Interactive Polar Plotting

Manipulate[PolarPlot[Sin[6t]^2,{t,0,i*Pi/6},
ColorFunction->Function[t,Hue[12*Sin[t]/i]],
                                  PlotStyle->Thick,ImageSize->Large],
{i,Range[1,12,1]}]

ContourPlot Examples

ContourPlot[{x1^2+x1*x2+x2^2+2*x1+6*x2+3==0,
                           x1^2+x1*x2-x2^2+2*x1+6*x2+3==0,
                           x1^2-x1*x2+x2^2+2*x1+6*x2+3==0,
             -x1^2+x1*x2+x2^2+2*x1+6*x2+3==0},
                     {x1,-10,10},{x2,-10,10},ContourStyle->"Pastel",
            GridLines->Automatic,ImageSize->Large,
PlotLegends->"Expressions"]

PolyhedronData Examples

Show[PolyhedronData["GreatRhombicosidodecahedron"] /.
        Polygon[p_]:> MapIndexed[{Glow[Hue[Mod[First[#2],64]/64]],
        Black,Opacity[0.8],Polygon[#1]} &,p],
    Boxed->False,ImageSize->Large]

3D Graphics Interactive Objects

Manipulate[Graphics3D[{Hue[n],EdgeForm[{Gray,Thickness[0.01]}],Opacity[.5],
                                                  PolyhedronData[p,"GraphicsComplex"]},
                                                 Boxed->False,ImageSize->Large],
          {p,PolyhedronData["Archimedean"]},{n,Range[0.1,0.9,0.1]}]

Interactive Parameters

Manipulate[Show[ParametricPlot[

{(Cos[a*t]^n+Sin[b*t]^m+1/c)*Sin[t],
      (Cos[a*t]^n+Sin[b*t]^m+1/c)*Cos[t]},{t,0,2\[Pi]}],
    Plot[Cos[a*t]^n+Sin[b*t]^m+1/c,{t,0,2\[Pi]},PlotStyle->Dashed],
    PlotRange->{{-2.2,2.2},{-2.2,2.2}},ImageSize->Large],
    {a,Range[2,4,1]},{b,Range[4,8,1]},{c,Range[4,8,1]},
    {n,Range[2,8,1]},{m,Range[2,8,1]}]

SuperFunctions: LearnDistribution

boston=ResourceData["Sample Data: Boston Homes"];
LDB=LearnDistribution[boston[All,{"RM","MEDV"}]]


syntheticboston=Table[SynthesizeMissingValues[LDB,
{n,Missing[]}],{n,Range[4,8,0.2]}];
syntheticboston[[1;;3]]

p1=ListPlot[boston[All,{"RM","MEDV"}],PlotLegends -> {"Data"}];
p2=ListPlot[syntheticboston,PlotStyle->Red,

PlotLegends -> {"Synthetic Data"}];
Show[p1,p2,GridLines->Automatic,ImageSize->Large]

Taylor Series

p1=Plot[Sin[z]Exp[-z],{z,0,5},PlotStyle->{Thickness[0.02],Opacity[0.5]}];
p2=Plot[Evaluate[Table[Normal[Series[Sin[z]Exp[-z],
{z,0,n}]],{n,1,24}]],{z, 0,5},PlotStyle->"BrightBands"];
Show[p1,p2,ImageSize->Large]

Parametric Drawing 3

PPRT3[n_,m_,col_,s_]:=Module[{a,b,c,u},
u=Pi/n; a=RandomInteger[{5,12}];
b=RandomInteger[{13,36}];c=2*RandomInteger[{3,6}];
PF1[i_,t_]:=Cos[u*t+2i*Pi/c]+Cos[a*u*t+2i*Pi/c]+Cos[b*u*t+2i*Pi/c];
PF2[i_,t_]:=Sin[u*t+2i*Pi/c]-Sin[a*u*t+2i*Pi/c]+Sin[b*u*t+2i*Pi/c];
Graphics[Table[{ColorData[col][1-i/(2c)],Opacity[.1],
Polygon[Table[{PF1[i,t], PF2[i,t]}, {t, 0,m*Pi,Pi/c}]]},{i,2c}],
Background->RGBColor["lightgrey"],
PlotLabel->{"a"->a,"b"->b,"c"->c},ImageSize->s]];
PPRT3[9,6,"CherryTones",600]

Parametric Drawing 2

PPRT2[n_,m_,col_,s_]:=Module[{a,b,c,u},
u=Pi/n; a=RandomInteger[{5,12}];
b=RandomInteger[{13,36}];c=2*RandomInteger[{3,6}];
PF1[i_,t_]:=Cos[u*t+2i*Pi/c]+Cos[a*u*t+2i*Pi/c]+Cos[b*u*t+2i*Pi/c];
PF2[i_,t_]:=Sin[u*t+2i*Pi/c]-Sin[a*u*t+2i*Pi/c]+Sin[b*u*t+2i*Pi/c];
Graphics[Table[{EdgeForm[ColorData[col][1-i/(2c)]],FaceForm[None],
Polygon[Table[{PF1[i,t], PF2[i,t]}, {t, 0,m*Pi,Pi/c}]]},{i,2c}],
Background->RGBColor["lightgrey"],
PlotLabel->{"a"->a,"b"->b,"c"->c},ImageSize->s]];
PPRT2[12,6,"DeepSeaColors",600]

Parametric Drawing

PPRT[n_,col_,s_]:=Module[{a,b,c,u},
u=Pi/n; a=RandomInteger[{5,12}];
b=RandomInteger[{13,24}];c=2*RandomInteger[{3,6}];
PF1[i_,t_]:=Cos[u*t+2i*Pi/c]+Cos[a*u*t+2i*Pi/c]+Cos[b*u*t+2i*Pi/c];
PF2[i_,t_]:=Sin[u*t+2i*Pi/c]-Sin[a*u*t+2i*Pi/c]+Sin[b*u*t+2i*Pi/c];
Show[Table[ParametricPlot[{PF1[i,t],PF2[i,t]},{t,0,12Pi},
PlotStyle->{ColorData[col][i/c],Thickness[.001]}],{i,c}],
PlotRange->All,Background->RGBColor["silver"],PlotPoints->200,
Axes->False,PlotLabel->{"a"->a,"b"->b,"c"->c},ImageSize->s]]
PPRT[11,"BrightBands",600]

AnglePath Examples 4

A=RandomReal[2,36]; T=Flatten[Table[A*Pi,{i,90}]];
Graphics[{RGBColor[{.5,.1,RandomReal[]}],Thickness[.001],
Line[AnglePath[T]]/.Line->BSplineCurve},ImageSize->400]

AnglePath Examples 3

Graphics[Table[{Hue[.05i],Thickness[.1^5],
Line[AnglePath[Array[ThueMorse,2^12](1.27+.0001i)Pi]]/.Line->BSplineCurve},
{i,13}],ImageSize->400]

Function Drawing in Polar Coordinates

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var('u,i'); f1(u,i)=i*(9+.9*cos(12*u))

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f2(u,i)=(1+.05*cos(36*u))

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f3(u,i)=(1+.05*cos(216*u))*(1+sin(u))

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sum([polar_plot(

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    f1(u,i)*f2(u,i)*f3(u,i),(u,-pi,pi),

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    color=(0,1-.3*i,0),thickness=.7,plot_points=200) 

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     for i in [.1,.2,..,3]]).show(axes=False)

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Inclination and Elevation

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u,v=var('u,v'); T1=Spherical('radius',['azimuth','inclination'])

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T2=SphericalElevation('radius',['azimuth','elevation'])

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p1=plot3d(v/6+cos(5*u),(u,0,2*pi),(v,0,2*pi),color='crimson',

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          opacity=.4,transformation=T1,frame=False,

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          plot_points=(50,50),mesh=True);

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p2=plot3d(v/6+cos(5*u),(u,0,2*pi),(v,0,2*pi),color='green',

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          opacity=.3,transformation=T2,frame=False,

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          plot_points=(30,30),mesh=True);

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(p1+p2).show()

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Spherical Plots 3d

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u,v=var('u,v'); f=cos(3*u)-cos(16*u)*sin(5*v)

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def color_f(u,v): return abs(sin(u+v))

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spherical_plot3d(f,(u,0,2*pi),(v,0,2*pi),

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                 frame=False,plot_points=50,

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                 color=(color_f,colormaps.hot))

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Histogram Examples

<script src='https://d3js.org/d3.v4.min.js'></script> was added in the page head.

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%%html

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<style>@import 'https://fonts.googleapis.com/css?family=Sancreek';

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.axis text,.bar text {fill:#3636ff; font-family:Sancreek; 

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                      font-size:90%; text-shadow:3px 3px 3px #aaa;}

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.bar rect {fill:#3636ff; opacity:.5;}

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</style><svg id='svg1' width='650' height='350' 

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style='background-color:lavender;'/><script>

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var data=d3.range(3000).map(d3.randomBates(5)),

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    formatCount=d3.format(',.0f');

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var svg=d3.select('#svg1'),

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    m=30,margin={top:m,right:m,bottom:m,left:m}, 

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    width=+svg.attr('width')-margin.left-margin.right,

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    height=+svg.attr('height')-margin.top-margin.bottom,

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    tr='translate('+margin.left+','+margin.top+')',

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    g=svg.append('g').attr('transform',tr);

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var x=d3.scaleLinear().rangeRound([0,width]);

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var bins=d3.histogram().domain(x.domain())

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           .thresholds(x.ticks(30))(data);

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var y=d3.scaleLinear().range([height,0])

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        .domain([0,1.1*d3.max(bins,function(d){return d.length;})]);

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var bar=g.selectAll('.bar').data(bins).enter()

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         .append('g').attr('class','bar')

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         .attr('transform',function(d){

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             return 'translate('+x(d.x0)+','+y(d.length)+')';});

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bar.append('rect').attr('x',0)

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   .attr('width',x(bins[0].x1)-x(bins[0].x0)-2)

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   .attr('height',function(d){return height-y(d.length);});

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bar.append('text').attr('text-anchor','middle')

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   .text(function(d){return formatCount(d.length);})

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   .attr('dy','.55em').attr('y',-10)

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   .attr('x',(x(bins[0].x1)-x(bins[0].x0))/2);

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g.append('g').attr('class','axis axis--x')

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 .call(d3.axisBottom(x))

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 .attr('transform','translate(0,'+height+')'); 

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g.append('text').attr('y',5)

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 .attr('font-family','Sancreek')

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 .text('The Example of D3 Histograms');

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Forms of Complex Numbers

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display(r.eval("""

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z<-complex(real=-sqrt(2-sqrt(3)),

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           imaginary=sqrt(2+sqrt(3)))

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# polar form

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zp<-Mod(z)*(cos(Arg(z))+sin(Arg(z))*sqrt(as.complex(-1)))

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# exponential form

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ze<-Mod(z)*exp(Arg(z)*sqrt(as.complex(-1)))

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cat(zp==ze); c(z,zp,ze)"""))

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One Neuron

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import numpy

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def sigmoid(x): return 1./(1+numpy.exp(-x))

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def sigmoid_derivation(x): return x*(1.-x)

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X=numpy.array([[.10,.05,.95],[.09,.03,.08],[.01,.09,.91],

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               [.04,.92,.07],[.05,.02,.04],[.07,.97,.05],

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               [.06,.02,.98],[.02,.06,.03],[.01,.09,.03],

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               [.02,.94,.01]])

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Y=numpy.array([[1,0,1,2,0,2,1,0,1,2]]).T

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synapse0=numpy.random.randn(3,1)

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for iter in range(100):

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    layer0=X; layer1=sigmoid(numpy.dot(layer0,synapse0))

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    layer1_error=layer1-Y

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    layer1_delta=layer1_error*sigmoid_derivation(layer1)

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    synapse0_derivative=numpy.dot(layer0.T,layer1_delta)/Y.shape[0]

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    synapse0-=synapse0_derivative

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print(numpy.hstack((numpy.hstack((X,Y)),(layer1.round(3)))))

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print('the 9th row wasn`t labeled rightly')

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print('the neuron has corrected this mistake')

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The Nth Prime

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p=Graphics()

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for i in range(2,9):

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    p+=plot(log(x^i),1,2,

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            fill=lambda x:nth_prime(x^i)/floor(x^i),

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            fillcolor=hue((i-2)/7))

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

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Messages