Buddhabrot, Mandelbrot’s older sibling

Many people have heard of Mandelbrot’s famous set. But far fewer have heard of its older sibling: the Buddhabrot. I find this object even more beautiful than the Mandelbrot set, and want to give it its proper appreciation here.

To start out with: What is the Mandelbrot set? It is defined as the set of complex values c that stay finite under arbitrary many repeated applications of the function fc(z) = z2 + c (where the first iteration is applied to z = 0).

So, for example, to check if the number 2 is in the Mandelbrot set, we just look at what happens when we plug in 0 to the function f2 and repeat:

0 becomes 02 + 2 = 2
2 becomes 22 + 2 = 6
6 becomes 62 + 2 = 38
38 becomes 382 + 2 = something large, and on and on.

You can probably predict that as we keep iterating, we’re going to eventually run off to infinity. So 2 is not in the set.

On the other hand, see what happens when we plug in -1:

0 becomes 02 + (-1) = -1
-1 becomes (-1)2 + (-1) = 0
0 becomes -1
-1 becomes 0
… and so on to infinity

With -1, we just bounce around back and forth between 0 and -1, so we clearly never diverge to infinity.

Now we can draw the elements of this set pretty easily, by simply shading in the numbers on the complex plane that are in the set. If you do so, you get the following visualization:


Cool! But what about all those colorful visualizations you’ve probably seen? (maybe even on this blog!)

Well, whether you’re in the Mandelbrot set or not is just a binary property. So by itself the Mandelbrot set doesn’t have a rich enough structure to account for all those pretty colors. What’s being visualized in those pictures is not just whether an element is in the Mandelbrot set, but also, if it’s not in the set (i.e. if the 0 diverges to infinity upon repeated applications of fc), how quickly it leaves the set!

In other words, the colors are a representation of how not in the set numbers are. There are some complex numbers that exist near the edge of the Mandelbrot that hang around the set for many many iterations before finally blowing up and running off to infinity. And these numbers will be colored differently from, say, the number “2”, which right away starts blowing up.

And that’s how you end up with pictures like the following:

Screen Shot 2019-05-22 at 3.37.11 PM.png

Now, what if instead of visualizing how in the set various complex numbers are, we instead look at what complex numbers are most visited on average when running through Mandelbrot iterations? Well, that’s how we get the Buddhabrot!

Specifically, here’s a procedure we could run:

  1. Choose a random complex number c.
  2. Define z = 0
  3. Update: z = fc(z)
  4. Give one unit of credit to whatever complex number z is now.
  5. Repeat 3 and 4 for some fixed number of iterations.
  6. Start over back at 1 with a new complex number.

As you run this algorithm, you can visualize the results by giving complex numbers with more credits brighter colors. And as you do this, a curious figure begins to appear on the (rotated) complex plane:

Screen Shot 2019-05-22 at 4.23.06 PM.png



That’s right, Buddha is hanging out on the complex plane, hiding in the structure of the Mandelbrot set!

Value beyond ethics

There is a certain type of value in our existence that transcends ethical value. It is beautifully captured in this quote from Richard Feynman:

It is a great adventure to contemplate the universe, beyond man, to contemplate what it would be like without man, as it was in a great part of its long history and as it is in a great majority of places. When this objective view is finally attained, and the mystery and majesty of matter are fully appreciated, to then turn the objective eye back on man viewed as matter, to view life as part of this universal mystery of greatest depth, is to sense an experience which is very rare, and very exciting. It usually ends in laughter and a delight in the futility of trying to understand what this atom in the universe is, this thing—atoms with curiosity—that looks at itself and wonders why it wonders.

Well, these scientific views end in awe and mystery, lost at the edge in uncertainty, but they appear to be so deep and so impressive that the theory that it is all arranged as a stage for God to watch man’s struggle for good and evil seems inadequate.

The Meaning Of It All

Carl Sagan beautifully expressed the same sentiment.

We are the local embodiment of a Cosmos grown to self-awareness. We have begun to contemplate our origins: starstuff pondering the stars; organized assemblages of ten billion billion billion atoms considering the evolution of atoms; tracing the long journey by which, here at least, consciousness arose. Our loyalties are to the species and the planet. We speak for Earth. Our obligation to survive is owed not just to ourselves but also to that Cosmos, ancient and vast, from which we spring.


The ideas expressed in these quotes feels a thousand times deeper and more profound than anything offered in ethics. Trolley problems seem trivial by comparison. If somebody argued that the universe would be better off without us on the basis of, say, a utilitarian calculation of net happiness, I would feel like there is an entire dimension of value that they are completely missing out on. This type of value, a type of raw aesthetic sense of the profound strangeness and beauty of reality, is tremendously subtle and easily slips out of grasp, but is crucially important. My blog header serves as a reminder: We are atoms contemplating atoms.

The spiritual and the scientific

There’s an Isaac Asimov quote that I love. It goes:

When people thought the Earth was flat, they were wrong. When people thought the Earth was spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.

I was recently reminded of this because I’m at an ashram this week, and in one of the talks, a swami brought up his beef with science.

He talked about how science is just another form of faith, and that therefore our intuition is a perfectly valid guide to understanding the universe. After all, all of our past scientific theories have turned out to be wrong, so we should expect that our current theories will also turn out to be wrong.

Thus the Asimov.


For various reasons, I’m often in spiritual places surrounded by spiritual people. These are the types of people that say “I believe in all religions” and go to yoga retreats and read books about sacred healing and ancient wisdom. When I’m at these places, people sometimes find out that I’m a physics student who is interested in things like Science and Rationality. The types of responses I get are interesting.

Usually the people I talk to are enthusiastic and eager to talk about the most recent scientific discoveries they’ve heard of. They’re also quick to point out that Science can’t tell us everything, and after all there are the virtues of faith to be considered. Other times I feel a subtle shift in attitude. This might be paranoia, but it’s as I’ve been registered as somebody belonging to the Other Team.

And after all, important swamis declare that science is just another form of faith, and spiritual people nod knowingly. And the Deepak Chopras of the world declare with relish that science cannot tell us objective truths, and that scientists are arrogant and dogmatic.

This is all very weird to me. Science is our best systematized attempt to understand the world we live in and to unearth the general principles that guide this world. Great scientists are guided by a fascination with the order of the universe and wonder at its comprehensibility. At their root they want to understand, in Einstein’s words, the mind of God.

And the spiritual tell me that “spiritual” means something like “interested in pondering the nature of reality at a deep level and appreciating the awe-inspiring and profound aspects of existence.”

If this is how I should understand these terms, then spirituality and science are two things that should definitely definitely not be enemies. In fact, if “spiritual” meant what the spiritual claim it means, then the best spiritual seekers should be the same people as the best scientists.

Look at this quote from Carl Sagan:

Science is not only compatible with spirituality; it is a profound source of spirituality. When we recognize our place in an immensity of light years and in the passage of ages, when we grasp the intricacy, beauty and subtlety of life, then that soaring feeling, that sense of elation and humility combined, is surely spiritual.

And from Neil Degrasse Tyson:

It’s quite literally true that we are star dust, in the highest exalted way one can use that phrase. I bask in the majesty of the cosmos.

Not only are we in the universe, the universe is in us. I don’t know of any deeper spiritual feeling than what that brings upon me.

Are these not expressions of an utmost appreciation for the spiritual, as defined above? Why don’t the spiritual embrace Neil Degrasse Tyson and his scientific colleagues with open arms as fellow earnest truth-seekers, and marvel at the beauty of the universe together? I mean, just look at the man – he’s practically overflowing with the type of joy and curiosity that the spiritual should love!

The spiritual will tell me: “Yes, some of the greatest scientists are very spiritual. Look at Einstein! He said that science without religion is lame, and that all serious scientists recognize a Spirit in the laws of nature! Science at its best can be and should be a deeply spiritual enterprise. But unfortunately, a lot of scientists out there are just too close-minded. This is why the spiritual can sometimes sound anti-science, because the scientists of the world dogmatically reject our reasonable beliefs, like that the spiritually enlightened can read minds and make objects levitate, or that the stars are sending us secret messages about our romantic prospects and whether we should change jobs, or that playing cards thrown randomly onto the ground can accurately tell us our future!”

Yes, scientists can be dogmatic, because scientists are humans. But it strikes me that perhaps part of the reason that the spiritual might claim that scientists are especially dogmatic has maybe something to do with the fact that scientists have repeatedly studied and disproved common spiritual beliefs and practices. More importantly, many of these beliefs are in direct conflict with the known laws of nature. As the saying goes: keep your mind open – but not so open that your brains fall out.

The spiritual: “But science too often tries to go too far and dismiss those things which it doesn’t understand!”

What, like the possible physical effects that the stars could have on the paths that our lives take? Or like the effects of diluting a chemical compound until not a single molecule remains on the potency of the final product as a medical instrument? Or the ways that the lines on your palm form, that really really have nothing to do with how rich you’ll be or how many kids you’ll end up having?

No, this won’t do. Science does not understand everything. There are plenty of mysteries out there, and we love that there are. They give scientists employment! But scientists are certainly not in the business of blindly dismissing those things that they actually do not understand.

Besides, are scientists really all that dogmatic? Look at the history of the scientific worldview. Consensus theories are constantly recycled as we make the long march towards understanding reality. Some of the strongest scientific consensuses are only a few decades old! Scientists are constantly updating and refurnishing their view of reality as the evidence changes.

Perfectly? No! But I’d hazard a guess that they do so better than the average person. Why? For one thing, they have a career incentive to do so. A scientist that sticks to the old phlogiston-theory of combustion can’t get published, and a scientist that discovers damning evidence of the falsity of an important consensus gets tenure, pay raises, and respect from their colleagues. The incentive structure of science is set up to reward those that can avoid becoming stuck in dogmatic patterns of belief.


Physicist and philosopher Tim Maudlin described a feature of truth-seeking enterprises as that they tend to be uniform across space and to vary across time. Ask a biologist in Bengal what they think about the structure of DNA, and you’ll get pretty much the same answer as a biologist at Oxford. And when new evidence comes in, the beliefs of scientists shift fairly uniformly.

Ask a spiritual seeker in India what they think about Shamanic healing, and you’ll likely get a different answer from a spiritual seeker in the UK.

Yes, science has problems and is definitely not perfect. But we’re not comparing it to an ideal perfected version of science conducted by perfect Bayesian epistemologists with infinite computing power, we’re comparing it to humanity’s status quo. With rampant climate change denial, young Earth creationism, disbelief in evolution and anti-vaccination conspiracies, it’d be hard to convince me that scientists are much worse than the average Joe at avoiding patterns of dogmatic thought.

I just don’t buy that the high epistemic standards and regard for truth held by the spiritual is the reason that they dismiss science. I’ve met too many spiritual people eager to have their charts read by astrologers or obtain homeopathic sugar pills or communicate with invisible spirits. And I don’t buy that scientists are not actually honest truth-seekers trying to understand the world.

Which is why I think that the word spiritual doesn’t actually mean what the spiritual claim it means. I’m not being a linguistic prescriptivist here; I’m saying that the definition that spiritual people provide of spirituality is the motte, and the bailey is something else, something that is apparently hostile to science and friendly to all sorts of pseudoscientific ideas.

The bailey is where the fertile and valuable ideological land is, and the motte is the easily defensible position that spiritual people can retreat to when their beliefs are questioned. The bailey is not actually fundamentally about the urge to understand nature. It’s not actually about the same type of wonder and joy that a scientist gets when they understand some important piece of how the world works. Based off of many of the interactions I’ve had with self-identified spiritual people, I would define it as something like “belief in the existence of some phenomenon for which there is no evidence, or evidence against, like Reiki, crystal healing, tarot cards, etc.”


Looking at what I’ve written so far, it sounds like I see nothing but conflict between spirituality and science. This is not so. I have focused on the aspects of spirituality that do come into conflict with science, mostly because I think that these play a large role in the anti-scientific attitudes among the spiritual. The spiritual are quite friendly towards science when it supports their beliefs.

And it often does! There are spiritual practices that science has found to be genuinely beneficial, more than predicted by placebo effects, and beneficial in many of the ways that the spiritual claim them to be. Meditation and yoga come to mind. Mindfulness practices also have an impressive evidence base. And things like a belief in a higher power and spiritual experiences can be genuinely uplifting and transformative.

I’ve talked about spiritual people as if they were all the same, harboring irrational beliefs and anti-scientific attitudes. But plenty of spiritual people I meet are genuinely appreciative of the sciences, and want their world-view to be as fully supported by the scientific evidence as possible. Some are even scientists themselves!

And anti-science attitudes are not at all ubiquitous across spiritual traditions. Buddhism is often praised for its friendliness towards the sciences, and its scientific approach to belief formation. The Dalai Lama says things like:

If scientific analysis were conclusively to demonstrate certain claims in Buddhism to be false, then we must accept the findings of science and abandon those claims.

I don’t know enough about the Dalai Lama’s personal epistemic habits to be confident that this is more than nice-sounding words. How does he think that this attitude affects Buddhist views on karma and reincarnation, for instance?

It is much easier to proclaim a science-friendly attitude than it is to actually accept the tough implications of such an attitude on beliefs central to one’s ideology. But attitudes like this seem like the right way forward in reconciling the actual meaning of spirituality with the meaning that the spiritual seem to want it to have.