The Constant Beneath the Hype
π, modern machine learning, and the mathematical infrastructure we keep forgetting
The oldest math in the room
On March 14, 1988, physicist Larry Shaw walked into the Exploratorium on the edge of San Francisco Bay and decided the date deserved a ceremony. Not because anything had happened that day. Because the date itself, written in American shorthand, read 3.14. The ratio of a circle’s circumference to its diameter. One of the most stubbornly irrational numbers in mathematics.
Shaw handed out fruit pies. Staff marched in a circle around a commemorative plaque. The Exploratorium, which has always understood that wonder is a legitimate pedagogical instrument, leaned in. By 1989 it was annual. By 2009 the U.S. House of Representatives had passed a resolution recognizing the day officially.
San Francisco inventing π Day is not a coincidence. For decades the Bay Area has organized itself around the belief that mathematics, engineering, and software can reshape how the world works. That conviction produced semiconductors, the internet, search engines, and the generation of large language models running in data centers right now. The relationship between that mathematical culture and the social consequences of what it created is a separate and unresolved conversation. But the commitment to mathematics as a practical instrument is real.
π sits at the center of that tradition in a way that is easy to overlook. It shows up in domains that feel completely unrelated to circles. Signal processing. Probability theory. Quantum mechanics. Rotational geometry. The positional encoding schemes inside the transformer architectures that power many modern language models. If you want to understand why mathematical foundations matter in these systems, π is a useful entry point, because it forces you to ask why a constant derived from the geometry of a circle would keep appearing in systems that have nothing obvious to do with circles.
The answer is structural, not mystical. And it tells you something important about what modern machine learning actually depends on.
Why π refuses to stay in geometry
Most people encounter π in grade school, in the context of circles. Those formulas are accurate as far as they go. They do not explain why π appears in the Gaussian distribution, or in the Fourier transform, or in the Schrodinger equation, or in the probability that two randomly chosen integers share no common factors, which happens to involve π squared. None of those contexts involves circles in any obvious way.
The reason π appears so broadly is that π is the signature of periodicity and rotational symmetry. Systems that oscillate, repeat, or rotate almost inevitably produce π in their governing equations. This is not coincidence. It is a structural feature of how the mathematics of continuous systems works.
Consider a pendulum. Its motion is periodic. The equations governing that motion involve trigonometric functions, which are defined in terms of the unit circle, which introduces π. Now scale that reasoning outward. Any system described by wave equations, any system that oscillates around an equilibrium, any system with rotational symmetry, will produce π in its structure. The natural world is full of such systems: electromagnetic waves, sound, quantum probability amplitudes, orbital trajectories, the patterns in data when you decompose it into frequency components.
This is why π keeps appearing in fields that seem disconnected from geometry. The connection is not decorative. It is required.
One of the cleaner ways to see this is through the Gaussian distribution, the probability distribution underlying measurement error, natural variation, noise in communication channels, and a substantial fraction of classical statistics and machine learning. Its probability density function contains π in the normalization constant, the term that ensures the distribution integrates to one across its entire range. Remove that constant and the mathematics breaks. The distribution does not represent a valid probability model.
Why does π appear in a probability distribution? Because the normalization constant derives from the geometry of continuous space. When mathematicians worked out how to normalize the Gaussian function over all real numbers, the integral they had to evaluate involved the geometry of high-dimensional space, and π emerges from that geometry exactly as it emerges from any calculation involving continuous rotational symmetry. The connection between geometry and probability runs deeper than it looks.
Once you recognize that pattern, the presence of π across scientific fields stops being surprising. Systems involving periodicity, geometry, or continuous probability produce π. Modern computing systems rely heavily on all three.
Where π actually lives in modern systems
The vague framing of this argument goes: neural networks rely on π-based calculations. That is true in roughly the same sense that bridges rely on arithmetic. Technically correct and useless as explanation. The more useful question is where π appears, why, and what that tells you about the structure of the systems involved.
Transformer architectures and the geometry of sequence
Transformers, the architecture underlying most modern large language models, process information in parallel rather than sequentially. This parallelism is what makes them computationally tractable at scale. But it introduces a problem: the architecture has no inherent notion of sequence. A transformer sees all tokens simultaneously. Without additional structure, it cannot distinguish between the same words in different orders.
The original solution, from the 2017 paper that introduced the architecture, was sinusoidal positional encoding. Each position in a sequence is assigned a vector built from sine and cosine functions at multiple frequencies. Those functions are periodic. Their period is defined in terms of radians. A full cycle is 2π. π is in the mathematical definition of the encoding.
Most leading models have since moved to rotary position embeddings, known as RoPE. The approach encodes sequence position by applying rotational transformations to the query and key vectors in the attention mechanism. Instead of adding a positional vector, RoPE rotates the feature vectors in a way that encodes relative position. The rotation is defined by sine and cosine functions. The reason those functions work for encoding position is that they capture rotational geometry, and rotational geometry involves π.
Llama 3 uses RoPE directly. Llama 4, released in April 2025, introduced iRoPE, which interleaves standard RoPE attention layers with NoPE layers that carry no positional encoding. This hybrid approach, combining explicit rotational position signals with layers that infer position from learned patterns, allowed Meta to reach a 10 million token context window in the Scout model, trained on sequences of only 256,000 tokens. The architecture is still built on RoPE as its positional foundation. The extensions modify how that foundation is applied, not what it is.
The attention mechanism that allows a language model to determine which tokens are relevant to which other tokens across long sequences is built on this rotational structure. π is not peripheral to that mechanism. It is in the definition of the transformations.
Fourier analysis and the frequency-domain perspective
When a speech recognition system processes audio, it does not analyze the raw waveform directly. It converts the signal into frequency space using a Fourier transform, identifying which frequencies are present and at what amplitudes. The same operation underlies image processing in computer vision, compression in data pipelines, and filtering in sensor fusion systems.
The Fourier transform represents a signal as a sum of sinusoidal components. π appears explicitly in the transform kernel. Every frequency analysis run in an audio pipeline, every convolutional operation processing image data in frequency space, every spectrogram feeding a speech model, carries π in its structure.
In systems that handle audio, sensor data, or signals of any kind, Fourier-related computations are operationally important all the way down to implementation. At the chip and systems level, the distinction between dense linear algebra and signal-oriented computation is not academic. It affects architecture, memory bandwidth requirements, and what can be implemented efficiently in hardware. For general-purpose accelerators, the dominant workload is tensor operations. But in systems handling audio, photonic signals, or sensor fusion, the Fourier mathematics persists into implementation concerns in ways that are hard to route around.
Probability, normalization, and generative systems
Probabilistic machine learning methods appear throughout the current generation of systems. Bayesian inference, Gaussian processes, and variational frameworks all rely on probability distributions that include π in their normalization. Diffusion models, which power much of current generative image and video production, are built on Gaussian noise processes. The mathematical structure of denoising diffusion probabilistic models depends directly on the properties of Gaussian distributions, which carry π in their foundational terms.
This does not mean π is the active ingredient in generative systems. It means the mathematical framework used to build and train these models inherits π from the probability theory it depends on, which inherits it from the geometry underlying continuous probability distributions. The ancestry runs deep.
The women who built the substrate
None of the figures in this section worked on machine learning as the field exists today. That is precisely the point. They built the mathematical and computational infrastructure that modern systems depend on, often without recognition, in several cases despite active institutional resistance. The machines are new. The foundations are not.
Ada Lovelace: the abstraction that made software possible
Ada Lovelace, working with Charles Babbage on his proposed Analytical Engine in the 1840s, is credited with writing the first algorithm intended for execution by a machine. Her notes on the Engine contain a method for computing Bernoulli numbers, a sequence with deep connections to π through the Basel problem and related infinite series.
The deeper contribution was conceptual. Lovelace recognized that a machine capable of manipulating symbols according to rules could operate on any domain expressible symbolically, including music, logic, and language. That recognition, written nearly two centuries ago, is the intellectual ancestor of every programming language, every compiler, and every language model that processes text. The abstraction she identified, separating the rule-following machine from the domain it operates in, is what makes general-purpose computation possible.
Katherine Johnson: precision as a life-safety requirement
Katherine Johnson spent her career at NASA calculating trajectories, first as a human computer in the segregated West Computing group at Langley, later as a mathematician whose verification was specifically requested by astronauts before they trusted the electronic machines. John Glenn’s team asked her to check the computer’s numbers before Friendship 7 launched in 1962.
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Orbital mechanics is saturated with π. Kepler’s laws involve π in the calculation of orbital periods. Angular velocity, elliptical orbit geometry, and the equations describing trajectories through gravitational fields involve trigonometric functions throughout.
The precision demanded by that work is worth holding. An error in a trajectory calculation did not produce a wrong number on paper. It produced a vehicle that missed its reentry corridor. Mathematical rigor was not an academic standard. It was a life-safety requirement.
That framing, where the cost of error is measured in real consequences rather than benchmark scores, is exactly how systems deployed in high-stakes environments need to be evaluated now. The benchmark is not the deployment.
Dorothy Vaughan: reading the transition before it arrived
Dorothy Vaughan led the West Area Computing section at NASA and later supervised the agency’s transition to electronic computers. When FORTRAN arrived, she taught herself the language and then taught her team, before management had formally decided to train anyone.
That is a specific kind of technical judgment: seeing past the immediate task to the structural change arriving behind it, and preparing before the institution was ready. The human computers who survived the transition were the ones who became machine programmers. Vaughan understood this and acted on it.
The organizations navigating the current infrastructure transition well are the ones developing observability, governance, and auditability systems before regulators require them, before failures make them unavoidable. Anticipatory technical positioning is a form of engineering.
Mary Jackson: where equations must match the wind tunnel
Mary Jackson worked in the Supersonic Pressure Tunnel at Langley, analyzing data on airflow around vehicles at high speed. Fluid dynamics is a domain where periodic structures are fundamental. Turbulence, wave propagation, and pressure distributions over curved surfaces involve differential equations whose solutions carry trigonometric functions throughout.
Jackson became NASA’s first Black female aerospace engineer. The standard her work held was that equations had to match physical reality. Discrepancies had to be understood, not averaged away.
That standard applies directly to deployed machine learning systems. Training accuracy is not deployment reliability. The gap between them is where infrastructure work lives.
Hedy Lamarr: signal security from an unexpected direction
Hedy Lamarr is the most underrated engineer on this list. She is primarily remembered as a film actress. She should also be remembered as the co-inventor of spread-spectrum frequency hopping, which she developed with composer George Antheil in 1942 to create jam-resistant radio guidance for torpedoes.
Frequency hopping works by rapidly switching the carrier frequency of a transmission according to a pattern synchronized between sender and receiver. The Fourier representation of a frequency-hopped signal involves π throughout. The switching patterns Lamarr and Antheil designed required understanding how signals behave in frequency space.
The patent was not commercially exploited during her lifetime. The technology became foundational to WiFi, Bluetooth, and GPS. She received the Electronic Frontier Foundation Pioneer Award in 1997. The gap between contribution and recognition was fifty-five years. That is not a footnote. It is a pattern worth naming explicitly.
Grace Hopper: the abstraction layer that made software scalable
Grace Hopper built the first compiler. Before her, programming meant writing directly in machine code or assembly, instructions specific to a particular machine’s architecture. The idea that a program could translate higher-level symbolic instructions into machine code was considered impractical. Her colleagues said it could not be done. She built it anyway.
The compiler is the abstraction layer that made software development scalable and portable. Every training run and inference request happening right now runs on code that was at some point translated from a higher-level language into machine instructions. Hopper built the first such translator. The abstraction principles she established are still load-bearing in every software stack built since.
What the math cannot provide
There is a version of the π Day argument that stops at the historical inventory: machine learning depends on old mathematics, that mathematics involves π, and therefore π is a reminder of how deep the foundations go. That version is accurate. It is also insufficient.
The more important argument is about what mathematical elegance does not guarantee.
Current machine learning research invests heavily in model performance: larger training datasets, improved architectures, more compute. The benchmark results are real. The capabilities are genuine. But systems deployed in real environments encounter constraints that models trained on curated data cannot resolve by themselves.
Deployed systems encounter data drift, where the distribution of inputs in production diverges from training. They encounter edge cases the loss function was not designed to handle. They operate on hardware with latency constraints and failure modes. They function inside organizations with audit obligations, regulatory requirements, and accountability structures that do not map cleanly onto loss functions.
A model may converge cleanly during training. That does not mean the system built around it will behave predictably when deployed at scale, under operational stress. Convergence is not governance. Optimization is not accountability. Mathematical elegance is necessary. It is not sufficient for reliability.
The history of engineering makes this point repeatedly. Aircraft design depends on mathematically elegant equations describing aerodynamics. Planes do not fly safely because the equations are beautiful. They fly safely because engineers built layered systems that monitor, verify, and correct real-world behavior: redundant sensors, independent control surfaces, failure protocols, accumulated operational knowledge about where the models are incomplete. The mathematical foundation is required. The infrastructure built on top of it is what produces safety.
Machine learning systems will face the same reality. The consequential infrastructure work over the next decade will not be discovering new mathematical constants or novel architectures. It will be building the observability, traceability, and auditability infrastructure that lets these systems behave reliably in the environments they are actually deployed in, rather than the idealized environments they were trained in.
The women in this article understood the gap between mathematical correctness and operational reliability because their work lived in that gap. Johnson’s calculations had to be right enough that a crew came home. Vaughan’s programs had to run correctly on machines whose behavior was not fully characterized. Jackson’s equations had to explain what the wind tunnel was actually measuring. Mathematical rigor was the foundation. Engineering judgment about where the models were incomplete was the work.
That framing applies directly to where machine learning infrastructure is now. The field has the mathematical foundations. What it is still building is the discipline to know when those foundations are insufficient and the infrastructure to detect it in production.
Old constants, new machines
The Exploratorium still celebrates π Day. The circular plaque Larry Shaw used for the original march is still there. The tradition he started, connecting mathematical ideas to physical experience, insisting that abstract structures deserve examination alongside application, continues.
San Francisco has changed considerably since 1988. The technology industry that concentrated here has produced wealth and dislocation simultaneously, in proportions that remain unresolved. The relationship between this city’s mathematical culture and the social consequences of the industry it generated is a complicated ongoing conversation.
But the mathematics has not changed. π is the same ratio it was when Archimedes bounded it using inscribed and circumscribed polygons around 250 BCE. It is the same constant that appears in Euler’s identity, in the Gaussian distribution, in quantum probability amplitudes, and in the rotary position encodings running in data centers right now. Llama 4’s iRoPE architecture, which reached a 10 million token context window in 2025, is built on rotational geometry that Archimedes would recognize as the geometry of circles.
The women in this article worked in different eras, under different constraints, in different technical domains. What they shared was mathematical seriousness: working at the level of precision the problems actually required, without shortcuts, without approximations that could not be defended, in fields where errors had consequences. That standard does not age. It does not become obsolete when the compute substrate changes.
Every language model running today depends on mathematical frameworks built decades or centuries before the field existed, developed in part by people whose names took too long to reach the history books, embedded in architectures that most users will never examine.
The newest machines still carry the fingerprints of very old mathematics.
Happy π Day from San Francisco.
This series covers infrastructure, governance, and the systems-level questions underneath modern machine learning. Previous articles address authority structures, identity observability, and the economics of open source infrastructure.
