Lavarand: A Mind-Bending Application for a Classic Symbol of the 1960s

How an icon of psychedelia became a key component of our digital security

A lava lamp display. (CCA, Dean Hochman)

“Randomness is the true foundation of mathematics.” — Gregory Chaitin

Over the years, I’ve come to understand that unpredictability, in the form of randomness or chaos, is not simply a mathematical concept or a pervasive characteristic of Nature, but also an essential component of aesthetic beauty. I can think of no object that embodies the faces of unpredictability better than the iconic Lava Lamp, whose flowing forms are both mesmerizing and, as we’ll see, technologically useful.

The Lava Lamp

First produced in 1963 and dubbed the “Astro Lamp," the Lava Lamp was conceived of and developed over the course of 15 years by British WWII pilot, accountant, and naturist Edward Craven-Walker. He had been inspired by a rudimentary egg-timer built using a cocktail shaker, which he saw at a Dorset pub. At the beginning of the Space Age and on the cusp of the psychedelic movement, the sleek-looking Astro Lamp arrived at a propitious moment.

As a kid, I was fascinated by lava lamps. Entranced by the dance of the amorphous, flowing blobs of wax through a viscous, colored fluid, I knew only that heat made the blobs rise, and when they cooled, they fell. If, by any chance, you’ve never seen a lava lamp in action, here’s an example:

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What I didn’t know is that they are a showcase for a fundamental phenomenon known as Rayleigh-Taylor instability. Rayleigh-Taylor instability describes the dynamics at the boundary between two fluids with different densities. When a denser fluid sits on top of a less dense fluid, instabilities that arise at the interface become amplified and cause them to mix in a non-linear manner. The phenomenon makes frequent appearances in Nature, in everything from the fingery filaments in the Crab Nebula, to the swirling clouds of cream poured into coffee.

Crab Nebula is a grand example of Rayleigh-Taylor instability. (Public domain, NASA)

In the case of our lamp, at room temperature, the wax is slightly denser than the fluid and sits at the bottom of the tapered bottle. When a 40-watt light bulb warms the fluid and wax, it begins the process of convection (because of this, the movement of the blobs is sometimes attributed to Rayleigh–Bénard convection). In the simplest terms, as the wax heats up, its density decreases until it is slightly less dense than the surrounding fluid. At that point, it begins to deform, and blob-like fingers detach and rise. As they rise, the heat source becomes more distant. The blobs cool, become denser, and start to descend.

Craven-Walker worked by trial and error, tinkering for over a decade to achieve an optimal formulation for the immiscible combination of wax and fluid. That formulation remains a trade secret. In fact, to this day, very few of the factory workers know the precise composition of the lamp's contents.

Part of the allure of watching the ballet of the blobs is that we correctly expect that no two will ever look precisely the same. But why is this true?

The Two Faces of Unpredictability

In common conversation, folks will often use the words “random” and “chaotic” interchangeably when describing a process that is unpredictable. However, in the world of mathematics, the two represent different phenomena.

A process that is truly random is what we call nondeterministic. This means that there is no underlying rule that can predict the next state or outcome. This is true, even if we have perfect information about the initial starting conditions of the process. The best we can do is to make predictions based on an underlying statistical distribution that allows us to assess the likelihood of a particular outcome. Classic examples are radioactive decay and the rolling of fair dice.

In contrast, a chaotic process is entirely deterministic. This means that there are strict rules that govern how the system evolves. Given identical starting conditions, a chaotic system will unfold in exactly the same manner. The hallmark of such a system is what we call “sensitive dependence on initial conditions” — for two identical systems, a tiny difference in starting points leads to enormous and unpredictable differences later in the process. Classic examples include the double pendulum, population dynamics in nature (as captured by the logistic map), and everybody's favorite, the weather.

A plot of the starting trajectory for the Lorenz attractor. It starts at the grean point and is truncated at the red point. (Image by Author)

In fact, it was meteorologist Edward Lorenz who first observed chaotic behavior while developing a model to predict the effects of convection in the atmosphere. At one point, he found what appeared to be an error when comparing the outputs of his model. What he discovered was that the “error” resulted from his failure to use all six digits of accuracy in his data (he assumed that 3-digits would suffice). Above is my rendering of what is now referred to as the Lorenz attractor. It’s generated from the set of three differential equations he used in his model, along with his particular values (not all initial conditions lead to chaos).

A lava lamp follows the strict laws of physics with regard to gravity, density/buoyancy, surface tension, and thermodynamics. However, as in the case of the weather, no amount of knowledge will allow us to replicate any set of initial conditions. Even with two identically manufactured lava lamps, the most minuscule variation in, for instance, ambient air temperature, wax starting position, glass texture, or wattage fluctuation will lead to different patterns in their respective fluid ballets.

The Computational Pursuit of Randomness

Randomness is the backbone of digital security. Encryption depends on the sheer difficulty of guessing keys and, in the most basic sense, the more random a key looks, the harder it is to guess (technically, we want keys with high entropy).

It’s a simple fact of life that the deterministic nature of computers makes them unsuitable for producing randomness on the basis of logic alone. The best they can do is to generate pseudo-random numbers. These are produced by an algorithm called a pseudo-random number generator (PRNG) that is designed so that its output can pass statistical tests for randomness. The problem with this approach is well-illustrated by an early PRNG called RANDU.

In the 1960s, IBM developed RANDU to run on its mainframe. It was super fast, very popular, and on the surface seemed to produce unpredictable numbers. It used what is called a linear congruential generator (LCG) to hop from one number to the next. This is the formula:

Here, the “mod” operator means we are interested only in the remainder after dividing by 2³¹. Plotting the output in non-overlapping pairs {x, y}, the output looks random:

A 2D plot of the PRNG called RANDU. (Image by Author)

In reality, RANDU was a disaster. Here is what the picture looks like when we plot non-overlapping triples in the form of (x, y, z).

A 3D spectral test of the PRNG RANDU. (Image by Author)

As researchers discovered, the output is highly structured, falling along 15 parallel planes. It's estimated that thousands of scientific research papers were invalidated by this finding, often because scientists used RANDU for their Monte Carlo simulations.

To address this problem, engineers realized that they needed not only better algorithms but also the input from true sources of randomness. This is where the lava lamp comes in.

Harvesting Chaos for Better Security

The concept of entropy in information theory was introduced by American polymath Claude Shannon in 1948. We can think of it as the amount of “surprise” in a sequence of data. A sequence with high entropy is totally unpredictable — it is impossible to guess the next value. Conversely, a sequence with low entropy contains patterns that make it predictable.

While mathematicians were continually pursuing better algorithms for PRNGs, in 1996, engineers at Silicon Graphics (SGI) realized they could harvest the high entropy of lava lamps to produce random numbers. Their scheme had four basic components.

1. Physical harvesting: A CCD camera (charge-coupled device) takes a photo of six lava lamps, capturing the physical positions of their unpredictably meandering blobs. A key feature here is that the image data includes thermal noise (true randomness) from the sensor. Two simultaneous photos would therefore contain different data.
2. Data transformation: The color and intensity values of the pixels are converted into a raw data stream. The data stream is whitened to make it statistically uniform.
3. Seed production: The whitened data is fed to the hash algorithm SHA-1 (Secure Hash Algorithm 1). A secure hash algorithm is a sort of digital fingerprint. It's a “one-way street” that takes a long string of characters and produces a unique string of a fixed length that looks like gibberish. Hashes are designed so that changing a single bit in the input flips roughly 50% of the bits in the output. The 921,600 bytes of image data are thereby scrambled and compressed into a unique 160-bit seed.
4. Random output: That unique seed, which can never be reproduced, becomes the starting point for the cryptographically secure pseudorandom number generator (CSPRNG), known as Blum-Blum-Shub.

The CSPRNG can then produce an unpredictable output based on a physical process that has never happened before and will never happen again.

Figure 3 from SGI’s patent for Lavarand. (Public domain)

In the mid-1990s, there were urgent security concerns revolving around a flaw in the design of the PRNG used by the Netscape browser for encryption. With the expiration of SGI’s patent, the giant infrastructure and security company Cloudflare picked up where they left off, creating LavaRand (with a capital “R”). Given that roughly 10% of internet traffic flows through its servers, Cloudflare had a pressing motivation to improve its security.

Every time your browser communicates with a website, it performs what’s referred to as a “handshake” using TLS (Transport Layer Security). That handshake allows the browser and server to verify each other’s identities and negotiate a unique, secret encryption key to use for that session. Its security depends on a solid source of randomness. Cloudflare, therefore, requires a truly enormous supply of random numbers.

Where SGI’s implementation was meant to be a true random number generator (TRNG), Cloudflare’s goal was to use its local Linux entropy pools, drawn from sources like thermal noise and network timings, and supplement them with a second, independent source of randomness.

The Wall of Entropy in their San Francisco office holds 100 lava lamps. Where SGI's six lava lamps were housed in an isolated environment, Cloudflare’s lamps sit in the corporate lobby, where passersby, selfies, and changing light and temperature variations add further unpredictability to the mix. The lamps are photographed by modern high-definition webcams, which add more electronic noise than the original SGI CCD.

The wall of lava lamps at Cloudflare in 2022. (Source: Wikimedia Commons)

In case this weren’t enough, Cloudflare has decentralized the concept of entropy pooling:

• Their London office contains a wall of double pendulums, which are videoed for the pool.
• The Austin office contains twirling translucent rainbow mobiles that are periodically photographed for the pool.
• The Singapore office contains a Geiger counter driven by a pellet of uranium. Timings between pulses are converted to a number sequence and fed into the pool.
• The Lisbon office features a wall of 50 custom-built, 18-inch (45 cm) rectangular “wave machines” arranged on a wall and periodically photographed for the pool.

But wait… there’s more. Cloudflare went on to found a collaborative effort called the League of Entropy (Yes, sign me up!). It's a consortium of organizations around the world that contribute sources of entropy to a shared pool that provides what they refer to as a “distributed randomness beacon.” From their description:

A decentralized randomness beacon combines randomness from multiple independent high entropy sources to generate a truly unbiased random number for anyone that may need a public source of randomness.

That beacon is fundamentally immune to any adversarial attempt to tamper with or control the entropy used to generate encryption keys. Even the complete failure of a system (e.g., a power failure at the San Francisco office) would have no impact on the beacon.

Coda

It’s safe to say that when Craven Walker first started selling lava lamps, neither he nor anyone else could imagine that his entertaining and downright groovy invention might one day help to power the sophisticated encryption that underpins our global economy.

One reason for the unfolding of this improbable path is the enduring popularity of the lamp. While sales did slump in the 70s, today millions of lava lamps are sold globally. Had the lamp been a passing fad, the LavaRand inventors might never have known of it.

I suspect that it’s still going strong because the languid and serene motions of the ever-morphing display provide a welcome balm to the relentless and enervating pace of our digital age. That sense of serenity is enhanced by its overall aesthetic charm, the study of which is called neuroesthetics (see link below).

The outlines or silhouettes of natural objects tend to have a relaxing effect on people. (Photo by Author)

While most of us strive for stability in our daily lives, research in recent decades points to the fact that we tend to resonate with the unpredictability of natural scenery and with art that in some way resembles the forms of nature. For instance, studies show that people find images with a fractal dimension of approximately 1.3–1.5 more pleasing than those with higher or lower dimensions, indicating a sweet spot between a boring line (one-dimensional) and a super complicated squiggle (which can be two-dimensional).

In essence, we are hardwired to enjoy the moderately chaotic shapes that mimic many natural processes. This predisposition is not limited to images. We also enjoy unpredictability in the aural world — the soothing sound of light rain is an example of *pink noise*, a self-similar type of sound that people generally find soothing.

The serendipitous journey from Craven-Walker’s spotting of the egg timer, to his roughly twelve-year search for the ideal fluids, to the lamp’s launch at the cusp of the psychedelic era, its rebirth in popularity after the 70s, and its notice by SGI engineers is, taken together, an appropriately unpredictable chain of events.

One thing that has remained constant, though, is that this iconic symbol of unpredictability is still manufactured at the original location in Poole, Dorset, UK.

Thank you for your time! I’m Harlan Brothers, and I share the stories behind how and why math and science work. If you want to see more content like this, please follow me. I reply to every thoughtful comment — I’m always happy to start a conversation!

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Down the Rabbit Hole

• An excellent story from mathematician and Science Spectrum writer Laurel W: Math Misconceptions: Chaos vs. Randomness
• Wikipedia: Edward Craven Walker
• Wikipedia: Rayleigh–Taylor instability
• Video: How Lava Lamps Are Made
• An excellent story on the Monte Carlo method from fellow Science Spectrum editor Cole Frederick: The Power of a Simple Simulation
• The original 1996 patent for Lavarand: Method for seeding a pseudo-random number generator with a cryptographic hash of a digitization of a chaotic system
• Cloudflare’s blog post: Harnessing chaos in Cloudflare offices
• Cloudflare: League of Entropy
• Wikipedia: Neuroesthetics
• Research paper: Reduction of Physiological Stress Using Fractal Art and Architecture

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