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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. == 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 == 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 == === 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 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]) |
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] * [:interact/web:Web Applications] * [:interact/bio:Bioinformatics] == 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) |
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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 === 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 == Miscellaneous Graphics == |
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 |
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== 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]
- [:interact/web:Web Applications]
- [:interact/bio:Bioinformatics]
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
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