution on a flat surface for all values of x and is the formula on a spherical surface when the radius of the spsherical surface is very large compared to the polar radius.

However this is not the solution on a hyperbolic surface.  This has the solution r=cosh^‑1(cosh(a)*cosh(b)).  When sin(0)=0 and cos(0)=1, this also has the solution r*eⁱ⁰=r +0*i and since eⁱ⁰=1, this also means r²=r²  and thus r=r.  But sin(π) is also zero, but cos(π)=-1 this means that as a complex number r*e^iπ=-r+0*i.    This can only be solved by the hyperbolic cosine and requires that r=ln(cosh(-r) ± sinh(-r)).  If -r is further replaced by –(x²+y²), this becomes ln(cosh(x²+y²) ± sinh(x²+y²)).  The single solution on a flat or spherical surface, when the coefficient of the imaginary axis is 0 and the angle of rotation of the imaginary axis is π, becomes a logarithm of cosh(x²+y²), the constant term, plus or minus an uncertainty, which is sinh(x²+y²), on a hyperbolic surface. However if the uncertainty is greater than the constant term, the logarithm is NOT imaginary.  It is undefined.

If a light cone is constructed which has three dimensions: space, time, and imagination, and space is measured on an absolute scale, i.e. there are no negative numbers, then the surface that light follows defines the edge of that volume.  The prohibition against negative numbers is merely an indication that the position in space is a vector, not a scalar.  The surface may not be flat, but may be hyperbolic. If that surface is hyperbolic, then the volume is not a cone connected to an inverted cone at the origin, but a two-sheeted hyperboloid where the two sheets connect at a single point at the origin.

The position in space as a vector is the velocity as a vector multiplied by time, v*t.  As a scalar this is often expressed as v*t, with the provision that v>0.  This effectively combines the dimensions of space, x,  with that of  time,  t.  Reality is defined as the position on the imaginary axis where i=0.  Thus the expression v*t+0*i is a statement of a position in real spacetime.  With v*t = a and with b=0, this is a complex number.  Using the conversion of a complex number in Cartesian coordinates, to one in polar coordinates this is re^ix=a+b*i where x is tan^-1(b/a) and r²=a²+b².  According to Euler’s Formula, there are two solutions when the coefient of the imaginary axis is 0: reⁱ⁰=v*t+0*i, where r²=(v*t)²+0² and tan^‑1(0/(v*t))=0; and re^iπ=-v*t+0*i where r²=(‑v*t)²+0² and  tan^-1(0/(-v*t))=π.  If r is on a flat, Euclidean, surface, r could be solved by Pythagoras’ Theorem.  However while t can be positive or negative, sin(x), the coefficient of the imaginary axis is cyclical, which means that while x can be positive or negative, it is properly x=n*π where n can be positive or negative and has a maximum and minimum value at integer multiples of π.

Einstein’s Theory of Realitivity can be derived from Einstein’s Triangle,  (mc²)²=(mvc)²+(m₀c²)², where v is the velocity of a particle, m is the relativistic mass of a particle, m₀ is the rest mass of that particle, and c is the speed of light.  This is conventionally solved on a flat, Euclidean, surface, but if Einstein’s Triangle is solved for a hyperbolic surface, the correct formulation would be

m=m_0* ln(cosh(1-v²/c²)±sinh(1-v²/c²))

or

 v=c*ln(cosh(1-m/m₀)±sinh(1-m/m₀))

If the Cartesian coordinates are converted to polar coordinates, and x is an integer multiple of π,  then the polar radius can be solved as

r=ln(cosh(-v*t))*cosh(n*π) ± sinh(-v*t) * sinh(n*π)).

Further if cosh(n*π)=0 and sinh(n*π)=-1, since sinh(-x)=-sinh(x), then this can be stated as 

r=ln(0±sinh(v*t)))

Since sinh(x)=-cos(i*x), while the radius can be defined as the natural logarithm of a regular cyclical trigonometric function, that function only repeats, is cyclical, in imaginary, not real, planes.  It is a single value in the real plane where the coefficient of the imaginary axis is zero.