Here's a tutorial and sequence of pictures showing how to approach the cardioid and reveal these lovely cloud digits (or bulbs as I sometimes call them). Any part of the Mandelbrot can be approached this way. Of course, there's no end to this progression. However the compute required grows very rapidly. This hobby is not for the impatient.
First image (above) the red arrow indicates the digit of interest, one of the eight cardinal compass points of the Mandelbrot set. Rectangle in white, as for all images below, outlines the region being shown next.. image below (NOTE - the period of the sine wave representing the number of iterations for these escape time 'shaded' images changes (increases , sometimes greatly) from picture to picture in the sequence. This is done to control the increasing complexity of the image. They are all escape time pictures however, despite the magnified image appearing different.)
Circle in white - Digit of interest. For this type of zoom, there's a binary decision to make.
Red arrows - We follow two's never ending attempt to reach unity.
Bulbs circled in blue - We must choose one of these two bulbs for the next zoom. This is the binary decision. This process is repeated, until ones patience expires
For this progression I'm attempting to select the digit wherein the three largest minibrots, circled in green, have nearly identical areas. As well as overall digit symmetry.
I choose the left bulb, image below
If will be noted the three largest minibrots, circled in green, are similar in size, as is the symmetry of this digits arms.
Let's have a closer look, below
Again, we must choose between the two digits, circled in blue, avoiding the asymptotes. As per selection criteria enumerated above digit number two is chosen.. below
For the next zoom, the left digit is deemed slightly (very slightly) more symmetrical. There's no regular pattern, from my observations, whether to choose the left or right digit for the next sequence. Only visual examination
For final digit we choose the right bulb this time - image below.
The computational expense of this zoom method is staggering. For the first image a maximum iteration (bailout) of ~500 was required for an accurate census. For the image above, the bailout was 500,000 to attain an accurate approximation of the Mandelbrot set. As can be seen the three largest minibrots (A, B, C) are relatively symmetrical and the cardioid limit is now approaching a straight line, 45 degree angle.