Quivers

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1

import pylab,numpy

2

from mpl_toolkits.mplot3d import Axes3D

3

t=1.1; cq=numpy.arange(0,1,.13)

4

cq=numpy.concatenate((cq,numpy.repeat(cq,2)))

5

xq=[1,-1,1,-1,1,-1,1,-1]

6

yq=[1,1,-1,-1,1,1,-1,-1]

7

zq=[1,1,1,1,-1,-1,-1,-1]

8

f=pylab.figure(figsize=(5,5))

9

ax=f.add_subplot(111,projection='3d')

10

q=ax.quiver(8*[0],8*[0],8*[0],xq,yq,zq,

11

            colors=pylab.cm.hsv(cq),lw=10,alpha=.5) 

12

ax.set_xlabel(r'$\mathscr{X}$',fontsize=15)

13

ax.set_ylabel(r'$\mathscr{Y}$',fontsize=15) 

14

ax.set_zlabel(r'$\mathscr{Z}$',fontsize=15)

15

ax.set_xlim(-t,t); ax.set_ylim(-t,t)

16

ax.set_zlim(-t,t); pylab.show()

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---

Messages

Interactive Vector Fields

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1

@interact

2

def _(n=[4,5,..,8]):

3

    f(u,v)=exp(-(u^n+v^n)/n)

4

    s1(u,t)=(u,-.1*t,.1*t)

5

    s2(v,t)=(v,-.1*t,.1*t)

6

    pvf=sum([plot_vector_field(

7

        f(u,v).gradient(),s1(u,i),s2(v,i),

8

        color=hue(i/3)) for i in [1..3]])

9

    pvf.show(figsize=(5,5),frame=False)

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---

Messages

Interactive 3D Plotting

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1

@interact

2

def _(N=[5,4,..,1],pp=[5,10,..,30]):

3

    f1(x,y,z,t)=x^2+y^2-.5*z*t^2

4

    f2(x,y,z,t)=x^2+y^2+.5*z*t^2

5

    p1=sum([implicit_plot3d(

6

        f1(x,y,z,i),(x,-i,i),(y,-i,i),(z,-2,2),

7

        color=hue(sin(.24*i)),opacity=1-.15*i,

8

        plot_points=pp,frame=False) for i in [1..N]])

9

    p2=sum([implicit_plot3d(

10

        f2(x,y,z,i),(x,-i,i),(y,-i,i),(z,-1,0),

11

        color=hue(sin(.24*i)),opacity=.3,

12

        plot_points=pp,frame=False) for i in [1..N]])

13

    p3=plot3d(lambda x,y:0,(-N,N),(-N,N),

14

              color='steelblue',opacity=.5)

15

    (p1+p2+p3).show()

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---

Messages

Min Spanning Trees

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1

import warnings; warnings.filterwarnings('ignore')

2

g=graphs.HoffmanSingletonGraph()

3

t=Graph()

4

t.add_edges(g.min_spanning_tree(

5

    starting_vertex=0))

6

[t.set_edge_label(u,v,u) for u,v,l in t.edges()]

7

t.show(layout='tree',tree_root=0,

8

       color_by_label=True,

9

       vertex_size=0,figsize=(5,5))

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---

Messages

Interactive Graph Exploration 2

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1

@interact

2

def _(n=[3,4,5,6]):

3

    g=Graph({},loops=True,multiedges=True,sparse=True)

4

    def add(g,x,y):

5

        g.add_edges([(x,y,'a'),(x,y,'a'),(x,y,'b'),

6

                     (x,y,'b'),(x,y,'c'),(x,y,'c'),

7

                     (x,y,'d'),(x,y,'d'),(x,y,'e'),

8

                     (x,y,'e'),(x,y,'f'),(x,y,'f')])

9

    [add(g,i,i+1) for i in [1..n-1]] 

10

    [add(g,i,i+5) for i in [1..n-1]]; add(g,n,1)

11

    g.graphplot(vertex_size=0,vertex_labels=False,

12

                color_by_label=True).show(figsize=(5,5))

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---

Messages

Interactive Graph Exploration

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1

import warnings; warnings.filterwarnings('ignore')

2

@interact

3

def _(n=[95,90,..,5]):

4

    g=graphs.CubeGraph(4); edge_colors={}; 

5

    c=[colormaps.hsv(n*i)[:3] for i in [0..3]]

6

    for i in [0..3]: edge_colors[c[i]]=[]

7

    for u,v,l in g.edges():

8

        [edge_colors[c[i]].append((u,v,l)) 

9

         for i in [0..3] if u[i]!=v[i]] 

10

    g=g.graphplot(

11

        edge_colors=edge_colors,edge_thickness=2.5,

12

        vertex_labels=False,vertex_size=0)

13

    g.show(figsize=(5,5))

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---

Messages

HTML Exercises

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1

%%html

2

<style>

3

text {font-size:500%;}

4

.trans,.comb {margin:0; color:transparent;

5

              text-shadow:0 0 5px #3636ff; 

6

              font-size:500%; position:relative;}

7

.trans::before {content:attr(title); position:absolute; 

8

                text-shadow:0 0 0 white;}   

9

.comb::before {content:attr(title); position:absolute; 

10

               text-shadow:0 0 0 slategray;}

11

</style>

12

<emoji class='trans' title='&#x1F985'>&#x1F985</emoji>

13

&nbsp;<emoji class='comb' title='&#x1F985'>&#x1F3A1</emoji>

14

&nbsp;<text>&#x1F985</text><br></br> 

15

<emoji class='trans' title='&#x1F3A1'>&#x1F3A1</emoji>

16

&nbsp;<emoji class='comb' title='&#x1F3A1'>&#x1F3A1</emoji>

17

&nbsp;<text>&#x1F3A1</text><br></br> 

18

<emoji class='trans' title='&#x1F3A0'>&#x1F3A0</emoji>

19

&nbsp;<emoji class='comb' title='&#x1F3A0'>&#x1F3A1</emoji>

20

<text>&#x1F3A0</text>

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---

Messages

String Replacements

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1

r.eval("""

2

replace_function<-function(w) {

3

    while (grepl('7777',w) | grepl('8888',w)) { 

4

        print(w)

5

        if (grepl('7777',w)) { w<-sub('7777','8',w) } 

6

        else { w<-sub('8888','7',w) } }   

7

    return(w) }

8

string<-paste(replicate(70,'7'),collapse='')

9

replace_function(string)""")

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---

Messages

Magic Functions

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1

import pylab; from skimage import io,color,measure

2

import warnings; warnings.filterwarnings('ignore')

3

from IPython.core.magic import register_line_magic

4

@register_line_magic

5

def vector(file):

6

    file_path='https://raw.githubusercontent.com/'+\

7

              'OlgaBelitskaya/data/main/decors/'

8

    img=io.imread(file_path+file)

9

    level=.7; img=color.colorconv.rgba2rgb(img) 

10

    gray_img=color.colorconv.rgb2gray(img) 

11

    contours=measure.find_contours(gray_img,level)

12

    pylab.figure(figsize=(5,5))

13

    pylab.gca().invert_yaxis()

14

    pylab.imshow(img,alpha=.1)

15

    [pylab.plot(c[:,1],c[:,0],lw=1) for c in contours]

16

    pylab.xticks([]); pylab.yticks([]); pylab.show()

17

%vector 00_01_001.png

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---

Messages

Interactive Region Plots

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1

@interact

2

def _(n=[4..8],b=[round(.25+.05*i,2) for i in range(11)]):

3

    var('x,y'); f=cos(x^n+y^n); t=3.

4

    p=region_plot(

5

        f<=b,(x,-t,t),(y,-t,t),

6

        incol='#3636ff',plot_points=220)

7

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

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---

Messages

Graph Partitions

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1

@interact

2

def _(N=[1,2,4,5,10,20]):

3

    D=graphs.DodecahedralGraph()

4

    p=[[i..i+N-1] for i in [0,N,..,20-N]]

5

    D.show(partition=p,figsize=(5,5))

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

---

Messages

Interactive Complex Plotting

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1

@interact

2

def _(n=[4..7]):

3

    var('z'); cf=z^n+1/z^n; d=2

4

    p=complex_plot(cf,(-d,d),(-d,d))

5

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

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---

Messages

Combine JavaScript Sources

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

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1

%%html

2

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

3

#svg002 {background-color:lavender;} 

4

text {fill:#3636ff; font-family:Monoton; 

5

      font-size:120%; text-shadow:3px 3px 3px slategray;} 

6

.line1 {fill:none; stroke:slategray; stroke-width:6;}

7

.line2 {fill:none; stroke:#3636ff; stroke-width:2;} 

8

.dot {fill:#3636ff; stroke:slategray; stroke-width:2;} 

9

</style><svg id='svg002'><script>

10

var n=20,m=35,margin={top:m,right:m,bottom:m,left:m},

11

    width=650-margin.left-margin.right,

12

    height=350-margin.top-margin.bottom,

13

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

14

var xScale=d3.scaleLinear().domain([0,n-1]).range([0,width]),

15

    yScale=d3.scaleLinear().domain([0,n]).range([height,0]); 

16

var line=d3.line().curve(d3.curveMonotoneX)

17

           .x(function(d,i) {return xScale(i);})

18

           .y(function(d) {return yScale(d.y);});

19

var data=d3.range(n).map(function(d) {

20

    return {'y':d3.randomUniform(n)()}})

21

var svg=d3.select('#svg002')

22

          .attr('width',width+margin.left+margin.right)

23

          .attr('height',height+margin.top+margin.bottom)

24

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

25

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

26

   .call(d3.axisBottom(xScale))

27

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

28

svg.append('g').attr('class','y axis')

29

   .call(d3.axisLeft(yScale));

30

svg.append('path').datum(data)

31

   .attr('class','line1').attr('d',line);

32

svg.append('path').datum(data)

33

   .attr('class','line2').attr('d',line);

34

svg.selectAll('.dot').data(data).enter()

35

   .append('circle').attr('class','dot').attr('r',3)

36

   .attr('cx',function(d,i) {return xScale(i)})

37

   .attr('cy',function(d) {return yScale(d.y)});

38

svg.append('text').attr('x',.4*width)

39

   .text('D3 & Google Fonts');

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---

Messages

Matrix Operations

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1

print(r.eval("""

2

A<-matrix(sample.int(100,size=3*3,replace=TRUE),

3

          nrow=3,ncol=3)

4

print(A); print(t(A)); print(solve(A))

5

cat("\n"); print(det(A)); qr(A)"""))

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---

Messages

Table Styling

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1

import numpy,pandas

2

style_dict={'font-family':'times','font-size':'120%',

3

            'text-shadow':'#aaa 2px 2px 2px'}

4

head_styler=(('color','slategray'),('font-size','200%'),

5

             ('text-shadow','#aaa 3px 3px 3px'))

6

index=[chr(i) for i in range(ord('a'),ord('h')+1)]; 

7

columns=['♔','♕','♖',

8

         '♗','♘','♙']

9

df=pandas.DataFrame(numpy.random.randn(8,6),

10

                    index=index,columns=columns)

11

df.style.set_properties(**style_dict)\

12

.background_gradient(cmap='Pastel1')\

13

.set_table_styles([dict(selector='th',props=head_styler)])

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---

Messages

Surface Plotting

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1

var('u,v')

2

x=(2+cos(v/2)*sin(u)-sin(v/2)*sin(4*u))*cos(v)

3

y=(2+cos(v/2)*sin(u)-sin(v/2)*sin(4*u))*sin(v)

4

z=30*abs(sin(v/2)*sin(u)+cos(v/2)*sin(4*u))

5

sum([parametric_plot3d(

6

    [2*i*x,2*i*y,z],(u,0,2*pi),(v,0,2*pi),

7

    opacity=.7,mesh=True,frame=False,color='#ff3636') 

8

     for i in [1..3]])

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---

Messages

Interactive Scatter Plotting 2

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1

@interact

2

def _(marker=selector(['alpha','beta','gamma',

3

                       'delta','epsilon']),

4

      color=['red','orange','yellow','green',

5

             'cyan','blue','purple']):

6

    m=r'$\%s$'%marker

7

    p=plot(cos(x),(x,0,10),plot_points=30,

8

           linestyle='',color=color,marker=m)

9

    p.show(figsize=(5,3),gridlines=True)

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---

Messages

Interactive Scatter Plotting

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1

@interact

2

def _(N=(1,7,1)):

3

    p=sum([point([cos(pi*i/100)^3+RR(j),sin(pi*i/100)],

4

                 marker='d',color=colormaps.hsv(i+10*j)[:3])

5

           for i in range(200) for j in [0..N-1]])

6

    p.show(xmin=-1.1,xmax=.1+N,ymin=-1,ymax=1,

7

           gridlines=True,figsize=(5,5))

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---

Messages

Interactive Polar Plotting

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1

@interact

2

def _(n=[36,33,..6],m=[7,6,..,2]):

3

    f(t,i)=(1+sin(n*t))*cos(n*t)+i; g(t,i)=.5*cos(n*t)+i

4

    p=sum([polar_plot(f(t,j),0,2*pi,color=hue(n/36),

5

                      fill=g(t,j),fillcolor=hue(n/36))

6

           for j in [1..m]])

7

    p.show(axes=False,figsize=5)

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---

Messages

Examples of Simple Highcharts 2

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1

%%javascript

2

Highcharts.chart('container',{

3

    xAxis:{categories:['Jan','Feb','Mar','Apr','May','Jun',

4

                       'Jul','Aug','Sep','Oct','Nov','Dec']},

5

    yAxis:{max:150,title:{text:'Temperature'}},

6

    title:{style:{'color':'#aa36aa','fontSize':'24px'},

7

           text:'Global Station Temperature'},

8

    series:[{name:'2017',color:'#aa3636',

9

             data:[113,141,137,115,122,76,108,110,96,115,105,115]},

10

            {name:'2018',color:'#3636aa',

11

             data:[103,114,121,120,100,86,106,90,89,111,97,121]}]

12

},function(chart){

13

    var point=chart.series.data,

14

    text=chart.renderer.text(point.plotX+chart.plotLeft+10,

15

                             point.plotY+chart.plotTop-10).add(),

16

    box=text.getBBox();});

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---

Messages

<script src='https://code.highcharts.com/highcharts.js'> </script> was added in the page head.

highcharts.js is launching here...

Regression Examples

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1

print(r.eval("""

2

svg(filename='Rplots.svg',width=6,height=5,onefile=TRUE,

3

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

4

library('MASS'); library('nnet')

5

data(Boston); n<-dim(Boston)[1]

6

model<-nnet(as.matrix(Boston[1:430,-14]),

7

            as.matrix(Boston[1:430,14]),

8

            size=65,trace=F,maxit=2*10^3,linout=T,decay=.1^3)

9

predictions<-predict(model,as.matrix(Boston[431:n,-14]),type='raw')

10

plot(as.matrix(Boston[431:n,14]),col='#3636ff',type='o',

11

     ylab='targets & predictions',yaxt='n'); par(new=TRUE)

12

plot(predictions,col='#ff3636',type='o',cex=1.3,

13

     ylab='targets & predictions',ylim=c(0,60)) 

14

legend(10,55,legend=c('targets','predictions'),

15

       col=c('#3636ff','#ff3636'),lty=c(1,1)); dev.off()

16

"""))

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---

Messages

Tachyon 3D Ray Tracers 2

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1

t=Tachyon(camera_center=(-5,-2,2),look_at=(1,0,0),

2

          xres=500,yres=500,)

3

t.texture('mirror',ambient=.01,diffuse=.01,

4

          specular=.9,opacity=.9,color=(.2,.2,.2))

5

t.texture('t1',color=(.7,0,0),texfunc=1)

6

t.texture('t2',color=(0,.2,.8),texfunc=5)

7

t.plane((0,0,0),(0,0,1),'t1')

8

t.plane((0,0,3),(0,0,4),'t2')

9

t.fcylinder((-1,1,1),(2,-2,1),.3,'mirror')

10

t.sphere((.5,-.5,1),.8,'mirror')

11

t.sphere((-1,1,1),.5,'mirror')

12

t.sphere((2,-2,1),.5,'mirror')

13

t.light((-6,-3,2),1,(2,2,1)); show(t)

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---

Messages

Simple Plotly Charts

<script src='https://cdn.plot.ly/plotly-latest.min.js'></script> was added in the page head.

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1

%%html

2

<div id='test_plotly' style='width:600px; height:600px;'/><script>

3

function plotlyChart(embed,n,m,q) {doc=document.getElementById(embed);

4

    var s1='ϱ = cos  '.concat(n).concat(' θ'),

5

        s2='ϱ = 1 + cos '.concat(n).concat(' θ');

6

    var x1=Array(m).fill(0).map((r,t)=>

7

        (r+Math.cos(n*q*t))*Math.cos(q*t)),

8

        y1=Array(m).fill(0).map((r,t)=>

9

        (r+Math.cos(n*q*t))*Math.sin(q*t));

10

    var x2=Array(m).fill(1).map((r,t)=>

11

        (r+Math.cos(n*q*t))*Math.cos(q*t)),

12

        y2=Array(m).fill(1).map((r,t)=>

13

        (r+Math.cos(n*q*t))*Math.sin(q*t));

14

    Plotly.plot(doc,[{x:x1,y:y1,line:{color:'#3636ff'},name:s1}]);

15

    Plotly.plot(doc,[{x:x2,y:y2,line:{color:'#ff3636'},name:s2}]); };

16

plotlyChart('test_plotly',24,640,.01);

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---

Messages

Examples of Simple GoogleCharts

<script src='https://www.gstatic.com/charts/loader.js'></script> was added in the page head.

xxxxxxxxxx

 

1

%%html

2

<div id='googlechart' style='width:530px; height:580px;'/><script>

3

google.charts.load('current',{'packages':['corechart']})

4

google.charts.setOnLoadCallback(drawChart);

5

var doc=document.getElementById('googlechart');

6

var n=24,q=.01,m=638; var s2='ϱ = 1 + cos '.concat(n).concat(' θ');

7

var s1='ϱ = cos  '.concat(n).concat(' θ');

8

function drawChart() {

9

    var xy=Array(2*m).fill(0).map((r,t)=>

10

        [(Math.floor(t/m)+r+Math.cos(n*q*t))*Math.cos(q*t),

11

         (r+Math.cos(n*q*t))*Math.sin(q*t),

12

         (r+Math.floor(t/m)+Math.cos(n*q*t))*Math.sin(q*t)]);

13

    for (var i=1; i<m-1; i++) {xy[m-1+i][1]=NaN; xy[i+1][2]=NaN;}

14

    xy.unshift(['x',s1,s2]); 

15

    var data=google.visualization.arrayToDataTable(xy),

16

        options={curveType:'function',legend:{position:'bottom'}};

17

    var chart=new google.visualization.LineChart(doc); 

18

    chart.draw(data,options); };

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---

Messages

The Python Variant

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1

import numpy,pylab; n,m=24,600

2

T=numpy.linspace(start=0,stop=2*numpy.pi,num=m)

3

XY=numpy.array([[(numpy.cos(n*T)+r)*numpy.cos(T),

4

                 (numpy.cos(n*T)+r)*numpy.sin(T)] for r in range(3)])

5

XYP=numpy.array([[T,abs(r+numpy.cos(n*T))] for r in range(3)])

6

fig=pylab.figure(figsize=(5,10)); ax1=fig.add_subplot(211)

7

ax2=fig.add_subplot(212,projection='polar')

8

colors=['#ff36ff','#ff3636','#3636ff']

9

labels=[u'ϱ = cos %sθ'%n,u'ϱ = 1 + cos %sθ'%n,u'ϱ = 2 + cos %dθ'%n]

10

[ax1.plot(XY[i][0],XY[i][1],color=colors[i],label=labels[i]) 

11

 for i in range(3)]

12

[ax2.plot(XYP[i][0],XYP[i][1],color=colors[i],label=labels[i]) 

13

 for i in range(3)]

14

lines,labels=ax1.get_legend_handles_labels() 

15

ax1.legend(lines,labels,bbox_to_anchor=(1,1.15),loc=5) 

16

ax1.grid(); ax1.set_xlim(-4,4); ax1.set_ylim(-4,4); 

17

ax1.set_xlabel('x'); ax1.set_ylabel('y'); pylab.show()

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Messages

Parametric Plots with Simple Basic Libraries

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print(r.eval("""

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

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    pointsize=12,onefile=TRUE,family='sans',bg='slategray',

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

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t<-seq(0,2*pi,len=360); n=12; d=.4

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x1<-cos(n*t)*cos(t); y1<-cos(n*t)*sin(t) 

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x2<-(cos(n*t)+d)*cos(t); y2<-(cos(n*t)+d)*sin(t)

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plot(x2,y2,type='o',cex=.8,col='#3636ff',

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     xlab='',ylab=''); par(new=T); 

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plot(x1,y1,type='o',cex=.8,col='#ff3636',

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     xlim=c(-1-d,1+d),ylim=c(-1-d,1+d),xlab='x',ylab='y',

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     fg='white',col.axis='white',col.lab='white'); 

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

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"""))

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

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Examples of Simple Highcharts

<script src='https://code.highcharts.com/highcharts.js'></script> was added in the page head.

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%%javascript

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var n=24,q=.01; 

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var s2='ϱ = 1 + cos '.concat(n).concat(' θ'),

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    s1='ϱ = cos  '.concat(n).concat(' θ');

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var xy1=Array(640).fill(0).map((r,t)=>

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    [(r+Math.cos(q*t))*Math.cos(n*q*t),

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     (r+Math.cos(n*q*t))*Math.sin(q*t)]);

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var xy2=Array(640).fill(1).map((r,t)=>

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    [(r+Math.cos(n*q*t))*Math.cos(q*t),

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     (r+Math.cos(n*q*t))*Math.sin(q*t)]);

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Highcharts.chart('container1',{

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    chart:{type:'area'},xAxis:{title:{text:'x'}},

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    yAxis:{title:{text:'y'}},

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    title:{text:'Parametric Area Plot'},credits:{enabled:false},

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    series: [{name:s2,color:'#3636ff',data:xy2},

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             {name:s1,color:'#ff3636',data:xy1}]});

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

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Messages

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