Sage Interactions

Post code that demonstrates the use of the interact command in Sage here. It should be easy for people to just scroll through and paste examples out of here into their own sage notebooks.

We'll likely restructure and reorganize this once we have some nontrivial content and get a sense of how it is laid out. If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected].

TableOfContents

Miscellaneous

Evaluate a bit of code in a given system

by William Stein (there is no way yet to make the text box big):

@interact
def _(system=selector([('sage0', 'Sage'), ('gp', 'PARI'), ('magma', 'Magma')]), code='2+2'):
    print globals()[system].eval(code)

attachment:evalsys.png

Graph Theory

Automorphism Groups of some Graphs

by William Stein (I spent less than five minutes on this):

@interact
def _(graph=['CycleGraph', 'CubeGraph', 'RandomGNP'],
      n=selector([1..10],nrows=1), p=selector([10,20,..,100],nrows=1)):
    print graph
    if graph == 'CycleGraph':
       print "n (=%s): number of vertices"%n
       G = graphs.CycleGraph(n)
    elif graph == 'CubeGraph':
       if n > 8:
           print "n reduced to 8"
           n = 8
       print "n (=%s): dimension"%n
       G = graphs.CubeGraph(n)
    elif graph == 'RandomGNP':
       print "n (=%s) vertices"%n
       print "p (=%s%%) probability"%p
       G = graphs.RandomGNP(n, p/100.0)

    print G.automorphism_group()
    show(plot(G))

attachment:autograph.png

Calculus

A contour map and 3d plot of two inverse distance functions

by William Stein

@interact
def _(q1=(-1,(-3,3)), q2=(-2,(-3,3)), 
      cmap=['autumn', 'bone', 'cool', 'copper', 'gray', 'hot', 'hsv', 
           'jet', 'pink', 'prism', 'spring', 'summer', 'winter']):
     x,y = var('x,y')
     f = q1/sqrt((x+1)^2 + y^2) + q2/sqrt((x-1)^2+(y+0.5)^2)
     C = contour_plot(f, (-2,2), (-2,2), plot_points=30, contours=15, cmap=cmap)
     show(C, figsize=3, aspect_ratio=1)
     show(plot3d(f, (x,-2,2), (y,-2,2)), figsize=5, viewer='tachyon')     

attachment:mountains.png

A simple tangent line grapher

by Marshall Hampton

html('<h2>Tangent line grapher</h2>')
@interact
def tangent_line(f = input_box(default=sin(x)), xbegin = slider(0,10,1/10,0), xend = slider(0,10,1/10,10), x0 = slider(0, 1, 1/100, 1/2)):
    prange = [xbegin, xend]
    x0i = xbegin + x0*(xend-xbegin)
    var('x')
    df = diff(f)
    tanf = f(x0i) + df(x0i)*(x-x0i)
    fplot = plot(f, prange[0], prange[1])
    print 'Tangent line is y = ' + tanf._repr_()
    tanplot = plot(tanf, prange[0], prange[1], rgbcolor = (1,0,0))
    fmax = f.find_maximum_on_interval(prange[0], prange[1])[0]
    fmin = f.find_minimum_on_interval(prange[0], prange[1])[0]
    show(fplot + tanplot, xmin = prange[0], xmax = prange[1], ymax = fmax, ymin = fmin)

attachment:tangents.png

Function tool

Enter symbolic functions f, g, and a, a range, then click the appropriate button to compute and plot some combination of f, g, and a along with f and g. This is inspired by the Matlab funtool GUI.

x = var('x')

@interact
def _(f=sin(x), g=cos(x), xrange=input_box((0,1)), yrange='auto', a=1,
      action=selector(['f', 'df/dx', 'int f', 'num f', 'den f', '1/f', 'finv',
                       'f+a', 'f-a', 'f*a', 'f/a', 'f^a', 'f(x+a)', 'f(x*a)', 
                       'f+g', 'f-g', 'f*g', 'f/g', 'f(g)'], 
             width=15, nrows=5, label="h = "),
      do_plot = ("Draw Plots", True)):

    try:
        f = SR(f); g = SR(g); a = SR(a)
    except TypeError, msg:
        print msg[-200:]
        print "Unable to make sense of f,g, or a as symbolic expressions."
        return 
    if not (isinstance(xrange, tuple) and len(xrange) == 2):
          xrange = (0,1)
    h = 0; lbl = ''
    if action == 'f':
        h = f
        lbl = 'f'
    elif action == 'df/dx':
        h = f.derivative(x)
        lbl = '\\frac{df}{dx}'
    elif action == 'int f':
        h = f.integrate(x)
        lbl = '\\int f dx'
    elif action == 'num f':
        h = f.numerator()
        lbl = '\\text{numer(f)}'
    elif action == 'den f':
        h = f.denominator()
        lbl = '\\text{denom(f)}'
    elif action == '1/f':
        h = 1/f
        lbl = '\\frac{1}{f}'
    elif action == 'finv':
        h = solve(f == var('y'), x)[0].rhs()
        lbl = 'f^{-1}(y)'
    elif action == 'f+a':
        h = f+a
        lbl = 'f + a'
    elif action == 'f-a':
        h = f-a
        lbl = 'f - a'
    elif action == 'f*a':
        h = f*a
        lbl = 'f \\times a'
    elif action == 'f/a':
        h = f/a
        lbl = '\\frac{f}{a}'
    elif action == 'f^a':
        h = f^a
        lbl = 'f^a'
    elif action == 'f^a':
        h = f^a
        lbl = 'f^a'
    elif action == 'f(x+a)':
        h = f(x+a)
        lbl = 'f(x+a)'
    elif action == 'f(x*a)':
        h = f(x*a)
        lbl = 'f(xa)'
    elif action == 'f+g':
        h = f+g
        lbl = 'f + g'
    elif action == 'f-g':
        h = f-g
        lbl = 'f - g'
    elif action == 'f*g':
        h = f*g
        lbl = 'f \\times g'
    elif action == 'f/g':
        h = f/g
        lbl = '\\frac{f}{g}'
    elif action == 'f(g)':
        h = f(g)
        lbl = 'f(g)'
      
      
    html('<center><font color="red">$f = %s$</font></center>'%latex(f))
    html('<center><font color="green">$g = %s$</font></center>'%latex(g))
    html('<center><font color="blue"><b>$h = %s = %s$</b></font></center>'%(lbl, latex(h)))
    if do_plot:
        P = plot(f, xrange, color='red', thickness=2) +  \
            plot(g, xrange, color='green', thickness=2) + \
            plot(h, xrange, color='blue', thickness=2)
        if yrange == 'auto':
            show(P, xmin=xrange[0], xmax=xrange[1])
        else:
            yrange = sage_eval(yrange)
            show(P, xmin=xrange[0], xmax=xrange[1], ymin=yrange[0], ymax=yrange[1])

attachment:funtool.png

Newton-Raphson Root Finding

by Neal Holtz

This allows user to display the Newton-Raphson procedure one step at a time. It uses the heuristic that, if any of the values of the controls change, then the procedure should be re-started, else it should be continued.

# ideas from 'A simple tangent line grapher' by Marshall Hampton
# http://wiki.sagemath.org/interact

State = Data = None   # globals to allow incremental additions to graphics

@interact
def newtraph(f = input_box(default=8*sin(x)*exp(-x)-1, label='f(x)'), 
             xmin = input_box(default=0), 
             xmax = input_box(default=4*pi), 
             x0 = input_box(default=3, label='x0'),
             show_calcs = ("Show Calcs",True),
             step = ['Next','Reset'] ):
    global State, Data
    prange = [xmin,xmax]
    state = [f,xmin,xmax,x0,show_calcs]
    if (state != State) or (step == 'Reset'):   # when any of the controls change
        X = [RR(x0)]                     # restart the plot
        df = diff(f)
        Fplot = plot(f, prange[0], prange[1])
        Data = [X, df, Fplot]
        State = state

    X, df, Fplot = Data
    i = len(X) - 1              # compute and append the next x value
    xi = X[i]
    fi = RR(f(xi))
    fpi = RR(df(xi))
    xip1 = xi - fi/fpi
    X.append(xip1)

    msg = xip1s = None          # now check x value for reasonableness
    is_inf = False
    if abs(xip1) > 10E6*(xmax-xmin):
        is_inf = True
        show_calcs = True
        msg = 'Derivative is 0!'
        xip1s = latex(xip1.sign()*infinity)
        X.pop()
    elif not ((xmin - 0.5*(xmax-xmin)) <= xip1 <= (xmax + 0.5*(xmax-xmin))):
        show_calcs = True
        msg = 'x value out of range; probable divergence!'
    if xip1s is None:
        xip1s = '%.4g' % (xip1,)

    def Disp( s, color="blue" ):
        if show_calcs:
            html( """<font color="%s">$ %s $</font>""" % (color,s,) )
    Disp( """f(x) = %s""" % (latex(f),)  +
          """~~~~f'(x) = %s""" % (latex(df),) )
    Disp( """i = %d""" % (i,)  +
          """~~~~x_{%d} = %.4g""" % (i,xi)  +
          """~~~~f(x_{%d}) = %.4g""" % (i,fi)  +
          """~~~~f'(x_{%d}) = %.4g""" % (i,fpi) )
    if msg:
        html( """<font color="red"><b>%s</b></font>""" % (msg,) )
        c = "red"
    else:
        c = "blue"
    Disp( r"""x_{%d} = %.4g - ({%.4g})/({%.4g}) = %s""" % (i+1,xi,fi,fpi,xip1s), color=c )

    Fplot += line( [(xi,0),(xi,fi)], linestyle=':', rgbcolor=(1,0,0) ) # vert dotted line
    Fplot += points( [(xi,0),(xi,fi)], rgbcolor=(1,0,0) )
    labi = text( '\nx%d\n' % (i,), (xi,0), rgbcolor=(1,0,0),
                 vertical_alignment="bottom" if fi < 0 else "top" )
    if is_inf:
        xl = xi - 0.05*(xmax-xmin)
        xr = xi + 0.05*(xmax-xmin)
        yl = yr = fi
    else:
        xl = min(xi,xip1) - 0.02*(xmax-xmin)
        xr = max(xi,xip1) + 0.02*(xmax-xmin)
        yl = -(xip1-xl)*fpi
        yr = (xr-xip1)*fpi
        Fplot += points( [(xip1,0)], rgbcolor=(0,0,1) )       # new x value
        labi += text( '\nx%d\n' % (i+1,), (xip1,0), rgbcolor=(1,0,0),
                 vertical_alignment="bottom" if fi < 0 else "top" )
    Fplot += line( [(xl,yl),(xr,yr)], rgbcolor=(1,0,0) )  # tangent

    show( Fplot+labi, xmin = prange[0], xmax = prange[1] )
    Data = [X, df, Fplot]

attachment:newtraph.png

Differential Equations

Euler's Method in one variable

by Marshall Hampton. This needs some polishing but its usable as is.

def tab_list(y, headers = None):
    '''
    Converts a list into an html table with borders.
    '''
    s = '<table border = 1>'
    if headers:
        for q in headers:
            s = s + '<th>' + str(q) + '</th>'
    for x in y:
        s = s + '<tr>'
        for q in x:
            s = s + '<td>' + str(q) + '</td>'
        s = s + '</tr>'
    s = s + '</table>'
    return s
var('x y')
@interact
def euler_method(y_exact_in = input_box('-cos(x)+1.0', type = str, label = 'Exact solution = '), y_prime_in = input_box('sin(x)', type = str, label = "y' = "), start = input_box(0.0, label = 'x starting value: '), stop = input_box(6.0, label = 'x stopping value: '), startval = input_box(0.0, label = 'y starting value: '), nsteps = slider([2^m for m in range(0,10)], default = 10, label = 'Number of steps: '), show_steps = slider([2^m for m in range(0,10)], default = 8, label = 'Number of steps shown in table: ')):
    y_exact = lambda x: eval(y_exact_in)
    y_prime = lambda x,y: eval(y_prime_in)
    stepsize = float((stop-start)/nsteps)
    steps_shown = max(nsteps,show_steps)
    sol = [startval]
    xvals = [start]
    for step in range(nsteps):
        sol.append(sol[-1] + stepsize*y_prime(xvals[-1],sol[-1]))
        xvals.append(xvals[-1] + stepsize)
    sol_max = max(sol + [find_maximum_on_interval(y_exact,start,stop)[0]])
    sol_min = min(sol + [find_minimum_on_interval(y_exact,start,stop)[0]])
    show(plot(y_exact(x),start,stop,rgbcolor=(1,0,0))+line([[xvals[index],sol[index]] for index in range(len(sol))]),xmin=start,xmax = stop, ymax = sol_max, ymin = sol_min)    
    if nsteps < steps_shown:
        table_range = range(len(sol))
    else:
        table_range = range(0,floor(steps_shown/2)) + range(len(sol)-floor(steps_shown/2),len(sol))
    html(tab_list([[i,xvals[i],sol[i]] for i in table_range], headers = ['step','x','y']))

attachment:eulermethod.png

Linear Algebra

Numerical instability of the classical Gram-Schmidt algorithm

by Marshall Hampton (tested by William Stein, who thinks this is really nice!)

def GS_classic(a_list):
    '''
    Given a list of vectors or a matrix, returns the QR factorization using the classical (and numerically unstable) Gram-Schmidt algorithm.    
    '''
    if type(a_list) != list:
        cols = a_list.cols()
        a_list = [x for x in cols]
    indices = range(len(a_list))
    q = []
    r = [[0 for i in indices] for j in indices]
    v = [a_list[i].copy() for i in indices]
    for i in indices:
        for j in range(0,i):
            r[j][i] = q[j].inner_product(a_list[i])
            v[i] = v[i] - r[j][i]*q[j]
        r[i][i] = (v[i]*v[i])^(1/2)
        q.append(v[i]/r[i][i])
    q = matrix([q[i] for i in indices]).transpose()
    return q, matrix(r)
def GS_modern(a_list):
    '''
    Given a list of vectors or a matrix, returns the QR factorization using the 'modern' Gram-Schmidt algorithm.    
    '''
    if type(a_list) != list:
        cols = a_list.cols()
        a_list = [x for x in cols]
    indices = range(len(a_list))
    q = []
    r = [[0 for i in indices] for j in indices]
    v = [a_list[i].copy() for i in indices]
    for i in indices:
        r[i][i] = v[i].norm(2)
        q.append(v[i]/r[i][i])
        for j in range(i+1, len(indices)):
            r[i][j] = q[i].inner_product(v[j])
            v[j] = v[j] - r[i][j]*q[i]
    q = matrix([q[i] for i in indices]).transpose()
    return q, matrix(r)
html('<h2>Numerical instability of the classical Gram-Schmidt algorithm</h2>')
@interact
def gstest(precision = slider(range(3,53), default = 10), a1 = input_box([1,1/1000,1/1000]), a2 = input_box([1,1/1000,0]), a3 = input_box([1,0,1/1000])):
    myR = RealField(precision)
    displayR = RealField(5)
    html('precision in bits: ' + str(precision) + '<br>')
    A = matrix([a1,a2,a3])
    A = [vector(myR,x) for x in A]
    qn, rn = GS_classic(A)
    qb, rb = GS_modern(A)
    html('Classical Gram-Schmidt:')
    show(matrix(displayR,qn))
    html('Stable Gram-Schmidt:')
    show(matrix(displayR,qb))

attachment:GramSchmidt.png

Linear transformations

by Jason Grout

A square matrix defines a linear transformation which rotates and/or scales vectors. In the interact command below, the red vector represents the original vector (v) and the blue vector represents the image w under the linear transformation. You can change the angle and length of v by changing theta and r.

@interact
def linear_transformation(theta=slider(0, 2*pi, .1), r=slider(0.1, 2, .1, default=1)):
    A=matrix([[1,-1],[-1,1/2]])
    v=vector([r*cos(theta), r*sin(theta)])
    w = A*v
    circles = sum([circle((0,0), radius=i, rgbcolor=(0,0,0)) for i in [1..2]])
    print jsmath("v = %s,\; %s v=%s"%(v.n(4),latex(A),w.n(4)))
    show(v.plot(rgbcolor=(1,0,0))+w.plot(rgbcolor=(0,0,1))+circles,aspect_ratio=1)

attachment:Linear-Transformations.png

Singular value decomposition

by Marshall Hampton

import scipy.linalg as lin
var('t')
def rotell(sig,umat,t,offset=0):
    temp = matrix(umat)*matrix(2,1,[sig[0]*cos(t),sig[1]*sin(t)])
    return [offset+temp[0][0],temp[1][0]]
@interact
def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= selector(['Off','On'],label='offset image from domain')):
    rf_low = RealField(12)
    my_mat = matrix(rf_low,2,2,[a11,a12,a21,a22])
    u,s,vh = lin.svd(my_mat.numpy())
    if ofs == 'On': 
        offset = 3
        fsize = 6
        colors = [(1,0,0),(0,0,1),(1,0,0),(0,0,1)]
    else: 
        offset = 0
        fsize = 5
        colors = [(1,0,0),(0,0,1),(.7,.2,0),(0,.3,.7)]
    vvects = sum([arrow([0,0],matrix(vh).row(i),rgbcolor = colors[i]) for i in (0,1)])    
    uvects = Graphics()
    for i in (0,1):
        if s[i] != 0: uvects += arrow([offset,0],vector([offset,0])+matrix(s*u).column(i),rgbcolor = colors[i+2])
    html('<h3>Singular value decomposition: image of the unit circle and the singular vectors</h3>')
    print jsmath("A = %s  = %s %s %s"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist())))) 
    image_ell = parametric_plot(rotell(s,u,t, offset),0,2*pi)
    graph_stuff=circle((0,0),1)+image_ell+vvects+uvects
    graph_stuff.set_aspect_ratio(1)
    show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize])

attachment:svd1.png

Algebra

Groebner fan of an ideal

by Marshall Hampton

@interact
def gfan_browse(p1 = input_box('x^3+y^2',type = str, label='polynomial 1: '), p2 = input_box('y^3+z^2',type = str, label='polynomial 2: '), p3 = input_box('z^3+x^2',type = str, label='polynomial 3: ')):
    R.<x,y,z> = PolynomialRing(QQ,3)
    i1 = ideal(R(p1),R(p2),R(p3))
    gf1 = i1.groebner_fan()
    testr = gf1.render()    
    html('Groebner fan of the ideal generated by: ' + str(p1) + ', ' + str(p2) + ', ' + str(p3))
    show(testr, axes = False, figsize=[8,8*(3^(.5))/2])

attachment:gfan_interact.png

Number Theory

Continued Fraction Plotter

by William Stein

@interact
def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]):
    c = list(continued_fraction(RealField(prec)(number))); print c
    show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2])

attachment:contfracplot.png

Illustrating the prime number thoerem

by William Stein

@interact
def _(N=(100,(2..2000))):
    html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N)
    show(plot(prime_pi, 0, N, rgbcolor='red') + plot(x/(log(x)-1), 5, N, rgbcolor='blue'))

attachment:primes.png

Computing Generalized Bernoulli Numbers

by William Stein (Sage-2.10.3)

@interact
def _(m=selector([1..15],nrows=2), n=(7,(3..10))):
    G = DirichletGroup(m)
    s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,m)
    s += '<table border=1>'
    s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \
           ''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>'
    for eps in G.list():
        v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]])
        s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%(
             eps, eps.conductor(), v)
    s += '</table>'
    html(s)

attachment:bernoulli.png

Fundamental Domains of SL_2(ZZ)

by Robert Miller

L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000, -1, -1)]
R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000)]
xes = [x/1000.0 for x in xrange(-500,501,1)]
M = [[x,abs(sqrt(x^2-1))] for x in xes]
fundamental_domain = L+M+R
fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
@interact
def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)):
    global fundamental_domain
    if gen == 't+1':
        fundamental_domain = [[x+1,y] for x,y in fundamental_domain]
    elif gen == 't-1':
        fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
    elif gen == '-1/t':
        new_dom = []
        for x,y in fundamental_domain:
            sq_mod = x^2 + y^2
            new_dom.append([(-1)*x/sq_mod, y/sq_mod])
        fundamental_domain = new_dom
    P = polygon(fundamental_domain)
    P.ymax(1.2); P.ymin(-0.1)
    P.show()

attachment:fund_domain.png

Computing modular forms

by William Stein

j = 0
@interact
def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40), 
      group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]):
    M = CuspForms(group(N),k)
    print j; global j; j += 1
    print M; print '\n'*3
    print "Computing basis...\n\n"
    if M.dimension() == 0:
         print "Space has dimension 0"
    else:
        prec = max(prec, M.dimension()+1)
        for f in M.basis():
             view(f.q_expansion(prec))
    print "\n\n\nDone computing basis."

attachment:modformbasis.png

Computing the cuspidal subgroup

by William Stein

html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')
@interact
def _(N=selector([1..8*13], ncols=8, width=10, default=10)):
    A = J0(N)
    print A.cuspidal_subgroup()

attachment:cuspgroup.png

A Charpoly and Hecke Operator Graph

by William Stein

# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots
@interact
def f(N = prime_range(11,400),
      p = selector(prime_range(2,12),nrows=1),
      three_d = ("Three Dimensional", False)):
    S = SupersingularModule(N)
    T = S.hecke_matrix(p)
    G = Graph(T, multiedges=True, loops=not three_d)
    html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p))
    show(T.charpoly().factor())
    if three_d:
        show(G.plot3d(), aspect_ratio=[1,1,1])
    else:
        show(G.plot(),figsize=7)

attachment:heckegraph.png

Demonstrating the Diffie-Hellman Key Exchange Protocol

by Timothy Clemans (refereed by William Stein)

@interact
def diffie_hellman(button=selector(["New example"],label='',buttons=True), 
    bits=("Number of bits of prime", (8,12,..512))):
    maxp = 2^bits
    p = random_prime(maxp)
    k = GF(p)
    g = k.multiplicative_generator()
    a = ZZ.random_element(10, maxp)
    b = ZZ.random_element(10, maxp)

    print """
<html>
<style>
.gamodp {
background:yellow
}
.gbmodp {
background:orange
}
.dhsame {
color:green;
font-weight:bold
}
</style>
<h2>%s-Bit Diffie-Hellman Key Exchange</h2>
<ol style="color:#000;font:12px Arial, Helvetica, sans-serif">
<li>Alice and Bob agree to use the prime number p=%s and base g=%s.</li>
<li>Alice chooses the secret integer a=%s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li>
<li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li>
<li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
<li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
</ol></html>
    """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p, 
       (g^ b)^a, g^a, b, p, (g^a)^b)

attachment:dh.png

Plotting an elliptic curve over a finite field

E = EllipticCurve('37a')
@interact
def _(p=slider(prime_range(1000), default=389)):
    show(E)
    print "p = %s"%p
    show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)

attachment:ellffplot.png

Bioinformatics

Web app: protein browser

by Marshall Hampton (tested by William Stein)

import urllib2 as U
@interact
def protein_browser(GenBank_ID = input_box('165940577', type = str), file_type = selector([(1,'fasta'),(2,'GenPept')])):
    if file_type == 2:
        gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&id='
    else:
        gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&dopt=fasta&id='
    f = U.urlopen(gen_str + GenBank_ID)        
    g = f.read()
    f.close()
    html(g)

attachment:biobrowse.png

Coalescent simulator

by Marshall Hampton

def next_gen(x, selection=1.0):
    '''Creates the next generation from the previous; also returns parent-child indexing list'''
    next_x = []
    for ind in range(len(x)):
        if random() < (1 + selection)/len(x):
            rind = 0
        else:
            rind = int(round(random()*(len(x)-1)+1/2))
        next_x.append((x[rind],rind))
    next_x.sort()
    return [[x[0] for x in next_x],[x[1] for x in next_x]]
def coal_plot(some_data):
    '''Creates a graphics object from coalescent data'''
    gens = some_data[0]
    inds = some_data[1]
    gen_lines = line([[0,0]])
    pts = Graphics()
    ngens = len(gens)
    gen_size = len(gens[0])
    for x in range(gen_size):
        pts += point((x,ngens-1), hue = gens[0][x]/float(gen_size*1.1))
    p_frame = line([[-.5,-.5],[-.5,ngens-.5], [gen_size-.5,ngens-.5], [gen_size-.5,-.5], [-.5,-.5]])
    for g in range(1,ngens):
        for x in range(gen_size):
            old_x = inds[g-1][x]
            gen_lines += line([[x,ngens-g-1],[old_x,ngens-g]], hue = gens[g-1][old_x]/float(gen_size*1.1))
            pts += point((x,ngens-g-1), hue = gens[g][x]/float(gen_size*1.1))
    return pts+gen_lines+p_frame
d_field = RealField(10)
@interact
def coalescents(pop_size = slider(2,100,1,15,'Population size'), selection = slider(-1,1,.1,0, 'Selection for first taxon'), s = selector(['Again!'], label='Refresh', buttons=True)):
    print 'Population size: ' + str(pop_size)
    print 'Selection coefficient for first taxon: ' + str(d_field(selection))
    start = [i for i in range(pop_size)]
    gens = [start]
    inds = []
    while gens[-1][0] != gens[-1][-1]:
        g_index = len(gens) - 1
        n_gen = next_gen(gens[g_index], selection = selection)
        gens.append(n_gen[0])
        inds.append(n_gen[1])
        coal_data1 = [gens,inds]
    print 'Generations until coalescence: ' + str(len(gens))
    show(coal_plot(coal_data1), axes = False, figsize = [8,4.0*len(gens)/pop_size], ymax = len(gens)-1)

attachment:coalescent.png

Miscellaneous Graphics

Catalog of 3D Parametric Plots

var('u,v')
plots = ['Two Interlinked Tori', 'Star of David', 'Double Heart',
         'Heart', 'Green bowtie', "Boy's Surface", "Maeder's Owl",
         'Cross cap']
plots.sort()

@interact
def _(example=selector(plots, buttons=True, nrows=2),
      tachyon=("Raytrace", False), frame = ('Frame', False),
      opacity=(1,(0.1,1))):
    url = ''
    if example == 'Two Interlinked Tori':
        f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*cos(u), 4+sin(v))
        f2 = (8+(3+cos(v))*cos(u), 3+sin(v), 4+(3+cos(v))*sin(u))
        p1 = parametric_plot3d(f1, (u,0,2*pi), (v,0,2*pi), color="red", opacity=opacity)
        p2 = parametric_plot3d(f2, (u,0,2*pi), (v,0,2*pi), color="blue",opacity=opacity)
        P = p1 + p2
    elif example == 'Star of David':
        f_x = cos(u)*cos(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        f_y = cos(u)*sin(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        f_z = sin(u)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, 0, 2*pi),opacity=opacity)
    elif example == 'Double Heart':
        f_x = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) + abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*sin(v)
        f_y = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) - abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*cos(v)
        f_z = sin(u)*(abs(cos(4*u/4))^1 + abs(sin(4*u/4))^1)^(-1/1)
        P = parametric_plot3d([f_x, f_y, f_z], (u, 0, pi), (v, -pi, pi),opacity=opacity)
    elif example == 'Heart':
        f_x = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
        f_y = sin(u) *(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
        f_z = v
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -1, 1), frame=False, color="red",opacity=opacity)
    elif example == 'Green bowtie':
        f_x = sin(u) / (sqrt(2) + sin(v))
        f_y = sin(u) / (sqrt(2) + cos(v))
        f_z = cos(u) / (1 + sqrt(2))
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -pi, pi), frame=False, color="green",opacity=opacity)
    elif example == "Boy's Surface":
        url = "http://en.wikipedia.org/wiki/Boy's_surface"
        fx = 2/3* (cos(u)* cos(2*v) + sqrt(2)* sin(u)* cos(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        fy = 2/3* (cos(u)* sin(2*v) - sqrt(2)* sin(u)* sin(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        fz = sqrt(2)* cos(u)* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, pi), plot_points = [90,90], frame=False, color="orange",opacity=opacity) 
    elif example == "Maeder's Owl":
        fx = v *cos(u) - 0.5* v^2 * cos(2* u)
        fy = -v *sin(u) - 0.5* v^2 * sin(2* u)
        fz = 4 *v^1.5 * cos(3 *u / 2) / 3
        P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, 1),plot_points = [90,90], frame=False, color="purple",opacity=opacity)
    elif example =='Cross cap':
        url = 'http://en.wikipedia.org/wiki/Cross-cap'
        fx = (1+cos(v))*cos(u)
        fy = (1+cos(v))*sin(u)
        fz = -tanh((2/3)*(u-pi))*sin(v)
        P = parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red",opacity=opacity)
    else:
        print "Bug selecting plot?"
        return


    html('<h2>%s</h2>'%example)
    if url:
        html('<h3><a target="_new" href="%s">%s</a></h3>'%(url,url))
    show(P, viewer='tachyon' if tachyon else 'jmol', frame=frame)

attachment:parametricplot3d.png

Interactive rotatable raytracing with Tachyon3d

C = cube(color=['red', 'green', 'blue'], aspect_ratio=[1,1,1],
         viewer='tachyon') + sphere((1,0,0),0.2)
@interact
def example(theta=(0,2*pi), phi=(0,2*pi), zoom=(1,(1,4))):
    show(C.rotate((0,0,1), theta).rotate((0,1,0),phi), zoom=zoom)

attachment:tachyonrotate.png

Interactive 3d plotting

var('x,y')
@interact
def example(clr=Color('orange'), f=4*x*exp(-x^2-y^2), xrange='(-2, 2)', yrange='(-2,2)', 
    zrot=(0,pi), xrot=(0,pi), zoom=(1,(1/2,3)), square_aspect=('Square Frame', False),
    tachyon=('Ray Tracer', True)):
    xmin, xmax = sage_eval(xrange); ymin, ymax = sage_eval(yrange)
    P = plot3d(f, (x, xmin, xmax), (y, ymin, ymax), color=clr)
    html('<h1>Plot of $f(x,y) = %s$</h1>'%latex(f))
    aspect_ratio = [1,1,1] if square_aspect else [1,1,1/2]
    show(P.rotate((0,0,1), -zrot).rotate((1,0,0),xrot), 
         viewer='tachyon' if tachyon else 'jmol', 
         figsize=6, zoom=zoom, frame=False,
         frame_aspect_ratio=aspect_ratio)

attachment:tachyonplot3d.png

Anchor(eggpaint)

Somewhat Silly Egg Painter

by Marshall Hampton (refereed by William Stein)

var('s,t')
g(s) = ((0.57496*sqrt(121 - 16.0*s^2))/sqrt(10.+ s))
def P(color, rng):
    return parametric_plot3d((cos(t)*g(s), sin(t)*g(s), s), (s,rng[0],rng[1]), (t,0,2*pi), plot_points = [150,150], rgbcolor=color, frame = False, opacity = 1)
colorlist = ['red','blue','red','blue']
@interact
def _(band_number = selector(range(1,5)), current_color = Color('red')):
    html('<h1 align=center>Egg Painter</h1>')
    colorlist[band_number-1] = current_color
    egg = sum([P(colorlist[i],[-2.75+5.5*(i/4),-2.75+5.5*(i+1)/4]) for i in range(4)])
    show(egg)

attachment:eggpaint.png