L'Chaim

 

To Life

May all your futures be pleasant ones
Not like our present ones
Drink, l'chaim, to life
To life, l'chaim
L'chaim, l'chaim, to life
It takes a wedding to make us say
Let's live another day
Drink, l'chaim, to life

Life appears to be reality plus imagination.

Euler’s Formula is e^ix=cos(x)+sin(x)i.  But this is true only on a Euclidean flat surface.

It is not true on a hyperbolic surface. On a  hyperbolic surface, Euler’s Formula is true when x=0, but it is not true when x=π.  Euler’s  Formula is true because it is the transformation of a complex number in polar coordinates to a complex number in Cartesian, rectangular, coordinates, when the real radius, r, is equal to 1. However coordinate transformation requires the use of a triangle.   A triangle behaves differently on flat and hyperbolic surfaces.  If that coordinate transformation is of a complex number with both a real and an imaginary axis, then it is re^ix=rcos(x)+rsin(x)i.  It is true that x=tan^‑1(rsin(x)/rcos(x)) on all surfaces.  However r²=(rcos(x))²+(rsin(x))² only is r=√((rcos(x))²+v(rsin(x))²) on a flat space.  On hyperbolic surfaces, if x=0, then this also appears to be correct, but for x=π, if the coefficient of the imaginary axis is not ignored, then this leads to r=cosh^‑1(cosh(rcos(x))*cosh(rsin(x))),  and for x=π, it leads to r=-r.  This is only true if the coefficient of the real axis, the radius, is always equal to 0.  Euler’s Formula works because of the trigonometric identity cos²+sin²=1.  However the hyperbolic trigonometric identity is cosh²‑sinh²=1.  On a hyperbolic surface, it is proposed that Euler’s formula should be restated as e^ix=cosh(x)-isinh(x).  In this case, you can take advantage of the fact that cosh(x) is symmetric about the zero axis, i.e. cosh(x)=cosh(-x), while in flat space it is sin(x) that is symmetric, i.e. sin(x)=sin(-x).  With the restatement of Euler’s Formula, r=ln(0±sinh(cosh(x))), and for x=π, r can be any value and is not limited to only zero.

So not just for zero, but for non-zero values of the radius of reality, if we live on a hyperbolic surface, life may be -r+0*i.  It has a coefficient of zero for imagination, but it is imagination nonetheless.