Fractals

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Contents

Sierpinsky Triangle

Sierpinksy Triangle
Sierpinksy Triangle
x = 0
y = 1.8
z = 0
for i = 1, 4000 do
a = math.random(1,3)

if a == 1 then 
x = x / 2
z = (z - 250)/2
end

if a == 2 then 
x = (x - 250)/2
z = (z + 250)/2
end

if a == 3 then 
x = (x + 250)/2
z = (z + 250)/2
end


p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

3D Sierpinsky Triangle

3D Sierpinsky Triangle
3D Sierpinsky Triangle

WARNING This can freeze up your computer. (My computer could only handle 5000 after about a half hour.) If you have a slow computer, reduce the "For" loop to a lower number, like 500, then work your way up to 1000.

For a better picture of what this is, see the wikipedia article on Sierpinsky triangles 

local x = 0
local y = 1.8
local z = 0

for i = 1, 5000 do
a = math.random(1,8)

if a == 1 then 
x = (x - 200)/2
y = (y - 200)/2
z = (z + 200)/2
end

if a == 2 then 
x = (x + 200)/2
y = (y - 200)/2
z = (z + 200)/2
end

if a == 3 then 
x = (x - 200)/2
y = (y - 200)/2
z = (z - 200)/2
end

if a == 4 then 
x = (x + 200)/2
y = (y - 200)/2
z = (z - 200)/2
end

if a == 5 then 
x = (x + 0)/2
y = (y + 200)/2
z = (z + 0)/2
end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,y,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

Sierpinksy

Sierpinsky
Sierpinsky

This won't make a mathematical recreation of the Sierpinksy fractal, but it will merely make a pretty visual representation of it. 

x=1
y=1

for i = 50, 1, -1 do

x=i
y=(50-i)
z=(50-i)

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1,i,1))
p.Size = Vector3.new(y,1,z)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(1)

end


for i = 50, 1, -1 do

x=i
y=(50-i)
z=(50-i)

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1+50,i,1+50))
p.Size = Vector3.new(y,1,z)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(1)

end

for i = 50, 1, -1 do

x=i
y=(50-i)
z=(50-i)

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1+50,i,1))
p.Size = Vector3.new(y,1,z)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(1)

end

for i = 50, 1, -1 do

x=i
y=(50-i)
z=(50-i)

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1,i,1+50))
p.Size = Vector3.new(y,1,z)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(1)

end

for i = 50, 1, -1 do

x=i
y=(50-i)
z=(50-i)

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1+25,i+50,1+25))
p.Size = Vector3.new(y,1,z)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(1)

end

Fern leaf

Fern leaf fractal
Fern leaf fractal
x = 1
y = 1.8
z = 1
for i = 1, 500 do
a = math.random(1,100)

if a == 1 then

x = 0
z = (0.16*z)

elseif a > 1 and a <= 8 then

x = ((0.20*x) - (0.26*z))
z= (0.23*x + 0.22*z + 16)

elseif a > 8 and a <= 15 then

x = ((-0.15*x) + 0.28*z)
z = (0.26*x + 0.24*z + 4.4)

elseif a > 15 and a <= 100 then

x = 0.85*x + 0.04*z
z = ((-0.04*x) + 0.85*z + 16)

end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
wait(.1)
end

Mandelbrot set

Mandelbrot set
Mandelbrot set

This image requires the Output window of Roblox studio. Thanks to Serveringhaus.org for this code. 

iter=100
esclim=2.0
ulx=-2.0
uly=1.0
lrx=1.0
lry=-1.0
width=70
height=40

function abs(x,y)
    return math.sqrt(x*x+y*y)
end

function escapeq(cx,cy)
    local zx=0.0
    local zy=0.0
    local i=0

    while i<iter and abs(zx,zy)<esclim do
        tmp=zx*zx-zy*zy
        zy=2.0*zx*zy
        zx=tmp

        zx=zx+cx
        zy=zy+cy
        i=i+1
    end

    return i<iter
end

for y=1,height do
    line=''
    for x=1,width do
        zx=ulx+(lrx-ulx)/width*x;
        zy=uly+(lry-uly)/height*y;
        if escapeq(zx,zy) then
            line=line..'.'
        else
            line=line..'#'
        end
    end
    print(line);
end

Mandelbrot Brick Set

Brick version of the Mandelbrot Fractal
Brick version of the Mandelbrot Fractal

This is almost identical to the script above, but it uses bricks instead of the Output menu. 

iter=100
esclim=2.0
ulx=-2.0
uly=1.0
lrx=1.0
lry=-1.0
width=70
height=40

function abs(x,y)
    return math.sqrt(x*x+y*y)
end

function escapeq(cx,cy)
    local zx=0.0
    local zy=0.0
    local i=0

    while i<iter and abs(zx,zy)<esclim do
        tmp=zx*zx-zy*zy
        zy=2.0*zx*zy
        zx=tmp

        zx=zx+cx
        zy=zy+cy
        i=i+1
    end

    return i<iter
end

for y=1,height do
    for x=1,width do
        zx=ulx+(lrx-ulx)/width*x;
        zy=uly+(lry-uly)/height*y;
        if escapeq(zx,zy) then
p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,y))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
        else
p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,y))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
--p.Color = Color3.new(2)
p.Parent = game.Workspace
--wait(.1)
        end
    end
end

Cantor set

2D Cantor set
2D Cantor set
x = 0
y = 1.8
z = 0
for i = 1, 3000 do
a = math.random(1,4)

if a == 1 then 
x = (x - 250)/3
z = (z - 250)/3
end

if a == 2 then 
x = (x - 250)/3
z = (z + 250)/3
end

if a == 3 then 
x = (x + 250)/3
z = (z + 250)/3
end

if a == 4 then 
x = (x + 250)/3
z = (z - 250)/3
end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

3D Cantor set

3D Cantor set
3D Cantor set

WARNING This can freeze up your computer. If you have a slow computer, reduce the "For" loop to a lower number, like 500, then work your way up to 1000. 

local x = 0
local y = 1.8
local z = 0

for i = 1, 2000 do
a = math.random(1,8)

if a == 1 then 
x = (x - 250)/3
y = (y - 250)/3
z = (z - 250)/3
end

if a == 2 then 
x = (x - 250)/3
y = (y - 250)/3
z = (z + 250)/3
end

if a == 3 then 
x = (x - 250)/3
y = (y + 250)/3
z = (z - 250)/3
end

if a == 4 then 
x = (x - 250)/3
y = (y + 250)/3
z = (z + 250)/3
end

if a == 5 then 
x = (x + 250)/3
y = (y + 250)/3
z = (z + 250)/3
end

if a == 6 then 
x = (x + 250)/3
y = (y - 250)/3
z = (z + 250)/3
end

if a == 7 then 
x = (x + 250)/3
y = (y + 250)/3
z = (z - 250)/3
end

if a == 8 then 
x = (x + 250)/3
y = (y - 250)/3
z = (z - 250)/3
end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,y,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

3D Cantor set 2

3D Cantor set
3D Cantor set

This is a proportional representation of the 3D Cantor set. 

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1,17,1))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1+66,17,1+66))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1+66,17,1))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1,17,1+66))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1+66,17+66,1+66))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1,17+66,1+66))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1+66,17+66,1))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1,17+66,1))
p.Size = Vector3.new(33,33,33)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace

Untitled

Untitled
Untitled
x = 68
y = 1.8
z = 56
for i = 1, 500 do
a = math.random(1,5)

if a == 1 then 
x = x / 3
z = (z - 57)/3
end

if a == 2 then 
x = (x + 68)/3
z = (z - 13)/3
end

if a == 3 then 
x = (x + 42)/3
z = (z + 58)/3
end

if a == 4 then 
x = (x - 42)/3
z = (z + 58)/3
end

if a == 5 then 
x = (x - 68)/3
z = (z - 13)/3
end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

Untitled2

Untitled
Untitled
x = 82
y = 1.8
z = 30
for i = 1, 500 do
a = math.random(1,6)

if a == 1 then 
x = (x - 185)/5
z = (z - 300)/5
end

if a == 2 then 
x = (x + 180)/5
z = (z - 300)/5
end

if a == 3 then 
x = (x + 345)/5
z = (z + 5)/5
end

if a == 4 then 
x = (x + 175)/5
z = (z + 300)/5
end

if a == 5 then 
x = (x - 180)/5
z = (z + 300)/5
end

if a == 6 then 
x = (x - 350)/5
z = (z + 5)/5
end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

Quadratic Koch, 2nd iteration

Quadratic Koch, 3D (type 1), 2nd iteration
Quadratic Koch, 3D (type 1), 2nd iteration

This doesn't get into the math behind it. See wikipedia for more info. 

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(1,16.2,0))
p.Size = Vector3.new(30,30,30)
p.Anchored = true
p.Color = Color3.new(0)
p.formFactor = "Symmetric"
p.Parent = game.Workspace


p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(0,21,0))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"


p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(0,16.2,20))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(21,16.2,0))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"


p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(-19,16.2,0))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"


p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(0,16.2,-20))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"


p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(0,6.2,30))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(-29, 6.2, 0))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(0, 6.2, -30))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(31, 6.2, 0))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(31, 6.2, -30))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(-30, 6.2, -30))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(-29, 6.2, 30))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(30, 6.2, 30))
p.Size = Vector3.new(10,10,10)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
p.formFactor = "Symmetric"

Vicsek Fractal

I don't know why the middle is kicked off at an angle
I don't know why the middle is kicked off at an angle
x = 50
y = 1.8
z = 50
for i = 1, 1000, 1  do
a = math.random(1,5)

if a == 1 then 
x = .33*x - 66
z = .33*z + 66
end

if a == 2 then 
x = .325*x +.325*z 
z = -.325*x+.325*z
end

if a == 3 then 
x = .33*x - 66
z = .33*z - 66
end

if a == 4 then 
x = .33*x + 66
z = .33*z - 66
end

if a == 5 then 
x = .33*x + 66
z = .33*z + 66
end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

Fractal from my old calculus book

I don't know the name of this one. It's lopsided for some reason.

fractal from my old calculus book
fractal from my old calculus book
x = 50
y = 1.8
z = 50
for i = 1, 2000, 1  do
a = math.random(1,5)

if a == 1 then 
x = (.25*x) - 25
z = .25*z + 25
end

if a == 2 then 
x = .288*x -(.167*z) - 50
z = .167*x +.288*z - 50
end

if a == 3 then 
x = (.25*x)-74.6
z = .25*z+74.6
end

if a == 4 then 
x = .5*x+50
z = .5*z-50
end

if a == 5 then 
x = .18*x+.174*z+50
z = (-.174*x)+.18*z+50
end

p = Instance.new("Part")
p.CFrame = CFrame.new(Vector3.new(x,1.8,z))
p.Size = Vector3.new(1,1,1)
p.Anchored = true
p.Color = Color3.new(1)
p.Parent = game.Workspace
--wait(.1)
end

Other Roblox Fractal images

3D Cantor dust Third iteration of 3D Koch quadratic fractal Square koch 3
Sierpinsky triangle fractal Variation of the Sierpinsky triangle fractal Vicsek fractal
Fractal T Fractal kite

See Also

Wikipedia article on Fractals

Beauty in Mathematics, an introductory article on fractals with helpful images

Retrieved from "http://wiki.roblox.com/index.php/Fractals"