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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.

[[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 ===		 Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected].

 * [:interact/graph_theory:Graph Theory]
 * [:interact/calculus:Calculus]
 * [:interact/diffeq:Differential Equations]
 * [:interact/linear_algebra:Linear Algebra]
 * [:interact/algebra:Algebra]
 * [:interact/number_theory:Number Theory]

== Web applications ==

=== Stock Market data, fetched from Yahoo and Google ===
Line 58: Line 16:
{{{
@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 ===
{{{
import urllib

class Day:
    def __init__(self, date, open, high, low, close, volume):
        self.date = date
        self.open=float(open); self.high=float(high); self.low=float(low); self.close=float(close)
        self.volume=int(volume)
    def __repr__(self):
        return '%10s %4.2f %4.2f %4.2f %4.2f %10d'%(self.date, self.open, self.high,
                   self.low, self.close, self.volume)

class Stock:
    def __init__(self, symbol):
        self.symbol = symbol.upper()

    def __repr__(self):
        return "%s (%s)"%(self.symbol, self.yahoo()['price'])
    
    def yahoo(self):
        url = 'http://finance.yahoo.com/d/quotes.csv?s=%s&f=%s' % (self.symbol, 'l1c1va2xj1b4j4dyekjm3m4rr5p5p6s7')
        values = urllib.urlopen(url).read().strip().strip('"').split(',')
        data = {}
        data['price'] = values[0]
        data['change'] = values[1]
        data['volume'] = values[2]
        data['avg_daily_volume'] = values[3]
        data['stock_exchange'] = values[4]
        data['market_cap'] = values[5]
        data['book_value'] = values[6]
        data['ebitda'] = values[7]
        data['dividend_per_share'] = values[8]
        data['dividend_yield'] = values[9]
        data['earnings_per_share'] = values[10]
        data['52_week_high'] = values[11]
        data['52_week_low'] = values[12]
        data['50day_moving_avg'] = values[13]
        data['200day_moving_avg'] = values[14]
        data['price_earnings_ratio'] = values[15]
        data['price_earnings_growth_ratio'] = values[16]
        data['price_sales_ratio'] = values[17]
        data['price_book_ratio'] = values[18]
        data['short_ratio'] = values[19]
        return data

    def historical(self):
        try:
            return self.__historical
        except AttributeError:
            pass
        symbol = self.symbol
        def get_data(exchange):
             name = get_remote_file('http://finance.google.com/finance/historical?q=%s:%s&output=csv'%(exchange, symbol.upper()),
                       verbose=False)
             return open(name).read()
        R = get_data('NASDAQ')
        if "Bad Request" in R:
             R = get_data("NYSE")
        R = R.splitlines()
        headings = R[0].split(',')
        self.__historical = []
        try:
            for x in reversed(R[1:]):
                date, opn, high, low, close, volume = x.split(',')
                self.__historical.append(Day(date, opn,high,low,close,volume))
        except ValueError:
             pass
        self.__historical = Sequence(self.__historical,cr=True,universe=lambda x:x)
        return self.__historical

    def plot_average(self, spline_samples=10):
        d = self.historical()
        if len(d) == 0:
            return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
        avg = list(enumerate([(z.high+z.low)/2 for z in d]))
        P = line(avg) + points(avg, rgbcolor='black', pointsize=4) + \
                 text(self.symbol, (len(d)*1.05, d[-1].low), horizontal_alignment='right', rgbcolor='black')
        if spline_samples > 0:
            k = 250//spline_samples
            spl = spline([avg[i*k] for i in range(len(d)//k)] + [avg[-1]])
            P += plot(spl, (0,len(d)+30), color=(0.7,0.7,0.7))
        P.xmax(260)
        return P

    def plot_diff(self):
        d = self.historical()
        if len(d) == 0:
            return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
        diff = []
        for i in range(1, len(d)):
             z1 = d[i]; z0 = d[i-1]
             diff.append((i, (z1.high+z1.low)/2 - (z0.high + z0.low)/2))
        P = line(diff,thickness=0.5) + points(diff, rgbcolor='black', pointsize=4) + \
                 text(self.symbol, (len(d)*1.05, 0), horizontal_alignment='right', rgbcolor='black')
        P.xmax(260)
        return P

symbols = ['bsc', 'vmw', 'sbux', 'aapl', 'amzn', 'goog', 'wfmi', 'msft', 'yhoo', 'ebay', 'java', 'rht', ]; symbols.sort()
stocks = dict([(s,Stock(s)) for s in symbols])

@interact
def data(symbol = symbols, other_symbol='', spline_samples=(8,[0..15])):
     if other_symbol != '':
         symbol = other_symbol
     S = Stock(symbol)
     html('<h1 align=center><font color="darkred">%s</font></h1>'%S)
     S.plot_average(spline_samples).save('avg.png', figsize=[10,2])
     S.plot_diff().save('diff.png', figsize=[10,2])

     Y = S.yahoo()
     k = Y.keys(); k.sort()
     html('Price during last 52 weeks:<br>Grey line is a spline through %s points (do not take seriously!):<br> <img src="cell://avg.png">'%spline_samples)
     html('Difference from previous day:<br> <img src="cell://diff.png">')
     html('<table align=center>' + '\n'.join('<tr><td>%s</td><td>%s</td></tr>'%(k[i], Y[k[i]]) for i in range(len(k))) + '</table>')

}}}

attachment:stocks.png

=== CO2 data plot, fetched from NOAA ===
Line 73: Line 138:
{{{
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

== 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

== 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,f)
    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

=== 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

== Web apps ==

=== Bioinformatics: protein browser ===
While support for R is rapidly improving, scipy.stats has a lot of useful stuff too. This only scratches the surface.
{{{
import urllib2 as U
import scipy.stats as Stat
co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines()
datalines = []
for a_line in co2data:
    if a_line.find('Creation:') != -1:
        cdate = a_line
    if a_line[0] != '#':
        temp = a_line.replace('\n','').split(' ')
        temp = [float(q) for q in temp if q != '']
        datalines.append(temp)
trdf = RealField(16)
@interact
def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009)):
    htmls1 = '<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>'
    htmls2 = '<h4>'+cdate+'</h4>'
    sel_data = [[q[2],q[4]] for q in datalines if start_date < q[2] < end_date]
    c_max = max([q[1] for q in sel_data])
    c_min = min([q[1] for q in sel_data])
    slope, intercept, r, ttprob, stderr = Stat.linregress(sel_data)
    html(htmls1+htmls2+'<h4>Linear regression slope: ' + str(trdf(slope)) + ' ppm/year; correlation coefficient: ' + str(trdf(r)) + '</h4>')
    var('x,y')
    show(list_plot(sel_data, plotjoined=True, rgbcolor=(1,0,0)) + plot(slope*x+intercept,start_date,end_date), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False)
}}}
attachment:co2c.png

=== Pie Chart from the Google Chart API ===
by Harald Schilly

{{{
# Google Chart API: http://code.google.com/apis/chart
import urllib2 as inet
from pylab import imshow
@interact
def gChart(title="Google Chart API plots Pie Charts!", color1=Color('purple'), color2=Color('black'), color3=Color('yellow'), val1=slider(0,1,.05,.5), val2=slider(0,1,.05,.3), val3=slider(0,1,.05,0.1), label=("Maths Physics Chemistry")):
    url = "http://chart.apis.google.com/chart?cht=p3&chs=600x300"
    url += '&chtt=%s&chts=000000,25'%title.replace(" ","+")
    url += '&chco=%s'%(','.join([color1.html_color()[1:],color2.html_color()[1:],color3.html_color()[1:]]))
    url += '&chl=%s'%label.replace(" ","|")
    url += '&chd=t:%s'%(','.join(map(str,[val1,val2,val3])))
    print url
    html('<div style="border:3px dashed;text-align:center;padding:50px 0 50px 0"><img src="%s"></div>'%url)
}}}
attachment:interact_with_google_chart_api.png



== Bioinformatics ==

=== Web app: protein browser ===
Line 365: Line 207:
=== 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
Line 366: Line 258:

=== 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
Line 385: Line 380:

== Miscellaneous ==

== Profile a snippet of code ==
{{{
html('<h2>Profile the given input</h2>')
import cProfile; import profile
@interact
def _(cmd = ("Statement", '2 + 2'),
      do_preparse=("Preparse?", True), cprof =("cProfile?", False)):
    if do_preparse: cmd = preparse(cmd)
    print "<html>" # trick to avoid word wrap
    if cprof:
        cProfile.run(cmd)
    else:
        profile.run(cmd)
    print "</html>"
}}}
attachment:profile.png


=== 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

=== A Random Walk ===

by William Stein

{{{
html('<h1>A Random Walk</h1>')
vv = []; nn = 0
@interact
def foo(pts = checkbox(True, "Show points"),
        refresh = checkbox(False, "New random walk every time"),
        steps = (50,(10..500))):
    # We cache the walk in the global variable vv, so that
    # checking or unchecking the points checkbox doesn't change
    # the random walk.
    html("<h2>%s steps</h2>"%steps)
    global vv
    if refresh or len(vv) == 0:
        s = 0; v = [(0,0)]
        for i in range(steps):
             s += random() - 0.5
             v.append((i, s))
        vv = v
    elif len(vv) != steps:
        # Add or subtract some points
        s = vv[-1][1]; j = len(vv)
        for i in range(steps - len(vv)):
            s += random() - 0.5
            vv.append((i+j,s))
        v = vv[:steps]
    else:
        v = vv
    L = line(v, rgbcolor='#4a8de2')
    if pts: L += points(v, pointsize=10, rgbcolor='red')
    show(L, xmin=0, figsize=[8,3])
}}}
attachment:randomwalk.png

=== 3D Random Walk ===
{{{
@interact
def rwalk3d(n=(50,1000), frame=True):
    pnt = [0,0,0]
    v = [copy(pnt)]
    for i in range(n):
        pnt[0] += random()-0.5
        pnt[1] += random()-0.5
        pnt[2] += random()-0.5
        v.append(copy(pnt))
    show(line3d(v,color='black'),aspect_ratio=[1,1,1],frame=frame)
}}}
attachment:randomwalk3d.png

Sage Interactions

Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected].

  • [:interact/graph_theory:Graph Theory]
  • [:interact/calculus:Calculus]
  • [:interact/diffeq:Differential Equations]
  • [:interact/linear_algebra:Linear Algebra]
  • [:interact/algebra:Algebra]
  • [:interact/number_theory:Number Theory]

Web applications

Stock Market data, fetched from Yahoo and Google

by William Stein 

import urllib

class Day:
    def __init__(self, date, open, high, low, close, volume):
        self.date = date
        self.open=float(open); self.high=float(high); self.low=float(low); self.close=float(close)
        self.volume=int(volume)
    def __repr__(self):
        return '%10s %4.2f %4.2f %4.2f %4.2f %10d'%(self.date, self.open, self.high, 
                   self.low, self.close, self.volume)

class Stock:
    def __init__(self, symbol):
        self.symbol = symbol.upper()

    def __repr__(self):
        return "%s (%s)"%(self.symbol, self.yahoo()['price'])
    
    def yahoo(self):
        url = 'http://finance.yahoo.com/d/quotes.csv?s=%s&f=%s' % (self.symbol, 'l1c1va2xj1b4j4dyekjm3m4rr5p5p6s7')
        values = urllib.urlopen(url).read().strip().strip('"').split(',')
        data = {}
        data['price'] = values[0]
        data['change'] = values[1]
        data['volume'] = values[2]
        data['avg_daily_volume'] = values[3]
        data['stock_exchange'] = values[4]
        data['market_cap'] = values[5]
        data['book_value'] = values[6]
        data['ebitda'] = values[7]
        data['dividend_per_share'] = values[8]
        data['dividend_yield'] = values[9]
        data['earnings_per_share'] = values[10]
        data['52_week_high'] = values[11]
        data['52_week_low'] = values[12]
        data['50day_moving_avg'] = values[13]
        data['200day_moving_avg'] = values[14]
        data['price_earnings_ratio'] = values[15]
        data['price_earnings_growth_ratio'] = values[16]
        data['price_sales_ratio'] = values[17]
        data['price_book_ratio'] = values[18]
        data['short_ratio'] = values[19]
        return data

    def historical(self):
        try:
            return self.__historical
        except AttributeError:
            pass
        symbol = self.symbol
        def get_data(exchange):
             name = get_remote_file('http://finance.google.com/finance/historical?q=%s:%s&output=csv'%(exchange, symbol.upper()), 
                       verbose=False)
             return open(name).read()
        R = get_data('NASDAQ')
        if "Bad Request" in R:
             R = get_data("NYSE")
        R = R.splitlines()
        headings = R[0].split(',')
        self.__historical = []
        try:
            for x in reversed(R[1:]):
                date, opn, high, low, close, volume = x.split(',')
                self.__historical.append(Day(date, opn,high,low,close,volume))
        except ValueError:
             pass
        self.__historical = Sequence(self.__historical,cr=True,universe=lambda x:x)
        return self.__historical

    def plot_average(self, spline_samples=10):
        d = self.historical()
        if len(d) == 0:
            return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
        avg = list(enumerate([(z.high+z.low)/2 for z in d]))
        P = line(avg) + points(avg, rgbcolor='black', pointsize=4) + \
                 text(self.symbol, (len(d)*1.05, d[-1].low), horizontal_alignment='right', rgbcolor='black')
        if spline_samples > 0:
            k = 250//spline_samples
            spl = spline([avg[i*k] for i in range(len(d)//k)] + [avg[-1]])
            P += plot(spl, (0,len(d)+30), color=(0.7,0.7,0.7))
        P.xmax(260)
        return P

    def plot_diff(self):
        d = self.historical()
        if len(d) == 0:
            return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
        diff = [] 
        for i in range(1, len(d)):
             z1 = d[i]; z0 = d[i-1]
             diff.append((i, (z1.high+z1.low)/2 - (z0.high + z0.low)/2))
        P = line(diff,thickness=0.5) + points(diff, rgbcolor='black', pointsize=4) + \
                 text(self.symbol, (len(d)*1.05, 0), horizontal_alignment='right', rgbcolor='black')
        P.xmax(260)
        return P

symbols = ['bsc', 'vmw', 'sbux', 'aapl', 'amzn', 'goog', 'wfmi', 'msft', 'yhoo', 'ebay', 'java', 'rht', ]; symbols.sort()
stocks = dict([(s,Stock(s)) for s in symbols])

@interact
def data(symbol = symbols, other_symbol='', spline_samples=(8,[0..15])):
     if other_symbol != '':
         symbol = other_symbol
     S = Stock(symbol)
     html('<h1 align=center><font color="darkred">%s</font></h1>'%S)
     S.plot_average(spline_samples).save('avg.png', figsize=[10,2])
     S.plot_diff().save('diff.png', figsize=[10,2])

     Y = S.yahoo()
     k = Y.keys(); k.sort()
     html('Price during last 52 weeks:<br>Grey line is a spline through %s points (do not take seriously!):<br> <img src="cell://avg.png">'%spline_samples)
     html('Difference from previous day:<br> <img src="cell://diff.png">')
     html('<table align=center>' + '\n'.join('<tr><td>%s</td><td>%s</td></tr>'%(k[i], Y[k[i]]) for i in range(len(k))) + '</table>')

attachment:stocks.png

CO2 data plot, fetched from NOAA

by Marshall Hampton

While support for R is rapidly improving, scipy.stats has a lot of useful stuff too. This only scratches the surface. 

import urllib2 as U
import scipy.stats as Stat
co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines()
datalines = []
for a_line in co2data:
    if a_line.find('Creation:') != -1:
        cdate = a_line
    if a_line[0] != '#':
        temp = a_line.replace('\n','').split(' ')
        temp = [float(q) for q in temp if q != '']
        datalines.append(temp)
trdf = RealField(16)
@interact
def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009)):
    htmls1 = '<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>'
    htmls2 = '<h4>'+cdate+'</h4>'
    sel_data = [[q[2],q[4]] for q in datalines if start_date < q[2] < end_date]
    c_max = max([q[1] for q in sel_data])
    c_min = min([q[1] for q in sel_data])
    slope, intercept, r, ttprob, stderr = Stat.linregress(sel_data)
    html(htmls1+htmls2+'<h4>Linear regression slope: ' + str(trdf(slope)) + ' ppm/year; correlation coefficient: ' + str(trdf(r)) + '</h4>')
    var('x,y')
    show(list_plot(sel_data, plotjoined=True, rgbcolor=(1,0,0)) + plot(slope*x+intercept,start_date,end_date), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False)

attachment:co2c.png

Pie Chart from the Google Chart API

by Harald Schilly 

# Google Chart API: http://code.google.com/apis/chart
import urllib2 as inet
from pylab import imshow
@interact
def gChart(title="Google Chart API plots Pie Charts!", color1=Color('purple'), color2=Color('black'), color3=Color('yellow'), val1=slider(0,1,.05,.5), val2=slider(0,1,.05,.3), val3=slider(0,1,.05,0.1), label=("Maths Physics Chemistry")):
    url = "http://chart.apis.google.com/chart?cht=p3&chs=600x300"
    url += '&chtt=%s&chts=000000,25'%title.replace(" ","+")
    url += '&chco=%s'%(','.join([color1.html_color()[1:],color2.html_color()[1:],color3.html_color()[1:]]))
    url += '&chl=%s'%label.replace(" ","|")
    url += '&chd=t:%s'%(','.join(map(str,[val1,val2,val3])))
    print url
    html('<div style="border:3px dashed;text-align:center;padding:50px 0 50px 0"><img src="%s"></div>'%url)

attachment:interact_with_google_chart_api.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

Miscellaneous

Profile a snippet of code

html('<h2>Profile the given input</h2>')
import cProfile; import profile
@interact
def _(cmd = ("Statement", '2 + 2'), 
      do_preparse=("Preparse?", True), cprof =("cProfile?", False)):
    if do_preparse: cmd = preparse(cmd)
    print "<html>"  # trick to avoid word wrap
    if cprof:
        cProfile.run(cmd)
    else:
        profile.run(cmd)
    print "</html>"

attachment:profile.png

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

A Random Walk

by William Stein 

html('<h1>A Random Walk</h1>')
vv = []; nn = 0
@interact
def foo(pts = checkbox(True, "Show points"), 
        refresh = checkbox(False, "New random walk every time"),
        steps = (50,(10..500))):
    # We cache the walk in the global variable vv, so that
    # checking or unchecking the points checkbox doesn't change
    # the random walk. 
    html("<h2>%s steps</h2>"%steps)
    global vv
    if refresh or len(vv) == 0:
        s = 0; v = [(0,0)]
        for i in range(steps): 
             s += random() - 0.5
             v.append((i, s)) 
        vv = v
    elif len(vv) != steps:
        # Add or subtract some points
        s = vv[-1][1]; j = len(vv)
        for i in range(steps - len(vv)):
            s += random() - 0.5
            vv.append((i+j,s))
        v = vv[:steps]
    else:
        v = vv
    L = line(v, rgbcolor='#4a8de2')
    if pts: L += points(v, pointsize=10, rgbcolor='red')
    show(L, xmin=0, figsize=[8,3])

attachment:randomwalk.png

3D Random Walk

@interact
def rwalk3d(n=(50,1000), frame=True):
    pnt = [0,0,0]
    v = [copy(pnt)]
    for i in range(n):
        pnt[0] += random()-0.5
        pnt[1] += random()-0.5
        pnt[2] += random()-0.5
        v.append(copy(pnt))
    show(line3d(v,color='black'),aspect_ratio=[1,1,1],frame=frame)

attachment:randomwalk3d.png

interact (last edited 2021-08-23 15:58:42 by anewton)