Today's image uses the accumulator colors method with a variation in the accumulation function.

The log-log smooth coloring method is designed to be as close to escape count coloring as possible. There is very little room to get creative. The accumulator colors method abandons the idea that the integer portion of the smooth color value should match the escape count. Since something less than one is added to the accumulator at each step, the accumulator value is almost certainly less than the iteration escape count. This is not a big deal. In fact after applying linear or nonlinear re-scaling, even escape count coloring does not coincide with the escape count.

The only requirement for smooth coloring with the accumulator colors method is that the amount added at each step must be zero on the escape boundary. That provides a lot of freedom for variations.

Notice how a halo like effect is generated around the fractal. For the fractal purists out there, there is no mathematical significance to this variation. It is done purely for artistic effect.

Here is a small zoom. The accumulator colors method is again provides smooth colors. A variation is used to add some interest to the background, without distracting from the main fractal shapes.

The image center is -1.055+0.261i. It is the period 2/7 bulb off the main 1/2 bulb. After looking at a few pictures like this you soon learn how to read these fraction numbers off the image. Notice that there are seven branches radiating from the center, and that the longest is the second clockwise from the root. If you zoom out to the full view, you see the black area on the bottom is part of the main 1/2 bulb (nose or head).

In the top left corner, notice how one stalk reaches a pinch point the seems to grow again. This is true of all of the branches in the picture. It is almost a degenerate case, but applying the same fraction counting scheme as for the 2/7 branch point, we get 1/2. That tells us that all of these are growing off the main 1/2 bulb.

Here is a zoom into a Misiurewicz point using accumulator colors, with linear palette re-scaling. The main fractal has a lot of "sparkles". This is the same situation as with escape count coloring, the rate of color change increases with the higher iteration count.

The accumulator color method can create color shading that more closely follows the fractal shape. With iteration counts or log-log smoothing the background is a single color, or gradually changes from the outside to the inside. See Smooth 1D Colors #6 and Smooth 1D Colors #7 for comparison.

Minor oops, I intended to use the same coordinates to contrast accumulator coloring and log-log coloring. I just realized that I have two different Misiurewicz points. I have this image prepared so I am going with it. I may follow up later with new colors for the images in the earlier posts.

Additional color tweaks on the same fractal shape. Added non-linear scaling (square root) to the accumulator color value to reduce the noise in the high-count areas. Other tweaks on the accumulator color add some highlight and shadow around the main fractal shape.

Here is the image from Smooth 1D Colors #6 reimagined with accumulator colors. Notice the "shadows" behind the fractal, and that the blue and white are gone from the background and pushed into the fractal fine detail.

Those are not real shadows. No 3D, no light, no shadow. I use the word just to describe the lighter and darker shades in the background. The different shades arise because accumulator colors is not simple escape counting, or log-log smoothing of escape counts. Orbits of nearby points may escape at the same time, but they behave differently along the way. They take different paths on the way out. Accumulator colors uses all the points in the orbit.

Another view of the Misiurewicz point, this time with accumulator colors and non-linear (square root) scaling.

Links to previous posts of this area: #6 #7 #15

Earlier, before I started this series of posts here, I had started a thread on Fractal Forums, Smooth Colors. This image is one of a series that I posted there demonstrating the visual effects that result from changing various parameters. I am not going to repost that whole series. However, I like to do color experiments on this fractal location. I will use it as a reference to jump into another coloring idea, tomorrow.

For those of you who possess a complex number plane, this picture was taken at -0.9125+.2732i, with width = 0.125.

OK, one more repost from my Smooth Colors series on Fractal Forums.