Slope Formula Tutorial for UltraFractal - Page 8
Apollonian Gasket
The Apollonian Gasket is probably the best known of the circle inversion fractals. It is described at the Mathworld web site and many other places on the web. It has a fractal dimension of approximately 1.3058... and corresponds to a limit set that is invariant under a Kleinian group (see the section on Kleinian Group Fractals). It is created by iteratively carrying out circle inversions on three mutually tangent circles of the same radius. There are four circles that are mutually tangent that do the inversions, three outside circles and one inner circle.
There are two Apollonian Gasket formulas that use that use the slope algorithm, one is in reb.ufm and the other is in reb.ucl. There is some redundancy between the two formulas, but both will be treated here. The two formulas are Slope Apollonian Gasket (the ufm) and Apollonian Gasket (the ucl). Both formulas require specialized gradients to get realistic ray trace-like coloring. In addition there is a transform that is intended to be used with both of these formulas that is in reb.uxf. It is also called Apollonian Gasket. The following parameters are in common with both formulas and the transform (only selected parameters are listed):
The following parameters are common to the ufm and the ucl:
(radiuspower - alpha*xpower - beta*ypower)1/power/gamma
(radiuspower + alpha*xpower - beta*ypower)1/power/gamma
(radiuspower + alpha*xpower + beta*ypower)1/power/gamma
(radiuspower - alpha*xpower + beta*ypower)1/power/gamma
The following parameters are for coloring the ucl:
The user is encouraged to explore any parameters not listed here.
The first two images show the gradient use for the ufm and the ucl. With the ufm, all gasket shapes, in this case spheres, are colored with the same color. With the ucl, an 8 color gradient is used.
ApolloBasic created with the ufm |
ApolloBasic2 created with the ucl |
Because the gradients are so critical to creating these images, uprs for the two images follow:
ApolloBasic { fractal: title="ApolloBasic" width=640 height=640 layers=1 gamma=1.77000000000000002 credits="Ronald Barnett;1/26/2007" layer: method=multipass caption="Background" opacity=100 mapping: center=0/0 magn=1 formula: maxiter=100 filename="reb.ufm" entry="Slope_Apollonian" p_version=200 p_solid=yes p_applyMapping=no p_percentInc=0.0 p_level=100 p_scircle=0.0001 p_base=no p_lace=no p_ptype=None p_pert=1 p_xfer=linear p_zscale=0.4 p_zscale2=0.025 p_fill="quartic 1" p_pwr=2 p_alpha=1 p_beta=1 p_gamma=1 p_gnoffset=0/0 p_gfbmf=0.1 p_gnscale=0 p_gfreq=1 p_giter=7 inside: transfer=none outside: density=.5 transfer=linear filename="reb.ucl" entry="3D_Texturizer_Enhanced_III" p_trap=no p_trapsum=no p_itrap=0.0/1.0 p_solidb=no p_noslope=no p_coloring=X p_spread=1.0 p_ltype="Infinite light" p_angle=120.0 p_elevation=60.0 p_lpointx=1.5 p_lpointy=2 p_lplane=5 p_lightx=0 p_lighty=0 p_lightz=0 p_ambient=0.0 p_texture=Lattice p_gridx=1.0 p_gridy=1.0 p_dx=0.0 p_dy=0.0 p_theight=1.0 p_weight=0.0 p_blend=0.5 p_toffset=1.0 p_smooth=no p_transfer=linear p_fbm=0.2 p_fbmf=0.5 p_noffset=0/0 p_nscale=0.0 p_toffsetII=2/2 p_tweight=0.0 p_ts=5.0 p_tc=7.0 p_pwr=1E-8 p_pmix=0.0 f_fn3=sin f_fn1=sqr f_fn2=acos gradient: comments="Simple grayscale gradient." smooth=yes rotation=-33 index=157 color=14691209 index=167 color=16777215 index=177 color=14756745 index=-33 color=0 opacity: smooth=no index=0 opacity=255 }
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The next three images are layered images which show off some of the capabilities of the ufm, the ucl and the transform. The first, ApolloOnApollo, layer the ufm and the ucl, taking advantage of the texturing abilities of the formulas. ApolloOuter10 and ApolloInner10 show some perturbation effects, some additional parameter settings, and use of the Apollonian Gasket transform. The user should study the uprs to determine how these images where created.
ApolloOnApollo | ApolloOuter10 |
ApolloInner10 |
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