### Author Topic: Fractals !  (Read 761 times)

#### PhoenixII54

• Newbie
• Posts: 37
##### Fractals !
« on: September 15, 2017, 11:10:58 am »
No, you didn't misread, what i'm about to give you is a series of scripts normally used in mathematics.
They don't desserve any porpose other than having fun with the engine, so enjoy !

1. the Mandelbrot set
Code: Lua
1. --[[
2. A useless-thus-indispensable Mandelbrot set renderer
3.
4. How it works :
5. the Mandelbrod set consists of a 2d grid of complex numbers,
6. which are an extension of real numbers and are written in, the form :
7. a+b*i, where a and b are two real numbers, and i is the square root of -1,
8. also called the imaginary nuimber.
9.
10. For each point of the grid, let(s call it c, we gererate a new complex, number called z by applying the following formula :
11. initial z value : 0
12. new z = (z to the power of <whatever you want>) + c
13. then we compare the new value to (2 to the power of <the same exponent as before>
14. then, we continue in one of the wollowing cases :
15.    if z is lower than the powered 2, then we do the above steps again, unless we reached the maximum number if iterations
16.    if, however, we are equal or higher than the powered 2,
17.    then we stop iterating since the next calculated numbers wil keep growing forever, namely, we escaped the 2-range stability area
18.
19. Anyway, once we have finished calculatins the final value for a given number, we color it according to it's behaviour :
20. -if we escaped, then we color the corresponding point using the number of cleared iterations to index the color from a palette
21. taking the mod from table size
22. -else, we color it pitch black
23.
24. this process goes on and on until all visible points are rendered.
25.
26. For more information, feel free to check on the Wiki or any math article
27.
28. How to use :
29. 1. create a map of the size of your choice
30. 2. copy/paste this whole code in the map script.
31. 3. setup the config table with your own parameters
32. 4. Run the map !
33. --]]
34.
35. local map = ...
36. local game = map:get_game()
37.
38.
39. -- the configuration table
40. local config={
41.   -- Size of the rendering area in game pixels.
42.   -- Must be lower than the map size to render properly
43.   grid={
44.     width=401,
45.     height=401
46.   },
47.   -- the maximum number of iterations for each point.
48.   maxIterations=1000,
49.   -- the calculation power. Defaulr : 2
50.   powerExponent=2,
51.   -- the starting complex numbers boundaries
52.   numberBoundaries={
53.     x={
54.       min=-2,
55.       max=2,
56.     },
57.     y={
58.       min=-2,
59.       max=2,
60.     },
61.   },
62.   step={},
63. }
64.
65. local start_time
66. -- returns the module of any complex number, aka the length of the vector represented by this number
67. local function get_module(c)
68.   return c.real*c.real+c.imag*c.imag
69. end
70.
71. --calculates z^N with z=real+i*imag and N>0
72. local function complex_power(c, exponent)
73.   local squaredModule=get_module(c)
74.   local newModule=math.pow(squaredModule,exponent/2)
75.   local argument=math.atan2(c.imag,c.real)
76.   return {
77.     real=math.cos(exponent*argument)*newModule,
78.     imag=math.sin(exponent*argument)*newModule,
79.   }
80. end
81.
82. local currentNumber={
83.   real=config.numberBoundaries.x.min,
84.   imag=config.numberBoundaries.y.max,
85. }
86.
87. local currentPoint={
88.   x=0,
89.   y=0,
90. }
91. -- a 1*1 surface used as a pixel, to compensate the lack of proper API pixel manipulatinf functions
92. local pixel=sol.surface.create(1,1)
93.
94. local fractalRender=sol.surface.create(config.grid.width,config.grid.height)
95. -- the actual color palette. Feel free to modify it with your own colors (any size will work)
96. local color_palette={
97.   {255,255,0},
98.   {255,223,31},
99.   {255,191,63},
100.   {255,159,95},
101.   {255,127,127},
102.   {255,95,159},
103.   {255,63,191},
104.   {255,31,223},
105.   {223,0,255},
106.   {255,31,255},
107.   {191,63,255},
108.   {159,95,255},
109.   {127,127,255},
110.   {95,159,255},
111.   {63,191,255},
112.   {31,223,255},
113.   {0,255,255},
114.   {31,255,223},
115.   {63,255,191},
116.   {95,255,159},
117.   {127,255,127},
118.   {159,255,95},
119.   {191,255,63},
120.   {223,255,31},
121. }
122. -- the rendering call
123. local function draw_point(x,y,i)
124.   local color
125.   if i==config.maxIterations then
126.     color={0,0,0}
127.   else
128.     local index=(i%#color_palette)+1
129.     color=color_palette[index]
130.   end
131.   pixel:fill_color(color)
132.   pixel:draw(fractalRender,x,y)
133. end
134. -- the core functions. Called once per map update.
135. -- i don't use for loops to get each point, but increase indices at the end of this function instead.
136. -- to save the engine from lagging/crashing
137. local function calculateNextPoint()
138.   local z={
139.     real=0,
140.     imag=0,
141.   }
142.   local real=currentNumber.real
143.   local imag=currentNumber.imag
144.   local i=0
145.   local e=config.powerExponent
146.   --The core calculation loop. see the explaination at the beginning of the script
147.   while (i<config.maxIterations and get_module(z)<=math.pow(2,e)) do
148.     local temp=complex_power(z,e)
149.     z.real=temp.real+real
150.     z.imag=temp.imag+imag
151.     i=i+1
152.   end
153.   --render point
154.   draw_point(currentPoint.x,currentPoint.y,i)
155.   --Prepare for next loop
156.   currentPoint.x=currentPoint.x+1
157.   currentNumber.real = real + config.step.x
158.
159.   --End of horizontal line
160.   if real > config.numberBoundaries.x.max then
161.     currentNumber.real = config.numberBoundaries.x.min
162.     currentNumber.imag = imag-config.step.y
163.     currentPoint.x=0
164.     currentPoint.y=currentPoint.y+1
165.   end
166.
167.   if imag<config.numberBoundaries.y.min then
168.   --All points have been processed. We can stop the function from starting again
169.
170.     --Debug lines for render time benchmarking? uncomment the next two lines to activate
171.     --print("done !")
172.     --print("time elapsed :"..os.clock()-start_time .."s")
173.
174.     return false
175.   end
176.   return true
177. end
178.
179. function map:on_started()
180.   map:get_hero():set_visible(false)
181. end
182.
183. function map:on_finished()
184.   map:get_hero():set_visible(true)
185. end
186. -- starting the whole process
187. function map:on_opening_transition_finished()
188.   config.step.x=(config.numberBoundaries.x.max-config.numberBoundaries.x.min)/(config.grid.width-1)
189.   config.step.y=(config.numberBoundaries.y.max-config.numberBoundaries.y.min)/(config.grid.height-1)
190.   --print("step X="..config.step.x..",step Y="..config.step.y)
191.
192.   --getting the current time for later usage, if you want to benchmark the rendering duration, just uncomment the next line
193.   --start_time=os.clock()
194.
195.   sol.timer.start(1, calculateNextPoint)
196. end
197. function map:on_draw(dst)
198.   local camera=map:get_camera()
199.   local x,y=camera:get_position()
200.   local w,h=sol.video.get_quest_size()
201.   fractalRender:draw_region(x,y,w,h,dst)
202. end
Here is a screenshot of a render using the default exponent (2) :

2. the Sierpinsky Triangle
Code: Lua
1. --[[
2. The Sierpinsky triangle.
3. On of the well known fractals, often used as a demonstration for calculators programming
4. When rendered, it shows a Triforce-like shape, with each triangle being a raduction of the base figure.
5. like this
6.
7.        /\
8.       /--\
9.      /\  /\
10.     /__\/__\
11.    /\      /\
12.   /__\    /__\
13.  /\  /\  /\  /\
14. /__\/__\/__\/__\
15.
16. For more information, feel free to check on the Wiki or any math article
17.
18. How to use :
19. 1. create a map of the size of your choice
20. 2. copy/paste this whole code in the map script.
21. 3. setup the config variables with your own parameters
22. 4. Run the map !
23. --]]
24.
25. local map = ...
26.
27. -- Must be lower or equel to map size to render properly
28. local grid={
29.   width=200,
30.   height=200
31. }
32. local maxIterations=10000
33. -- the outer triangle summits
34. local summits={
35.    {0,0},
36.    {grid.width,0},
37.    {grid.width/2,grid.height},
38. }
39. local point
40.
41. -- initial position
42. local x=grid.width/2
43. local y=grid.height/2
44.
45. local iter
46. local fractalRender= sol.surface.create(grid.width,grid.height)
47. local pixel=sol.surface.create(1,1)
48. pixel:fill_color({0,0,0})
49. local game = map:get_game()
50.
51. -- Event called at initialization time, as soon as this map becomes is loaded.
52. function map:on_started()
53.   map:get_hero():set_visible(false)
54.   iter=0
55.   --print("initial point : X="..x..", Y="..y)
56.
57. end
58.
59. function map:on_finished()
60.   map:get_hero():set_visible(true)
61. end
62.
63. function map:on_update()
64.
65.   if iter <maxIterations then
66.     --these three lines are all the calculation. Yes, this fractal is very easy to setup.
67.     --Note, they directly come from internet, so feel free to change to your own.
68.     point=summits[math.random(3)]
69.     x=(x+point[1])/2
70.     y=(y+point[2])/2
71.
72.     --[[
73.
74.     -- my calculator's example script from the instruction book. Kept here for historical reasons, feel free to remove.
75.     random=math.random(999)
76.     --local random
77.     --print("random number : "..random)
78.     if random<=333 then
79.       x=x/2
80.       y=y/2
81.     elseif random >333 and random <=666 then
82.       x=(x+(base_size/2))/2
83.       y=(y+base_size)/2
84.     else
85.       x=(x+base_size)/2
86.       y=y/2
87.     end
88.   --]]
89.   --print("new point on cycle "..count..": X="..x..", Y="..y)
90.   pixel:draw(fractalRender, x,grid.height-y)
91.   iter=iter+1
92.   end
93.
94.
95. end
96.
97. function map:on_draw(dst)
98.   fractalRender:draw(dst)
99. end
Screenshot of the render :

That's all for now! More to come later.... maybe !
Feel free to test, but be warned : it can, and will, take some time to get a full render, depending of your computer power and the parameters you chose.
if you have suggestions for improvements, feel free to.
Enjoy !

Edit (2017/09/21) : Added screenshots of the renders.
« Last Edit: September 21, 2017, 10:47:28 am by PhoenixII54 »

#### Diarandor

• Hero Member
• Posts: 1008
• Cats are cool! （ΦωΦ）
##### Re: Fractals !
« Reply #1 on: September 15, 2017, 05:26:44 pm »
Thanks a lot for sharing these scripts. Fractals are very interesting and beautiful. Although we are a bit unlucky that pixel art is really bad to draw a fractal...
But now we can easily draw triforces with your Sierpinsky script
“If you make people think they're thinking, they'll love you. But if you really make them think, they'll hate you.”

#### froggy77

• Full Member
• Posts: 105
##### Re: Fractals !
« Reply #2 on: September 19, 2017, 07:43:54 pm »

#### MetalZelda

• Hero Member
• Posts: 551
##### Re: Fractals !
« Reply #3 on: September 24, 2017, 11:36:16 pm »
Interesting concept, I've always been amazed by fractals

#### froggy77

• Full Member
• Posts: 105
##### Re: Fractals !
« Reply #4 on: November 03, 2017, 07:13:58 pm »
Indeed, it is an interesting concept. Thank you for having added images.
I tested the first script; ouch! "time elapsed :95.834s" to display the result.