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We'll likely restructure and reorganize this once we have some nontrivial content and get a sense of how it is laid out. | 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]] |
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=== 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 |
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=== 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; (needs sage-2.11 or higher, with gfan-0.3 interface) {{{ @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 |
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=== 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 |
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s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,f) | s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,m) |
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=== 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 |
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g = k.multiplicative_generator() | if bits>100: g = k(2) else: g = k.multiplicative_generator() |
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== Web apps == === Bioinformatics: protein browser === |
== Bioinformatics == === Web app: protein browser === |
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=== 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 |
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=== 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 |
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].
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; (needs sage-2.11 or higher, with gfan-0.3 interface)
@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) if bits>100: g = k(2) else: 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
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