HTML Exercises

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%%html

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<style>

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@import 'https://fonts.googleapis.com/css?family=Sancreek&effect=3d';

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</style>

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<p style='font-family:Sancreek; font-size:200%; padding:20px;' 

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class='font-effect-3d'>

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<text style='color:#ff3636;'>Sancreek Font</text> & 

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<text style='color:#3636ff;'>3D Effect</text></p>

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Messages

Matrix Norms

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print(r.eval("""

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A<-matrix(rexp(25,rate=.1),nrow=5,ncol=5); 

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norm_type<-c('O','I','F','M','2')

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for (t in norm_type) {cat(t,norm(A,type=t),'\n')}

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A"""))

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Artificial Clusters

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# 1000 data points, 4 clusters

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import pylab; from sklearn import datasets,mixture

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from sklearn.model_selection import train_test_split

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bd=datasets.make_blobs(

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    n_samples=1000,cluster_std=.5,

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    centers=[[1,1],[-1,-1],[1,-1],[-1,1]])

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X_train,X_test,y_train,y_test=\

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train_test_split(bd[0],bd[1],test_size=.2,random_state=1)

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usl=mixture.BayesianGaussianMixture(n_components=4,n_init=4)

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usl.fit(X_train,y_train); y_predict=usl.predict(X_test)

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pylab.figure(figsize=(6,5)); pylab.grid()

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pylab.scatter(X_test[:,0],X_test[:,1],c=y_test,

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              s=50,alpha=.5,cmap=pylab.cm.winter)

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pylab.scatter(X_test[:,0]+.03,X_test[:,1]+.03,c=y_predict,

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              s=30,alpha=.5,cmap=pylab.cm.cool)

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pylab.scatter([1,-1,1,-1],[1,-1,-1,1],c='#3636ff',

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              marker='*',s=300,alpha=.7)

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pylab.show()

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Messages

Logical Equations

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bc=Words(alphabet=[0,1],length=10); c=0

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def lf(x):

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    eq1=(x[0] | (not x[1])) & (x[2] | (not x[3]))

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    eq2=(x[2] | (not x[3])) & (x[4] | (not x[5]))

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    eq3=(x[4] | (not x[5])) & (x[6] | (not x[7]))

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    eq4=(x[6] | (not x[7])) & (x[8] | (not x[9]))    

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    return (eq1 & eq2 & eq3 & eq4)

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for el in bc: c+=int(lf(el))

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pretty_print(html('<p>true values: %d</p>'%c))

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string=str(html(table([[(bc[i*10+j],lf(bc[i*5+j])) 

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                for j in range(5)] for i in range(10)])))

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html(string[:33]+'style="font-size:11px;" '+string[33:])

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Messages

Mesh Surfaces

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var('x,y,z')

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f1=x^2/9+y^2/16-z^2/25-1; f2=x^2/4+y^2/9-z^2/16

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implicit_plot3d(f1,(x,-7,7),(y,-7,7),(z,-7,7),contour=0,

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                color='#ff3636',opacity=.3,

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                plot_points=20,mesh=True)+\

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implicit_plot3d(f2,(x,-7,7),(y,-7,7),(z,-7,7),contour=0,

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                color='#3636ff',opacity=.5,frame=False)

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Messages

Color Schemes

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@interact

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def _(N=(1,12,1)):

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    p=sum(plot(sin(x-2*k)-x^2,x,hue=sin(k/N)) 

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          for k in [-1.4,-1.3,..,1.5])

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    p.show(figsize=(5,5),gridlines=True)

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Messages

Function Plotting by Segments

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@interact

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def _(N=(47,0,-1)):

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    var('x,y')

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    p=implicit_plot(sin(x)-cos(x),(0,4*pi),(-1,1),linestyle=':')

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    p+=sum([plot((sin(x),cos(x)),(pi*n/12,pi*(n+1)/12),

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                 thickness=12,alpha=.7,

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                 color=colormaps.jet(6*n)[:3]) 

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            for n in [0..N]])

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    p.show(aspect_ratio=4,gridlines=True,figsize=5)

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Messages

Random Plotting as Art Objects 4

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display(r.eval("""

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svg(filename='Rplots.svg',width=6,height=6,pointsize=10,

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    onefile=TRUE,family='times',bg='lightgray',

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    antialias=c('default','none','gray','subpixel'))

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a<-(.5+runif(1))*sample(c(-1,1),1); b<-sample(6:18,1) 

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c<-.001*sample(1:99,1)*sample(c(-1,1),1)

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t<-seq(0,16*b*pi,len=36*360)

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col<-rgb(.7*runif(1),.7*runif(1),.7*runif(1))

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x<-sin(t/6)-a*sin(b*t)*cos(t)-c*sin(16*b*t) 

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y<-cos(t/6)-a*sin(b*t)*sin(t)-c*cos(16*b*t)

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plot(x,y,type='o',lwd=.1,cex=.1^5,col=col,xlab='x',ylab='y',

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     fg='slategray',col.axis='slategray',col.lab='slategray')

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grid(col='slategray'); dev.off()

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c(paste0('a = ',a,'; b = ',b,'; c = ',c,'; color:',col))"""))

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Messages

Random Plotting as Art Objects 3

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display(r.eval("""

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svg(filename='Rplots.svg',width=6,height=6,onefile=TRUE,

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    pointsize=10,family='times',bg='lightsteelblue',

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    antialias=c('default','none','gray','subpixel'))

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a<-(.5+runif(1))*sample(c(-1,1),1); b<-sample(6:12,1)

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c<-.001*sample(1:99,1)*sample(c(-1,1),1); d<-sample(4:8,1)

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t<-seq(0,24*pi,len=24*18)

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col<-rgb(runif(1),runif(1),runif(1))

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x<-sin(t/6)+a*sin(b*t)*cos(t)+c*sin(b*t)

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y<-cos(t/6)+a*sin(b*t)*sin(t)+c*cos(d*b*t)

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plot(x,y,type='o',cex=.8,col='white',lwd=3); par(new=TRUE)

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plot(x,y,type='o',cex=.4,col=col,lwd=1,xlab='x',ylab='y',

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     fg='slategray',col.axis='slategray',col.lab='slategray')

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grid(col='slategray'); dev.off()

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c(paste0('a = ',a,'; b = ',b,'; c = ',c,'; d = ',d,'; color:',col))"""))

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Messages

Surface Drawing as Art Objects 2

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var('u,v'); a,b=7,4

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u_min,u_max=-pi,pi; v_min,v_max=-pi,pi

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f_x=cos(a*u)*sinh(v)/(1+cosh(u)*cosh(v))

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f_y=sin(b*u)*sinh(v)/(1+cosh(u)*cosh(v))

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f_z=sinh(u)*cosh(v)/(1+cosh(u)*cosh(v))

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parametric_plot3d(

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    [f_x,f_y,f_z],(u_min,u_max),(v_min,v_max),

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    plotpoints=100,color='steelblue',opacity=.7,

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    mesh=True,frame=False)

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Messages

Surface Drawing as Art Objects

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var('u,v'); u_min,u_max=0,2*pi; v_min,v_max=0,pi

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f_x=(2+cos(u))*cos(v)^3*sin(v)

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f_y=(2+cos(u+2*pi/3))*cos(v+2*pi/3)^2*sin(v+2*pi/3)^2 

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f_z=(2+cos(u-2*pi/3))*cos(v+2*pi/3)^2*sin(v+2*pi/3)^2

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sum([parametric_plot3d(

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    [f_x*i,f_y*j,f_z*k],(u_min,u_max),(v_min,v_max),color='cyan',

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    plotpoints=100,opacity=.3,mesh=True,frame=False) 

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     for i in [-1,1] for j in [-1,1] for k in [-1,1]])

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Messages

Complex Numbers as Art Objects 2

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x,y,k=var('x,y,k') 

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f(x,y)=cos(81*arg(sum((sin(3*(x+I*y))*\

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      (x+I*y))^factorial(k),k,1,4)))

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fl=[f(x,y) for x in [-1.,-.98,..,1.] 

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    for y in [-1.,-.98,..,1.]]

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list_plot(fl,axes=False,size=2,figsize=(6,6))

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Messages

Complex Numbers as Art Objects

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z,k=var('z,k')

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f=cos(81*arg(sum((sin(3*z)*z)^factorial(k),k,1,4)))

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complex_plot(f,(-1,1),(-1,1),

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             axes=False,plot_points=400,

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             figsize=(6,6),aspect_ratio=1)

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Messages

Polar Plots as Art Objects 2

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var('u,v'); n=10;

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f1=exp(cos(u)^2+sin(u))-3*cos(4*u) 

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f2=-floor(4*u)*abs(sin(4*u))/2

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s1=sum([spherical_plot3d(i*f1/n,(u,0,2*pi),(v,1,1.01),

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       color=(i/n,0,1-i/n)) for i in [1..n]])

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s2=sum([spherical_plot3d(i*f2/n,(u,0,2*pi),(v,1,1.01),

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       color=(1-i/n,i/n,0)) for i in [1..n]])

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(s1+s2).show(frame=False)

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Messages

Polar Plots as Art Objects

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var('t'); n=10

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f1=exp(cos(t)^2+sin(t))-3*cos(4*t)

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f2=-floor(4*t)*abs(sin(4*t))/2

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s1=sum([polar_plot(i*f1/n,(t,0,2*pi),

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                   color=Color([i/n,0,1-i/n])) 

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        for i in [1..n]])

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s2=sum([polar_plot(i*f2/n,(t,0,2*pi),

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                   color=Color([1-i/n,i/n,0])) 

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        for i in [1..n]])

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(s2+s1).show(axes=false)

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Messages

Random Plotting as Art Objects 2

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a=(.5+random())*sample([-1,1],1)[0] 

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b=randint(6,12); c=.001*randint(1,100) 

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col=['silver',Color([random() for i in range(3)])]

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fx(t)=sin(t/6)-a*sin(b*t)*cos(t)-c*sin(16*b*t)

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fy(t)=cos(t/6)-a*sin(b*t)*sin(t)-c*cos(16*b*t)

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st='<p>a, b, c: %.3f, %d, %.3f</p>'

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pretty_print(html(st%(a,b,c)))

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sum([parametric_plot(

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    [fx(t),fy(t)],(t,0,12*pi),axes=False,plot_points=400,

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    color=col[i],thickness=1.2-i,adaptive_recursion=100) 

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     for i in [0,1]]).show()

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xy=[[fx(t),fy(t)] for t in [0,.003,..,12*pi+.01]]

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list_plot(xy,color=col[1],axes=False,

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          size=1,aspect_ratio=1).show()

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Messages

Random Plotting as Art Objects

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a=(.5+random())*sample([-1,1],1)[0] 

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c=.01*randint(1,29)*sample([-1,1],1)[0]; 

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b=randint(6,18); d=randint(3,7)

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col=['silver',Color([random() for i in range(3)])]

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st='<p>a, b, c, d: %.2f, %d, %.2f, %d</p>'

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fx(t)=sin(t/6)+a*sin(b*t)*cos(t)+c*sin(b*t)

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fy(t)=cos(t/6)+a*sin(b*t)*sin(t)+c*cos(d*b*t)

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pretty_print(html(st%(a,b,c,d)))

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sum([parametric_plot(

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    [fx(t),fy(t)],(t,0,12*pi+.01),plot_points=300,

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    thickness=1.2-i,axes=False,color=col[i],

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    adaptive_recursion=100) for i in [0,1]]).show()

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xy=[[fx(t),fy(t)] for t in [0,.003,..,12*pi]]

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list_plot(xy,color=col[1],axes=False,size=1,

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          aspect_ratio=1).show()

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Messages

JuliaSetPlot Examples 2

RJSP2[s_,col_]:=Module[{a,n},a=.1*Random[];n=RandomInteger[{10,15}] ;
JuliaSetPlot[z^2/(z^n-.5-a),z,ImageSize->s,

ColorFunction->col,PlotLabel->{a,n}]];
{RJSP2[400,"CherryTones"],RJSP2[400,"BrightBands"]}

JuliaSetPlot Examples

RJSP1[c_,d_,s_]:=Module[{a,b},a=.1*Random[]; b=.1*Random[];
JuliaSetPlot[c+a +(d+b)*I,ColorFunction->"CoffeeTones",
PlotLabel->{c+a,d+b},ImageSize->s]]
{RJSP1[-.9,.2,400],RJSP1[.3,.4,400]}

Hilbert & Sierpinski Curves

RFCP1[f_,s_]:=Module[{k,n,t,x0,y0},
k=RandomInteger[{2,5}]; n=RandomInteger[{5,11}];
x0=Random[]; y0=Random[]; t=RandomInteger[{2,4}];
Graphics[Table[{Hue[i/n],Thickness[.1^t],
GeometricTransformation[f[k]/.Line->BSplineCurve,
RotationTransform[i*Pi/n,{x0,y0}]]},{i,2n}],ImageSize->s,
Background->RGBColor["silver"],
PlotLabel->{k,n,t,x0,y0}]]
{RFCP1[HilbertCurve,400],RFCP1[SierpinskiCurve,400]}

Koch Curve Examples

RCCP1[s_]:=Module[{a,k,n,t,x0,y0},
k=RandomInteger[{3,6}]; n=RandomInteger[{3,15}];

x0=Random[]; y0=Random[];
a=RandomInteger[{10,170}];t=RandomInteger[{2,4}];
Graphics[{Table[{Hue[2*i/n],Thickness[.1^t],
GeometricTransformation[KochCurve[

k,{0,a \[Degree],-a \[Degree],
-a \[Degree],a \[Degree]}]/.Line->BSplineCurve,
RotationTransform[i*Pi/n,{.5,0}]]},{i,2n}],

Text[{a,k,n},{0,-.7}],Text[{t,x0,y0},{.9,.7}]},
ImageSize->s,Background->RGBColor["silver"]]]
{RCCP1[400],RCCP1[400]}

LaplaceTransform Examples

F[u,v]:=LaplaceTransform[Cos[x + y],{x,y},{u,v}]
Plot3D[{Cos[u+v],Evaluate[F[u,v]]},{u,-Pi,Pi},{v,-Pi,Pi},
Axes->False,PlotStyle->Opacity[.7],

Mesh->{50,50},MeshStyle->Gray,
ImageSize->500,Boxed->False,AspectRatio->1]

Surfaces as Exercises 6

MSPP3[s_]:=Module[{x,y,z},
x=u - u^3/3 + u *v^2; y=v- v^3/3 + v *u^2; z=u^2-v^2;
ParametricPlot3D[Table[{i*x,j*y,k*z},{i,{-2,1}},{j,{-2,1}},{k,{-2,1}}],
{u,-2,2},{v,-2,2},AspectRatio->1,PlotRange->All,PlotPoints->30,
PlotStyle->Directive[Cyan,Opacity[.3]],

Axes->False,Boxed->False,Mesh->{15,25},

MeshStyle->{RGBColor["mintcream"],RGBColor["steelblue"]},
ImageSize->s,Background->RGBColor["silver"],

ViewPoint->{.1,.1,1.4}]];
MSPP3[600]

Surfaces as Exercises 5

MSPP4[a_,b_,c_,s_]:=Module[{x,y,z},
x=v*Cos[u]-a*v^4Cos[b*u];

y=v*Sin[u]+a*v^4Sin[b*u];

z=Exp[c*Log[v]]*Cos[c*u];
ParametricPlot3D[Table[{i*x,j*y,k*z},{i,{-2,2}},{j,{1,2}},{k,{1,2}}],
{u,0,4Pi},{v,0,1},AspectRatio->1,

PlotPoints->30,Axes->False,Boxed->False,
PlotStyle->Directive[Cyan,Opacity[.4]]

Mesh->{30,15},MeshStyle->{White,Cyan},
ImageSize->s,Background->RGBColor["silver"],

ViewPoint->{.1,.4,.9}]];
MSPP4[.5,2,1.5,600]