# The following equations came form Introduction to Mathematical Modeling Using Discrete Dynamical Systems by Frederick Marotto.
# P was generated randomly.
def BarnsleyX(x,y,rand):
p=rand
if(p<=.01):
return .5
elif(p<=.11):
return -.15*x+.28*y+.575
elif(p<=.23):
return .2*x-.26*y+.4
return .85*x+.04*y+.075
def BarnsleyY(x,y,rand):
p=rand
if(p<=.01):
return .16*y
elif(p<=.11):
return .26*x+.24*y-.086
elif(p<=.23):
return .23*x+.22*y+.045
return -.04*x+.85*y+.18
Array=[]
pointx=1
pointy=1
Array.append([pointx, pointy])
for x in range(50000):
r=random()
x0=pointx
pointx=Barnsleyx(pointx, pointy, r)
pointy=Barnsleyy(x0, pointy, r)
Array.append([pointx, pointy])
show (scatter_plot(Array, markersize=.0028))
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# The following code adjusts the y-values from above, thereby allowing for the leaf to curve differently.
def BarnsleyX(x,y,rand):
p=rand
if(p<=.01):
return .5
elif(p<=.11):
return -.15*x+.17*y+.575
elif(p<=.23):
return .2*x-.25*y+.4
return .85*x+.04*y+.075
def BarnsleyY(x,y,rand):
p=rand
if(p<=.01):
return .16*y
elif(p<=.11):
return .26*x+.24*y-.086
elif(p<=.23):
return .23*x+.09*y+.045
return -.04*x+.85*y+.18
Array=[]
pointx=1
pointy=1
Array.append([pointx, pointy])
for x in range(50000):
r=random()
x0=pointx
pointx=BarnsleyX(pointx, pointy, r)
pointy=BarnsleyY(x0, pointy, r)
Array.append([pointx, pointy])
show (scatter_plot(Array, markersize=.0028))
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# This fractal is an example of a Mandelbrot Set.
def mandelbrot(imaginary,real):
w=complex(real,imaginary)
z=complex(0,0)
for r in range(1,99):
z=(z*z)+w
if(abs(z)>2):
return r
return 0
Array=[]
for s in range(100):
Array.append([])
realMin=-1
realMax=.5
imaginaryMin=-1
imaginaryMax=1
for y in range(200):
w_imaginary=imaginaryMin+y*(imaginaryMax-imaginaryMin)/200
for x in range(300):
w_real=realMin+x*(realMax-realMin)/300
mand=mandelbrot(w_imaginary,w_real)
if(mand>0):
Array[mand].append([x,y])
mandPlot=scatter_plot(Array[0],markersize=.395,facecolor='green',edgecolor='green')
for k in range(1,99):
colors=10*k/100
from sage.plot.colors import green
if(len(Array[k])>0):
mandPlot=mandPlot+scatter_plot(Array[k],markersize=.35,facecolor=(green.lighter(fraction=colors)).html_color(),edgecolor=(green.lighter(fraction=colors)).html_color())
show(mandPlot)
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# Here, the parameters on the graph and the values have been adjusted to provide a wider view of the above graph.
def mandelbrot(imaginary,real):
w=complex(real,imaginary)
z=complex(0,0)
for r in range(1,99):
z=(z*z)+w
if(abs(z)>2):
return r
return 0
Array=[]
for s in range(100):
Array.append([])
realMin=-2
realMax=1
imaginaryMin=-1.5
imaginaryMax=1.5
for y in range(250):
w_imaginary=imaginaryMin+y*(imaginaryMax-imaginaryMin)/250
for x in range(250):
w_real=realMin+x*(realMax-realMin)/250
mand=mandelbrot(w_imaginary,w_real)
if(mand>0):
Array[mand].append([x,y])
mandPlot2=scatter_plot(Array[0],markersize=.71,facecolor='gray',edgecolor='gray')
for t in range(1,99):
colors=4*t/100
from sage.plot.colors import gray
if(len(Array[t])>0):
mandPlot2=mandPlot2+scatter_plot(Array[t],markersize=.75,facecolor=(gray.lighter(fraction=colors)).html_color(),edgecolor=(gray.lighter(fraction=colors)).html_color())
show(mandPlot2)
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# The following is a seirpeinksi triangle which was introduced in class.
def midFormula(Coord1, Coord2):
return [(Coord1[0]+Coord2[0])/2,(Coord1[1]+Coord2[1])/2]
def Triangle(tri1,tri2,tri3,x0,iterations):
seirpinski=[tri1,tri2,tri3]
points=[x0]
x=x0
for i in range(iterations):
rand=floor(3*random())
x=midFormula(seirpinski[rand],x)
points.append(x)
return points
T=Triangle([-.5,0],[3,2],[0,5],[.2,.4],100000)
show(list_plot(T,size=1))
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# The following was derived from a seirpinski triangle. The midpoint formula subtracted the coordinates as opposed to adding them as in the previous example. This resulted in the following shape.
def midFormula(Coord1, Coord2):
return [(Coord1[0]-Coord2[0])/2,(Coord1[1]-Coord2[1])/2]
def Snowflake(snow1,snow2,snow3,x0,iterate):
InverseSeirpinski=[snow1,snow2,snow3]
startPoint=[x0]
x=x0
for i in range(iterate):
rand=floor(3*random())
x=midFormula(InverseSeirpinski[rand],x)
startPoint.append(x)
return startPoint
S=Snowflake([-.5,0],[3,2],[0,5],[.2,.4],100000)
show(list_plot(S,size=1))
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