The Buddhaset (or Buddhabrot) is a special way of rendering the Mandelbrot-Set.

A program enters random starting-values into the basic formula z–>z²+c instead of just drawing the final result as described here each iteration point is drawn. This is repeated over and over – the more frequent an area is visited by iteration points, the brighter it will be coloured.

Wikipedia

Even more details here and here

Nice freeware tool to create your own Buddhabrot-Renders.

All images below are pure buddhabrot-pictures with no editing besides minor contrast&colour adjustment.

*Related*

I’ve always been fascinated by Buddabrot renders, but do to the nature of how the Buddabrot is rendered, deep zoom exploration doesn’t seem possible.

At the current state of development & CPU-power it definitely is impossible. But who would have thought 10 years ago that we could fly through fractal 3d-landscapes.. Let’s hope there will be more development in this area.

I can’t help but wondering if the Buddhabrot-technique somehow is the “holy grail” of a true 3d-Mandelbrot-Rendering, and we just haven’t yet found a way to properly ‘map’ it..

The holy grail of 3D-Mandelbrots has kind of been covered by the Mandelbulber community. I think one big issue however, is the fact it is difficult or impossible to do deep zoom exploration on the scale that the Mandelbrot is explored. Most Mandelbulber videos use things like the traditional Mandelbulb (perpendicular Mandelbrot on the vertical; classic Mandelbrot on the horizontal) or the 3D Burning Ship (Celtic on the horizontal; BS on the vertical).

It is my belief that these 3D variants of the Mandelbrot contain the same amount of fractal detail as the traditional 2D Mandelbrot and ABS() variations which I helped to catalog, but the fractal iteration bands are like the layers of an union. Deep zooms create intricate patterns in a 2D plane which is visible to the eye. A deep zoom of the 3D Mandelbulb or BS fractal will have similar layers, but our perspective in 3D lies within the fractal rather than outside it. The mandelbulbs generally use distance estimation rather than iteration depth, but we are only scraping the outer surface of the fractal. It is like looking at an intricate 2D render with fixed bailout and no coloring. This creates the all too common “whipped cream” effect when viewed in 3D. Lastly, all the mandelbulb programs seem to be using GPU floats to render the fractals, which limits depth to floating precision, making truly deep exploration all but impossible.

Back to Buddabrot which is colored based upon all of the points the iteration orbits strike rather than bailout of a specific pixel. In order to compute a zoomed in region of the Buddabrot, rays from everywhere within the fractal must be rendered. So a small zoomed in section of a Buddabrot will take the same length of time to compute as the entire image. And the further zoomed in, the more noise in the image data requiring ever more orbits to be computed. As a result, defining the Buddabrot beyond a few gigapixels of resolution is impossible or extremely difficult with current CPU technology as well as the amount of RAM present within the system. Maybe someday people will render deep zoom images of Buddabrot beyond float precision, but for now I don’t see that happening.

Thank you for explaining this in such detail – This really cleared some things up I wasn’t aware of.

This really sounds like it will be impossible to do really deep zooms, no matter how fast our computers get.

The only hope would be that quantum computing, works for calculating the Mandelbrot-set – calculating all solutions at the same time..

also your analogy of our limited, smooth view of the surface of 3d fractals was great, really helped my understanding. We’re basically not zooming in but just moving along the x,y,z-axis in all those 3d-videos..

Yes, with 3D renders, the perspective is inside the xyz plane; with 2D renders the perspective is outside the xy plane. Since you can’t see all the layers of an onion without taking a slice, much of Mandelbulb exploration just explore the outer exterior of the fractal from different perspectives because we are limited by what we can see from within the plane. That said, the bulbs on an order 8 Mandelbulb, or the masts on the 3D Burning Ship fractal are still very stunning to look at.