分形/复平面迭代/次级尾迹

如何找到任何曼德勃罗集分量（可通过有限次边界穿越从主心形 (M0) 访问）的根点上的外射线的角度？

关键词

• 调谐
• 外射线
  • 以二进制数形式的外角

算法 = Douady 调谐

"$M_{p/q}$ 中的 r/s 内射线是外射线 $\theta _{\pm }(p/q,r/s)$ 的着陆点，这些外射线是从 $\theta _{\pm }(r/s)$ 中得到的，方法是将

• 数字 0 替换为 $\theta _{-}(p/q)$ 的重复块（长度为 q）
• 数字 1 替换为 $\theta _{+}(p/q)$ 的重复块（长度为 q）"

"通过重复相同的过程（称为“调谐”），我们可以计算外射线的幅度，这些外射线着陆在任何可通过有限次边界穿越从 $M_{0}$ 访问的分量的边界上。"（Shaun Bullett）

Douady 调谐

内部角数示例

曼德勃罗集周期 1 大陆附近的尾迹。使用距离估计（外部和内部）呈现曼德勃罗集边界。用周期（蓝色）、内部角和射线（绿色）以及外部角和射线（红色）标记。

1 个角

• 1/2 族

2 个角

有角的内部地址:

 $1\quad {\xrightarrow {p/q}}\ q\quad {\xrightarrow {r/s}}\ s*q$

描述一种方法

• 从周期 1 双曲分量 c=0 的中心
• 沿着角度为 p/q 的内射线走向根点（p/q 键）
• 沿着角度为 r/s 的内射线走向 s*q 双曲分量的分量（核）的中心

其中

• 1、q、s 是双曲分量的周期
• p/q、r/s 是内部角

(1/2, 1/3)

有角的内部地址

 $1\quad \xrightarrow {1/2} \ 2\quad \xrightarrow {1/3} \ 3*2=6$

首先计算p/q = 1/2 和 r/s=1/3 尾迹 的外角 

$\theta _{-}(r/s)=\theta _{-}(1/3)=0.(001)$
$\theta _{+}(r/s)=\theta _{+}(1/3)=0.(010)$

$\theta _{-}(p/q)=\theta _{-}(1/2)=0.({\color {Blue}01})$
$\theta _{+}(p/q)=\theta _{+}(1/2)=0.({\color {Red}10})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

使用此 c 程序，可以得到

input string = sIn = 0.(001)
Input Length = 3

replace string for digit 0 = sR0 = 0.(01)
Length of sR0 = 2

replace string for digit 1 = sR1 = 0.(10)
Length of sR1 = 2
output string in plain form sOut = 0.(010110)
Output Length = 6

output string in wikipedia math formula form = sOutf = 
 0.(\ {\color{Blue}01}\ {\color{Blue}01}\ {\color{Red}10})
sR0 displayed as a blue and sR1 as a red font

input string = sIn = 0.(010)
Input Length = 3

replace string for digit 0 = sR0 = 0.(01)
Length of sR0 = 2

replace string for digit 1 = sR1 = 0.(10)
Length of sR1 = 2
output string in plain form sOut = 0.(011001)
Output Length = 6

output string in wikipedia math formula form = sOutf = 
 0.(\ {\color{Blue}01}\ {\color{Red}10}\ {\color{Blue}01})
sR0 displayed as a blue and sR1 as a red font

结果

$\theta _{-}(p/q,r/s)=\theta _{-}(1/2,1/3)=0.(\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10})$

$\theta _{+}(p/q,r/s)=\theta _{+}(1/2,1/3)=0.(\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01})$

使用沃尔夫·容的曼德尔程序进行检查

The angle  22/63  or  p010110
has  preperiod = 0  and  period = 6.
The conjugate angle is  25/63  or  p011001 .
The kneading sequence is  ABABA*  and
the internal address is  1-2-6 .
The corresponding parameter rays land
at the root of a satellite component of period 6.
root c = -1.125000000000000  +0.216506350946110 i    period = 10000
It bifurcates from period 2.
Center = c = -1.138000666650965  +0.240332401262098 i    period = 6

(1/3,1/25)

有角的内部地址

 $1\quad {\xrightarrow {1/3}}\ 3\quad {\xrightarrow {1/25}}\ 3*25=75$

首先计算 p/q 和 r/s 尾迹 的外部角

$\theta _{-}(r/s)=\theta _{-}(1/25)=0.(0000000000000000000000001)$
$\theta _{+}(r/s)=\theta _{+}(1/25)=0.(0000000000000000000000010)$

$\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$
$\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

使用此 c 程序，可以得到

input string = sIn = 0.(0000000000000000000000001)
Input Length = 25

replace string for digit 0 = sR0 = 0.(001)
Length of sR0 = 3

replace string for digit 1 = sR1 = 0.(010)
Length of sR1 = 3
output string in plain form sOut = 0.(001001001001001001001001001001001001001001001001001001001001001001001001010)
Output Length = 75

output string in wikipedia math formula form = sOutf = 
 0.(\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Blue}001}\ {\color{Red}010})

所以结果是

$\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/25)=0.(\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010})$
$\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/25)=0.(\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001})$

使用 不能 使用  

• 沃尔夫·容的曼德尔程序，因为周期 75 超过了曼德尔程序（5.14 版本）的容量
• knowledgedoor 计算器  : “新数字有超过 100000 个小数位。很抱歉，我们必须中止计算以控制服务器负载”
• 来自 xploringbinary 的二进制到十进制转换器，因为  : ***错误：输入中包含无效字符***

浮点小数可以计算（xploringbinary 计算器）

0.00100100100100100100100100100100100100100100100100100100100100100100100101001001001001001001001001001001001001001001001001001001001001001001001001010010010010010010010010010010010010010010010010010010010010010010010010010100100100100100100100100100100100100100100100100100100100100100100100100101000100100100100100100100100100100100100100100100100100100100100100100100101000100100100100100100100100100100100100100100100100100100100100100100100101000100100100100100100100100100100100100100100100100100100100100100100100101 = 0.14285714285714285714288739403383051072639529883365108929843563075583153775763414823768752002069960029227911125999721923628800596414804821292109009060239475901499810101181257672784693799296426200506484998155193353019205064595746824097168836030939675768822757699065155639759014879516081816465598503673993914613650270765449619097486325642647311484333459622592887076714167693559186268814330532241007104844902043043591808926221035405535394453260989380025345642382995407618853788922160842622677279223353252746164798736572265625

注意

1/7 = 0.(001) = 0.(142857)= 0.1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428...

所以差异在小数点后第 23 位

使用 mandelbrot-symbolics

 ./m-binangle-to-rational ".(001001001001001001001001001001001001001001001001001001001001001001001001010)"
 5396990266136737387082/37778931862957161709567

 ./m-binangle-to-rational ".(001001001001001001001001001001001001001001001001001001001001001001001010001)"
 5396990266136737387089/37778931862957161709567

或使用 web 界面，输入  : 1_1/3->3_1/25->75

(1/3,1/2)

有角的内部地址

 $1\quad {\xrightarrow {1/3}}\ 3\quad {\xrightarrow {1/2}}\ 6$

首先计算 p/q 和 r/s 的外部角 尾迹  

$\theta _{-}(r/s)=\theta _{-}(1/2)=0.(01)$
$\theta _{+}(r/s)=\theta _{+}(1/2)=0.(10)$

$\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$
$\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

结果是  

$\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/2)=0.({\color {Blue}001}\ {\color {Red}010})$
$\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/2)=0.({\color {Red}010}\ {\color {Blue}001})$

可以使用 沃尔夫·容的曼德尔程序 进行检查  

The angle  10/63  or  p001010
has  preperiod = 0  and  period = 6.
The conjugate angle is  17/63  or  p010001 .
The kneading sequence is  AABAA*  and
the internal address is  1-3-6 .
The corresponding parameter rays are landing
at the root of a satellite component of period 6.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3,1/3)

首先计算 p/q 和 r/s 的外部角 尾迹（此处 p/q=r/s）  

$\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$
$\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

结果是  

$\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/2)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010})$
$\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/2)=0.({\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001})$

可以使用 沃尔夫·容的曼德尔程序 进行检查  

The angle  74/511  or  p001001010
has  preperiod = 0  and  period = 9.
The conjugate angle is  81/511  or  p001010001 .
The kneading sequence is  AABAABAA*  and
the internal address is  1-3-9 .
The corresponding parameter rays are landing
at the root of a satellite component of period 9.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3,1/4)

首先计算 p/q 和 r/s 的外部角 尾迹  

$\theta _{-}(r/s)=\theta _{-}(1/4)=0.(0001)$
$\theta _{+}(r/s)=\theta _{+}(1/4)=0.(0010)$

$\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$
$\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

结果是  

$\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/4)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010})$
$\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/4)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001})$

可以使用 沃尔夫·容的曼德尔程序 进行检查  

The angle  586/4095  or  p001001001010
has  preperiod = 0  and  period = 12.
The conjugate angle is  593/4095  or  p001001010001 .
The kneading sequence is  AABAABAABAA*  and
the internal address is  1-3-12 .
The corresponding parameter rays are landing
at the root of a satellite component of period 12.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3,3/4)

首先计算 p/q 和 r/s 的外部角 尾迹  

$\theta _{-}(r/s)=\theta _{-}(3/4)=0.(1101)$
$\theta _{+}(r/s)=\theta _{+}(3/4)=0.(1110)$

$\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$
$\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

结果是  

$\theta _{-}(p/q,r/s)=\theta _{-}(1/3,1/2)=0.({\color {Red}010}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Red}010})$
$\theta _{+}(p/q,r/s)=\theta _{+}(1/3,1/2)=0.({\color {Red}010}\ {\color {Red}010}\ {\color {Red}010}\ {\color {Blue}001})$

可以在 沃尔夫·容的曼德尔程序 中进行检查  

The angle  1162/4095  or  p010010001010
has  preperiod = 0  and  period = 12.
The conjugate angle is  1169/4095  or  p010010010001 .
The kneading sequence is  AABAABAABAA*  and
the internal address is  1-3-12 .
The corresponding parameter rays are landing
at the root of a satellite component of period 12.
It is bifurcating from period 3.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3,1/89)

具有 5 条外部射线落在曼德尔布罗特集上的参数平面的部分。

有角的内部地址

 $1\quad \xrightarrow {1/3} \ 3\quad \xrightarrow {1/89} \ 3*89=267$

(1/4, 1/5)

输入是  

$(p/q,r/s)=(1/4,1/5)$

首先计算 p/q 和 r/s 的外部角 尾迹  

$\theta _{-}(r/s)=\theta _{-}(1/5)=0.(00001)$
$\theta _{+}(r/s)=\theta _{+}(1/5)=0.(00010)$

$\theta _{-}(p/q)=\theta _{-}(1/4)=0.({\color {Blue}0001})$
$\theta _{+}(p/q)=\theta _{+}(1/4)=0.({\color {Red}0010})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

结果是  

$\theta _{-}(p/q,r/s)=\theta _{-}(1/4,1/5)=0.({\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Red}0010})$
$\theta _{+}(p/q,r/s)=\theta _{+}(1/4,1/5)=0.({\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Blue}0001}\ {\color {Red}0010}\ {\color {Blue}0001})$

可以使用 沃尔夫·容的曼德尔程序 进行检查  

The angle  69906/1048575  or  p00010001000100010010
has  preperiod = 0  and  period = 20.
The conjugate angle is  69921/1048575  or  p00010001000100100001 .
The kneading sequence is  AAABAAABAAABAAABAAA*  and
the internal address is  1-4-20 .
The corresponding parameter rays are landing
at the root of a satellite component of period 20.
It is bifurcating from period 4.
Do you want to draw the rays and to shift c
to the corresponding center?

(4/5, 1/17)

曼德尔布罗特集 - 尾迹 1-( 4/5)->5-(1/17)->85

角度内部地址是  

• $1\quad {\xrightarrow {4/5}}\ 5\quad {\xrightarrow {1/17}}\ 5*17=85$
• 1-(4/5)-> 5 -(1/17)-> 85

输入是  

$(p/q,r/s)=(4/5,1/17)$

首先计算 p/q 和 r/s 的外部角 尾迹  

$\theta _{-}(r/s)=\theta _{-}(1/17)=0.(00000000000000001)$
$\theta _{+}(r/s)=\theta _{+}(1/17)=0.(00000000000000010)$

$\theta _{-}(p/q)=\theta _{-}(4/5)=0.({\color {Blue}11101})$
$\theta _{+}(p/q)=\theta _{+}(4/5)=0.({\color {Red}11110})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 为 $\theta _{-}(p/q)$ 的长度为 q 的块
• 数字 1 为 $\theta _{+}(p/q)$ 的长度为 q 的块

结果是  

 $\theta _{-}(p/q,r/s)=\theta _{-}(4/5,1/17)=0.(\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Red}11110})$
$\theta _{+}(p/q,r/s)=\theta _{+}(4/5,1/17)=0.(\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Blue}11101}\ {\color {Red}11110}\ {\color {Blue}11101})$

使用 沃尔夫·容的曼德尔程序 无法进行检查，因为周期过大。它只提供周期不超过 64 的答案

可以使用克劳德的 web 界面 在线进行检查

• .(1110111101111011110111101111011110111101111011110111101111011110111101111011110111110)
• .(1110111101111011110111101111011110111101111011110111101111011110111101111011111011101)

3 个角度

(1/2, 1/3, 1/4)

我们从右到左遍历角度列表

首先计算 (1/3,1/4) 尾迹，它将用作新的 r/s 尾迹  

 $\theta _{-}(1/3,1/4)=0.(001001001010)=\theta _{-}(r/s)$
 $\theta _{+}(1/3,1/4)=0.(001001010001)=\theta _{+}(r/s)$

之后计算 1/2 尾迹（最左侧），它将用作 p/q 尾迹  

 $\theta _{-}(p/q)=\theta _{-}(1/2)=0.({\color {Blue}01})$
 $\theta _{+}(p/q)=\theta _{+}(1/2)=0.({\color {Red}10})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 由从 $\theta _{-}(p/q)$
• 数字 1 由从 $\theta _{+}(p/q)$

结果是  

$\theta _{-}(1/2,1/3,1/4)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01})$
$\theta _{+}(1/2,1/3,1/4)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10})$

以纯文本格式（用于复制） 

 theta_minus(1/2, 1/3, 1/4) = 0.(010110010110010110011001)
 theta_plus( 1/2, 1/3, 1/4) = 0.(010110010110010110010110)

可以使用 Wolf Jung 的 Mandel 程序 中的射线到点命令（Ctrl+e）来检查这种唤醒 

The angle  5858713/16777215  or  p010110010110010110011001
has  preperiod = 0  and  period = 24.
The conjugate angle is  5858902/16777215  or  p010110010110011001010110 .
The kneading sequence is  ABABAAABABAAABABAAABABA*  and
the internal address is  1-2-6-24 .
The corresponding parameter rays are landing
at the root of a satellite component of period 24.
It is bifurcating from period 6.
Do you want to draw the rays and to shift c
to the corresponding center?

(1/3, 1/4, 1/5)

输入是一个列表 

(1/3, 1/4, 1/5)

我们从右到左遍历角度列表，并将列表分成两个子列表 

$p/q=1/3$

$r/s=(1/4,1/5)$

首先计算 (1/4,1/5) 唤醒，它将用作新的 r/s 唤醒 

$\theta _{-}(1/4,1/5)=0.(00010001000100010010)=\theta _{-}(r/s)$
$\theta _{+}(1/4,1/5)=0.(00010001000100100001)=\theta _{+}(r/s)$

然后计算 1/3 唤醒（最左边的），它将用作 p/q 唤醒 

 $\theta _{-}(p/q)=\theta _{-}(1/3)=0.({\color {Blue}001})$
 $\theta _{+}(p/q)=\theta _{+}(1/3)=0.({\color {Red}010})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 由从 $\theta _{-}(p/q)$
• 数字 1 由从 $\theta _{+}(p/q)$

结果是  

$\theta _{-}(1/3,1/4,1/5)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001})$
$\theta _{+}(1/3,1/4,1/5)=0.({\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Blue}001}\ {\color {Red}010})$

可以使用 沃尔夫·容的曼德尔程序 进行检查  

The angle  164984615799661137/1152921504606846975  or  p001001001010001001001010001001001010001001001010001001010001
has  preperiod = 0  and  period = 60.
The conjugate angle is  164984615799689802/1152921504606846975  or  p001001001010001001001010001001001010001001010001001001001010 .
The kneading sequence is  AABAABAABAAAAABAABAABAAAAABAABAABAAAAABAABAABAAAAABAABAABAA*  and
the internal address is  1-3-12-60 .
The corresponding parameter rays are landing
at the root of a satellite component of period 60.
It is bifurcating from period 12.
Do you want to draw the rays and to shift c
to the corresponding center?

4 个角度列表

(1/2, 1/3, 1/4, 1/5)

输入是一个列表 

(1/2, 1/3, 1/4, 1/5)

所以内部加法器应该是 

1-2-6-24-120

无法使用 Mandel 程序进行检查，因为它限制在周期 64。

我们从右到左遍历输入角度列表，并将列表分成两个子列表 

$p/q=1/2$

$r/s=(1/3,1/4,1/5)$

首先计算 (1/3, 1/4, 1/5) 唤醒，它将用作新的 r/s 唤醒 

$\theta _{-}(1/3,1/4,1/5)=0.(001001001010001001001010001001001010001001001010001001010001)=\theta _{-}(r/s)$
$\theta _{+}(1/3,1/4,1/5)=0.(001001001010001001001010001001001010001001010001001001001010)=\theta _{+}(r/s)$

然后计算 1/2 唤醒（最左边的），它将用作 p/q 唤醒 

 $\theta _{-}(p/q)=\theta _{-}(1/2)=0.({\color {Blue}01})$
 $\theta _{+}(p/q)=\theta _{+}(1/2)=0.({\color {Red}10})$

然后在 $\theta (r/s)$ 中替换

• 数字 0 由从 $\theta _{-}(p/q)$
• 数字 1 由从 $\theta _{+}(p/q)$

结果是（要检查！ !!!!） 

$\theta _{-}(1/2,1/3,1/4,1/5)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10})$$\theta _{+}(1/2,1/3,1/4,1/5)=0.({\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01}\ {\color {Red}10}\ {\color {Blue}01})$

theta_minus(1/2, 1/3, 1/4, 1/5) = 0.(010110010110010110011001010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110)
theta_plus(1/2, 1/3, 1/4, 1/5)  = 0.(010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110010110010110010110011001)

可以使用克劳德·海兰德-艾伦的book 程序 通过视觉检查。

size 640 360
view 53 -1.113644126576409e+00 2.5205986428803329e-01 3.9234950282896473e-04
ray_in 2000 .(010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110010110010110010110011001)
ray_in 2000 .(010110010110010110011001010110010110010110011001010110010110010110011001010110010110010110011001010110010110011001010110)
text 53 -1.1152327443471231e+00 2.5276283972645397e-01 1/4
text 53 -1.1136201098499858e+00 2.5201617701965662e-01 1/5
text 53 -1.1152327443471231e+00 2.5276283972645397e-01 1/4
text 53 -1.1138472738947567e+00 2.5348331923684125e-01 24

多角度

周期 776

1-(1/2)-> 2 -(1/3)-> 6 -(1/2)-> 12 -(1/3)-> 36 -(1/2)-> 72 -(2/3)-> 216 -(1/2)-> 432 -(1/3)-> 1296 -(1/2)-> 2592 -(2/3)-> 7776   -> main misiurewicz point -> right branch

尝试 mandelbrot-web

周期 20

来自克劳德的Mandelbrot-web 的信息

• 外部角度（抖动外部射线）
  • 二进制 = .(00000000001111111111)
  • 十进制有理数形式 = 1/1025
  • 周期 = 20
• 揉捏 1111111111000000000★
• 内部地址： 1→11→12→13→14→15→16→17→18→19→20
• 登陆 .00000000001111111111 1/1025 .00000000010000000000 1024/1048575

有角度的内部地址

$1\xrightarrow {1/11} 11\xrightarrow {1/2} 12\xrightarrow {1/2} 13\xrightarrow {1/2} 14\xrightarrow {1/2} 15\xrightarrow {1/2} 16\xrightarrow {1/2} 17\xrightarrow {1/2} 18\xrightarrow {1/2} 19\xrightarrow {1/2} 20$

来自 Wolf Jung 的程序 Mandel 的信息

• 角度 1023/1048575 或 p00000000001111111111
• 具有前周期 = 0 和周期 = 20。
• 共轭角度为 1024/1048575 或 p00000000010000000000。
• 揉捏序列为 AAAAAAAAAABBBBBBBBB* 且
• 内部地址为 1-11-12-13-14-15-16-17-18-19-20。
• 相应的参数射线落在周期为 20 的原始分量根部，中心为 c = 0.329617350093832 +0.042415693708911 i

代码

/*

------------ git --------------------
https://gitlab.com/adammajewski/c-string-replaceing

cd existing_folder
git init
git remote add origin [email protected]:adammajewski/c-string-replace.git
git add .
git commit
git push -u origin master

-------------- asprintf --------------------------------------
Using asprintf instead of sprintf or snprintf by james
http://www.stev.org/post/2012/02/10/Using-saprintf-instead-of-sprintf-or-snprintf.aspx

http://ubuntuforums.org/showthread.php?t=279801


gcc c.c -D_GNU_SOURCE -Wall // without #define _GNU_SOURCE

gcc c.c -Wall
----------- run ----------------------

./a.out



*/


#define _GNU_SOURCE // asprintf
#include <stdio.h>
#include <stdlib.h>
#include <string.h> // strlen
 
int main() {
    // output = theta+(p/q, r/s) or theta-(p/q, r/s)
    char *sOut  = ""; // in plaint text format
    char *sOutf = ""; // formatted for wikipedia math formula 
    // input
    char *sIn = "00000000000000010";  // rs+ or rs-
    
    // strings which will replace 0 and 1 digit in the input 
    // length(sR0) = length(sR1)
    char *sR0 = "11101"; // pq-
    char *sR1 = "11110"; // pq+


   int iMax; // length of the input string
   int i; 


   iMax = (int) strlen(sIn);

   for (i=0; i<iMax; i++){

   // printf("i = %d, sIn[i] = %c = %d\n", i, sIn[i],  sIn[i] - '0');
    // create sOut by replaceing digit (0 or 1) from aIn by a block of digits :  sR0 or sR1
    if (((int) sIn[i] -'0') == 0 ) // http://stackoverflow.com/questions/868496/how-to-convert-char-to-integer-in-c
       {asprintf(&sOut, "%s%s", sOut, sR0);
       asprintf(&sOutf, "%s\\ {\\color{Blue}%s}", sOutf, sR0); }
       
       else // if sIn[i]==1 
       {asprintf(&sOut, "%s%s", sOut, sR1);
       asprintf(&sOutf, "%s\\ {\\color{Red}%s}", sOutf, sR1);}
   }
      


    printf("input string = sIn = 0.(%s)\n", sIn);
    printf("Input Length = %d\n\n", iMax);
    
    
    printf("replace string for digit 0 = sR0 = 0.(%s)\n", sR0);
    printf("Length of sR0 = %d\n\n", (int) strlen(sR0));
    
    printf("replace string for digit 1 = sR1 = 0.(%s)\n", sR1);
    printf("Length of sR1 = %d\n", (int) strlen(sR1));
    
    
    
    printf("output string in plain form sOut = 0.(%s)\n", sOut);
    printf("Output Length = %d\n\n",  (int) strlen(sOut));
    printf("output string in wikipedia math formula form = sOutf = \n 0.(%s)\n", sOutf);
    printf("sR0 displayed as a blue and sR1 as a red font\n");

    free(sOut); 
    free(sOutf); 
    return 0;
}

输出

input string = sIn = 0.(00000000000000010)
Input Length = 17

replace string for digit 0 = sR0 = 0.(11101)
Length of sR0 = 5

replace string for digit 1 = sR1 = 0.(11110)
Length of sR1 = 5
output string in plain form sOut = 0.(1110111101111011110111101111011110111101111011110111101111011110111101111011111011101)
Output Length = 85

output string in wikipedia math formula form = sOutf = 
 0.(\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Blue}11101}\ {\color{Red}11110}\ {\color{Blue}11101})
sR0 displayed as a blue and sR1 as a red font

另请参阅

• 岛屿
• Claude Heiland-Allen 的浏览器中的 Mandelbrot 符号
• Claude Heiland-Allen 的 automatically_finding_external_angles
• Claude Heiland-Allen 的书籍程序
• Virpi K a u k o 的 Mandelbrot 集中可见分量的树木 - 使用双曲分量的内部地址，使用 Mandelbrot 集 M 的子肢的树结构