Joined 11 November 2007
  " Mathematics takes place at different time-scales. If you can solve a problem in 55 minutes that others need an hour to solve, you can probably get a good job. If you can solve a problem in a month that others might need a year to solve, you will probably do well as a graduate student. But if you can solve a problem in 10 years that nobody else can solve in a lifetime, you could be a great mathematician." Robert Israel


  " anyone who wishes to study this topic should earn a Ph.D. in number theory and spend several years researching the relevant topics in depth, with the guidance of a world class expert." Alon Amit, PhD in Mathematics; Mathcircler.


  "The author apologises wholeheartedly to those who dare read the source code." Freddie R. Exall

  A BELIEF IS NOT A PROOF.

  "Category theory is ... the most, abstract fields of mathematics" Robb Seaton

For the sake of completeness, here is the entire code I used:"



## icc

  convert image_rgb.tiff -profile "RGB.icc" -profile "CMYK.icc" image_cmyk.tiff


## some notes

Some notes from | wikipedia

$Z^{2}$ $Real=x^{2}-y^{2}$ $Imag=2*x*y$ $Z^{3}$ $Real=x^{3}-3*y^{2}*x$ $Imag=3*x^{2}*y-y^{3}$ $Z^{4}$ $Real=x^{4}-6*x^{2}*y^{2}+y^{4}$ $Imag=4*x^{3}*y-4*x*y^{3}$ $Z^{5}$ $Real=x^{5}-10*x^{3}*y^{2}+5*x*y^{4}$ $Imag=5*x^{4}*y-10*x^{2}*y^{3}+y^{5}$ $Z^{6}$ $Real=x^{6}-15*x^{4}*y^{2}+15*x^{2}*y^{4}-y^{6}$ $Imag=6*x^{5}*y-20*x^{4}*y^{2}+6*x*y^{5}$ $Z^{7}$ $Real=x^{7}-21*x^{5}*y^{2}+35*x^{3}*y^{4}-7*x*y^{6}$ $Imag=7*x^{6}*y-35*x^{4}*y^{3}+21*x^{2}*y^{5}-y^{7}$ ,

$Exp(Z)$ $Real=Exp(x)*Cos(y)$ $Imag=Exp(x)*Sin(y)$ $Ln(Z)$ $Real=0.5*Ln(x^{2}+y^{2})$ $Imag=Arctan(y/x)$ $Sin(Z)$ $Real=Sin(x)*((Exp(y)+Exp(-y))/2)$ $Imag=Cos(x)*((Exp(y)-Exp(-y))/2)$ $Cos(Z)$ $Real=Cos(x)*((Exp(y)+Exp(-y))/2)$ $Imag=-Sin(x)*((Exp(y)-Exp(-y))/2)$ $SinH(Z)$ $Real=Cos(y)*((Exp(x)-Exp(-x))/2)$ $Imag=Sin(y)*((Exp(x)+Exp(-x))/2)$ $CosH(Z)$ $Real=Cos(y)*((Exp(x)+Exp(-x))/2)$ $Imag=Sin(y)*((Exp(x)-Exp(-x))/2)$ ### parabolic/hyperbolic/elliptic

The meaning of the terms "elliptic, hyperbolic, parabolic" in different disciplines in mathematics

### curves

https://www.mathcurve.com/courbes2d.gb/rosace/rosace.shtml rose curve = n-folium: The curve is composed of a n base patterns. The pattern is called : the petal or branch / leaf / lobe - symmetrical about Ox obtained for angle between -pi/(2n) and pi/(2n)

osculating circle of a sufficiently smooth plane curve

#### curvature

Interesting curves involning the curvature concept by Xah Lee:

• Evolute curve (the centers of osculating circles)
• Radial curve (locus of osculating circle normals)
• circle = curve with constant curvature everywhere
• line = curve with curvature of 0 everywhere)
• Clothoid = spiral cirve of linearly increasing curvature)

### computer graphic

www.pling.org.uk/cs/cgv.html
www.tutorialspoint.com/computer_graphics/computer_graphics_curves.htm
github.com/jagregory/abrash-black-book
pages.mtu.edu/~shene/COURSES/cs3621/NOTES/model/b-rep.html
pages.mtu.edu/~shene/COURSES/cs3621/NOTES/notes.html


## test

$1/237142198758023568227473377297792835283496928595231875152809132048206089502588927\approx 4.21687917729220928973942962050800760308398455294740302003110521004325771638790385468222014406044810316697454753662...*10^{-}81$ ${\frac {1}{2^{267}-1}}={\frac {1}{237142198758023568227473377297792835283496928595231875152809132048206089502588927}}\approx 4.216879177292209*10^{-81}$ ( land on the root point of period 267 component : c267 = 0.250137369683480-0.000003221184145 i with angled internal adress : $1{\xrightarrow {1/267}}267$ ( land on the root point of period 268 component c268 = 0.250137369683480-0.000003221184145i period = 10000 i with angled internal adress : $1{\xrightarrow {267/268}}268$ ## mandelbrot set

the Mandelbrot set for the function 1/z - z∙(1 + 0.001∙z)/(1 - 0.002∙z + 0.001∙z2) = "1 -0.002 - 0.999 -0.001 0 1 -0.002 0.001":

 $f(z)={\frac {1}{z}}-{\frac {z*(1+0.001*z)}{(1-0.002*z+0.001*z^{2})}}$ # gradient line of the 2D scalar field

Key words:

• "gradient line" 2d "scalar field"

flow :

• level curves and gradient vector
• flow across continuously-spaced level curves
• gradient is the direction of steepest ascent in the zz-direction, the reverse of the flow is the path of an object as it rolls on the surface, starting from a high place and rolling down to a lower place (in the exact opposite direction as the gradient vectors point).
• def from Streamline Tracing on Irregular Grids by H˚akon Hægland
• "The instantaneous curves that are at every point tangent to the direction of the velocity at that point are called streamlines of the flow"
• "A pathline of a fluid particle is the locus of its position in space as time passes. It is thus the trajectory of a particle of fixed identity"

## Khan

the gradient points in the direction which increases the value of f most quickly. There are two ways to think about this direction:

• Choose a fixed step size, and find the direction such that a step of that size increases fff the most. Given steps of a constant size away from a particular point, the gradient is the one which increases f the most.
• Choose a fixed increase in fff, and find the direction such that it takes the shortest step to increase fff by that amount. Given steps which increase f by a given size, the gradient direction is the shortest among these.

Either way, you're trying to :

• maximize the rise over run,
• either by maximizing the rise, or minimizing the run.

# spiral

## The Golden Ratio and the Golden Angle

In disc phyllotaxis, as in the sunflower and daisy, the mesh of spirals occurs in Fibonacci numbers because divergence (angle of succession in a single spiral arrangement) approaches the golden ratio. The shape of the spirals depends on the growth of the elements generated sequentially. In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals—ideally. That is because Fermat's spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979 is

$r=c{\sqrt {n}},$
$\theta =n\times 137.508^{\circ },$

where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.

Illustration of Vogel's formula of the pattern of sunflower florets (see article) for n from 1 to 500, using the polar coordinates equations $r=c{\sqrt {n}}$  and $\theta =n\times {\frac {2\pi }{\phi +1}}$ . Can be produced using the following MATLAB code:

n=1:500;
r=sqrt(n);
t=2*pi/((sqrt(5)+1)/2+1)*n;
plot(r.*cos(t),-r.*sin(t),'o')


# Orthogonal

tangent to circle in the polar form

 z= r(t)

 y = x*tan(t)


Tangent to cardioid in polar form :

 z = 2a( 1 - cos(t))


is line :

 // y = (x*(-1+2*cos(t))*sin(t)+(-3+3*cos(t))*sin(t))/(-1+2*cos(t)^2-cos(t))
y = (x*(-1+2*cos(t))*sin(t)+(-2a+2a*cos(t))*sin(t))/(-1-cos(t)+2*cos(t)^2)


where t is changing from 0 to 2*pi

## ellipse

$(x,y)=(a\cos t,b\sin t),\ 0\leq t<2\pi \ .$

slope m

$m=-{\frac {b}{a}}\cot t\quad$

The equation of the tangent at point ${\vec {c}}_{\pm }(m)$  has the form $y=mx+n$

$y=mx\pm {\sqrt {m^{2}a^{2}+b^{2}}}\;.$

Implicit

• function
• implicit differentiation = differentiation of the implicit function

# smooth curve from points

"curve fitting is a set of techniques used to fit a curve to data points "

Fit method

• linear
• join points with segments = concatenated linear segments
• straight line using linear regression
• nonlinear
• polynomial
• cubic spline
• Smooth Bézier Spline Through Prescribed Points

# petal

"An attracting petal, P + , for a map M at zero is an open simply connected forward invariant region with 0 ∈ ∂P + , that shrinks down to the origin under iteration of M . More precisely, P + is an attracting petal if M (P + ) ⊂ P + ∪ {0} and n≥0 M n (P + ) = {0}. "

## example

http://mathoverflow.net/questions/104482/parabolic-immediate-basins-always-simply-connected?rq=1 "An example is f(z)=z+1−1/zf(z)=z+1−1/z. There is one petal for the neutral point at infinity. Let AA be the dmain of attraction of ∞∞. Critical points are ±i±i. Everything is symmetric with respect to the real line, because the function is real. One critical point is in AA, so by symmetry the other one is also in AA. The map f:A→Af:A→A is 2-to-1 (because ff is of degree 22), so Riemann and Hurwitz tell us that AA is infinitely connected."

shareciteeditflag answered Aug 12 '12 at 13:38

Alexandre Eremenko

## cylinder

What is the difference between the cylinder $\mathbb {C} \setminus 0$  and cylinder $\mathbb {C} /\mathbb {Z}$  ?

## topology

"A topologist is someone who doesn't know the difference between a cup of coffee and a donut."