
Technical Reference 
Complex Formulae / Roots and Powers Chart / Important Mathematical Constants / Polyhedra / Spherical Harmonics /
1) z = x + iy where x = Real part of z and y = Imaginary part of z
2) c = a + ib where a = Real part of c and b = Imaginary part of c
3) z = re^iq = (sqrt(x^2
+ y^2)) (cos q + i sin q)
where q = arctan (y / x), r = sqrt(x^2 + y^2) and "sqrt" means square
root
4) z^n = r^n*e^inq = (sqrt(x^2 + y^2))^n (cos nq + i sin nq) ; r and q as above
5) sqrt(z) = (sqrt(r)sqrt(e^iq))
= (sqrt(sqrt(x^2 + y^2))) [cos (.5 arctan (y / x))
+ i sin (arctan (y / x))]
6) ln z = ln[sqrt(x^2 + y^2)] + i arctan (y / x)
7) e^z = e^x(cos y + i sin y)
8) sin z = sin x cosh y + i cos x sinh y = i sinh iz = (e^iz  e^iz) / 2i
9) cos z = cos x cosh y  i sin x sinh y = cosh iz = (e^iz + e^iz) / 2
10) sinh z =  i sinh iz = (e^z  e^z) / 2
11) cosh z = cos iz = (e^z + e^z) / 2
12) tanh z =  i tan (iz) = (e^z  e^z) / (e^z + e^z)
13) sech z = sech (iz) = [cosh z] ^ 1
14) csch z = i csc (iz) = [sinh z] ^ 1
15) arcsinh z = ln(z + sqrt(z^2 + 1))
16) arccosh z = ln(z + sqrt(z^2  1)) , ln(z  sqrt(z^2  1))
17) arctanh z = .5 * ln[(1 + z) / (1  z)]
18) arcsech z = ln[(1 + sqrt(z^2 + 1)) / z]
19) arccsch z = ln[(1 + sqrt(1  z^2 )) / z] , ln[(1  sqrt(1  z^2 )) / z]
20) arccoth z = .5 * ln[(z + 1) / (z  1)]
21) sin^2(z) + cos^2(z) = 1
22) cosh^2(z)  sinh^2(z) = 1
23) tan z = (sin 2x + i sinh 2y) / (cos 2x + cosh 2y)
24) cot z = (sin 2x  i sinh 2y) / (cosh 2y  cos 2x)
25) nth root of z = [nth root of (x^2 + y^2)](cos (q / n) + i sin (q / n))
26) Newton's Method z(n+1) = z(n)  [f(z(n)) / f '(z(n))]
27) Henon Attractor: (for z(n) = x(n) + iy(n)) , x(n+1) = ax(n) + y(n) and y(n+1)= bx(n)
28) Halley Map: z(n+1) = z(n)  L[(2f(z(n))f '(z(n))) / (2(f '(z(n)))^2  f' '(z(n))f(z(n)))]
29) Lorenz Attractor: dx / dt = a(y  x) dy / dt = x(r  z)  y dz / dt = xy  bz
N  N^2  N^3  sqrt(N)  N  N^2  N^3  sqrt(N)  
1  1  1  1  21  441  9261  4.583  
2  4  8  1.414  22  484  10648  4.690  
3  9  27  1.732  23  529  12167  4.796  
4  16  64  2  24  576  13824  4.899  
5  25  125  2.236  25  625  15625  5  
6  36  216  2.449  26  676  17576  5.099  
7  49  343  2.646  27  729  19683  5.196  
8  64  512  2.828  28  784  21952  5.292  
9  81  729  3  29  841  24389  5.385  
10  100  1000  3.162  30  900  27000  5.477  
11  121  1331  3.317  31  961  29791  5.568  
12  144  1728  3.464  32  1024  32768  5.657  
13  169  2197  3.606  33  1089  35937  5.745  
14  196  2744  3.742  34  1156  39304  5.831  
15  225  3375  3.873  35  1225  42875  5.916  
16  256  4096  4  36  1296  46656  6  
17  289  4913  4.123  37  1369  50653  6.083  
18  324  5832  4.243  38  1444  54872  6.164  
19  361  6859  4.359  39  1521  59319  6.245  
20  400  8000  4.472  40  1600  64000  6.325 
Important Mathematical Constants
1) Pi  The ratio of the circumference of a circle to its diameter, supposedly first discovered by Archimedes (287212 BC). He surmised that pi was
3 10/17 < Pi < 3 1/7
The first hundred digits of pi are given here though I understand that 50 billion digits (!) have been calculated already:
Pi = 3. 1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...
Probably the most famous formula for determining pi is Leibnitz' formula=
Pi = 4  (4/3) + (4/5)  (4/7) + (4/9)  (4/11) =
Summation (from n=0 to infinity) of [(1)^n][4/(2n+1)]
Another famous summation involving pi was discovered by Euler as:
(Pi^2)/6 = 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 +.... + (1/n)^2
2) e  The natural logarithm base, supposed named after the great mathmatician Leonhard Euler. The first hundred digits of e are given here as well:
e = 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 ...
I offer my students two ways to remember how to calculate the value of e:
e = limit (as n > infinity) of (1 + 1/n)^n
e= Summation (from n=0 to infinity) of simply (1/n!)
3) Feigenbaum's Number  This number, first shown by Becker and Dorfler, was demonstrated by Mitchell Feigenbaum to be a fundamental constant of nature having to do with the ratio of intervals of growth rate versus the doubling of up and down cycles characteristic of that rate. Keith Briggs, a scientist from the University of Melbourne, Australia, has calculated the most precise Feigenbaum number to date:
F = 4. 6692016091 0299067185 3203820466
2016172581 8557747576 8632745651
3430041343 3021131473 7138689744 0239480138 17165984855 1898151344
0862714202 7932522312 4429888908 9085994493 5463236713 4115324817 1421994745
5644365823 7932020095 6105833057 5458617652 2220703854 1064674949 4284981453
3917262005 6875566595 2339875603 825637225
4) Square Root of two = 1. 41421 35623 73095 0488...
5) Square Root of three = 1. 73205 08075 68877 2935...
6) Square Root of five = 2. 23606 79774 99789 6964...
7) Square Root of pi = 1.77245 38509 05516 02729 8167... (also known as Gamma(.5))
8) Square Root of e = 1. 64872 12707 00128 1468...
9) The Golden Mean, phi = (1 + sqrt(5)) / 2 = 1.61803 39887 99894...
10) e ^ pi = 23. 14069 26327 79267 006...
11) pi ^ e = 22. 45915 77183 61045 47342 715...
12) e ^ e = 15. 15426
13) Euler's constant (usually given as lower case gamma) = .57721 56649 01532 86060 6512...
= limit (as n goes from 1 to infinity) of (1/n  ln n)
14) 1 radian = the number of degrees that are subtended when the length of a radius is traced along the circumference of a circle.
1 radian = 180 / pi = 57. 29577 95130 8232...
15) ln 2 = .69315... = 1  1/2 + 1/3  1/4 + 1/5  1/6 + ...
= Summation (from n = 1 to infinity) of (1)^(n+1) * (1/n)
16) ln 10 = 2.30259...
There is a library of more obscure mathematical constants HERE.
Polyhedra, the plural of polyhedron, are threedimensional solid figures with many geometrical faces to them. There are five commonly known regular polyhedra, regular meaning all faces are congruent and all edges and angles are congruent. They are:
Tetrahedron  4 faces  equilateral triangle 
Hexahedron (Cube)  6 faces  square 
Octahedron  8 faces  equilateral triangle 
Dodecahedron  12 faces  pentagon 
Icosahedron  20 faces  equilateral triangle 
There is information regarding formulas to find the volumes, surface areas, inscribed radii, and circumscribed radii of the above polyhedra HERE.
There are also the Archimedean solids, solid shapes whose faces are all regular polygons of two or more kinds, and whose vertices are all identical. There are 13 different kinds. Two (the snub cube and snub dodecahedron) come in paired mirrorimage forms. Eleven of these solids can be formed by truncating (chopping the corners off) simpler solids. They have pleasingly symmetrical crystalline shapes, and are described below. These eleven are:
Truncated Tetrahedron  8 faces (4 triangles, 4 hexagons) 
Truncated Cube  14 faces (8 triangles, 6 octagons) 
Truncated Octahedron  14 faces (6 squares, 8 hexagons) 
Cuboctahedron  14 faces (8 triangles, 6 squares) 
Truncated Dodecahedron  32 faces (20 triangles, 12 dodecagons)  soccer ball pattern 
Truncated Icosahedron  32 faces (12 pentagons, 20 hexagons)  soccer ball / fullerene shape 
Icosidodecahedron  32 faces (20 triangles, 12 pentagons) 
Small Rhombicuboctahedron  26 faces (8 triangles, 18 squares) 
Great Rhombicuboctahedron  26 faces (12 squares, 8 hexagons, 6 octagons) 
Small Rhombicosidodecahedron  62 faces (20 triangles, 30 squares, 12 pentagons) 
Great Rhombicosidodecahedron  62 faces (30 squares, 20 hexagons, 12 dodecagons) 
Thanks to Grant Hutchison for the info.
Spherical harmonics are expressions in threedimensional spherical coordinates which are primarily used to describe the theoretical hybrid electron orbital shapes in molecules. The three coordinates are r (for radius), theta (degrees in the traditional xy plane), and phi (degrees in the yz plane). You may also recognize this way of laying out spatial coordinates from Star Trek's "210 mark 45" designation for navigation as the degrees in theta and phi. As with the rectangular coordinates, x, y, and z, we can describe any point in three dimensional space using such a coordinate system. All types of scientists use spherical and cylindrical (rho, theta, and z) coordinate systems to analyze various physical phenomena.
Here is the general formula used to produce these mathematical "flying saucers" with some examples below...
r = (cos (theta))^2 + (cos(2 * theta))^4 + sin(4 * phi)
r = (cos(12 * theta))^5 + (cos(8 * theta))^3 + cos(6 * theta)
r = 2 * (cos(6 * theta))^6  4 * (cos(4 * theta))^4  2 * (cos(2 * theta))^2
rho = (sin(theta))^4 + (sin(2 * theta))^2 + e ^ (1  sin(z))
rho = 4 * (cos(4 * theta))^4  2 * (cos(2 * theta))^2 + (1 + cos (z))^2
You can experiment with an infinite number of possibilities. You will soon discover what each coefficient, exponent, and function does to the overall shape of the object. Happy Hunting!
For a very large and extended study of over 640 images, go HERE.
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