ln (phi) dalam Fungsi Hiperbolik

Bentuk lain $\ln (\phi)$ dalam fungsi hiperbolik, tepatnya pada inverse hyperbolic cosecant dari 2, atau inverse hyperbolic sine dari 1/2:

$\displaystyle \boxed {\ln(\phi)=\mathrm{csch}^{-1}(2)= \mathrm{sinh}^{-1}\left ( \frac{1}{2} \right )= 0.481211825....}$

Formula kunci kemunculan $\ln(\phi)$ dalam fungsi hiperbolik dari adanya masalah ini  :

$\displaystyle \boxed {e^k-e^{-k}=1}$ atau $\displaystyle \boxed {e^{2k}-e^k-1=0}$,

$\boxed {k=\ln(\phi)}$ atau $\boxed {e^{k}=\phi}$

dengan $e$ adalah konstanta Euler dan $\phi$ adalah golden ratio.

Ini juga tak kalah menarik, bisa dibuktikan pula bahwa $\ln (\phi)$ dapat dinyatakan dalam perluasan deret berikut ini :

$\displaystyle \boxed {\ln(\phi)=\sum_{n=0}^{\infty} \frac{(-1)^n(2n)!}{2^{4n+1}(2 n+1)(n!)^2}= 0.481211825....}$

Boy’s Surfaces

Boy’s surface is a nonorientable surface found by Werner Boy in 1901. Boy’s surface is an immersion of the real projective plane in 3-dimensional without infinities and singularities, but it meets itself in a triple point (self-intersect). The images below are generated using 3D-XplorMath, and with the “optimal” Bryant-Kusner parametrization when $a= 0.5$, $-1.45 < u < 0$, and $0 < v < 2\pi$ :

Other viewpoints :

Projective Plane

There are many ways to make a model of the Boy’s surface using the projective plane, one of them is to take a disc, and join together opposite points on the edge with self intersection (note : In fact, this can not be done in three dimensions without self intersections); so the disc must pass through itself somewhere. The Boy’s surface can be obtained by sewing a corresponding band (Möbius band) round the edge of a disc.

The $\mathbb{R}^{3}$ Parametrization of Boy’s surface

Rob Kusner and Robert Bryant discovered the beautiful parametrization of the Boy’s surface on a given complex number $z$, where $\left | z \right |\leq 1$, so that giving the Cartesian coordinates $\left ( X,Y,Z \right )$ of a point on the surface.

In 1986 Apéry gave  the analytic equations for the general method of nonorientable surfaces. Following this standard form, $\mathbb{R}^{3}$ parametrization of the Boy’s surface can also be written as a smooth deformation given by the equations :

$x\left ( u,v \right )= \frac {\sqrt{2}\cos\left ( 2u \right ) \cos^{2}\left ( v \right )+\cos \left ( u \right )\sin\left ( 2v \right ) }{D},$

$y\left ( u,v \right )= \frac {\sqrt{2}\sin\left ( 2u \right ) \cos^{2}\left ( v \right )-\sin \left ( u \right )\sin\left ( 2v \right ) }{D},$

$z\left ( u,v \right )= \frac {3 \cos^{2}\left ( v \right )}{D}.$

where

$D= 2-a\sqrt{2}\sin\left ( 3u \right )\sin\left ( 2v \right )$,

$a$ varies from 0 to 1.

Here are some links related to Boy’s surface :

Parametric Breather Pseudospherical Surface

Parametric breather surfaces are known in one-to-one correspondence with the solutions of a certain non-linear wave-equation, i.e., the so-called Sine-Gordon Equation. It turns out, solutions to this equation correspond to unique pseudospherical surfaces, namely soliton. Breather surface corresponds to a time-periodic 2-soliton solution.

Parametric breather surface has the following parametric equations :

$x = -u+\frac{2\left(1-a^2\right)\cosh(au)\sinh(au)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)}$

$y = \frac{2\sqrt{1-a^2}\cosh(au)\left(-\sqrt{1-a^2}\cos(v)\cos\left(\sqrt{1-a^2}v\right)-\sin(v)\sin\left(\sqrt{1-a^2}v\right)\right)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)}$

$z = \frac{2\sqrt{1-a^2}\cosh(au)\left(-\sqrt{1-a^2}\sin(v)\cos\left(\sqrt{1-a^2}v\right)+\cos(v)\sin\left(\sqrt{1-a^2}v\right)\right)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)}$

Where $0, $u$ controls how far the tip goes, and $v$ controls the girth.

When $a=0.4$, $-13, and $-38 :

With orthographic projection :

Surface in $\mathbb{R}^{3}$ having constant Gaussian curvature $K= -1$ are usually called pseudospherical surfaces.

If $X : M \subset \mathbb{R}^{3}$ is a surface with Gaussian curvature $K= -1$ then it is known that there exists a local asymptotic coordinate system $(x,t)$ on $M$ such that the first and second fundamental forms are:

$dx^{2}+dt^{2}+2 \cos{q}~dx~dt$, and $2 \sin{q}~dx~dt$,

where $q$ is the angle between asymptotic lines (the x-curves and t-curves). The Gauss-Codazzi equations for$M$ in these coordinates become a single equation, the sine-Gordon equation (SGE) :

$q_{xt}= \sin{q}$

The SGE is one of the model soliton equations.

• Chuu-Lian Terng. 2004. Lecture notes on curves and surfaces in $\mathbb{R}^{3}$, available here.