## Thursday, January 22, 2009

### Integral Apollonian Packings with Sage

At the national math meetings this year I heard about some really interesting and fun work on integral Apollonian circle packings. The AMS has a nice introductory article about them. I couldn't resist trying to compute and draw some in Sage. Carl Witty greatly improved the speed of my first attempt, so what follows can be considered joint work of ours. (Sage can compute nicer PDF output of these, but blogger doesn't embed PDFs.)

def kfun(k1,k2,k3,k4):    """    The Descartes formula for the curvature of an inverted tangent circle.    """    return 2*k1+2*k2+2*k3-k4def 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    if newk > 0:        newr = 1/newk    elif newk < 0:        newr = -1/newk    else:        newr = Infinity    return [newx, newy, newr, newk]def mcircle(circdata, label = False, thick = 1/10):    """    Draws a circle from the data.  label = True    """    if label==True and circdata[3] > 0 and circdata[2] > 1/2000:        lab = text(str(circdata[3]),(circdata[0],circdata[1]), fontsize = \500*(circdata[2])^(.95), vertical_alignment = 'center', horizontal_alignment \= 'center', rgbcolor = (0,0,0))    else:        lab = Graphics()    circ = circle((circdata[0],circdata[1]), circdata[2], rgbcolor = (0,0,0), \thickness = thick)    return circ+labdef 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)        add_circs(newcirc, c2, c3, c1)        add_circs(newcirc, c3, c1, c2)zst1 = [0,0,1/2,-2]zst2 = [1/6,0,1/3,3]zst3 = [-1/3,0,1/6,6]zst4 = [-3/14,2/7,1/7,7]circlist = [zst1,zst2,zst3,zst4]add_circs(zst1,zst2,zst3,zst4)add_circs(zst2,zst3,zst4,zst1)add_circs(zst3,zst4,zst1,zst2)add_circs(zst4,zst1,zst2,zst3)sum([mcircle(q, label = True, thick = 1/2) for q in \circlist]).save('./Apollonian3.png',axes = False, figsize = [12,12], xmin = \-1/2, xmax = 1/2, ymin = -1/2, ymax = 1/2)

CS-imas said...

Nice article!

I have a problem. I need to put 15 balls inside one ball so that the balls take the maximum volume.

I guess that the procedure in your article may work.

Ignacio Gallo said...

Thanks, I wouldn't have known where to start!