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},

dimana solusi riilnya adalah:

\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 :

Boy's Surface (Bryant-Kusner)

Other viewpoints :

Boy's Surface (Bryant-Kusner) 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<a<1, u controls how far the tip goes, and v controls the girth.

When a=0.4, -13<u<13, and -38<v<38 :

With orthographic projection :

About Pseudospherical Surfaces :

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 forM 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.

References and Readings :

  • Chuu-Lian Terng. 2004. Lecture notes on curves and surfaces in \mathbb{R}^{3}, available here.
  • Chuu-Lian Terng. 1990s. About Pseudospherical Surfaces, available here.
  • Richard S Palais. 2003. A Modern Course on Curves and Surfaces, available here.

About 3D-XplorMath :

3D-XplorMath is a Mathematical Visualization program. The older original version, written in Pascal, runs only on Macintosh computers, but there is also a newer cross-platform Java version, called 3D-XplorMath-J which is written in the Java programming language; to use it, you must have Java 5.0 or higher installed on your computer.