Question

In: Math

Illustrate with a picture showing that the following are simply not true in Hyperbolic Geometry a)....

Illustrate with a picture showing that the following are simply not true in Hyperbolic Geometry

a). The area of a triangle can be made arbitrarily large

b). The angle sum of all triangles is a constant.

Solutions

Expert Solution

a)

In hyperbolic geometry, the sum of the angles of a triangle is always less than 180 degrees (PI radians). The amount less than 180 is called the defect. In 1794 Gauss discovered this formula for the area of a triangle in hyperbolic geometry: ; where alpha, beta, and gamma are the interior angles of the triangle.

Thus the area is proportional to the defect, with the above proportionality constant (k is 1 for the model of the hyperbolic plane we're using). Now it's easy to see why there is an upper limit to the area of all triangles; namely, the defect measures how much the angle sum is less than 180. Since the angle sum can never get below 0, the defect can never get above 180. Therefore, the area of a triangle in hyperbolic geometry is bounded.

b)Using the above theorem we can safely say that

Again if we take any point P on AD,

If we keep on doing this infinitely we get infinite triangles with a different defect or infinite triangles with different angle sum (), where is the defect. Hence angle sum of a triangle in hyperbolic geometry is not constant.


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