8,1,2,3,1,3,2,1,2,3,3,1,3,2,1,3,2,3,4,2,1,2,1,2,7,2,3,1 ,5,1,3,3,2,3,3,1,5,1,2,1,6,6,2,1,2,3,1,5,3,3,3,1,3,2,1, 3,2,7,2,1,2,3,4,3,5,1,2,3,1,3,3,3,2,3,1,3,2,4,5,1,5,1,2 ,1,2,3,4,2,1,2,6,4,2,1,3,2,3,6,1,2,1,6,3,2,3,3,3,1,3,5, 1,2,3,1,3,3,2,1,5,1,5,1,2,3,3,1,3,3, 2,3,4,3,2,1,3,2,3,4 ,2,1,3,2,4,3,2,4,2,3,4,5,1,5,1,3,2,1,2,1,5,1,5,1,2,1,2, 7,2,1,2,3,3,1,3,2,4,5,4,2,1,2,3,4,3,2,3,3,3,1,3,3,2,1,2 ,3,1,5,1,2,1,5,1,5,1,3,2,4,3,2,1,2,3,3,4,2,1,3,5,4,2,1, 3,2,4,5,3,1,2,4,3,3,2,1,2,3,1,3,2,3,1,5,6,1,2,1,2,3,1,3 ,2,1,2,6,1,3,3,5, 3,4,2,1,2,1,2,4,3,6,2,3,1,6,2,1,2,3,4, 2,1,2,1,6,5,1,2,1,2,3,1,5,1,2,3,4,3,2,1,3,2,3,4,2,3,1,2 ,4,3,2,3,1,2,3,1,3,3,2,3,3,4,3,2,1,5,1,5,1,2,1,5,1,3,2, 1,5,3,1,3,2,1,3,2,3,4,3,2,1,6,5,3,1,2,3,1,6,2,1,2,4,3,2 ,1,2,1,5,1,5,3,1,2,3,1,3,2,1,5,3,1,3,2,6,3,4,3,2,1,2,13. The Gudermannian Angles 4, 3,2,3,1,2,3,4,3,3,2,3,1,3,2,1,2,1,5,6,1,2,6,1,3,2,1,2,3 ,3,1,6,3,2,9,1,2,1,2,4,3,2,3,1,2,4,3,3,2,1,2,3,1,3,2,1, 2,6,1,6,3,2,3,1,3,2,3,3,3,1,3,2,1,3,2,3,4,2,1,2,1,2,7,2 ,3,1,5,1,3,3,2,1,5,1,5,1,2,7,5,1,2,1,2,3,1,3,5,3,3,1,5, 1,3,2,3,4,2,1,2,3,4,3,5,1,2,3,1,3, 3,2,1,2,3,1,3,2,1,3,5 ,1,5,3,1,2,3,4,2,1,2,6,1,3,2,1,3,2,3,6,1,2,1,2,4,3,2,3, 1,5,1,3,5,3,3,1,3,2,1,2,1,5,1,6,2,3,3,1,6,2,3,3,1,3,2,1 ,3,2,7,2,1,3,2,4,3,2,3,1,2,3,4,3,3,5,1,3,2,3,1,5,1,5,1, 2,1,2,4,3,2,1,2,3,3,4,2,4,2,3,4,2,1,2,1,6,3,2,3,3,3,1,3 ,3,2,1,2,3,1,3, 3,2,1,5,1,5,1,3,2,3,1,3,2,1,2,3,7,2,1,3, 5,4,2,1,2,1,2,4,5,4,2,4,3,5,1,2,3,1,3,2,3,1,5,1,5,1,2,1 ,2,3,4,2,1,2,3,3,1,3,3,3,5,4,2,1,2,3,4,3,2,4,2,3,1,3,3, 2,1,2,3,6,1,2,1,5,1,5,1,2,1,2,4,5,1,2,3,4,3,2,1,3,2,3,4 ,2,4,2,4,3,2,3,1,2,3,1,3,3,2,3,3,1,3,3,2,1,5,6,1,2, 1,2, 3,1,3,2,1,8,1,3,2,1,5,3,4,2,1,2,1,6,3,5,1,2,3,1,6,2,1,2 ,4,3,2,1,2,1,6,5,3,1,2,3,1,3,2,1,2,3,3,1,3,2,1,5,3,4,5, 1,2,4,3,2,3,1,2,3,1,3,3,3,2,3,4,2,1,2,1,5,6,1,2,1,5,1,3 ,2,1,2,3,3,1,5,1,3,2,7,3,2,1,2,4,5,3,1,2,3,1,3,3,2,1,2, 4,3,2,1,2,6,1,6,2,1,2,3,1,3,2,1,“Wonder5,3,1,3,2,4,2,3,4,2,1,2 is the beginning of wisdom” Greek Proverb ,1,2,7,2,3,1,5,4,3,2,1,2,3,1,5,1,2,1,6,5,1,2,3,3,1,3,2, 3,3,3,1,3,3,3,2,3,6,1,2,3,4,3,5,1,2,4,3,3,2,1,2,3,1,3,2 ,1,3,5,1,5,1,3,2,3,4,2,3,6,1,3,2,1,3,2,3,6,1,2,1,2,7,2, 3,1,2,3,1,3,5,1,5,1,3,2,1,2,6,1,5,1,2,3,3,1,3,3,2,3,3,1 ,5,1,3,2,3,4,2,1,3,2,4,3,2,3,1,2,3,4,3,2,1,5,1,3,2,1,3, 5,1,5,3,1,2,4,3,2,1,2,3,3,1,3,2,4,2,3,4,3,4,2,1,2,3,2,3 Complex Arithmetic – Patterns Behind Numbers 274 Page Copyright © N. Trif – Ottawa, Canada, 2010 Complex Arithmetic – Patterns Behind Numbers 1. The Gudermannian Angles The gudermannian of a hyperbolic angle express in hyperbolic radians is equal with the circular radians measure of the angle of the corresponding circular sector. This angle is called the gudermannian angle. This angle was introduced by Gudermann and was called the longitude of u. Historically its inverse is in fact the first angle introduced by Lambert and called the transcendental angle. Let’s start from a rectangular hyperbola and consider the hyperbolic angle 푢. The hyperbolic sine, hyperbolic cosine and hyperbolic tangent are defined by (see Figure 13-01): 푂퐻1 = sinh(푢) 푂퐻2 = cosh(푢) 푂퐻3 = tanh(푢) If from 퐻2, the foot of the ordinate of N, we draw 퐻2푀 tangent to the unit circle with the center in O, then the angle AOM is the gudermanina of the hyperbolic angle u. 퐻2푀 ⊥ 푂푀 ′ ′ 퐻2푀 ⊥ 푂푀 푂퐶1 = sin(푣) 푂퐶2 = cos(푣) 푂퐶3 = tan(푣) If v is the circular angle and v the corresponding hyperbolic angle the following relationships are also true: 1 cosh(푢) = = sec(푣) cos⁡(푣) sinh(푢) = tan⁡(푣) tanh(푢) = sin⁡(푣) 275 Page Copyright © N. Trif – Ottawa, Canada, 2010 Complex Arithmetic – Patterns Behind Numbers Figure 13-01: The definition of gudermannian and lambertian functions. The connection between the circular and hyperbolic functions These are some of the most important properties of the gudermannian angle: 1. The bisector of angle AOM also bisects the sector AON, and the line AN. 퐴푂푋 = 푋푂푀 퐴푂푌 = 푌푂푁 퐴푍 = 푍푁 276 Page Copyright © N. Trif – Ottawa, Canada, 2010 Complex Arithmetic – Patterns Behind Numbers Figure 13-02: The bisector of angle AOM also bisects the hyperbolic angle AON, and the line AN. 2. The bisector of angle AOM is parallel to MN, and the points M,N and A’ are in line. 푂푍 ∥ 푀푁 퐴′푀푁 − 푛⁡푙푛푒 277 Page Copyright © N. Trif – Ottawa, Canada, 2010 Complex Arithmetic – Patterns Behind Numbers Figure 13-03: The bisector of angle AOM is parallel to MN, and the points M,N and A’ are in line 3. The tangent at N passes through the foot of the ordinate of M, and intersects MH2 on the tangent at A. 푀퐶2 ⊥ 푂퐻2 퐶2푁 − 푡푎푛푒푛푡⁡푡표⁡푡ℎ푒⁡ℎ푦푝푒푟푏표푙푎 278 Page Copyright © N. Trif – Ottawa, Canada, 2010 Complex Arithmetic – Patterns Behind Numbers Figure 13-04: The tangent at N passes through the foot of the ordinate of M, and intersects MH2 on the tangent at A. 4. The angle 퐴푁퐻2 is half of the gudermannian angle AOM. 1 퐴푁퐻 = 퐴푂 2 2 279 Page Copyright © N. Trif – Ottawa, Canada, 2010 Complex Arithmetic – Patterns Behind Numbers 2. Geometrical Interpretation of the Gudermannian This section is based on the Introduction of the Smithsonian Mathematical Tables “Hyperbolic Functionsi book “Geometrical Illustrations of Hyperbolic Functions. To better understand the connection between the circular action and hyperbolic action, it is worth presenting the connection that exists between the circle, the rectangular hyperbola and a specific ellipse. The general equation of the ellipse is: 푢2 푣2 + = 1 훽2 훼2 Figure 13-05: The unit circle, the rectangular hyperbola and the corresponding ellipse.