We have felt;

Infinity depends on definition. Or, it might be better to understand as an undefined term. In any case, it is imposible to make infinity as a mathematical subject. Models simulate hyperbolic geometry perfectly, but not infinity itself. Lobachevsky established hyperbolic geometry with postulate perfectly. But he relied on his own supposition about infinity itself. Also he judged the geometry stands up by insight without evidence. Riemann said that we can go into any world by choosing a certain metric. But to give metric sounds intentional and not inevitable. It is the most reasonable that we use projective geometry following Klein's Erlangen Programme. Be that as it may, we still feel something kile a conjuring trick. We have not found a necessity to get into group theory. According to pologists, almost everything around us has the hyperbolic geometry. |

The following is histrical men and their works for the source of our story.

Euclid: He wrote that Euclid's Elements. (around BC 300)

His logic has been so strong for more than 2000 years!

Pappus: He found the cross ratio is invariant. (around 320)

Mercator: He made the world map by using conformal mapping. (1538)

Napier: He invented the logarithm. (1614)

Newton and Leibniz: They invented differential and integral

calculus. (around 1670)

Huygenes, Leibniz and Johann Bernoulli: They invented hyperbolic

function. (1691)

Poncelet: He made projective geometry. (1822)

Lobachevsky, Bolyai and Gauss: They discovered hyperbolic geometry.

(around 1826)

Gauss: He developed differential geometry. (since around 1827)

Laguerre: He expressed an angle with cross ratio. (1853)

Riemann: His idea was to construct a geometry by giving metric. (1854)

Beltrami: He gave us Pseudo-sphere. He pointed out that it has

constant negative curvature. (1868)

Kline: He made a disc model (1871), and announced Erlangen Programme. (1872)

Poincare: He invented Upper Half Plane and disc models. (1882)

Thurston: He found that the Hyperbolic Non-Euclidean World is

fundamental point of topology. (around 1975)

Fig. 1 above is the closing entertainment for our mysterious sightseeing. It is made by Hayakawa, and named Hyperbolic Jungle Jim.

Well, What's next? We will deal with so-called figure-8 knot that is closely related to hyperbolic geometry. And we are interested in Minkowski space, too.

Fig. 2 below shows connections among hyperbolic geometry and other three fiels. It is copied from Hyperbolic Geometry by J.W. Cannon, W.J. Floyd, and W.R. Parry. Red marks that we added are our interest.

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The End of Part 2 of Hyperbolic Non-Euclidean World