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 , , and :
Other viewpoints :
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 Parametrization of Boy’s surface
Rob Kusner and Robert Bryant discovered the beautiful parametrization of the Boy’s surface on a given complex number , where , so that giving the Cartesian coordinates of a point on the surface.
In 1986 Apéry gave the analytic equations for the general method of nonorientable surfaces. Following this standard form, parametrization of the Boy’s surface can also be written as a smooth deformation given by the equations :
varies from 0 to 1.
Here are some links related to Boy’s 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 :
Where , controls how far the tip goes, and controls the girth.
When , , and :
With orthographic projection :
About Pseudospherical Surfaces :
Surface in having constant Gaussian curvature are usually called pseudospherical surfaces.
If is a surface with Gaussian curvature then it is known that there exists a local asymptotic coordinate system on such that the first and second fundamental forms are:
, and ,
where is the angle between asymptotic lines (the x-curves and t-curves). The Gauss-Codazzi equations for in these coordinates become a single equation, the sine-Gordon equation (SGE) :
The SGE is one of the model soliton equations.
References and Readings :
- Chuu-Lian Terng. 2004. Lecture notes on curves and surfaces in , 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.
Lewat blognya Colleen Young via Twitter awalnya saya dapatkan info mengenai software ini. Formulator Tarsia ialah sebuah editor/tool yang bisa digunakan oleh guru matematika untuk membuat suatu aktivitas belajar siswa di kelas. Aktivitas belajar ini diformulasikan dalam bentuk Puzzle seperti Jigsaw, domino, follow-me cards, rectangular cards, matching rectangular cards, student discussion circle, dll. Masing-masing Puzzle telah disediakan dalam berbagai bentuk template dan guru bisa memilih template yang cocok untuk dikembangkan.
Formulator Tarsia Templates
Formulator Tarsia ini hanya bisa bekerja di Windows (2000/XP/2003 Server/Vista), diproduksi oleh Hermitech Lab. Untuk mengunduhnya silahkan kunjungi halaman ini (scroll ke bawah sampai ada bagian dengan judul Formulator Tarsia). Dalam paket download/instalasi juga disertakan beberapa contoh Puzzle dari Bryan Dye (mathsnet.net) dan Craig Barton (mrbartonmaths.com).