Beyond Grains of Sand: From the Planck Length to the Edge of Infinity

(Note: You'll want to read this on my website for the math to render.)

As a fan of science journalism, I have a recurring pet peeve. If you read enough popular science, you will run into something like this: "There are more possible games of chess than there are atoms in the observable universe." Or perhaps you’ve heard it applied to a deck of cards, or the number of grains of sand on Earth.

As a physicist, this framing has always frustrated me because it treats two fundamentally different concepts as if they were the same thing. In the case above, it mixes up the physical tallying of actual, tangible matter with mathematical calculations of possibility. Comparing a shuffled deck of cards to a galaxy doesn't teach you anything about cards, and it certainly doesn't teach you anything about astrophysics.

But, what if we truly want to understand how astoundingly small and unfathomably large things can get? How would we do that? Well, we have to map how numbers scale across domains. While we can start by counting things in the physical universe, that can't be our final destination—it must be merely a stepping stone. Once we run out of physical stuff to count, we have to give the numbers some room to breathe. We transition into the math of probability and statistics. When those numbers get too immense to manage, we hand them over to the explosive growth of computer algorithms. And finally, when even the most powerful algorithms break down, we must step into the boundless realm of pure infinity.

To make sense of it all, we need a roadmap. Here is a proper tour through five distinct phases of increasing magnitude—a progression that allows us to genuinely understand just how truly small, and how unfathomably big, things can actually get.

Phase 1: The Sub-Unity Regime (Approaching Zero)

Before we climb upward, we have to look at the absolute bottom. In the physical universe, the infinitesimal is strictly bounded by the Planck scale. Formulated using the fundamental constants of nature, the Planck length in meters is exactly

$$\ell_P = \sqrt{\hbar G / c^3} \approx 1.616 \times 10^{-35}$$

Below this scale, spacetime dissolves into a quantized foam. To actually "see" or measure something that unimaginably tiny, the Heisenberg uncertainty principle tells us we would need to concentrate an enormous amount of energy into that tiny spot. But if you pack that much energy into a space that small, the immense gravity would cause the region to immediately collapse into a microscopic black hole. So, that's as small as things get!

But what happens if we shift our focus from physical size to the mathematics of probability? Strange things happen when physical systems get incredibly close to absolute zero energy. According to the Poincaré Recurrence Theorem, if you leave a closed physical system alone for a long enough time, it will eventually cycle back to exactly how it started. If our universe keeps expanding forever, it will eventually cool down into a dark, empty vacuum. But over trillions upon trillions of years, random, tiny quantum vibrations in that empty vacuum will theoretically bump particles together into every possible arrangement.

This leads to a famous thought experiment: the Boltzmann Brain Paradox. It would take a wildly lucky coincidence (a drop in entropy of roughly $\Delta S \approx 10^{66} k_B$) for those random vibrations to accidentally assemble a fully functioning, self-aware human brain floating in the void, complete with fake memories. However, it takes an even more ridiculous coincidence to accidentally assemble an entire, fully functioning universe like ours. Therefore, statistically speaking, randomly assembled brains floating in empty space should vastly outnumber biological brains living in a real universe. This is a nightmare for scientists: if it's true, you are almost certainly just a random brain hallucinating this exact moment, meaning none of our memories or scientific observations can be trusted.

Fortunately, our thinking on this has evolved. Modern quantum physics offers a much simpler escape. Current theories suggest that an eternally empty, still quantum vacuum simply doesn't have the active, ticking-clock mechanics required to actually spark a conscious entity into existence. We are saved by the stillness of the void.

Phase 2: The Macroscopic Physical Inventory

Having stared into the probabilistic void at the absolute bottom of scale, let's reverse direction and zip all the way up through the physical universe. This is the macroscopic inventory—the domain of whole numbers, where we map the absolute upper boundaries of the material cosmos we inhabit.

• Total Atoms: If we want to find the biggest numbers, a good place to start is with the stuff we can touch, see, and interact with. There are roughly $10^{80}$ atoms of baryonic matter in the observable universe. Baryonic matter is just the standard "normal" matter that makes up you, me, your computer, and every star and galaxy in the sky (primarily protons and neutrons). While $10^{80}$ sounds like a massive tally, in the grand scheme of the cosmos, normal matter is just a rounding error.
• Total Cosmic Relics: If our goal is to find the absolute most numerous physical things we can possibly count, we have to look past normal matter and turn to massless (or nearly massless) particles. When the Big Bang occurred, it flooded the universe with radiation that is still cooling, stretching, and propagating today. For every single physical atom of normal matter in existence, there are roughly one billion relic Big Bang photons ($\approx 1.5 \times 10^{89}$) and one billion relic neutrinos ($\approx 1.2 \times 10^{89}$). The particle inventory of our universe isn't dominated by stars; it is dominated almost entirely by this silent, invisible ocean of cosmic relics.
• The Ultimate Volumetric Pixel Count: What if we stop counting particles and start counting the physical "space" itself? If we take the entire spatial volume of the observable universe and divide it by a single cubic Planck volume (the absolute smallest unit of space before geometry breaks down into quantum foam), we find that 3D reality has a maximum volumetric resolution of roughly $10^{185}$ "pixels."

This $10^{185}$ limit represents a hard, physical ceiling. There is simply no physical stuff—and no physical space—left to count past this number.

However, as I explored in a recent post about testing quantum gravity using a galactic microquasar, the Holographic Principle dictates something far stranger. The maximum information capacity of a region of space isn't determined by its 3D volume, but is actually constrained by its 2D surface area. Holographic Duality suggests that the gravitational physics happening inside a 3-dimensional volume can be perfectly described by a 2-dimensional quantum system living on the flat boundary surrounding it. The surface area of our observable universe is only roughly $10^{124}$ Planck areas. This staggering discrepancy—that the 3D "bulk" contains vastly more pixels than the 2D boundary can mathematically encode—is exactly why physicists suspect our 3D reality is actually an emergent, lower-dimensional projection.

Phase 3: The Combinatorial Explosion

In order to count things bigger than $10^{185}$, we have to completely change our approach. We must stop counting objects and start counting states. Think of it this way: if Phase 2 was about counting every single physical grain of sand in the entire cosmos, Phase 3 is about calculating the total number of ways you could possibly arrange them. This is the realm of the combinatorial explosion, where game theory, probability, and cryptography take over.

In this third phase, numbers no longer grow by simply tallying more physical stuff. Instead, they scale violently through factorials and exponentials.

Let's look at cryptography, which relies on these massive state spaces to guarantee strong protection. A standard symmetric keyspace like AES-256 gives us $2^{256}$ (roughly $1.15 \times 10^{77}$) possible combinations—a number right on the heels of the total atom count of the universe. When we step up to asymmetric cryptography, an RSA-2048 public key pushes the integer magnitude all the way to $2^{2048}$, or roughly $3.23 \times 10^{616}$. And for the cryptography nerds looking ahead to the post-quantum (PQ) transition, lattice-based algorithms like ML-KEM (Kyber) construct even wilder mathematical spaces. By operating on polynomials in hundreds or thousands of dimensions, finding the "shortest vector" in these cryptographic lattices requires navigating a state space so unfathomably vast that even a theoretical quantum computer cannot brute-force its way through the haystack.

These combinatorial explosions happen in incredibly familiar places, too. If you shuffle a standard deck of 52 playing cards, the number of possible arrangements is $52!$ (52 factorial), which is roughly $8.06 \times 10^{67}$ unique permutations. To put that in perspective, every time you give a deck a truly randomized, clean shuffle, you are almost certainly holding an exact sequence of cards that has never existed in the history of the universe, and will likely never exist again.

If we look at the game tree complexity of the game Go, the board is only a 19x19 grid, but the number of legal paths a game could possibly take branches exponentially, eventually scaling to an astonishing $10^{700}$.

These numbers completely dwarf the physical particle counts of Phase 2, but they still share a fundamental limitation: they are finite, computable states that map to distinct probabilistic realities. If we want to count things even bigger, and push past $10^{700}$, we are going to have to break standard arithmetic entirely and invent a fundamentally new way to write numbers.

Phase 4: The Fast-Growing Hierarchy

Ok, what's bigger than $10^{700}$?

As a physicist, this is the point where I start to get dizzy. At a certain scale, standard scientific notation ($10^n$) completely fractures. Why? Because if we want to build even bigger numbers, we have to stop just counting combinations and start writing algorithms. We have to use recursion—creating mathematical rules where the output of one massive calculation is immediately fed back in as the starting input for the next. To write down the numbers this kind of runaway algorithmic logic can generate, you would need exponents stacked on top of exponents stacked on top of exponents. The "power tower" of tiny superscript numbers would quickly reach the moon. Exponentiation is simply too slow to keep up.

To chart this, mathematicians use the Fast-Growing Hierarchy, a system that categorizes how violently different algorithms explode.

Instead of multiplication or simple exponents, we step up to iterated exponentiation. The legendary computer scientist Donald Knuth invented up-arrow notation to handle these massive power towers.

• A single arrow is standard exponentiation: $3 \uparrow 3 = 3^3 = 27$.
• Two arrows means "tetration", or stacking power towers. $3 \uparrow\uparrow 3$ doesn't mean $3 \times 27$; it means $3 \uparrow (3 \uparrow 3)$, which evaluates to $3^{27}$, or roughly $7.6 \times 10^{12}$.
• Three arrows ($3 \uparrow\uparrow\uparrow 3$) means you create a power tower of 3s that is exactly $7.6 \times 10^{12}$ levels tall. Let that sink in for a moment. That is $3^{3^{3^{3^{\dots}}}}$ continuing upward for over seven trillion levels. If you tried to physically write out that equation, and each "3" was just one inch tall, the stack of exponents would reach from the Earth all the way past the Sun. And remember, that solar-system-spanning tower just represents the equation to calculate the number, not the final number itself.
• Four arrows ($3 \uparrow\uparrow\uparrow\uparrow 3$) means you evaluate that entire sun-spanning power tower, and use that mind-bending, incalculable number just to dictate the height of your next power tower.

With that terrifying engine in mind, let's look at the monsters of this phase:

• Graham’s Number: An upper bound in Ramsey theory, this number is constructed using a recursive feedback loop of Knuth's arrows. You start with four arrows: $3 \uparrow\uparrow\uparrow\uparrow 3$. Let's call that unimaginable number $g_1$. To calculate the next step, $g_2$, you evaluate 3 [some number of arrows] 3. But how many arrows do you use? Exactly $g_1$ arrows. So, $g_2$ is a 3, followed by a mind-melting $g_1$ number of up-arrows, followed by another 3. You repeat this recursive feedback loop 64 times, with each new layer using the previous layer's incomprehensibly massive result merely to dictate the number of arrows in the current calculation. The final output ($g_{64}$) is Graham's Number. It is so unfathomably dense that if you tried to encode its individual digits into your brain, the information density would exceed the Bekenstein bound, collapsing your head into a black hole.
• TREE(3): Derived from Kruskal's Tree Theorem in graph theory, the TREE sequence is governed by a simple rule: no smaller tree can be structurally "inf-embedded" into a subsequent larger tree (meaning you cannot trace the shape of a previous tree inside the branches of a new one). Because of how this logic works, the sequence must eventually terminate, but the combinatorial explosion is violently fast. Why? Because while Graham's number uses simple recursive operations (stacking arrows), the TREE sequence forces the math to explore an unimaginably vast space of branching tree structures before it finds a repeating pattern. It grows so fast that it makes Graham's number look like a rounding error. It is so large that if you converted the entire observable universe into a supercomputer, it would lack the physical states required to compute even the number of digits in TREE(3), let alone the number itself.
• Rayo’s Number: Finally, we hit the absolute ceiling of computability. Every number we have discussed so far, including TREE(3), can theoretically be calculated by a Turing Machine—a hypothetical computer with infinite memory and infinite time. But theoretical computers have limits; there are numbers that grow so fast that no step-by-step algorithm can ever compute them without getting stuck in an infinite loop (a boundary known as the Busy Beaver limit). To transcend Turing limits entirely, Rayo's Number abandons computer algorithms and utilizes formal set theory. It is defined as the smallest integer strictly greater than any number uniquely namable in the language of first-order set theory using at most $10^{100}$ symbols. It completely subsumes the Fast-Growing Hierarchy by relying on the absolute truth of the mathematical universe rather than the mechanical steps of a computer.

Phase 5: The Transfinite Regime

Once we pass Rayo's Number, the integer number line terminates. We can no longer build numbers using step-by-step algorithms. Instead, scale is measured by cardinality—the structural size of infinite sets. To navigate this, we have to rely on the absolute foundational operating system of modern mathematics: Zermelo-Fraenkel set theory (ZFC). We care about ZFC because when you run out of computable numbers, the fundamental axioms of math are the only tools left to measure reality.

We often think of infinity as a single, final destination. It isn't. In the late 19th century, mathematicians discovered that infinities come in different sizes. For example, the infinity of continuous decimals (like the real number line) is strictly, uncountably larger than the infinity of the standard whole numbers. This revelation proved that infinity isn't a hard ceiling; it is an endless ladder.

But how big can infinity get? To climb higher, mathematicians postulate Large Cardinal Axioms. These declare the existence of infinities so unimaginably vast—like Inaccessible cardinals—that they form completely self-contained mathematical universes. Because they are entirely self-contained, our standard mathematical operating system (ZFC) isn't powerful enough to even prove they exist.

The absolute, theoretical ceiling of this transfinite scale is the Reinhardt Cardinal. It proposes a function that perfectly maps the entire infinite mathematical universe into a smaller copy of itself.

In 1971, Kenneth Kunen proved that this is mathematically impossible, as it creates a fatal logical contradiction. To tie this back to the physical universe, Kunen's Inconsistency Theorem is a beautiful, logical mirror to the Holographic Principle we discussed in Phase 2. Just as you cannot physically compress the total information of a universe into a restricted physical boundary without destroying it, you cannot compress the absolute infinity of the mathematical universe into a restricted projection of itself. Standard mathematics simply breaks down under the paradox of absolute self-containment.

That is the true edge of scale. Not a beach full of sand, but the precise boundary where the logic of mathematics refuses to allow a universe to swallow itself.