Joined 11 March 2007

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 The Photographer's Barnstar foto

# color

Shades of Color Notice that since (0,0,0) is black and (1,1,1) is white, shades of any particular color are created by moving closer to black or to white. You can use the parametric equation for a linear relationship between two values to make shades of a color darker or lighter.

A parametric equation to calculate a linear change between values A and B:

  C = A + (B-A)*t;  // where t varies between 0 and 1


To change a color (r,g,b) to make it lighter, move it closer to (1,1,1).

  newR = r + (1-r)*t;  // where t varies between 0 and 1
newG = g + (1-g)*t;  // where t varies between 0 and 1
newB = b + (1-b)*t;  // where t varies between 0 and 1


To change a color (r,g,b) to make it darker, move it closer to (0,0,0).

  newR = r + (0-r)*t;  // where t varies between 0 and 1
newG = g + (0-g)*t;  // where t varies between 0 and 1
newB = b + (0-b)*t;  // where t varies between 0 and 1
// or
newR = r*t;  // where t varies between 1 and 0
newG = g*t;  // where t varies between 1 and 0
newB = b*t;  // where t varies between 1 and 0


## Intermediate Colour Values

Smooth colouring requires the ability to obtain a colour in between two colours picked from a discrete palette:

function:   getIntermediateColourValue
parameters: c1(R,G,B), c2(R',G',B'), m: 0 <= m <= 1
return:     [R + m * (R' - R),
G + m * (G' - G),
B + m * (B' - B)]


# Gimp

• flatpak run org.gimp.GIMP

# other Julia

• θ = 1/(10 + 1/(10 + · · ·)) Consider a polynomial P : C → C of degree d ≥ 2, with an irrationnally indifferent fixed point at the origin: P(0) = 0, P′ (0) = e 2iπθ, θ ∈ R \ Q. ( from On the linearization of degree three polynomials and Douady’s conjecture by Arnaud Ch´eritat (joint work with Xavier Buff)
• (*Koch center 3 cycle Julia*)\[IndentingNewLine]f[z_]=(-0.187-0.421*I)z^3+(-2.332-0.239*I)z^2+(-0.285+2.879*I)z+1; Roger Lee BagulaSelf-Similarity and Fractals An example of a "Koch" Julia: I was looking for an image for my new music album and found this unique Julia and re-did the picture with different colors and higher resolution: https://www.wolframcloud.com/.../j_koch9p1example2000...

## cubic

• A 4k cubic Julia by ‎Chris Thomasson‎ . Here is the formula: z = pow(z, 3) - (pow(-z, 2.00001) - 1.0008875); link
• cubic
• "Let c = (.387848...) + i(.6853...). The left picture shows the filled Julia set Kc of the cubic map z3 + c, covered by level 0 of the puzzle. The center of symmetry is at 0, the point where the rays converge is α and the other fixed points are marked by dotted arrows. In this example the rotation number around α is ρα = 25 and the ray angles are 5 121 7→ 15 121 7→ 45 121 7→ 14 121 7→ 42 121 7→ 5 121 . The right picture illustrates level 1 of the puzzle for the same map. "A New Partition Identity Coming from Complex Dynamics

## rational

Julia sets of rational maps ( not polynomials )

• McMullen Maps

### Christopher Williams

Family:

${\displaystyle z_{k+1}=z_{k}^{P}+c-\lambda z_{k}^{-Q}}$

Examples:

• Julia set of ${\displaystyle z^{2}-0.0625z^{-2}}$  The most obvious feature is that it's full of holes! The fractal is homemorphic to (topologically the same as) the Sierpinski carpet
• Julia set of z2 - 1 - 0.005z-2
• f = z^2 - 0.01z^-2
• f = z2 - 0.01z-2

Phoenix formula

• z5 - 0.06iz-2

### Robert L. Devaney

• Singular perturbations of complex polynomials

${\displaystyle G_{\lambda ,c}(z)=z^{n}+c+{\frac {\lambda }{z^{n}}}}$

# Julia sets for fc(z) = z*z + c

const double Cx=-0.74543; const double Cy=0.11301;

-0.808 +0.174i;

-0.1 +0.651i; ( beween 1 and 3 period component of Mandelbrot set )

-0.294 +0.634i

## spirals

• on the parameter plane
• part of M-set near Misiurewicz points
• on the dynamic plane
• Julia set near cut points
• critical orbits
• external rays landing on the parabolic or repelling perriodic points

 ${\displaystyle (r,t)\to \lambda \to c}$

 ${\displaystyle t={\frac {p}{q}}+\epsilon }$

 ${\displaystyle \lambda =re^{2\pi ti}}$


In period 1 component of Mandelbrot set :

 ${\displaystyle c:{\frac {\lambda }{2}}-{\frac {\lambda ^{2}}{4}}}$


In period 2 component of Mandelbrot set :

### table

Caption text
Period c r 1-r t p/q p/q - t image author address
1 0.37496784+i*0.21687214 0.99993612384259 0.000063879203489 0.1667830755386747 1/6 0.00011640887201 Cr6spiral.png 1
1 -0.749413589136570+0.015312826507689*i. 0.9995895293978963 0.00041047060211000001 0.4975611481254812 1/2 -0.00243885187451881 png 1
2 -0.757 + 0.027i 0.977981594918841 0.02201840508115904 0.01761164373863864 0/1 -0.01761164373863864 pauldelbrot 1 -(1/2)-> 2
2 -0.752 + 0.01i 0.9928061240745848 0.007193875925415205 0.006414063302849116 0/1 -0.006414063302849116 pauldelbrot 1 -(1/2)-> 2
10 -1.2029905319213867188 + 0.14635562896728515625 i 0.979333 0.02490599999999998 0.985187275828761422 0/1 -0.01481272417123857 marcm200 1 -(1/2)-> 2 -(2/5)-> 10
14 -1.2255649566650390625 0.1083774566650390625 0.951928 0.048072 0.992666114460366900 1/1 0.0073338855396331 marcm200 1 -(1/2)-> 2 -(3/7)-> 14
14 -1.2256811857223510742 +0.10814088582992553711 i 0.955071 0.044929 0.984062994677356362 1/1 0.01593700532264363 marcm200 1 -(1/2)-> 2 -(3/7)-> 14
14 -0.8422698974609375 -0.19476318359375 i 0.952171 0.04782900000000001 0.935491618649184731 1/1 0.06450838135081527 marcm200 1 -(1/2)-> 2 -(6/7)-> 14

### pauldelbrot

"c=0.027*%i-0.757"

period =  1
z= 0.01345178808414596*%i-0.5035840525848648
r = |m(z)| = 1.007527366616821
1-r = -0.007527366616821407
t = turn(m(z))  =  0.4957496478171055
p/q = 1/2
p/q-t = -0.004250352182894435

z= 1.503584052584865-0.01345178808414596*%i
r = |m(z)| = 3.007288448945452
1-r = -2.007288448945452
t = turn(m(z))  =  0.9985761611087214
p/q = 1/2
p/q-t = 0.4985761611087214

period =  2
z= (-0.1022072682395012*%i)-0.3679154600985363
r = |m(z)| = 0.977981594918841
1-r = 0.02201840508115904
t = turn(m(z))  =  0.01761164373863864
p/q = 1/2
p/q-t = -0.4823883562613613

z= 0.1022072682395012*%i-0.6320845399014637
r = |m(z)| = 0.9779815949188409
1-r = 0.02201840508115915
t = turn(m(z))  =  0.01761164373863864
p/q = 1/2
p/q-t = -0.4823883562613613



c=0.01*%i-0.752"

period =  1
z= 0.0049949453016411*%i-0.501011962705025
r = |m(z)| = 1.002073722342039
1-r = -0.00207372234203973
t = turn(m(z))  =  0.498413323518715
p/q = 0
p/q-t = 0.498413323518715

z= 1.501011962705025-0.0049949453016411*%i
r = |m(z)| = 3.002040547136002
1-r = -2.002040547136002
t = turn(m(z))  =  0.9994703791038458
p/q = 0
p/q-t = 0.9994703791038458

period =  2
z= (-0.06402358560400053*%i)-0.4219037804142045
r = |m(z)| = 0.9928061240745848
1-r = 0.007193875925415205
t = turn(m(z))  =  0.006414063302849116
p/q = 0
p/q-t = 0.006414063302849116

z= 0.06402358560400053*%i-0.5780962195857955
r = |m(z)| = 0.9928061240745848
1-r = 0.007193875925415205
t = turn(m(z))  =  0.006414063302849116
p/q = 0
p/q-t = 0.006414063302849116

(%o296) "/home/a/Dokumenty/periodic/MaximaCAS/p1/p.mac"
(%i297)


### marcm200

the input point was -1.2029905319213867e+00 + +1.4635562896728516e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 1.21186e+00 to the east of the input point
the input point is exterior to this component at
radius 1.41904e+00 and angle 0.486382891412633800 (in turns)
the multiplier is -1.41385e+00 + +1.21263e-01 i
a point in the attractor is -7.0694e-01 + +6.06308e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are -0.20299 + +0.14636 i
the atom domain coordinates in polar form are 0.25025 to the north-west
the nucleus is 2.50250e-01 to the south-east of the input point
the input point is exterior to this component at
radius 1.00100e+00 and angle 0.400579159596292533 (in turns)
the multiplier is -8.11962e-01 + +5.85423e-01 i
a point in the attractor is +1.81557e-01 + -1.07368e-01 i

- a period 4 circle
with nucleus at -1.310703e+00 + +3.761582e-37 i
the component has size 1.17960e-01 and is pointing west
the atom domain has size 2.34844e-01
the atom domain coordinates of the input point are +0.507 + +0.53837 i
the atom domain coordinates in polar form are 0.73952 to the north-east
the nucleus is 1.81719e-01 to the south-west of the input point
the input point is exterior to this component at
radius 1.00200e+00 and angle 0.801158319192584956 (in turns)
the multiplier is +3.16563e-01 + -9.50682e-01 i
a point in the attractor is +1.815562e-01 + -1.073687e-01 i
external angles of this component are:
.(0110)
.(1001)

- a period 10 circle
with nucleus at -1.2103996e+00 + +1.5287483e-01 i
the component has size 2.02739e-02 and is pointing north-west
the atom domain has size 4.09884e-02
the atom domain coordinates of the input point are +0.16767 + -0.13713 i
the atom domain coordinates in polar form are 0.2166 to the south-east
the nucleus is 9.86884e-03 to the north-west of the input point
the input point is interior to this component at
radius 9.79333e-01 and angle 0.985187275828761422 (in turns)
the multiplier is +9.75094e-01 + -9.10160e-02 i
a point in the attractor is +7.0332348e-02 + -8.243835e-02 i
external angles of this component are:
.(0110010110)
.(0110011001)


the input point was -1.2255649566650391e+00 + +1.0837745666503906e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 1.23035e+00 to the east of the input point
the input point is exterior to this component at
radius 1.43387e+00 and angle 0.490097175551864883 (in turns)
the multiplier is -1.43109e+00 + +8.91595e-02 i
a point in the attractor is -7.15546e-01 + +4.45805e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are -0.22556 + +0.10838 i
the atom domain coordinates in polar form are 0.25025 to the west-north-west
the nucleus is 2.50250e-01 to the east-south-east of the input point
the input point is exterior to this component at
radius 1.00100e+00 and angle 0.428714049282459153 (in turns)
the multiplier is -9.02260e-01 + +4.33510e-01 i
a point in the attractor is +1.94019e-01 + -7.80792e-02 i

- a period 4 circle
with nucleus at -1.310703e+00 + +0e+00 i
the component has size 1.17960e-01 and is pointing west
the atom domain has size 2.34844e-01
the atom domain coordinates of the input point are +0.38418 + +0.4137 i
the atom domain coordinates in polar form are 0.56457 to the north-east
the nucleus is 1.37819e-01 to the south-west of the input point
the input point is exterior to this component at
radius 1.00200e+00 and angle 0.857428098564918195 (in turns)
the multiplier is +6.26142e-01 + -7.82277e-01 i
a point in the attractor is +1.940184e-01 + -7.80797e-02 i
external angles of this component are:
.(0110)
.(1001)

- a period 14 circle
with nucleus at -1.2299714e+00 + +1.1067143e-01 i
the component has size 1.06543e-02 and is pointing west-north-west
the atom domain has size 1.95731e-02
the atom domain coordinates of the input point are +0.16376 + -0.15414 i
the atom domain coordinates in polar form are 0.22489 to the south-east
the nucleus is 4.96796e-03 to the west-north-west of the input point
the input point is interior to this component at
radius 9.51928e-01 and angle 0.992666114460366900 (in turns)
the multiplier is +9.50917e-01 + -4.38495e-02 i
a point in the attractor is +7.5747371e-02 + -5.129233e-02 i
external angles of this component are:
.(01100110010110)
.(01100110011001)

the input point was -1.2256811857223511e+00 + +1.0814088582992554e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 1.23044e+00 to the east of the input point
the input point is exterior to this component at
radius 1.43394e+00 and angle 0.490119703062896594 (in turns)
the multiplier is -1.43118e+00 + +8.89616e-02 i
a point in the attractor is -7.15576e-01 + +4.44813e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are -0.22568 + +0.10814 i
the atom domain coordinates in polar form are 0.25025 to the west-north-west
the nucleus is 2.50253e-01 to the east-south-east of the input point
the input point is exterior to this component at
radius 1.00101e+00 and angle 0.428881674271762436 (in turns)
the multiplier is -9.02725e-01 + +4.32564e-01 i
a point in the attractor is +1.9408e-01 + -7.79018e-02 i

- a period 4 circle
with nucleus at -1.310703e+00 + +0e+00 i
the component has size 1.17960e-01 and is pointing west
the atom domain has size 2.34844e-01
the atom domain coordinates of the input point are +0.38355 + +0.41286 i
the atom domain coordinates in polar form are 0.56353 to the north-east
the nucleus is 1.37561e-01 to the south-west of the input point
the input point is exterior to this component at
radius 1.00202e+00 and angle 0.857763348543524873 (in turns)
the multiplier is +6.27801e-01 + -7.80972e-01 i
a point in the attractor is +1.940823e-01 + -7.790208e-02 i
external angles of this component are:
.(0110)
.(1001)

- a period 14 circle
with nucleus at -1.2299714e+00 + +1.1067143e-01 i
the component has size 1.06543e-02 and is pointing west-north-west
the atom domain has size 1.95731e-02
the atom domain coordinates of the input point are +0.15716 + -0.16242 i
the atom domain coordinates in polar form are 0.22601 to the south-east
the nucleus is 4.98108e-03 to the west-north-west of the input point
the input point is interior to this component at
radius 9.55071e-01 and angle 0.984062994677356362 (in turns)
the multiplier is +9.50287e-01 + -9.54765e-02 i
a point in the attractor is +6.2976569e-02 + -5.7543144e-02 i
external angles of this component are:
.(01100110010110)
.(01100110011001)

the input point was -8.422698974609375e-01 + -1.9476318359375e-01 i
the point didn't escape after 10000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at +0e+00 + +0e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are nan to the east
the nucleus is 8.64495e-01 to the east-north-east of the input point
the input point is exterior to this component at
radius 1.11403e+00 and angle 0.526643305764455283 (in turns)
the multiplier is -1.09846e+00 + -1.85625e-01 i
a point in the attractor is -5.49225e-01 + -9.28116e-02 i
external angles of this component are:
.(0)
.(1)

- a period 2 circle
with nucleus at -1e+00 + +0e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are +0.15773 + -0.19476 i
the atom domain coordinates in polar form are 0.25062 to the south-east
the nucleus is 2.50622e-01 to the north-west of the input point
the input point is exterior to this component at
radius 1.00249e+00 and angle 0.858340235783291439 (in turns)
the multiplier is +6.30920e-01 + -7.79053e-01 i
a point in the attractor is -1.0771e-01 + +2.48238e-01 i

- a period 14 circle
with nucleus at -8.4076071e-01 + -1.9927227e-01 i
the component has size 1.01164e-02 and is pointing south-east
the atom domain has size 1.66560e-02
the atom domain coordinates of the input point are +0.13324 + -0.2159 i
the atom domain coordinates in polar form are 0.25371 to the south-south-east
the nucleus is 4.75495e-03 to the south-south-east of the input point
the input point is interior to this component at
radius 9.52171e-01 and angle 0.935491618649184731 (in turns)
the multiplier is +8.75023e-01 + -3.75452e-01 i
a point in the attractor is +4.338561e-02 + +7.777828e-02 i
external angles of this component are:
.(10101010100110)
.(10101010101001)


## basilica

• San Marco Fractal = Basilica : c = - 3/4

## Siegel Disk

Julia set = Jordan curve Irrational recurrent cycles

• 0.59...+i0.43...
• 0.33...+i0.07...
• C= -0.408792866 -0.577405 i
• c=-0.390541-0.586788i

## Other

Feigenbaum: C= -1.4011552 +0.0 i

Tower: C= -1 + 0.0 i

Cauliflower: C= 0.25 +0.0 i

## Dendrite

Critical point eventually periodic 0 > -2 > 2 (fixed).

C= i

c^3 + 2c^2 +2c +2 =0

### 3

The core entropy for polynomials of higher degree Yan Hong Gao, Giulio Tiozzo The Julia set of fc(z) = z → z3 + 0.22036 + 1.18612i

To show the non-uniqueness, let us consider the following example, which comes from [Ga]. We consider the postcritically finite polynomial fc(z) = z3 + c with c ≈ 0.22036 + 1.18612i. The critical value c receives two rays with arguments 11/72 and 17/72. Then, Θ := { Θ1(0) := {11/216, 83/216} , Θ2(0) := {89/216, 161/21

## Other

Circle: C= 0.0 +0.0 i

Segment: C= -2 +0.0 i

## centers

• by ‎Chris Thomasson‎
• (0.355534, -0.337292) it is a center of period 85 componnet. Adress 1->5 -> 85,
• 29 cycle power of two Julia set at point: (-0.742466, -0.107902) address 1 -> 29
• 38-cycle power of 3 Julia at point (0.388823, -0.000381453)
• c = -0.051707765779845 +0.683880135777732 i period = 273, 1 -(1/3)-> 3 -(1/7)-> 21 -(1/13)-> 273 https://www.math.stonybrook.edu/~jack/tune-b.pdf
• https://arxiv.org/pdf/math/9411238.pdf see Figure.2
• 1- (1/3) → 3 -(2/3) → 7
• 1 - (1/2) -> 2 - (1/2) → 4 -(1/3) → 7
• 1-(1/3) → 3- (1/2) → 4
• 1-(1/3) → 3 - (1/2) → 5-(1/2) → 6
• 1-(1/3) → 3-(1/2) → 5-(1/2) → 7
• 1-(1/3) → 3-(1/3) → 7
• 1-(3/4)-> 4 -(?/5)-> 20 : c = 0.300078079301992 -0.489531524188048 i period = 20
• http://wrap.warwick.ac.uk/35776/1/WRAP_THESIS_Sharland_2010.pdf

### minibrot

• c = 0.284912968784722 +0.484816779093857 i period = 84

### Superattracting per 3 (up to complex conjugate)

C= -1.75488 (airplane)

C= -0.122561 + 0.744862 i (rabbit) Douady's Rabbit Rabbit: C= -0.122561 +0.744862 i = ( -1/8+3/4 i ??? ) whose critical point 0 is on a periodic orbit of length 3

### Superattracting per 4 (up to complex conjugate)

C= -1.9408

C= -1.3107

C= -1.62541

C= -0.15652 +1.03225 i

C= 0.282271 +0.530061 i

#### Kokopelli

• p = γM (3/15)
• p(z) = z^2 0.156 + 1.302*i
• The angle 3/15 or p0011 has preperiod = 0 and period = 4.
• The conjugate angle is 4/15 or p0100 .
• The kneading sequence is AAB* and the internal address is 1-3-4 .
• The corresponding parameter rays are landing at the root of a primitive component of period 4.
• c = -0.156520166833755 +1.032247108922832 i period = 4

### Superattracting per 5 (up to complex conjugate)

C= -1.98542

C= -1.86078

C= -1.62541

C= -1.25637 +0.380321 i

C= -0.50434 +0.562766 i

C= -0.198042 +1.10027 i

C= -0.0442124 +0.986581 i

C= 0.359259 +0.642514 i

C= 0.379514 +0.334932 i

### A superattracting per 15

C= -0.0384261 +0.985494 i

## dense

Density near the cardoid 3 by DinkydauSet Mandelbrot set

Again a location that's not deep but super dense.

Magnification: 2^35.769 5.8552024369543422761426995117521 E10

Coordinates: Re = 0.360999615968828800 Im = -0.121129382033034400

XaoS coordinates

(maxiter 50000) (view -0.775225602760841 -0.136878655029377 7.14008235131944E-11 7.14008235674045E-11)

https://arxiv.org/pdf/1703.01206.pdf "Limbs 8/21, 21/55, 55/144, 144/377, . . . scale geometrically fast on the right side of the (anti-)golden Siegel parameter, while limbs 5/13, 13/34, 34/89, 89/233, . . . scale geometrically fast on the left side. The bottom picture is a zoom of the top picture."

## a

https://plus.google.com/u/0/photos/115452635610736407329/albums/6124078542095960129/6124078748270945650 frond tail Misiurewicz point of the period-27 bulb of the quintic Mandelbrot set (I don't have a number ATM, but you can find that)﻿

# SVG

<?xml version="1.0" encoding="utf-8" standalone="no"?>
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">


# ps

"Obrazy w indywidualnym wkładzie, mam z samodzielnie napisanymi programami PASCAL (FREEPASCAL) i / lub XFIG (program do rysowania pod LINUX) początkowo tworzone jako pliki EPS. Niestety bezpośrednia integracja plików EPS z Wikipedią nie jest możliwa. Przydatna jest konwersja plików EPS na pliki XFIG za pomocą PSTOEDIT. Konwersja plików EPS na pliki SVG jest możliwa dzięki INKSCAPE, a także eksportowi plików xfig do plików SVG. Jednak nie znalazłem zamiennika dla etykiety LATEX, włączając ją do pliku LATEX w środowisku rysunkowym z plikiem psfrag. Także dla licznych możliwości manipulowania krzywymi za pomocą xfig wiem w inkscape (wciąż) brak odpowiednika. Z plików PS tworzę pliki SVG z ps2pdf i pdf2svg i używam programu inkscape do dostosowania stron (marginesów) do rysunków." de:Benutzer:Ag2gaeh

# Help

differences between :

<gallery> </gallery>
{{SUL Box|en|wikt}}



[[1]]

## compare with


==Compare with==

<gallery caption="Sample gallery" widths="100px" heights="100px" perrow="6">

</gallery>

</nowiki>


Change in your preferences : Show hidden categories

## syntaxhighlight

== c source code==
<syntaxhighlight lang="c">
</syntaxhighlight>

==text output==

==references==
<references/>



### GeSHi

description:

<syntaxhighlight lang="c" enclose="none">
g;
a=2;
</syntaxhighlight>


### source templetate

deprecated:

<syntaxhighlight lang="gnuplot">ssssssssssssss</syntaxhighlight>


### prettytable

Mathematical Function Plot
Description Function displaying a cusp at (0,1)
Equation ${\displaystyle y={\sqrt {|x|}}+1}$
Co-ordinate System Cartesian
X Range -4 .. 4
Y Range -0 .. 3
Derivative ${\displaystyle {\frac {dy}{dx}}={\frac {1}{2{\sqrt {x}}}}}$

Points of Interest in this Range
Minima ${\displaystyle \left(0,1\right)\,}$
Cusps ${\displaystyle \left(0,1\right)\,}$
Derivatives at Cusp ${\displaystyle \lim _{x\to 0^{+}}f'(x)=+\infty }$ ,

${\displaystyle \lim _{x\to 0^{-}}f'(x)=-\infty }$