Differences between revisions 29 and 53 (spanning 24 versions)
⇤ ← Revision 29 as of 2009-07-24 20:00:17 →
Size: 31062
Editor: MarshallHampton
Comment:
← Revision 53 as of 2012-01-17 14:56:34 → ⇥
Size: 12144
Editor: mhampton
Comment:
Deletions are marked like this. Additions are marked like this.
Line 5: Line 5:
<<TableOfContents>>

=== Everywhere continuous, nowhere differentiable function ===
Line 15: Line 18:
[[http://sage.math.washington.edu/home/wdj/art/cool-sage-pic-small1.png|cool pic 1]] {{http://sage.math.washington.edu/home/wdj/art/cool-sage-pic-small1.png|cool pic 1}}
Line 33: Line 36:
[[http://sage.math.washington.edu/home/wdj/art/balls-mirrored-sage-tachyon1a.png|cool ray tracing pic]] {{http://sage.math.washington.edu/home/wdj/art/balls-mirrored-sage-tachyon1a.png|cool ray tracing pic}}
Line 60: Line 63:
[[http://sage.math.washington.edu/home/wdj/art/boothby-tachyon1.png|cool pic 2]] {{http://sage.math.washington.edu/home/wdj/art/boothby-tachyon1.png|cool pic 2}}
Line 77: Line 80:
[[http://sage.math.washington.edu/home/wdj/art/boothby-tachyon2.png|cool pic 3]] {{http://sage.math.washington.edu/home/wdj/art/boothby-tachyon2.png|cool pic 3}}
Line 94: Line 97:
[[attachment:fourspheres.png]] {{attachment:fourspheres.png}}
Line 119: Line 122:
{{http://sage.math.washington.edu/home/wdj/art/hypotrochoid_R_equals_7,_r_equals_2,_d=3.gif}}

                     Inspiration:
                                                                               
 But a Sage newbie & out to do projects & learn more, one day I was surfing
 Wikipedia and hit < http://en.wikipedia.org/wiki/Hypotrochoid >, saw
 the animated graph, and thought, ... "I bet I can make Sage do that."
 Never one to back from a challenge, I did it. A few mistakes & wrong turns,
 at times some strong language & threatening the computer with violence, a few
 questions to Sage support groups, but, I finally pounded out code that
 worked.
                                                                                                                                                                                                 
 I named it "Hypotrochoid," as animating this was the original inspiration,
 but the code easily animates other graphs.
                           
                                                    
 What this program does:
                                                                               
 This program animates (not just "draws," but "animates") graphs of several
 relations, the hypotrochoid, the hypocycloid, the limacon (or "limacon of
 Pascal"; Sage doesn't like the French character in the original), the
 cardioid, the epitrochoid, and the epicycloid.
                                                 
                              
 The parametric equations that define a hypotrochoid follow; the parameter
 is *t*; for hypotrochoid we have R, r, d > 0, R > r > 0:
             
{{{#!python numbers=none
 x-coordinate: x = (R - r)*cos(t) + d*cos(((R - r)/r)*t)
 y-coordinate: y = (R - r)*sin(t) - d*sin(((R - r)/r)*t)
}}}
                                                                            
 For a epitrochoid, the equations are:

{{{#!python numbers=none
 x = (R + r)*cos(t) - d*cos(((R + r)/r)*t)
 y = (R + r)*sin(t) - d*sin(((R + r)/r)*t)
}}}
 The parametric equations are important in computing the period of the
 relation (below).
                                                                               
 These parametric equations live all over the Internet; Wikipedia has:
 
  ** for the hypochotroid: < http://en.wikipedia.org/wiki/Hypotrochoid >.
 See also < http://linuxgazette.net/133/luana.html >
                           
  ** For the epitrochoid, see < http://en.wikipedia.org/wiki/Epitrochoid >.
     
  ** For the Limacon (Sage completely choked and spewed error messages on the
 French character in the original, even in a comment) of Pascal, see
 < http://en.wikipedia.org/wiki/Lima%C3%A7on >.
     
  ** For the cardioid, see < http://en.wikipedia.org/wiki/Cardioid >.
                                                                               
  ** For the epitrochoid, see < < http://en.wikipedia.org/wiki/Epitrochoid >
                                                                               
  ** For the hypocycloid, see < http://en.wikipedia.org/wiki/Hypocycloid >
        
  ** For the Deltoid Curve, see < http://en.wikipedia.org/wiki/Deltoid_curve >.

 A few further comments: Some curves of the "roulette" family (see
 < http://en.wikipedia.org/wiki/Roulette_%28curve%29 > may be easily
 animated with this program, as follows:

                                                                                  
 1) The epitrochoid (use the negative of "small" radius *r*, easy, puts
    rotating circle on the outside.

 2) The hypocycloid, by setting 0 < r < R, d = r.

 3) The limacon (or "limacon of Pascal"; Sage doesn't like the French
    character in the original), use r < 0, R = abs(r).

 4) The epicycloid, use r < 0 (put rotating circle on outside), d = r

 5) The cardioid, r < 0 d = r
                                                                                                

To draw different graphs, vary that parameters *R* (fixed circle's radius), *r*
(rotating circle's radius), and *d* (length of "bar" from rotating circle),
below; other parameters may be tweaked at will.


Of some note, the *epitrochoid* is the "epicycle" curve of Ptolemaic system
astronomy; one project is to animate some of the Ptolemaic system, but this
is for the future.

Program commences:


{{{#!python numbers=off
#*****************************************************************************
# Copyright (C) 2007 Dean Moore < [email protected] >
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
# Parameters that define the image:

R = 5 # Fixed circle's radius.
r = 3 # Rotating circle's radius, rotates about fixed circle's radius; make
        # this negative to place rotating circle on outside.
d = 5 # Length of the "bar" extending from the center of the rotating circle.

step = 0.025 # Size of "step" in below loops; the smaller the step, the more
                # "frames" in the final "movie" & the better the image, but the
                # slower the program runs -- and the more bytes to the image.

figuresize = 4 # A constant, regulates size of final picture

delayBetweenImages = 0 # A constant, delay between images in final "movie."

colorOfFixedCircle = (0, 0, 1) # Colors of final image,
colorOfRotatingCircle = (0, 1, 0) # all described by
colorOfCenterPoint = (0, 0, 0) # names.
colorOfBar = (0, 0, 0)
colorOfCurve = (1, 0, 0)

thicknessOfFixedCircle = 1 # Thickness of "fixed" circle that never moves.
thicknessOfRotatingCircle = 1 # Thickness of circle that rotates.
thicknessOfCenterOfRotatingCircle = 15 # Size of small circle's center.
thicknessOfBar = 1 # Thickness of "bar" from rotating circle.
penSize = 10 # Size of "pen" at end of the "bar."
thicknessOfCurve = 0.5 # Thickness of final curve, really a lot of line segments.

# End of parameters user can realistically vary.

origin = (0, 0) # Maybe irrelevant, but NO MAGIC NUMBERS!! See
                # < http://en.wikipedia.org/wiki/Magic_number_(programming) >.
                # Comes from my days as a C/C++ programmer.

# Graphic parameters, defined here:

L = Graphics() # Graphics for the curve -- really a lot of segments of straight
               # lines. If the image looks choppy, drop variable *step* above.

v = [] # Holds the curve from *L*, later converted to variable *curve*.

# More parameters, all of which are defined by earlier parameters:

rDiff = (R-r) # The rotating circle may exceed fixed circle;
                           # values may be negative, so we may have a
                           # negative value. We use this so much, we give
                           # it a name.

sizeOfGraph = max(R, abs(r) + abs(d), abs(R) - (r) + abs(d)) # Big as it can get -- may be
                                                             # a liberal estimate.

circle2PI = (abs(rDiff)/r).denom() # Number of rotations about 2*pi. It
                                   # takes a bit of thought, but see the above
                                   # parametric equations. An example is best:
                                   # picture R=8, r=6. We have
                                   # (rDiff)/r = (8-6)/6 = 2/6 = 1/3, reduced. For
                                   # the argument *((rDiff)/r)t* to get back to 2*pi
                                   # (i.e., zero) in cos(((R - r)/r)t) & sin(((R - r)/r)t),
                                   # we need to go 2*(3*pi). A pen & paper & some
                                   # scribbling makes this easy, be it this not
                                   # clear. The *abs* is probably irrelevant.

limit = ceil(2*circle2PI/step) # Loop about circle this many times, by *step*.
                                   # Parameter *circle2PI* never is again used. Factor
                                   # of 2 is unavoidable, circle 2*pi; see below loops.

}}}

First: I use the same trig functions, over an over again. Why re-invent the
wheel & waste computer time? Put the needed trig functions in arrays & "recycle" them as needed.

{{{#!python numbers=off
sin1 = [ 0 for i in range(limit) ] # First
sin2 = [ 0 for i in range(limit) ] # define
cos1 = [ 0 for i in range(limit) ] # arrays, ...
cos2 = [ 0 for i in range(limit) ]

increment = 0 # Adding a bunch of numbers is more efficient than multiplying
              # *i*pi*step, but it is a trivial difference.

multiplier = pi*rDiff/r # Efficiency! Don't repeat the same math
                        # countless times.

for i in srange(limit): # Note above definition of parameter *limit*, the
                        # factor of 2, and the below trig functions. All
                        # this program's loops go *limit*.

    sin1[i] = sin(increment*pi) # Now fill
    cos1[i] = cos(increment*pi) # arrays
    sin2[i] = sin(increment*multiplier) # with our trig
    cos2[i] = cos(increment*multiplier) # functions.

    increment += step # End *i*-loop.

}}}

The next are described by names, but a few notes:

  ** The parameter *fixedCircle* is merely a circle of radius *R* centered at the origin.
I animated it, as that makes it appear in all frames. It never moves.

  ** The parameter *rotatingCircle* is more computation-intense. It's center is on the
circle at *rDiff* from the edge of the fixed circle; its radius is *r*, and it
rotates about the fixed circle a total of *circle2PI* times. Find any internet
reference on these curves to see it.

  ** The parameter *centerPoints* is solely to make the rotating circle look nicer,
a clear center.

  ** The parameter *bar* is the "bar" extending from the center of the rotating circle,
that follows its rotation around the fixed circle. It always emanates from the
same center as *fixedCircle*, -- see *fixedCircle* documentation, above -- and its
terminal point comes from the parametric equation of a hypotrochoid (top).

  ** The parameter *pointAtPen* makes a point where the end of the bar draws the figure,
of the same color as the final curve. Note it is at the same point as the terminal point of the "bar."

{{{#!python numbers=off
fixedCircle = animate([circle(origin, R, # Centered at origin, radius *R*.
                           rgbcolor=colorOfFixedCircle,
                           thickness = thicknessOfFixedCircle)
                           for i in srange(limit) # Animating loop.
                          ],
                           xmin=-sizeOfGraph, ymin=-sizeOfGraph,
                           xmax= sizeOfGraph, ymax= sizeOfGraph,
                           figsize=[figuresize, figuresize])

rotatingCircle = animate([circle((rDiff*cos1[i],rDiff*sin1[i]), r, # Note center, radius *r*
                           thickness = thicknessOfRotatingCircle,
                           rgbcolor=colorOfRotatingCircle)
                           for i in srange(limit) # Animating loop
                          ],
                           xmin=-sizeOfGraph, ymin=-sizeOfGraph,
                           xmax= sizeOfGraph, ymax= sizeOfGraph,
                           figsize=[figuresize,figuresize])

centerPoints = animate([point((rDiff*cos1[i],rDiff*sin1[i]), # Center of *fixedCircle*.
                           rgbcolor=colorOfCenterPoint,
                           pointsize=thicknessOfCenterOfRotatingCircle)
                           for i in srange(limit) # Animating loop
                          ],
                           xmin=-sizeOfGraph, ymin=-sizeOfGraph,
                           xmax= sizeOfGraph, ymax= sizeOfGraph,
                           figsize=[figuresize,figuresize])

bar = animate(line([(rDiff*cos1[i], rDiff*sin1[i]), # Starts here, and ...
                                (d*cos2[i] + rDiff*cos1[i], # Ends at
                                  rDiff*sin1[i] - d*sin2[i]) # this point.
                               ],
                           thickness = thicknessOfBar,
                           rgbcolor=colorOfBar)
                           for i in srange(limit)) # Animating loop

pointAtPen = animate([point((d*cos2[i]+rDiff*cos1[i], # End of pen, same x & y
                                 (rDiff)*sin1[i] - d*sin2[i]), # coordinates as endpoints of *bar*
                           rgbcolor=colorOfCurve,
                           pointsize=penSize)
                           for i in srange(limit) # Animating loop
                           ],
                           xmin=-sizeOfGraph, ymin=-sizeOfGraph,
                           xmax= sizeOfGraph, ymax= sizeOfGraph,
                           figsize=[figuresize,figuresize])

# Prepare to draw the curve; we use four "important" variables: x0, x1, y0 & y1;
# the line segment goes from (x0, y0) to (x1, y1):

x0 = rDiff + d # Initial values of these variables; bumped up at bottom
                   # of next loop -- see a graph or look at parametric equations
                   # at zero to understand why it has this value.

y0 = 0 # The "pen" out the small circle starts at this (x0, y0).

# Now to compute the curve, in the next loop. Starts at zero, even if
# this seems to make the first a junk computation; otherwise the output
# looks weird:

for i in srange(limit): # Parameter *circle2PI* is number of times we rotate
                        # the "fixed" circle; limit=2*circle2PI; see above
                        # documentation.

    x1 = rDiff*cos1[i] + d*cos2[i] # Initial, starting point of
                                    # the line; see parametric
                                    # equations.

    y1 = rDiff*sin1[i] - d*sin2[i] # Again, see the parametric
                                  # equation at top.
    # Now make the line:

    L += line([(x0, y0),(x1, y1)], rgbcolor = colorOfCurve , thickness=thicknessOfCurve)

    L.xmin(-sizeOfGraph) # Some nonsense
    L.ymin(-sizeOfGraph) # that needs done.
    L.xmax( sizeOfGraph)
    L.ymax( sizeOfGraph)

    x0 = x1 # Bump up these variables, start of
    y0 = y1 # next line segment is end of last.

    v.append(L) # *** End of loop ***
               # Stash curve *L* in the variable *v*.

curve = animate(v) # Animate the curve, and ...

# Show the entire "movie":

(fixedCircle + rotatingCircle + pointAtPen + bar + centerPoints + curve).show(delay=delayBetweenImages)

}}}
We've shown the final image; done with program.		 This animation was moved to the section on the animate command : [[http://wiki.sagemath.org/animate#AhypotrochoidanimationbyDeanMoore]]
Line 428: Line 126:
The following animates a hypotrochoid much to the same effect as the previous script, but much more concisely.

{{{#!python numbers=off
import operator

# The colors for various elements of the plot:
class color:
    stylus = (1, 0, 0)
    outer = (.8, .8, .8)
    inner = (0, 0, 1)
    plot = (0, 0, 0)
    center = (0, 0, 0)
    tip = (1, 0, 0)
# and the corresponding line weights:
class weight:
    stylus = 1
    outer = 1
    inner = 1
    plot = 1
    center = 5
    tip = 5

scale = 1 # The scale of the image
animation_delay = .1 # The delay between frames, in seconds

# Starting and ending t values
t_i = 0
t_f = 2*pi
# The t values of the animation frames
tvals = srange(t_i, t_f, (t_f-t_i)/60)

r_o = 8 # Outer circle radius
r_i = 2 # Inner circle radius
r_s = 3 # Stylus radius

# Coordinates of the center of the inner circle
x_c = lambda t: (r_o - r_i)*cos(t)
y_c = lambda t: (r_o - r_i)*sin(t)

# Parametric coordinates for the plot
x = lambda t: x_c(t) + r_s*cos(t*(r_o/r_i))
y = lambda t: y_c(t) - r_s*sin(t*(r_o/r_i))

# Maximum x and y values of the plot
x_max = r_o - r_i + r_s
y_max = find_maximum_on_interval(y, t_i, t_f)[0]

# The plots of the individual elements. Order is important; plots
# are stacked from bottom to top as they appear.
elements = (
    # The outer circle
    lambda t_f: circle((0, 0), r_o, rgbcolor=color.outer, thickness=weight.outer),
    # The plot itself
    lambda t_f: parametric_plot((x, y), t_i, t_f, rgbcolor=color.plot, thickness=weight.plot),
    # The inner circle
    lambda t_f: circle((x_c(t_f), y_c(t_f)), r_i, rgbcolor=color.inner, thickness=weight.inner),
    # The inner circle's center
    lambda t_f: point((x_c(t_f), y_c(t_f)), rgbcolor=color.center,pointsize=weight.center),
    # The stylus
    lambda t_f: line([(x_c(t_f), y_c(t_f)), (x(t_f), y(t_f))], rgbcolor=color.stylus, thickness=weight.stylus),
    # The stylus' tip
    lambda t_f: point((x_c(t_f), y_c(t_f)), rgbcolor=color.tip, pointsize=weight.tip),
)

# Create the plots and animate them. The animate function renders an
# animated gif from the frames provided as its first argument.
# Though avid python programmers will find the syntax clear, an
# explanation is provided for novices.
animation = animate([reduce(operator.add, (f(t) for f in elements))
                     for t in tvals],
                    xmin=-x_max, xmax=x_max,
                    ymin=-y_max, ymax=y_max,
                    figsize=(x_max*scale, y_max*scale * y_max/x_max))

animation.show(delay=animation_delay)

# The previous could be expressed more pedagogically as follows:
#
# Evaluate each function in the elements array for the provided t
# value:
#
# plots = lambda t: f(t) for f in elements
#
# Join a group of plots together to form a single plot:
#
# def join_plots(plots):
# result = plots[0]
# for plot in plots[1:]:
# result += plot
# return result
#
# or
#
# add = lambda a, b: a + b
# join_plots = lambda plots: reduce(add, plots)
#
# Create an array of plots, one for each provided t value:
#
# frames = [join_plots(plots(t)) for t in tvals]
#
# Finally, animate the frames:
#
# animation = animate(frames)
}}}

== The witch of Maria Agnesi ==		 This animation was moved to the section on the animate command : [[http://wiki.sagemath.org/animate#Asimplerhypotrochoid]]

=== The witch of Maria Agnesi ===
Line 536: Line 131:
{{{#!python numbers=none
xtreme = 4.1
myaxes = line([[-xtreme,0],[xtreme,0]],rgbcolor = (0,0,0))
myaxes = myaxes + line([[0,-1],[0,2.1]],rgbcolor = (0,0,0))
a = 1.0
t = var('t')
npi = RDF(pi)
def agnesi(theta):
    mac = circle((0,a),a,rgbcolor = (0,0,0))
    maL = line([[-xtreme,2*a],[xtreme,2*a]])
    maL2 = line([[0,0],[2*a*cot(theta),2*a]])
    p1 = [2*a*cot(theta),2*a*sin(theta)^2]
    p2 = [2*a*cot(theta)-cot(theta)*(2*a-2*a*sin(theta)^2),2*a*sin(theta)^2]
    maL3 = line([p2,p1,[2*a*cot(theta),2*a]], rgbcolor = (1,0,0))
    map1 = point(p1)
    map2 = point(p2)
    am = line([[-.05,a],[.05,a]], rgbcolor = (0,0,0))
    at = text('a',[-.1,a], rgbcolor = (0,0,0))
    yt = text('y',[0,2.2], rgbcolor = (0,0,0))
    xt = text('x',[xtreme + .1,-.1], rgbcolor = (0,0,0))
    matext = at+yt+xt
    ma = mac+myaxes+maL+am+matext+maL2+map1+maL3+map2
    return ma

def witchy(theta):
    ma = agnesi(theta)
    agplot = parametric_plot([2*a*cot(t),2*a*sin(t)^2],[t,.001,theta], rgbcolor = (1,0,1))
    return ma+agplot

a2 = animate([witchy(i) for i in srange(.1,npi-.1,npi/60)]+[witchy(i) for i in srange(npi-.1,.1,-npi/60)], xmin = -3, xmax = 3, ymin = 0, ymax = 2.3, figsize = [6,2.3], axes = False)

a2.show()
}}}

{{attachment:witch.gif}}

== p-adic Seasons Greetings ==		 This animation was moved to the section on the animate command : [[http://wiki.sagemath.org/animate#ThewitchofMariaAgnesi]]

=== p-adic Seasons Greetings ===
Line 644: Line 205:
=== "three famous plots of chaos" by Pablo Angulo === === Feigenbaum diagram ===
Author: Pablo Angulo
Posted to sage-devel 2008-09-13. See also https://sage.math.washington.edu:8101/home/pub/3
#Note: Mandelbrot set moved to interact/fractals
Line 646: Line 211:
# Author: Pablo Angulo

Posted to sage-devel 2008-09-13. See also https://sage.math.washington.edu:8101/home/pub/3
-----------------------------------
Line 671: Line 231:

---------------------------------------
#Lorentz attractor
#plots the orbit of the point (1,1,1) using the simplest euler method

h=0.01; # time step
k=2000 # number of iterations (time k*h will be reached)

sigma=10; #parameters
rho=28;
beta=8/3;

x=1; y=1; z=1; # initial data

puntos=[]
for i in range(k):
   x,y,z=x+h*( sigma*(y-x) ), y+h*( x*(rho - z) - y ), z+h*( x*y - beta*z )
   puntos.append((x,y,z))
point3d(puntos)

--------------------------------------------

#Mandelbrot set: the final plot is a subset of the complex plane;
#the color at point c is porportional to the number of iterations that
#the discrete dynamical system z->z^2+c takes to leave a circle around
#the origin when z0=0

N=100 #resolution of the plot
L=50 #limits the number of iterations
x0=-2; x1=1; y0=-1.5; y1=1.5 #boundary of the region plotted
R=3 #stop after leaving the circle of radius R

m=matrix(N,N)
for i in range(N):
   for k in range(N):
       c=complex(x0+i*(x1-x0)/N, y0+k*(y1-y0)/N)
       z=0
       h=0
       while (h<L) and (abs(z)<R):
           z=z*z+c
           h+=1
       m[i,k]=h
matrix_plot(m)

}}}		 }}}

{{attachment:feigenbaum.png}}


=== Sierpinski triangle ===
Line 731: Line 252:
    tlist_new = [x for x in tlist_old]     tlist_new = tlist_old[:]
Line 737: Line 258:
       tlist_old = [x for x in tlist_new]        tlist_old = tlist_new[:]
Line 754: Line 275:

=== The Tamer and the Lion by Provencal and Labbe ===

This animation was moved to the section on the animate command : [[http://wiki.sagemath.org/animate#TheTamerandtheLionbyProvencalandLabbe]]

=== Integral Curvature Apollonian Circle Packing ===
by Marshall Hampton and Carl Witty

{{{
def kfun(k1,k2,k3,k4):
    """
    The Descartes formula for the curvature of an inverted tangent circle.
    """
    return 2*k1+2*k2+2*k3-k4


colorlist = [(1,0,1),(0,1,0),(0,0,1),(1,0,0)]

def circfun(c1,c2,c3,c4):
    """
    Computes the inversion of circle 4 in the first three circles.
    """
    newk = kfun(c1[3],c2[3],c3[3],c4[3])
    newx = (2*c1[0]*c1[3]+2*c2[0]*c2[3]+2*c3[0]*c3[3]-c4[0]*c4[3])/newk
    newy = (2*c1[1]*c1[3]+2*c2[1]*c2[3]+2*c3[1]*c3[3]-c4[1]*c4[3])/newk
    newcolor = c4[4]
    if newk > 0:
        newr = 1/newk
    elif newk < 0:
        newr = -1/newk
    else:
        newr = Infinity
    return [newx, newy, newr, newk, newcolor]

def mcircle(circdata, label = False, thick = 1/10, cutoff = 2000, color = ''):
    """
    Draws a circle from the data. label = True
    """
    if color == '':
        color = colorlist[circdata[4]]
    if label==True and circdata[3] > 0 and circdata[2] > 1/cutoff:
        lab = text(str(circdata[3]),(circdata[0],circdata[1]), fontsize = \
500*(circdata[2])^(.95), vertical_alignment = 'center', horizontal_alignment \
= 'center', rgbcolor = (0,0,0),zorder=10)
    else:
        lab = Graphics()
    circ = circle((circdata[0],circdata[1]), circdata[2], rgbcolor = (0,0,0), \
thickness = thick)
    circ = circ + circle((circdata[0],circdata[1]), circdata[2], rgbcolor = color, \
thickness = thick, fill=True, alpha = .4, zorder=0)
    return lab+circ

def add_circs(c1, c2, c3, c4, cutoff = 300):
    """
    Find the inversion of c4 through c1,c2,c3. Add the result to circlist,
    then (if the result is big enough) recurse.
    """
    newcirc = circfun(c1, c2, c3, c4)
    if newcirc[3] < cutoff:
        circlist.append(newcirc)
        add_circs(newcirc, c1, c2, c3, cutoff = cutoff)
        add_circs(newcirc, c2, c3, c1, cutoff = cutoff)
        add_circs(newcirc, c3, c1, c2, cutoff = cutoff)

zst1 = [0,0,1/2,-2,0]
zst2 = [1/6,0,1/3,3,1]
zst3 = [-1/3,0,1/6,6,2]
zst4 = [-3/14,2/7,1/7,7,3]

circlist = [zst1,zst2,zst3,zst4]
add_circs(zst1,zst2,zst3,zst4,cutoff = 500)
add_circs(zst2,zst3,zst4,zst1,cutoff = 500)
add_circs(zst3,zst4,zst1,zst2,cutoff = 500)
add_circs(zst4,zst1,zst2,zst3,cutoff = 500)

circs = sum([mcircle(q, label = True, thick = 1/2) for q in \
circlist[1:]])
circs = circs + mcircle(circlist[0],color=(1,1,1),thick=1)
circs.save('./Apollonian3.png',axes = False, figsize = [12,12], xmin = \
-1/2, xmax = 1/2, ymin = -1/2, ymax = 1/2)
}}}

{{attachment:Apollonian.png}}

Pictures drawn by Sage

These pictures and images were drawn by Sage.

Contents

  1. Pictures drawn by Sage
    1. Everywhere continuous, nowhere differentiable function
    2. Mirrored balls in tachyon
    3. Math art by Tom Boothby
    4. Twisted cubic in tachyon
    5. Reflections from four spheres in tachyon
    6. A cone inside a sphere
    7. A cylinder inside a cone
    8. A hypotrochoid animation by Dean Moore
    9. A simpler hypotrochoid
    10. The witch of Maria Agnesi
    11. p-adic Seasons Greetings
    12. Lorentz butterfly
    13. Feigenbaum diagram
    14. Sierpinski triangle
    15. The Tamer and the Lion by Provencal and Labbe
    16. Integral Curvature Apollonian Circle Packing

Everywhere continuous, nowhere differentiable function

  • Everywhere continuous, nowhere differentiable function (in the infinite limit, anyway): 

p = Graphics()
for n in range(1,20):
  f = lambda x: sum([sin(x*3^i)/(2^i) for i in range(1,n)])
  p += plot(f,0,float(pi/3),plot_points=2000,rgbcolor=hue(n/20))

p.show(xmin=0, ymin=0,dpi=250)

cool pic 1

Mirrored balls in tachyon

t = Tachyon(camera_center=(8.5,5,5.5), look_at=(2,0,0), raydepth=6, xres=1500, yres=1500)
t.light((10,3,4), 1, (1,1,1))
t.texture('mirror', ambient=0.05, diffuse=0.05, specular=.9, opacity=0.9, color=(.8,.8,.8))
t.texture('grey', color=(.8,.8,.8), texfunc=7) ## try other values of texfunc too!
t.plane((0,0,0),(0,0,1),'grey')
t.sphere((4,-1,1), 1, 'mirror')
t.sphere((0,-1,1), 1, 'mirror')
t.sphere((2,-1,1), 0.5, 'mirror')
t.sphere((2,1,1), 0.5, 'mirror')
show(t)

cool ray tracing pic

Math art by Tom Boothby

# Author: Tom Boothby
# This is a remake of an old art piece I made in POVRay


t = Tachyon(xres=1000,yres=600, camera_center=(1,0,5), antialiasing=3)
t.light((4,3,2), 0.2, (1,1,1))
t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,1,1))
t.texture('t1', ambient=0.5, diffuse=0.5, specular=0.0, opacity=1.0, color=(0,0,0))
t.texture('t2', ambient=0.2, diffuse=0.7, specular=0, opacity=0.7, color=(.5,.5,.5))
t.texture('t3', ambient=.9, diffuse=5, specular=0,opacity=.1, color=(1,0,0))
t.sphere((1,0,0), 30, 't2')



k=0
for i in srange(-pi*10,0,.01):
  k += 1
  t.sphere((cos(i/10)-.1, sin(i/10)*cos(i), sin(i/10)*sin(i)), 0.1, 't0')
  t.sphere((cos(i/10) + 2.1, sin(i/10)*cos(i), sin(i/10)*sin(i)), 0.1, 't1')

t.show(verbose=1)

cool pic 2

Twisted cubic in tachyon

t = Tachyon(xres=512,yres=512, camera_center=(5,0,0))
t.light((4,3,2), 0.2, (1,1,1))
t.texture('t0', ambient=0.1, diffuse=0.9, specular=0.5, opacity=1.0, color=(1.0,0,0))
t.texture('t1', ambient=0.1, diffuse=0.9, specular=0.3, opacity=1.0, color=(0,1.0,0))
t.texture('t2', ambient=0.2, diffuse=0.7, specular=0.5, opacity=0.7, color=(0,0,1.0))
k=0
for i in srange(-5,1.5,0.1):
    k += 1
    t.sphere((i,i^2-0.5,i^3), 0.1, 't%s'%(k%3))

t.show()

cool pic 3

Reflections from four spheres in tachyon

t6 = Tachyon(camera_center=(0,-4,1), xres = 800, yres = 600, raydepth = 12, aspectratio=.75, antialiasing = True)
t6.light((0.02,0.012,0.001), 0.01, (1,0,0))
t6.light((0,0,10), 0.01, (0,0,1))
t6.texture('s', color = (.8,1,1), opacity = .9, specular = .95, diffuse = .3, ambient = 0.05)
t6.texture('p', color = (0,0,1), opacity = 1, specular = .2)
t6.sphere((-1,-.57735,-0.7071),1,'s')
t6.sphere((1,-.57735,-0.7071),1,'s')
t6.sphere((0,1.15465,-0.7071),1,'s')
t6.sphere((0,0,0.9259),1,'s')
t6.plane((0,0,-1.9259),(0,0,1),'p')
t6.show()

fourspheres.png

A cone inside a sphere

sage: u,v = var("u,v")
sage: p1 = parametric_plot3d([cos(u)*v, sin(u)*v, 3*v/2-1/3], (u, 0, 2*pi), (v, 0, 0.95),plot_points=[20,20])
sage: p2 = sphere((0,0,2/3), color='red', opacity=0.5, aspect_ratio=[1,1,1])
sage: show(p1+p2)

http://sage.math.washington.edu/home/wdj/art/cone-inside-sphere.jpg

A cylinder inside a cone

sage: u,v = var("u,v")
sage: p1 = parametric_plot3d([cos(u)*v, sin(u)*v, 3/2-3*v/2], (u, 0, 2*pi), (v, 0, 1.5), opacity = 0.5, plot_points=[20,20])
sage: p2 = parametric_plot3d([cos(u)/2, sin(u)/2, v-3/4], (u, 0, 2*pi), (v, 0, 3/2), plot_points=[20,20])
sage: show(p1+p2)

http://sage.math.washington.edu/home/wdj/art/cylinder-inside-cone.jpg

A hypotrochoid animation by Dean Moore

Hypotrochoid. Written by Dean Moore, February 2008

This animation was moved to the section on the animate command : http://wiki.sagemath.org/animate#AhypotrochoidanimationbyDeanMoore

A simpler hypotrochoid

This animation was moved to the section on the animate command : http://wiki.sagemath.org/animate#Asimplerhypotrochoid

The witch of Maria Agnesi

by Marshall Hampton

This animation was moved to the section on the animate command : http://wiki.sagemath.org/animate#ThewitchofMariaAgnesi

p-adic Seasons Greetings

  • I know this is early, but thanks to Robert Bradshaw's p-adic plot function, here is a p-adic Seasons Greetings:

http://sage.math.washington.edu/home/wdj/art/padic-seasons-greetings.png

This is the code: 

sage: P1 = Zp(3).plot(rgbcolor=(0,1,0))
sage: P2 = Zp(7).plot(rgbcolor=(1,0,0))
sage: P3 = text("$Seasons$ $Greetings$ ",(0.0,1.8))
sage: P4 = text("$from$ $everyone$ $at$ sagemath.org!",(0.1,-1.6))
sage: (P1+P2+P3+P4).show(axes=False)

Lorentz butterfly

"""
Draws Loretz butterfly using matplotlib (2d) or jmol (3d).
Written by Matthew Miller and William Stein.

"""

def butterfly2d():
    """"
    EXAMPLE:
        sage: butterfly2d()
    """    
    g = Graphics()
    x1, y1 = 0, 0
    from math import sin, cos, exp, pi
    for theta in srange( 0, 10*pi, 0.01 ):
        r = exp(cos(theta)) - 2*cos(4*theta) + sin(theta/12)^5
        x = r * cos(theta)  # Convert polar to rectangular coordinates
        y = r * sin(theta)
        xx = x*6 + 25       # Scale factors to enlarge and center the curve.
        yy = y*6 + 25
        if theta != 0: 
            l = line( [(x1, y1), (xx, yy)], rgbcolor=hue(theta/7 + 4) )
            g = g + l
            x1, y1 = xx, yy
    g.show(dpi=100, axes=False)

def butterfly3d():
    """"
    EXAMPLE:
        sage: butterfly3d()
    """    
    g = point3d((0,0,0))
    x1, y1 = 0, 0
    from math import sin, cos, exp, pi
    for theta in srange( 0, 10*pi, 0.05):
        r = exp(cos(theta)) - 2*cos(4*theta) + sin(theta/12)^5
        x = r * cos(theta)  # Convert polar to rectangular coordinates
        y = r * sin(theta)
        xx = x*6 + 25       # Scale factors to enlarge and center the curve.
        yy = y*6 + 25
        if theta != 0:
            l = line3d( [(x1, y1, theta), (xx, yy, theta)],
            rgbcolor=hue(theta/7 + 4) )
            g = g + l
            x1, y1 = xx, yy
    g.show(dpi=100, axes=False)

http://sage.math.washington.edu/home/wdj/art/butterfly2d.png

http://sage.math.washington.edu/home/wdj/art/butterfly3d.png

Feigenbaum diagram

Author: Pablo Angulo Posted to sage-devel 2008-09-13. See also https://sage.math.washington.edu:8101/home/pub/3 #Note: Mandelbrot set moved to interact/fractals 

#Plots Feigenbaum diagram: divides the parameter interval [2,4] for mu
#into N steps. For each value of the parameter, iterate the discrete
#dynamical system x->mu*x*(1-x), drop the first M1 points in the orbit
#and plot the next M2 points in a (mu,x) diagram

N=200
M1=200
M2=200
x0=0.509434

puntos=[]
for t in range(N):
   mu=2.0+2.0*t/N
   x=x0
   for i in range(M1):
       x=mu*x*(1-x)
   for i in range(M2):
       x=mu*x*(1-x)
       puntos.append((mu,x))
point(puntos,pointsize=1)

feigenbaum.png

Sierpinski triangle

  • This was a black+white Sierpinski triangle coded by Marshall Hampton, with some slight tweeking by David Joyner to add colors: 

def sierpinski_seasons_greetings():
    """
    Code by Marshall Hampton.
    Colors by David Joyner.
    General depth by Rob Beezer.
    Copyright Marshall Hampton 2008, licensed
    creative commons, attribution share-alike.
    """
    depth = 7
    nsq = RR(3^(1/2))/2.0
    tlist_old = [[[-1/2.0,0.0],[1/2.0,0.0],[0.0,nsq]]]
    tlist_new = tlist_old[:]
    for ind in range(depth):
       for tri in tlist_old:
           for p in tri:
               new_tri = [[(p[0]+x[0])/2.0, (p[1]+x[1])/2.0] for x in tri]
               tlist_new.append(new_tri)
       tlist_old = tlist_new[:]
    T = tlist_old
    N  = 4^depth
    N1 = N - 3^depth
    q1 = sum([line(T[i]+[T[i][0]], rgbcolor = (0,1,0)) for i in range(N1)])
    q2 = sum([line(T[i]+[T[i][0]], rgbcolor = (1,0,0)) for i in range(N1,N)])
    show(q2+q1, figsize = [6,6*nsq], axes = False)

GIMP was used to add a Season's greetings message:

http://sage.math.washington.edu/home/wdj/art/sierpinski-seasons-greetings-from-sage.png

Also (thanks to Rob Beezer) available in poster form in pdf format: http://sage.math.washington.edu/home/wdj/art/seasons-greetings-sage.pdf, and in A4 size: http://sage.math.washington.edu/home/wdj/art/seasons-greetings-sage-a4.pdf.

The Tamer and the Lion by Provencal and Labbe

This animation was moved to the section on the animate command : http://wiki.sagemath.org/animate#TheTamerandtheLionbyProvencalandLabbe

Integral Curvature Apollonian Circle Packing

by Marshall Hampton and Carl Witty 

def kfun(k1,k2,k3,k4):
    """
    The Descartes formula for the curvature of an inverted tangent circle.
    """
    return 2*k1+2*k2+2*k3-k4


colorlist = [(1,0,1),(0,1,0),(0,0,1),(1,0,0)]

def circfun(c1,c2,c3,c4):
    """
    Computes the inversion of circle 4 in the first three circles.
    """
    newk = kfun(c1[3],c2[3],c3[3],c4[3])
    newx = (2*c1[0]*c1[3]+2*c2[0]*c2[3]+2*c3[0]*c3[3]-c4[0]*c4[3])/newk
    newy = (2*c1[1]*c1[3]+2*c2[1]*c2[3]+2*c3[1]*c3[3]-c4[1]*c4[3])/newk
    newcolor = c4[4]
    if newk > 0:
        newr = 1/newk
    elif newk < 0:
        newr = -1/newk
    else:
        newr = Infinity
    return [newx, newy, newr, newk, newcolor]

def mcircle(circdata, label = False, thick = 1/10, cutoff = 2000, color = ''):
    """
    Draws a circle from the data.  label = True
    """
    if color == '':
        color = colorlist[circdata[4]]
    if label==True and circdata[3] > 0 and circdata[2] > 1/cutoff:
        lab = text(str(circdata[3]),(circdata[0],circdata[1]), fontsize = \
500*(circdata[2])^(.95), vertical_alignment = 'center', horizontal_alignment \
= 'center', rgbcolor = (0,0,0),zorder=10)
    else:
        lab = Graphics()
    circ = circle((circdata[0],circdata[1]), circdata[2], rgbcolor = (0,0,0), \
thickness = thick)
    circ = circ + circle((circdata[0],circdata[1]), circdata[2], rgbcolor = color, \
thickness = thick, fill=True, alpha = .4, zorder=0)
    return lab+circ

def add_circs(c1, c2, c3, c4, cutoff = 300):
    """
    Find the inversion of c4 through c1,c2,c3.  Add the result to circlist,
    then (if the result is big enough) recurse.
    """
    newcirc = circfun(c1, c2, c3, c4)
    if newcirc[3] < cutoff:
        circlist.append(newcirc)
        add_circs(newcirc, c1, c2, c3, cutoff = cutoff)
        add_circs(newcirc, c2, c3, c1, cutoff = cutoff)
        add_circs(newcirc, c3, c1, c2, cutoff = cutoff)

zst1 = [0,0,1/2,-2,0]
zst2 = [1/6,0,1/3,3,1]
zst3 = [-1/3,0,1/6,6,2]
zst4 = [-3/14,2/7,1/7,7,3]

circlist = [zst1,zst2,zst3,zst4]
add_circs(zst1,zst2,zst3,zst4,cutoff = 500)
add_circs(zst2,zst3,zst4,zst1,cutoff = 500)
add_circs(zst3,zst4,zst1,zst2,cutoff = 500)
add_circs(zst4,zst1,zst2,zst3,cutoff = 500)

circs = sum([mcircle(q, label = True, thick = 1/2) for q in \
circlist[1:]])
circs = circs + mcircle(circlist[0],color=(1,1,1),thick=1)
circs.save('./Apollonian3.png',axes = False, figsize = [12,12], xmin = \
-1/2, xmax = 1/2, ymin = -1/2, ymax = 1/2)

Apollonian.png

pics (last edited 2017-03-26 02:08:05 by mrennekamp)