### Hyperbolic Ornaments

Here is the interactive and theoretical part of the subject..

The Hyperbolic Applet

The basic construction consists of the hyperbolic triangle with the angles α, β and γ given by natural numbers k, n, m as fractions of 180°.

α = 180°/k, β = 180°/n and

γ = 180°/m with the condition

1/k + 1/m + 1/n < 1

The traingle has two straight lines AB and AC and an arc side BC. You controll the situation of the points A, B by moving the small yellow circles in the drawing window. The applet then completes the construction. The point C is calculated under the condition of the given angles. α is measured normally and the angles β and γ are measured between the straight lines and the circle line tangents. The sum of the three angle values has to be less than 180°. As a second condition the circle line of BC has to cut the circle with the centre A vertically.

Therefor the angle δ and the centre M of the construction circle is given. The blue fixed circle cuts the construction circle around M vertically. If the angles of the hyperbolic triangle are given by natural numbers k, m, n, so that the conditions are like noticed left then we have some interesting results:

- All picture triangles produced as iterated reflections at the triangle sides of the basic triangle itself, give a complete tesselation of the blue circle. They don't cover each other and only contact at its borders.
- The resulting triangle corners are from the same order as the corners of the basic hyperbolic triangle.

The applet is easy to use. Chose the natural numbers k, n und m without hurting the condition above.
If the image looks to "dotty", let draw again or increase the values for dots and iterations. A lot of experiments could be done by changing the generators. A generator is a short list of the allowed numbers 1, 2 or 3 representing the mirror operations at the triangle sides AB, AC or BC. The default is to have these three basic operations. Combining numbers for each generator p. e. 13 will change the resulting points: The new picture point is produced by the mirror operation at the side AB followed by the mirror operation at the side BC. You have to consider that a combination of p. e. 22 will do nothing, because two identical mirror operations will bring the first point back.