The story of two female programmers running the simulations behind chaos theory's discovery

The life of Edward Lorenz, who originally used a seagull in his 'butterfly effect' metaphor

By accidentally finding that rounding one variable to its third decimal place and re-running a weather simulation produced a drastically different two-month forecast, the American mathematician established the field of chaos theory. His paper—"Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?"—introduced the well-known metaphor for the behavior of chaotic systems. (Some readers may experience a paywall.)

An interview with James Yorke, the man who coined the mathematical term 'chaos'

Along with co-author Tien-Yien Li, the former University of Maryland mathematics department chair published the paper, "Period Three Implies Chaos" in 1975. "Chaos" then became the terminological placeholder for systems with sensitive dependence on initial conditions, which had been well known to mathematicians for over a decade.

Grace Hopper was inspired to join the armed forces after the attack on Pearl Harbor

Despite being too old to enlist and weighing below the required minimum to serve, Hopper was granted an exemption and assigned to work on the first electromechanical computer produced in the US. She and her team spent three months using the device to solve equations needed to design the implosion mechanism for the atomic bomb.

English mathematician Ada Lovelace wrote the first computer algorithm a century before computers existed and hypothesized that computing devices would one day be able to compose music—foreseeing the generative AI systems of today.

The origin of chaos theory, colloquially known as the 'butterfly effect'

The theory explains that systems following well-understood physical laws can still produce unpredictable outcomes due to small variations in their initial conditions. Originally identified in weather data, elements of chaos have been observed in stock market fluctuations, cardiac arrhythmias, and the evolution of online social movements.

View the Lorenz attractor, a graphical solution to chaotic equations that resembles a butterfly

The discovery of chaos theory came from Edward Lorenz's computer model of the atmosphere, which used three equations to quantify atmospheric circulation. However, the chaotic evolution of this system creates wandering figure-8-like shapes—some resembling butterfly wings—as it repeatedly attempts to settle on one of two values along unique trajectories.

The 'butterfly effect' is the informal name for chaos theory

In mathematics, chaos theory involves complex systems that are highly sensitive to their initial conditions. Originally discovered when analyzing weather data, slight variations in the data input into such systems create drastically different results, despite following known physical laws.

FLOW-MATIC was the first programming language resembling the English language

As part of the team working on UNIVAC—the first commercially produced electronic digital computer in the US—Grace Hopper helped develop the system that enabled nonprogrammers and businesspeople without a strong mathematics background to use the machine effectively, thereby enhancing the computer's commercial viability.

Explore interactive simulations of strange attractors, the graphs that help visualize chaos

Although simple, linear equations reach an explicit value for a given input, such as *f(x) \= x* + 1 equaling 4 when *x* equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

Jacqueline Jenkins-Nye—Bill Nye's mother—worked as a code-breaker in World War II

While in the Navy, she was recruited to work on Enigma to help decode messages from the German military. Historians estimate that the efforts of code-breakers may have collectively shortened the war by as much as two years.

Grace Hopper warned against prioritizing hardware purchases in future computing

Foreseeing the rising volume of data and the increasing demand for its processing and access, she advocated that organizations first map the total flow of information they require and then choose infrastructure to support it, rather than wastefully spending on the fastest computing technology and forcing software to compensate for unnecessary hardware or incorrect data.

The first compiler to facilitate communication with computers was developed by Grace Hopper

The A-0 program linker automatically translated mathematical symbols into machine-readable binary code. Although it took two years for computer executives to get on board with the idea, the technology was foundational to future programming languages that allowed computers to understand commands written in word-based languages.

Later in life, the clock in Grace Hopper's office ran counterclockwise

Acknowledging that most people have "an enormous tendency to resist change," Hopper's clock served as a reminder for the computer scientist that just because things have been done a certain way does not mean they should forever remain that way: "They love to say, 'We've always done it this way.' I try to fight that." (Some users may experience a paywall.)

Computational and biological systems share fundamental principles, like randomness

Proposed by Alan Turing and John von Neumann, the concept originates from how DNA and computers run by executing encoded instructions in parallel. The advent of AI and the temperature parameter has seen neural networks mimic the randomness of protein encoding.

At the age of seven, Grace Hopper dismantled alarm clocks

Foreshadowing a career in computers, Hopper exhibited a curiosity for gadgets at a young age and took apart seven clocks, determined to understand how they worked. Despite having a dollhouse, Hopper built an elevator for it and showed a greater interest in building toy furniture than playing with dolls.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

Grace Hopper won the Data Processing Management Association's 'Man of the Year' award

The inaugural recipient of the award—symbolic of the male-dominated nature of computer science—was honored in 1969 for her contributions to the development of COBOL and the first compiler. Hopper would also become the first woman from any country and the first American to be recognized as a Distinguished Fellow of the British Computer Society.

Designed in 1937 by Harvard graduate student Howard H. Aiken, the 'Harvard Calculator' was used by the US Navy to solve advanced physics problems.

Mathematics proves turbulence's chaos creates super diffusion

Turbulent flow is responsible for the rapid scattering of dandelion seeds, volcanic ash, and untied balloons. Its swirling behavior accelerates particle scattering faster than other mixing methods, as particles acquire energy from the interactions of eddies.

Learn how natural systems exhibit characteristics of chaos theory

In biology, some mathematical models of ecosystem populations and heart irregularities show strong sensitivity to initial conditions, which scientists can simulate to determine quantitative tipping points at which these systems become unstable. Within the social sciences, crowd behavior and the virality of social media posts have shown how human behavior fits within the framework of chaos theory.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

The story of Susy Thunder, the female hacker who disappeared

Here's the story of Susan Thunder, a woman who entered the hacker underground scene in the late 1970s and one day disappeared completely. She had a troubled past, facing abuse and hardship, and found solace in manipulating phone systems and later computer systems. She became involved with prominent hackers and her ability to control systems mirrored her quest for personal control. Her story highlights the role of social engineering in hacking.

The work of Claude Shannon, for whom Anthropic's AI chatbot is named

Known as the father of information theory, Claude Shannon applied Boolean algebra—a subset of mathematics that quantifies true or false statements as 1 or 0, respectively—to circuit design to develop the concept of a binary digit. His paper "A Mathematical Theory of Communication" (1948) laid the groundwork for all modern digital systems.

Quantum computers can simulate nature more effectively than classical computers

Ordinary computers struggle to simulate quantum fields due to entanglement, but systems built from those elements can model the complexity. Systems using particle in more than two states—qudits—have successfully modeled 2D electromagnetic fields and done so more efficiently.

Computer pioneer Alan Turing was also an elite runner

Alan Turing is credited with helping to found modern computer science, in addition to cracking the Nazi code known as Enigma. Beyond these world-historical achievements, the British mathematician also nearly represented his country in the Olympics in the marathon, running just 11 minutes slower than the eventual gold recipient at the 1948 games. Read more on this lesser-known aspect of the man here.

Noise and instability made quantum computing seem out of reach for decades

Recent advances in error suppression, mitigation, and correction are allowing quantum computers to perform practical tasks that challenge classical machines. IBM’s 2016 launch of its first cloud-accessible quantum computer attracted over 22,000 users in its first month alone.

Grace Hopper was named the 1959 Time magazine 'Woman of the Year'

The honor was part of the 2019 series "100 Women of the Year," which spotlighted an influential woman for each of the years the publication had previously awarded a "Man of the Year." In its write-up, Hopper was described as having "paved the way for women everywhere to thrive in math, computer science, and service to their countries."

The first algorithmic random number generator was developed in 1946

Although previous methods of randomness involved physical tools with limited outcomes, such as dice, or relied on chaotic variables for number generation, such as electronic noise, John Von Neumann's middle-square method turned the problem into a mathematical computation. It has since been replaced by methods that generate numbers from unique initial values, such as the time on a clock.

How a competition honoring a king's birthday uncovered history's first chaotic system

A committee established a contest for King Oscar II's 60th birthday, featuring four unsolved problems in mathematics, including one on proving the stability of a system of more than two objects held together by gravity. Focusing on three celestial objects, Henri Poincaré won the prize for his proof, which he later learned was incorrect and, upon correcting it, instead proved these systems to be unstable.

The world's first chatbot, ELIZA, was developed in the 1960s as a technical demonstration exploring human-computer communication, with one of its scripts allowing it to emulate a psychotherapist.

The foundation for computer viruses was published after its creator's death

In the late 1940s, Hungarian American mathematician, scientist, and engineer John von Neumann gave a series of lectures where he explained the theoretical framework for a computer program that could replicate itself—the von Neumann Universal Constructor. His lectures were later compiled and published in 1966, nine years after his death from cancer.

Jim Simons, mathematician turned government codebreaker turned trader, revolutionized quantitative investing

The late Jim Simons is remembered as one of the most successful (and profitable) hedge fund managers of all time. He went from physically gathering microfilm on commodities pricing decades ago to overseeing a firm that led Wall Street in daily trading volume.

The Grace Hopper Celebration honors women in computing every year

Since 1994, the world's largest gathering of women technologists has been produced by The Anita Borg Institute, a nonprofit driving intersectional equity in the global technical workforce. The event honors the legacy of Grace Hopper by highlighting the research and impact of women in technology.

A mathematical and visual analysis distinguishing chaos from randomness

Random systems lack any underlying pattern or causal relationships, while chaos is mathematically deterministic because the behavior follows established laws of nature. In 3D plots, chaotic systems produce outputs that approximate a mathematical function, while random points appear as statistical noise.

Quantum computers harness weird physics to explore unsolvable problems

Classical computers manipulate bits as either 0 or 1, but quantum computers use qubits to hold combinations of both, enabling new algorithms. This new approach could one day help with breakthroughs in medicine, chemistry, and AI.

Visualize chaos with the orbit of three or more objects bound by gravity

In this simulation, users can simulate randomized and preset orbital configurations for up to 10 celestial bodies to observe the system's evolution in an interactive 3D space. The same configurations can produce different orbital behavior—a feature of chaos—by making small changes to an object's mass under advanced settings.

Read an oral history of the Y2K panic

This oral history brings together programmers, engineers, and officials who first identified the date flaw and then spent years alerting the public to the danger and racing to repair millions of lines of code.

IBM developed a benchmark to measure the usable performance of a quantum computer

Dubbed "quantum volume," the benchmark tests how well a quantum computer runs complex tasks. It stops when the computer can no longer correctly select the most likely answers, called "heavy outputs," more than two-thirds of the time.

Watch how the Enigma machine encrypts messages and learn about its critical design flaw

Using settings selected from a series of rotors and a plugboard that let operators swap letters with any of their choosing, the machine lit up the output letter for each letter typed. While this system could generate messages requiring any of 150 quintillion potential keys, the machine's reflector never encrypted a letter as itself, providing an entry point for exploiting the system.

Claude Shannon, for whom Anthropic's AI tools are named, proved the existence of an unbreakable cipher

In the one-time pad approach, the encryption key is randomly generated and of equal length to the plaintext. This randomness ensures that every possible plaintext is equally likely—RBBHN can mean HELLO, SOLVE, CODES, and so on, using a different key—while matching the message and key lengths prevents the appearance of patterns that can be exploited.

Microsoft's initial successes came from promising products that did not yet exist

In discussions with teams responsible for MITS's Altair 8800 and IBM's PC, Bill Gates and Paul Allen claimed to have created ready-to-use software. Code for the former was not completed until Paul Allen's flight to MITS' headquarters, where the software was successfully tested for the first time.

The terms 'bug' and 'debugging' were popularized by Grace Hopper's team

Engineers working on the Mark II computer at Harvard University identified a moth in one of its components in 1947, which caused it to malfunction. Describing the incident as the "first actual case of bug being found," the term first used by Thomas Edison in 1878 to describe faults in an apparatus caught on.

Play an incremental game that teaches the history of particle physics

Developed during the CERN Summer Student Webfest 2014, this game tasks you with building the reputation of your particle accelerator as you discover particles and hire staff to collect data. A historical breakdown accompanies each discovery, allowing you to recreate the timeline of the field.

Bill Gates and Paul Allen were childhood friends before founding Microsoft

The two bonded over a teletype terminal at their school, which required a $40 hourly fee for computing time. They were banned from using the device after exploiting a bug that allowed them to bypass the fee, only to be later offered free computing time for finding more bugs.

Since computers rely on the distribution of electrons in microchips to represent binary code, cosmic rays and ionizing radiation from radioactive elements can introduce software errors by detaching electrons from atoms.

Explore (digitally) the Computer History Museum

When IBM’s Deep Blue supercomputer won its famous chess rematch with then world champion Garry Kasparov in May 1997, the victory was hailed far and wide as a triumph of artificial intelligence. But John McCarthy—the man who coined the term and pioneered the field of AI research—didn’t see it that way. An obituary about McCarthy—who passed in 2001—and his ideas.

Lego's Women of NASA set honors four NASA pioneers: computer scientist Margaret Hamilton, astronaut Sally Ride, astronaut Mae Jemison, and astronomer Nancy Grace Roman.

Efforts to find Majorana particles may finally lead to reliable quantum computers

Though the particle—which is its own antiparticle—remains theoretical, scientists from Microsoft to Google continue developing systems to mimic this property towards building error-resistant quantum computers. Scientists at UCSB have been attempting to get the particle using just pencil lead and Scotch tape.

An explanation of Turing machines—the foundations of digital algorithms and computers

The device reads data, overwrites or ignores it, and then moves on to the next piece of data based on a set of programmed rules. By imagining this data as zeros and ones—binary that makes up computer code—Turing machines became the theoretical models for how any algorithm could be run on a computer if the procedure could be coded into a set of digital rules.

In 2000, the Clay Mathematics Institute placed million-dollar bounties on seven unsolved problems in mathematics—the Millennium Prize Problems—in the hopes of spurring solutions.