Preface

      What is hyperbolic geometry? Our purpose of this webpage is to enjoy seeing the Hyperbolic Non-Euclidean World with our own eyes. Seeing is believing. Not only observation, strong imagination is necessary for our adventure. Information and knowledge are somewhat useful, but imagination is the most powerful weapon we have.

      In the Hyperbolic Non-Euclidean World we can see a panoramic view of much more than 360. The area of an infinitely large triangle is only (the ratio of the circumference of a circle to its diameter) and the sum of interior angles is zero. Angles and lengths are not of different natures, but they depend on each other. Many great mathematicians did not believe these facts though they themselves proved these theories.

      First, we will look briefly at what is called the Elliptic Non-Euclidean World, where we will be able to draw infinity into our hands. Then, we will enter the Hyperbolic Non-Euclidean World discovered by Nicolai Ivanovitch Lobachevsky and two other men, where we will be able to see many mysterious pictures. Before long, we will be in projective geometry which is more weird. For example, a tangent to a circle passes the center and the length of an infinitely stretched straight line is zero. Last, we will observe the compliment space of the figure-eight knot. We will experience the function of hyperbolic geometry. A space is characterized by its function. We will change "impossible" into "possible". All faces of a tetrahedron are glued to another tetrahedron without changing the shape of either. You will meet your clone. Morning coffee is the universe, and we drink it up.

      I, the author, am not a mathematician but a simple hobbyist. I had thought previously that Non-Euclidean geometry was old-fashioned. Indeed, Felix Klein wiped all mysterious matters from hyperbolic geometry. Later on, however, William P. Thurston dug out new mysteries. Today, hyperbolic geometry is not only an essential part of topology and knot theory, but it is applied also to physics, chemistry, biology and even the arts.

      You know that we can enjoy a masterpiece of painting even though we can not paint it ourselves. Everybody has the right to enjoy true mathematics even if one is not familiar with math. It is not necessary for sightseers to know the laws or rules of the region being visited. All we have to do is to see how the mountains look and how the rivers run.
      I began writing this webpage without any knowledge of hyperbolic geometry. I pursued whatever came to my mind on a given occasion. Sometimes I accepted a mathematician's idea without question. Anyway, the Hyperbolic Non-Euclidean World is very mysterious and captivating, so have fun! Let's take off on an academic sight-seeing flight, and enjoy panoramic views of infinity!