The post The Internet Chronicles – Part 2 of 12: Packet Switching and the First Message originally appeared on the HLFF SciLogs blog.

]]>*From the initial concepts to the development of the global network, we see today, each part of this 12-part series will explore the pivotal moments and key figures who turned the dream of a connected world into a reality. We will journey back to the early days of computer science and the internet, uncovering the ideas and collaborative efforts that paved the way for the digital age.*

*We previously looked at the audacious vision that started it all. After that, the stage was set for the first true iteration of the internet.*

In the late 1960s, the United States Department of Defense’s Advanced Research Projects Agency (ARPA) embarked on an ambitious project to develop a decentralized communication network. This initiative aimed to ensure that communication could be maintained, even if parts of the network were compromised.

This approach was built on the work of J. C. R. Licklider to enable resource sharing between remote computers. The resulting project, called Advanced Research Projects Agency Network (ARPANET), turned out to be the precursor to the modern internet – and in our “Internet Chronicles,” we will be spending quite a bit of time on ARPANET.

At the core of ARPANET’s initial success was the concept of packet switching.

Traditional circuit-switched networks, like those used in telephone systems, require a dedicated communication path between two points. This is not only inefficient but also vulnerable to disruptions; it is not practical at all to build a large-scale network with this approach.

In contrast, packet switching divides data into smaller packets transmitted independently across the network and reassembled at the destination.

In packet switching, data is divided into smaller units called packets. Each packet contains a header (which includes the source and destination address, sequencing units, and error-checking data) and a payload (the actual data being transmitted). Each packet is assigned a sequence number, allowing for reassembly in the correct order at the destination. A defined protocol defines how the data is switched and reassembled.

This approach makes much more efficient use of available bandwidth resources as it allows multiple communications to share the same network paths. By breaking data into smaller packets and dynamically routing each packet based on current network conditions, it avoids the need for dedicated circuits, reducing idle times and enhancing flexibility. This method ensures that the network can adapt to varying traffic loads and reroute around congestion or failures. Devices called packet switches (now commonly referred to as routers) use information from the headers to decide the optimal path from source to destination and can adapt in real-time. As packets traverse the network, routers examine the destination address in the packet header and each router uses a routing table and dedicated algorithms to determine the next hop for the packet.

To this day, packet switching is the dominant technology for data communications in computer networks worldwide. At the time, however, packet switching was anything but intuitive.

Two people were independently working on the approach. Paul Baran, an American engineer and internet pioneer, was one of them. Baran joined the RAND Corporation in 1959, as the corporation was focusing on resilient communication systems. It was a time when RAND, like many in the US, was concerned about the prospect of the Cold War escalating into a nuclear conflict. At the time, American military communications used high-frequency connections, which would have been wiped out for hours in the case of a nuclear attack as high-frequency communications are more vulnerable to the intense ionizing radiation that would form in the ionosphere.

Baran showed that a distributed relay node architecture was more survivable in the face of such an attack, and also had other advantages as well.

His model proposed breaking messages into smaller pieces, or “packets,” each containing a portion of the data along with addressing information. These packets could take different paths to their destination, ensuring that even if some paths were disrupted, the message could still be reassembled from the packets that arrived. This approach not only increased the reliability of the network but also optimized the use of available bandwidth.

After some work and optimization, Baran’s ideas were picking up steam at RAND. Unbeknownst to him, however, another researcher was working on the same principles.

Donald Davies, a Welsh computer scientist working at the National Physical Laboratory (NPL) in the United Kingdom, independently developed the concept of packet switching around the same time as Paul Baran in the United States. Davies was motivated by efficiency and reliability. At the time, time-sharing computer systems had to keep a phone connection open for each user, which was resource-intensive.

Davies also proposed dividing computer messages into very “short messages in fixed format” and routing them independently to be reassembled at the destination – yes, the same idea. He also coined the word “packet” for this task, after consulting with a linguist. It was a clever idea as “packet” can be converted into different languages without changing its meaning and Davies suspected this idea was of international importance.

His independent research not only paralleled but also complemented the work of his American counterparts, helping conceptualize the practical implementation of packet-switched networks and contributing to a broader understanding and acceptance of packet switching as the foundation for modern data communication systems.

Then, ARPANET came in.

In 1967, some of Davies’ work, presented by one of his colleagues, caught the attention of ARPANET. Larry Roberts, another internet pioneer, was making key decisions on how ARPANET should be built. Roberts was looking for solutions like the ones Davies and Baran were proposing. Curiously, Roberts had met with Baran but the two reportedly did not discuss networks.

Roberts was intrigued by Davies’ work and wanted to build ARPANET around the idea of packet switching. Ultimately, the initial design for the computer network was revised several times to incorporate elements from both Davies and Baran. Roberts also brought in one more key person to work on the project: Leonard Kleinrock.

Kleinrock was working in a relatively obscure field of mathematics called queuing theory. Queuing theory aims to understand and predict queue lengths and waiting times, providing insights into system performance under various conditions. This theory uses models to represent the flow of items through a process, where items can be anything from data packets in a network to customers in a bank.

Kleinrock had published his PhD thesis as a book in 1964, which included mathematical models useful for packet switching. He was an expert theoretician and he started working on ARPANET, first informally and then as a fully-fledged member.

However, having a theory and a working model is one thing, but implementing a workable system raised major engineering challenges. Most people working at ARPANET did not believe packet switching was scalable in a way that was economically feasible. Baran had met the same objections at RAND – ironic, given the scale of the internet today.

Nevertheless, the project continued to receive support. Then, in 1969, the network was set up and good to go.

The initial ARPANET network, the first (rudimentary) iteration of the internet, included four nodes: the University of California, Los Angeles (UCLA), the Augmentation Research Center (ARC), The University of California, Santa Barbara (UCSB), and the University of Utah School of Computing. These locations were chosen strategically, both from a line cost perspective and because all institutions had something to contribute.

It was on October 29, 1969 (10:30 p.m. PST) that the first packet-switched message was sent. It was a historic but rather humorous moment, showing just how undeveloped the technology still was. The plan was to send over a command: “Login.” However, only two characters were successfully transmitted (“Lo”) and then the system crashed.

“Hence, the first message on the Internet was ‘LO’ – as in ‘Lo and behold!’” Kleinrock said. “We didn’t plan it, but we couldn’t have come up with a better message: succinct, powerful and prophetic.”

There was a lot of work to do, but by December, the four-node network was relatively stable and could send commands. Work on packet switching was not done (far from it), but ARPANET was starting to take off. Ethernet, Internet Protocol (IP), and the Transmission Control Protocol (TCP) would all build on packet switching, but in 1969, they were still a few years away from being invented. The original packet switches, which we now call routers, still had some ways to go.

We shall explore their journey in the next installments.

*This is part 2 of 12 of a monthly series on the development of the internet. In the next part, we will look at how Transmission Control Protocol / Internet Protocol (TCP/IP) established a set of rules enabling disparate networks to communicate, thereby unifying them into a global “network of networks.”*

The post The Internet Chronicles – Part 2 of 12: Packet Switching and the First Message originally appeared on the HLFF SciLogs blog.

]]>The post Can Machine Learning Help Us Build an Animal Translator? originally appeared on the HLFF SciLogs blog.

]]>In 2023, humanity had its first contact with a non-human civilization; well, sort of. Scientists from the SETI Institute, University of California Davis, and the Alaska Whale Foundation teamed up to communicate with a humpback whale named Twain in Southeast Alaska.

The team used pre-recorded calls and were surprised to see Twain approach and circle their boat in a “greeting” behavior. Throughout a 20-minute window, the team broadcast the calls to Twain and matched the interval variations between each playback call. Essentially, they communicated in “humpback” tongue.

This is, of course, a rudimentary form of communication. The team demonstrated that whales can participate in turn-taking vocal exchanges and essentially broadcast some messages, interacting with a different species. Yet, although the general context was understandable, the scientists understood very little about what they were actually saying and what Twain was saying back.

This begs the question, are we able to properly communicate with animals?

The short answer is ‘not yet’, but thanks to advancements in machine learning, we are getting closer than ever to understanding other species – and in the process, improve our understanding of communication itself.

Whales have excellent levels of cooperation and social interaction, and they also emit vocalizations that can be picked up from hundreds or even thousands of miles away. This makes them excellent candidates for interspecies communication.

Sperm whales, for instance, are highly communicative creatures. They do not produce long, melodic calls like humpback whales, but they have a system of clicks called *codas*. Since 2005, researchers led by Shane Gero from the University of Ottawa have been following a pod of 400 sperm whales, recording their vocalizations and trying to decipher as much as possible about them.

A subset of the codas that sperm whales use have been shown to encode information about caller identity; in other words, the whales introduce themselves when talking. However, almost everything else is a message that whales are communicating to one another, and the contents of that message are challenging to decipher.

There are many features to communication that we rarely consider. The intensity and pitch of the sounds are obviously important but there are also more subtle elements, like rhythm and alternation, whether the tone is going up or down, and whether they exhibit any tonal ornamentation. All of this makes for a lot of data to understand and analyze.

In recently published research, Gero and his colleagues try to lay the basis for understanding the “whale alphabet.” They claim whales are much more expressive than previously thought and have a complex phonetic alphabet that is not very dissimilar to that of humans.

Jacob Andreas, one of the study’s co-authors, thinks a big data approach can be used for this purpose. He is training machine learning algorithms to recognize features of whale communication, and maybe even understand what the whales are saying.

The sheer volume and complexity of the data makes machine learning a suitable approach. Gero’s work has provided an extensive, high-quality dataset to work with (and he is far from the only biologist looking at whale communication). Using this, AI models will be able to work to identify and classify different types of whale clicks, distinguishing between echolocation clicks used for navigation and communication clicks used for social interaction.

The algorithms cluster codas based on their inter-click intervals, helping to identify distinct patterns and structures, representing the complex acoustic signals in high-dimensional feature spaces, which enables a more detailed analysis of the communication structures.

The results are incipient, but promising. They suggest that whale communication has a hierarchical structure, similar to sentences in human languages. The results also hint at the smallest meaning-carrying units and their combinations, something akin to human words. There is also a type of partial validation with this type of data.

To validate the AI-generated models, the researchers conducted playback experiments, where recorded whale sounds were played back to live whales to observe their responses. This approach helped test hypotheses about the meanings and functions of different codas and, while it is not perfect, it is still a useful way to test hypotheses and ground the AI-derived models in real-world whale behavior.

Does this mean we can talk to whales at the moment? Not really. But it does mean we are getting a more fundamental understanding at how some whales communicate, and we can maybe even pick up a few whale “words” and “sentences” here and there.

However, we are unsure whether whales communicate like us or whether their language is more like music. It is nonetheless possible that we soon start talking to whales, even as we lack a perfect understanding of what we are communicating.

From the vast expanse of the ocean, we now dive into the pens of farm animals. We kill and eat (or discard) around 80 billion land animals every year. While the livestock industry is not really interested in inter-species communication, it does care about information for practical decisions – and here too, AI can be of help.

In 2017, Peter Robinson, who was working on teaching computers to recognize emotions in human faces, had an idea: He thought he could do the same thing with sheep. Remarkably, the core of this approach can be traced back to Darwin, who argued that humans and animals show emotion through remarkably similar behaviors. In this spirit, Robinson and his colleagues thought there could be an opportunity to develop an AI that measures whether sheep are in pain.

“The interesting part is that you can see a clear analogy between these actions in the sheep’s faces and similar facial actions in humans when they are in pain – there is a similarity in terms of the muscles in their faces and in our faces,” said co-author Marwa Mahmoud in a press release from 2017.

They worked with a dataset of 500 photos of sheep taken by veterinarians as they were providing treatment. The veterinarians also estimated the pain levels of the sheep. After training, the model was able to estimate sheep pain with around 80% accuracy. More datasets can improve its performance even more.

For farmers, this could be important. Identifying discomfort or pain in sheep could direct an early intervention and farmers could provide them with early medical attention.

“I do a lot of walking in the countryside, and after working on this project, I now often find myself stopping to talk to the sheep and make sure they’re happy,” said Robinson.

Another approach presented in a different study also deals with sounds, specifically pig grunts. In this context, the goal was to discern between positive pig grunts (associated with joy or playfulness) and grunts indicative of negative emotions (like pain or being afraid).

They used a dataset of over 7,000 acoustic recordings gathered throughout the various life stages of 411 pigs, from their births to slaughter. After training the algorithm, the team claimed they were able to successfully attribute 92% of the calls to the correct emotion. The emotions of pigs were defined based on how they naturally react to various positive and negative external stimuli.

Other efforts are also taking an approach similar to the whale research. James Chen, an animal data sciences researcher and assistant professor at Virginia Tech, is building a database of cow vocalizations. Here too, the main focus is animal welfare, although Chen also believes he can spot which cows burp less (cows are an important source of methane emissions, and methane is a potent greenhouse gas).

Ultimately, communicating with farm animals is more about understanding when they are in trouble and less about two-way communication. AI is once more a great enabler in this area.

Everyone who has ever had pets at some point must have wanted to communicate directly with them. You may or may not be happy to hear what your cat (or dog) thinks of you, but several teams are working on it. Most such research is taking a similar approach to the pig grunts: detecting the emotions behind the vocalizations. This emotion prediction approach is also used on cats.

Recently, there has been a great deal of progress and there are algorithms that can estimate pet emotions and see whether a bark is playful or aggressive (at least for some breeds), but we are not yet able to talk to pets directly.

For now, we are not able to truly talk to any other species. However, we may be witnessing the incipient steps in the quest for decoding animal communication, and AI is playing a key role.

Most such efforts, be it for whales, sheep, or cats, follow the same general approach: you start from a reliable database (usually of vocalizations), tag and classify the data, cluster it, and attempt to decode communication patterns and elements. There is an important caveat, however: AI has a strong human bias.

Bees communicate where food sources are through dances; birds have exquisite, elaborate songs and chirps; insects use chemical signals; whales “sing” or click. The animal world has a near-infinite diversity in communication, and our models make simplifying assumptions, as they are built on a structure for human communication. We (and our models) try to interpret everything through our lens, but this is not always accurate. Quite possibly, it is never accurate.

Several groups of researchers are attempting to decode animal communication through AI, but they face important, foundational challenges.

In a recent paper, researchers Yossi Yovel and Oded Rechavi from Tel Aviv University explored the potential for AI to understand non-human animal communication, drawing inspiration from the fictional “Doctor Dolittle” who could converse with animals. They argue that communication with animals involves far more than directly translating their sounds into human language, emphasizing the intricate context and multifaceted nature of such interactions.

They note three main obstacles in this quest. First, the context in which animals communicate is crucial. For example, while AI can replicate animal sounds, understanding the intent behind these sounds (such as whether a bird is singing to attract a mate or to signal territory) requires contextual information.

Second, eliciting natural responses from animals is fraught with challenges. Animals’ behaviors are influenced by various factors, including their physiological state and environmental conditions. Capturing authentic communication without conditioning the animals requires diverse observational techniques. Because of its biases and limitations, AI may misinterpret subtle animal behaviors as responses, leading to inaccurate conclusions. Finally, the limited range of contexts in which animals typically communicate, such as alarm signals or courtship behaviors, constrains the breadth of potential interspecies communication.

Some of these challenges can be overcome. The key is to not simply rely on AI algorithms to decode everything but to work interdisciplinarily, with specialists in the field potentially tagging contextual information and interpreting the findings in their correct context.

Ultimately, the dream of communicating directly with other species is still not yet real, but we have a plausible roadmap towards achieving it. At least in some cases, understanding and communicating with another species does not seem impossible. Which begs the question: If we really could talk to another species, what would we say?

The post Can Machine Learning Help Us Build an Animal Translator? originally appeared on the HLFF SciLogs blog.

]]>The post A Puzzling Square originally appeared on the HLFF SciLogs blog.

]]>The trick involves trying to fit this square piece of wood into the fabric bag shown – and yes, the bag really is just a piece of fabric of the size shown, folded in half and stitched together at the ends, leaving a long side open.

The video states that you have to put the square *inside the bag*, “in such a way that all four sides of the square are covered by the bag” – along with the usual constraints of not being allowed to cut or destroy either of the objects in order to achieve this.

If you would like to go away and think about this now, please do. Then, before I discuss the solution, I would like to explain to you why I found this puzzle so interesting.

Back in January, I wrote about a set of coincidental recent discoveries that all involved a particular number, which was the titular Ubiquitous Surd known as the square root of three. But this is not my favourite surd, and there is another one which claims the crown, for many reasons.

The square root of two, a value of around 1.4142, has many enjoyable properties. As well as not being expressible as a fraction (and it being good fun to prove this fact in a variety of ways), it has many practical uses.

You may have seen Ben Sparks’ article back in August 2022, in which he explains that the A-ratio format of paper, used in most countries as the international standard, uses the ratio of 1:√2 to give paper a useful property – if you fold a sheet of A-ratio paper in half, the result will continue to be A-ratio, no matter how many times you do this.

This means we can easily copy two pages of a document onto a single page of the same size, and that the area (and thus the weight) of paper will neatly double if you increase the size to the next standard size up – making life easier for printers everywhere. But there is another cool fact hiding in a piece of A4 paper, which shows off even more of the cleverness of the square root of 2.

If you have a rectangular piece of paper and want to create a square one, there is a well-known method to make this happen: make a diagonal fold, taking one corner of the page and bringing it across until it touches the long side opposite. This fold will create a right-angled triangle which is exactly half of a square, and it will be sitting on top of a piece exactly the same shape, which is the other half of the same square – so cutting off the section to the right of it will leave a perfectly square piece of paper.

This square we have created from an A4 page has a fun secret. To see it, first note that if your original paper had a short side of length 1 unit, the long side, by our knowledge of paper ratios, would have been the square root of 2.

But if we consider the diagonal of this square – the length of that fold we just made – it is the diagonal of a square measuring one unit on each side. By Pythagoras’ theorem, we know this length will be the square root of 1² + 1², or the square root of two.

If you have a piece of A-ratio paper handy, you can try this – make the diagonal fold, and compare the length of this fold to the long edge of a piece of paper the same size: It should be exactly the same. This happens because the ratio we use for the paper, giving it all its wonderful halving-to-the-same-shape properties, is the same as the length of the diagonal – by definition. And if you have never noticed this pleasing coincidence, you can add it to your increasingly long list of reasons why you like the square root of two.

Now you might be wondering why I am talking about this, given that I started off showing you a puzzle by a magician about putting a square in a bag. Well, when I first saw the video, and took a moment to think about how this might work, I noticed something interesting about the length of the rectangular bag relative to the side of the square.

The width of the bag obviously was not enough to fit the square inside; and the length of the bag, while it was longer than the side of the square, was not – for example – twice the length of the square, which might allow me to do something clever by folding the bag.

But there was an important and familiar ratio there: If the square of wood measured 10cm, overlaying lines on the video and measuring their ratios implies the bag is around 14.6cm long – suspiciously, just a shade over the square root of 2 times the side length of the wooden square.

This was a strong hint to me as to how the puzzle might be solved – I will not give it away, but if you want to go back and watch the rest of the video you will see how this comes into play. Sometimes this kind of mathematical insight is enough to point you in the right direction to a solution, and for me it was deeply satisfying to have used this realisation to figure it out.

Hopefully my obsession with the square root of two has become more understandable: This is only scratching the surface of the cool things you can do with it. If you need further confirmation that I love it maybe a little too much, you might be pleased to hear that 516 days (aka 1.414 years, rounded to the nearest day) after our wedding, my partner and I celebrated our √2 wedding anniversary by going out for triangular sandwiches – whose long diagonal side is approximately √2 times the length of the shorter side. Irrationally delicious!

The post A Puzzling Square originally appeared on the HLFF SciLogs blog.

]]>The post The Internet Chronicles – Part 1 of 12: The Vision That Started It All originally appeared on the HLFF SciLogs blog.

]]>*From the initial concepts to the development of the global network we see today, each part of this 12-part series will explore the pivotal moments and key figures who turned the dream of a connected world into a reality. We will journey back to the early days of computer science and the internet, uncovering the ideas and collaborative efforts that paved the way for the digital age.*

*In this episode, we look at the audacious vision that started it all.*

Humans are a very communicative species. Our ability and desire to communicate is, perhaps, one of the defining features that enabled us to thrive on this planet. We talk to our family, we talk to our friends, we talk to random people we have never met before. However, direct communication only gets you so far.

In the late 19th century, the invention of the electrical telegraph brought a revolution in communication. For the first time, people had access to a system of communication that could cross large distances essentially instantaneously. Radiotelegraphy and telephones gradually became more common, but such systems were limited to point-to-point communication between two end devices. Furthermore, they only allowed the same type of mediated spoken communication. But what if you wanted to send someone a document, or some different kind of information?

To most people in 1960, this would have seemed unimaginable. After all, the first computers had only just been invented, and they were about as big as a room – the thought of linking such devices remotely and using them for communication purposes belonged rather to the realm of science fiction.

But not for J.C.R. Licklider.

Licklider, a psychologist by training, envisioned a global network that would allow people to share information and work together regardless of their physical location. In 1960, he published what would become a groundbreaking paper called “Man-Computer Symbiosis.”

In the paper, Licklider argued that computers should not be regarded as performing separate activities to humans. Computers and humans should work together, not in parallel. Looking back, some of the ideas he presented in this paper seem strikingly prescient.

“In the anticipated symbiotic partnership, men will set the goals, formulate the hypotheses, determine the criteria, and perform the evaluations. Computing machines will do the routinizable work that must be done to prepare the way for insights and decisions in technical and scientific thinking. Preliminary analyses indicate that the symbiotic partnership will perform intellectual operations much more effectively than man alone can perform them,” the researcher wrote in the seminal paper.

Licklider’s core idea was that computers would do routine work effectively and quickly. For this, you would need processing speed and easy user interfaces that would allow humans to interact with computers with ease. While this may seem normal to us now, at the time users would interact with computers using punch cards. However, Licklider wanted users to interact directly with the interface and obtain results immediately.

To this day, the paper is often cited as a foundational text in the field of computer science and human-computer interaction.

However, publishing a paper is one thing, but how do you get this type of idea off the ground?

The early days of the internet are tightly connected to the Advanced Research Projects Agency (ARPA). ARPA (later rebranded as the Defense Advanced Research Projects Agency, or DARPA) was essentially the research branch of the U.S. Department of Defense; it was where the most ambitious and audacious technology programs of the military were developed. It was also where Licklider was employed as the director of ARPA’s Information Processing Techniques Office.

As director, he commissioned the funding of various projects. From 1962 to 1964, he funded Project MAC at MIT. Project MAC had the objective of exploring the possibilities of time-sharing, where multiple users could simultaneously access a central computer. This was a departure from the batch processing systems of the time, where computers executed tasks sequentially, often leading to inefficiencies and long wait times.

The project became famous for its groundbreaking research in a number of fields, including operating systems and computation theory. MAC also featured a large mainframe computer that could be shared by up to 30 users simultaneously, each with their own terminal. This approach would go on to become what we now call a *server*. But let us not get ahead of ourselves.

In parallel to this project (and other work carried out with ARPA), Licklider kept pushing his visionary idea. In a series of memos sent out around 1962, he promoted what he called the “Intergalactic Computer Network.”

As grandiose as that name sounds, what Licklider was describing was essentially a proto-internet. He described the network as “an electronic commons open to all, ‘the main and essential medium of informational interaction for governments, institutions, corporations, and individuals.'”

Licklider was advocating for more than just technology. Rather, his intention was to enhance human capabilities with computers and then interlink those computers in a universally connected network. This “intergalactic” network began catching on at ARPA and Licklider’s ideas were reaching all the right people. His support for time-sharing systems and interactive computing also highlighted the advantages of a networked world; suddenly, this crazy idea of interconnectedness did not seem all that crazy anymore.

In a 1968 paper he co-authored, called “The Computer as a Communication Device,” Licklider described another incentive of this type of global network.

“Take any problem worthy of the name, and you find only a few people who can contribute effectively to its solution. Those people must be brought into close intellectual partnership so that their ideas can come into contact with one another. But bring these people together physically in one place to form a team, and you have trouble, for the most creative people are often not the best team players, and there are not enough top positions in a single organization to keep them all happy. Let them go their separate ways, and each creates his own empire, large or small, and devotes more time to the role of emperor than to the role of problem solver. The principals still get together at meetings. They still visit one another. But the time scale of their communication stretches out, and the correlations among mental models degenerate between meetings so that it may take a year to do a week’s communicating. There has to be some way of facilitating communication among people [without] bringing them together in one place.”

While this matter of remote communication is still not completely solved (as we all witnessed during the recent COVID-19 pandemic ), Licklider was once again very right about how the internet can turbo-charge collaboration and make working relationships much more efficient.

Licklider was not the only one to have developed theories of networking around this time. Independently, Paul Baran at the RAND Corporation was proposing a distributed network system around the same time. Simultaneously in the UK, Donald Davies of the National Physical Laboratory was proposing a national commercial data network. This was not the unique vision of one single man, yet Licklider’s push proved to be instrumental. He was in the right place, at the right time, convincing the right people.

His work convinced ARPA to start looking into this type of project more carefully and ultimately, in 1969, ARPA awarded contracts for the development of ARPANET. This single decision would prove to be instrumental for the development of the internet and ultimately, for our society as a whole. This story would go on to include the technologies proposed by Davies and Baran, as well as several other key internet pioneers.

The stage was set for the first iteration of the internet to emerge.

But that is a story for our next installment.

*This is part 1 of 12 of a monthly series on the development of the internet. In the next part, we will look at how the first true iteration of the internet – ARPANET – came to be. We will see how the mathematical framework for packet switching became a cornerstone for data transmission across networks and how the first email was sent.*

The post The Internet Chronicles – Part 1 of 12: The Vision That Started It All originally appeared on the HLFF SciLogs blog.

]]>The post On Trial: P versus NP and the Complexity of Complexity originally appeared on the HLFF SciLogs blog.

]]>Yet, there may be more merit to bringing the case to court than first impressions would suggest. The relative difficulty of tasks like these is the fundamental question underlying one of the most pernicious problems in mathematics and computer science that has remained unsolved since its formulation in 1971: the P versus NP problem. The solution to this problem has huge ramifications in the real world, impacting medicine, artificial intelligence, internet security and a host of other areas. For these reasons, P versus NP is one of seven Millenium Prize Problems selected by the Clay Mathematical Institute as the most important to solve in our time.

The “P” of P versus NP stands for “polynomial time.” A computer program runs in polynomial time if, when you increase the size of the input, the (idealised version of a) computer takes a proportionately longer time to complete its given task. List-sorting is a perfect example of a P problem, where there are known and straightforward methods of sorting the list and verifying that the list is sorted correctly that don’t scale at some absurd rate as the length of the list increases.

Expanding the “NP” of P versus NP is less informative (for the record, it stands for “nondeterministic polynomial time”), but in essence, NP problems are hard to solve and easy to verify. A generalised version of Sudoku – where instead of having a 9 × 9 grid you allow the grid to be arbitrarily large (*n*^{2} x *n*^{2}) – is an NP problem. There is no known way to solve it that is significantly faster than brute-force checking of every possible combination of numbers to fill the board. As a result, when you increase *n*, the time to solve the puzzle scales faster than exponentially; i.e. it gets really hard to solve, really quickly. However, verifying a solution remains polynomial in time, as you can run a relatively simple and quick algorithm to check that the digits entered in all rows, columns and squares follow all the rules; i.e. it remains easy to verify.

Relating what you now know about the P versus NP problem to list-sorting and taking on a massive Sudoku puzzle, the answer to the question of which is more difficult to solve seems even more obvious – list-sorting, hands-down! If you then applied the same train of thought to a large selection of P problems and NP problems and presented your findings in a civil court of law, it would be easy to show that – based on the *balance of probabilities* – all list-sorting-type problems are not equivalent to all massive Sudoku-like problems, i.e. P ≠ NP, and you would win your case easily.

But mathematicians and computer scientists require a higher standard of proof than that. In fact, society demands a higher standard of proof than that; because if P does turn out to equal NP, in principle, currently intractable problems will become easy. And this fact could be used for good – such as optimising transport routing and finding new medicines – or ill, including hacking bank accounts and government websites. Like juries in a criminal court determining the guilt of defendants, we all need to know *beyond reasonable doubt* whether or not P = NP. And that is a much tougher proposition.

Towards this aim, researchers have shown that nearly all seemingly hard NP problems are “NP-complete,” which means that if they had an efficient solution to one NP-complete problem, it could be adapted to solve every other NP problem quickly too. For example, generalised Sudoku is NP-complete, because it can be reduced to other classic complex problems that are known to be NP-complete, including the (clearly similar) Latin square completion problem and the (on the face of it, very dissimilar) Hamiltonian cycle problem. They are – in a precise mathematical sense – equivalent.

Therefore, to prove whether P does or does not equal NP, all a plucky researcher has to do is discover a clever algorithmic trick that solves an NP-complete problem in polynomial time (P = NP) or, alternatively, find just one NP-complete problem that they can prove is not solvable quickly by any computer program (P ≠ NP). However, despite half a century of effort from some of the brightest minds, proof – one way or another – remains elusive.

In recent years, many researchers have taken a step back from attempting to directly solve P versus NP. Instead, they have been questioning why solving P versus NP and similar problems is difficult in the first place. They have been asking questions such as “how hard is it to determine the difficulty of various computational problems?” and “why is it hard to determine how difficult they are?” These and other “meta” problems are the foundations for an area of mathematics and computer science called “meta-complexity”; both a topic of study and a tool that researchers are attempting to use in order to attack problems such as P versus NP.

An important focus for meta-complexity researchers at the moment is a particular problem with a long history that continues to defy clear classification: the minimum circuit size problem (MCSP). MCSP is interesting for several reasons, not least because it is one of the few challenging computational complexity problems left where it remains unclear whether it is NP-complete or not.

Found to play a central role in diverse areas such as cryptography, proof complexity, learning theory and circuit lower bounds, MCSP asks: Can you determine whether a black-box hypothetical computer used to compute a Boolean function has high or low circuit complexity? A Boolean function takes as input a certain number of zeros or ones and outputs zero or one (true or false). From this you can construct a truth table: a tabular representation of all the combinations of values for inputs and their corresponding outputs. Essentially, the truth table provides the input-output behaviour of the function, but gives no information about the function’s computational complexity. This complexity is represented by circuit complexity, defined as the total number of logic gates needed to build the smallest circuit that can compute the given function.

With these clarifications, the problem can be presented more precisely: MCSP takes as input the description of a Boolean function *f* as a truth table as well as a circuit size parameter *s*, and asks “is there a circuit that computes *f* of size ≤*s*?” As MCSP is both computationally complex and about computational complexity, it is unusual in being a meta-, meta-complexity problem, or a meta^{2}-complexity problem. The fact that researchers are trying to figure out why it is hard to determine how difficult MCSP is to solve, means it could be argued to be even more meta; a meta^{3}-complexity problem perhaps.

For the simplest Boolean functions, MCSP can be solved by an algorithm that runs in polynomial time. But the vast majority of functions seem to require an exponentially increasing number of gates as the functions get larger. Like generalised Sudoku, MCSP seems impossible to solve unless using a brute-force search, but also easy to verify if you are given a solution. You can guess a circuit, run every possible input and see if the output matches that of the given Boolean function. It is therefore clearly an NP problem. But is it also in P? Is it an NP-complete problem, i.e. would a fast way of solving it mean proving all problems in NP are actually in P? Many in the community suspect MCSP is not in P, but is NP-complete, which, if proven, would definitively mean that P ≠ NP.

In recent years, researchers have made significant progress in showing MCSP is NP-complete. For example, a huge number of proofs, many in the past six years, have emerged showing variants of MCSP are NP-complete. Perhaps some of the most interesting progress has been made by Shuichi Hirahara from the National Institute of Informatics in Tokyo.

In 2018, Hirahara derived a worst-case to average-case reduction for MCSP. Here, “average-case” complexity is a gauge of how easy or hard a problem is on average for most inputs, whereas “worst-case” complexity is the standard approach, considering the maximum complexity of the problem over all possible inputs. Problems where average-case and worst-case complexity may differ are interesting, as they can provide insights into the foundational complexity of these problems. Hirahara’s result implies that a hypothetical average-case MCSP algorithm would actually be powerful enough to solve a slightly different version of MCSP that restricts the types of circuits allowed without making any mistakes. This result is exciting because no other NP-complete problem has a known worst-case to average-case reduction. Given all NP-complete problems’ equivalence, it therefore provides a new avenue for research into the P versus NP problem.

Next, in 2022, using a blend of complexity theory and advanced information theoretic cryptography, Hirahara proved that MCSP on partial Boolean functions is NP-hard (NP-hard problems are as hard as NP-complete problems but do not necessarily have to be in NP). A partial Boolean function has a truth table that contains the usual zeros and ones, but also some “don’t know” symbols – some could be either zero or one, where you do not care what the circuit does. Somehow, adding this “don’t know” ambiguity allows a host of useful techniques to be deployed to resolve the partial-MCSP problem, but so far, it is unclear whether these same techniques can be applied to resolve the original MCSP problem.

Another researcher, Rahul Ilango from Massachusetts Institute of Technology, has been working to prove MCSP’s NP-completeness in multiple ways, looking at both simpler and more complicated versions of MCSP as entry points from which to attack the main problem. His most recent meta-complexity result succeeds in linking MCSP with the seemingly disparate set of Boolean satisfiability problems (SATs), where you are given a representation of a Boolean function as a formula or circuit and asked information about its function (e.g. whether it has a one in its truth table, or if it outputs a certain pattern of zeros and ones, etc.).

MCSP is what is known as a “black box problem”, because you want to know something about the representation, given black box access. SATs are completely different. Known as “white box problems”, SATs aim to say something about a function, given its representation. In 2023, Ilango and colleagues made the startling discovery that SAT reduces to MCSP with a random oracle, i.e. with a perfectly random function computable in P and accessible to all circuits and computers. As SAT is well known to be NP-complete, MCSP with a random oracle must also be NP-complete.

These and other recent results add to the case that the original MCSP is in fact NP-complete too. And, if this is true, MCSP could hide the missing piece of evidence that proves beyond reasonable doubt whether P does or does not equal NP.

Much like the problem itself, the trial of P versus NP has certainly been complex and lengthy, but progress is being made, and it feels like researchers are finally inching closer to a verdict.

The post On Trial: P versus NP and the Complexity of Complexity originally appeared on the HLFF SciLogs blog.

]]>The post Agriculture 5.0: Combating Weeds Using Machine Learning and Robotics originally appeared on the HLFF SciLogs blog.

]]>Making agriculture sustainable is one of the biggest challenges of our time. Not only does agriculture generate almost a third of our total greenhouse gas emissions, it uses up enormous amounts of water and fertilizer and contributes to soil erosion and degradation. At the same time, the planet’s population continues to grow, meaning agricultural practices must keep up with that as well. Thankfully, promising new technologies are coming in to help.

The integration of such new technologies is sometimes referred to as Agriculture 5.0, and it represents the latest paradigm shift in agriculture, integrating elements such as machine learning (ML), the Internet of Things (IoT), and robotics.

We have previously written about how the IoT and ML can help with agriculture and in fact, several young researchers who have attended the Heidelberg Laureate Forum (HLF) work in this field. However, the range of applications for this cutting-edge technology is extensive. Cheap sensors can help monitor things like water content and fertilizer usage, thus optimizing resource usage and reducing waste; smart algorithms can detect pests or health problems; and then, there are the weed robots.

It is remarkable how well agriculture, one of the oldest practices (arguably a defining one for our species), and cutting-edge technology work together so well. Take, for instance, weed control. Weeds have been one of the biggest challenges for farmers for thousands of years, and are also one of the main focuses of Agriculture 5.0.

In case you have lost track of your “agricultures”, Agriculture 4.0, also known as precision farming, introduced data-driven approaches to optimize agricultural practices. It focused on using telematics, remote data, and precision agriculture to enhance crop quality and yield. However, Agriculture 5.0 goes beyond data-driven practices to incorporate autonomous and unmanned technologies. In Agriculture 5.0, robots and AI play a central role, facilitating tasks such as soil analysis, crop monitoring, and weed management. This terminology is not ubiquitously accepted, but it has become commonplace in the scientific literature as it makes it easier to know the range of technologies and approaches that are most used.

However, in all agriculture iterations, however you may call them, weeds and pests have always been a problem.

Traditional weed management methods, such as the uniform application of herbicides, are often inefficient and environmentally harmful. These approaches can lead to the overuse of chemicals in some areas and under-treating in others. The excessive use of herbicides can harm non-target plants, degrade soil health, contaminate water sources, and contribute to the development of herbicide-resistant weed species.

This is where the new robots come in.

The approach of using machine-learning robots to identify and deal with weeds has already been discussed several times in peer-reviewed research. The “dealing with” part is relatively easy: Either a laser that zaps the weed or a small drop of pesticide strategically placed is all that is required to get the job done. However, identifying the weeds in a real-life situation with many unknowns is much more difficult.

The steps required to train such a weed-detecting robot are the already familiar ones for these algorithms. First, you need to acquire data. In this case, this is essentially a set of high-quality images of agricultural fields, either in the visual range or at different wavelengths with multispectral cameras. These images then undergo pre-processing, such as removing background noise, enhancing relevant features, and potentially using vegetation indices or other indicators.

“To train an effective model for weed detection and evaluate the performance of the learned model, the dataset is usually divided into training data and test data,” write the authors of a 2022 review on existing technologies. “The training set is used to train and tune the hyper-parameters of the model. The test set is used to provide an unbiased evaluation of a final model fitted by the training set. Sometimes, we also need to split a validation set from the training samples to observe the performance of the learned model to help in the final model selection.”

Afterwards, features that can be used to distinguish between crops and weeds are extracted. These features can be based on color, shape, texture, or relevant spectral properties. For instance, weeds may look relatively similar to crops, but they may have a higher (or lower) reflectivity, which can be detected (it is particularly here that multispectral cameras are useful, as they can show features not visible to the human eye). Subsequently, various ML algorithms are used to classify the extracted features into weed and crop categories.

Common algorithms include k-Nearest Neighbors (KNN), Support Vector Machines (SVM), and Convolutional Neural Networks (CNNs). Each algorithm has its strengths, with CNNs particularly excelling in image recognition tasks due to their ability to automatically learn and extract relevant features from raw images. The selected algorithms are trained – commonly using labelled datasets – and then validated and tested.

The major challenge is that even if such a model works, it will likely only work for the specific instance in which it was trained. Not all weeds are alike and not all crops are alike, so you cannot just train a model for every weed and every crop, you need to do it with specific plants in mind. Several datasets are already publicly available for training algorithms, but there is no “silver bullet” algorithm that works best in all cases.

A recent study from December 2023 summarized the challenge thusly:

“The key challenge in crop-weed classification for annotation-based techniques is the construction of the classification model and the optimization of the model parameters. The classifier model, like any other image classification issue, is created for specific applications, and its parameters must be fine-tuned and optimized. Classifier optimization necessitates the use of several algorithms in order to attain a high classification rate while minimizing false positives and data overfitting.”

“The biggest problem is how the technology is used, followed by the ways used owing to crop and weed morphology. As a result, determining which techniques are preferable on an “apple-to-apple” basis is extremely challenging.”

Automated weeding is more than just a good idea. Several such systems have already been developed in university settings and are starting to become commercialized. Autoweeders (armed with either lasers or poison) are routinely achieving a weed elimination rate of around 70%, and while they still have to be followed by a “manual” verification, this enables farmers to reduce the use of pesticides and make more sustainable decisions.

In addition, ML models can be used with either autonomous robots or IoT devices to create fully integrated weed management systems. For instance, you could continuously monitor fields and detect weeds, eliminating them without human intervention. Another approach would be to use drones to monitor larger areas and detect weeds and then have a separate tool destroy the weeds.

There are still important challenges, particularly when it comes to scaling the approach to different types of crops and different types of weeds. This may need to be done on a case-by-case basis, which will be time and resource-consuming. Ultimately, however, automated weeding has come a long way in only a few years and improvements in imaging technology, sensors, and machine learning algorithms are all pushing the boundaries of automated weeders.

Future iterations of these technologies could integrate more sophisticated AI models capable of learning and adapting to new weed species and evolving field conditions. Additionally, as the approach becomes implemented in real situations, collaborative efforts between researchers and farmers can also make automated weeders more robust and adaptable.

For millennia, agriculture has been a cornerstone of our society – and will continue to be so. Feeding the world sustainably is no easy feat, but we have new “weapons” at our disposal. By leveraging these technologies, we can enhance crop yields, reduce farming’s environmental impact, and contribute to the greener future of agriculture.

The post Agriculture 5.0: Combating Weeds Using Machine Learning and Robotics originally appeared on the HLFF SciLogs blog.

]]>The post Who Wants to Be Normal? originally appeared on the HLFF SciLogs blog.

]]>But one word which has received more abuse than many in mathematics is the word **normal**. Chambers’ Dictionary defines it as ‘regular, typical, ordinary’ and ‘not deviating from the standard’; but in mathematics we use it for several things, only some of which fit with this definition. Here are my top normals in maths: Hopefully you will find them all to be completely normal.

Probably the closest in meaning to the… well, normal use of the word normal, a **normal distribution** is a statistical distribution which represents how you might expect data values to be distributed across a range. For example, if you measure the heights of a large group of people, you might expect the bulk of people to be around average height, with maybe a few outliers who are extremely short or tall. The distribution of heights overall will follow a** bell curve** – which has a peak in the middle and drops smoothly off to either side, like a bell shape.

The centre of the bell will be found at the mean value, µ (which, for a normal distribution is equal to the median and mode averages of the data), and the width of the bell is described by the standard deviation of the data, denoted σ². If the values are clustered together, the bell will be narrower, and if they are more spread out, it will be a wider shape. It is completely symmetrical around the centre of the curve, and flattens out to horizontal at each side.

The standard normal distribution – shown above in red – is the particular curve resulting from a mean of 0, and a standard deviation of 1. Every other normal distribution will be some version of this, depending on the values of the mean and standard deviation. This kind of shape turns up in all kinds of datasets – often, real-world physical data follows this type of curve.

The normal distribution is useful to statisticians, because if you are expecting to see something behaving normally, and thus creating normally distributed data, you can use it to determine if an observation you have made was likely to have happened by chance, or if (for example) a medical intervention, or experimental change you have made, has had a significant effect.

A less obvious use of the word normal is in describing something that is pointing in completely the wrong direction. Imagine you have a curved surface (a real one, or something more abstract), and pick a point on the surface. If you want to describe the direction at that point which is moving directly away from the surface, that is called a **normal vector**.

If your surface is a 2D shape like a circle or a curve, you can draw a tangent at that point, and construct a line that is perpendicular to the tangent – meeting it at right angles. The normal vector will lie in this line.

If your shape is more of a 3D surface, you can instead construct a **tangent plane** – a small patch of flat surface, which touches the surface only at that point, and contains all the tangent lines at that point in every direction. The vector sticking perpendicularly up out of this plane is the normal vector.

This object is useful in mechanics for defining how forces are going to act on an object, and is used in computer graphics to understand how light will reflect off an object, to help create realistic 3D renderings.

The word ‘normal’ is also used to describe a particular type of number – one with an infinite decimal expansion, which additionally has the property that every digit occurs with an equal frequency – we say the digits are distributed uniformly. So, if the number is written in base 10 (decimal), each of the digits 0 to 9 will occur exactly one tenth of the time in the decimal expansion.

A number is **normal in a given base b,** if every digit from 0 to b occurs with equal frequency, and every finite string of digits of a given length occurs with equal frequency too. This means if a number is normal in base 10, it is guaranteed to contain every possible finite string of decimal digits somewhere in its infinite decimal expansion. If a number is normal in every possible number base, it is **absolutely normal **(and that is absolutely actually what it is called).

The fascinating thing about normal numbers is, it can be quite tricky to find them, but also completely trivial. If I wanted to give you an example of a normal number in base 10, I could write:

0.1234567891011121314151617181920212223242526272829…

Known as **Champernowne’s constant**, this number consists of all the numbers from 1 upwards, written in order one after the other. This will, by definition, contain all the possible strings of digits, and will satisfy the conditions of being normal – but it is also pretty boring, because I designed it that way. While it is known that plenty of normal numbers exist, nobody has yet managed to show that a given number is normal unless it has been specifically constructed to have that property.

There are numbers which are conjectured to be normal – one example is π, the circle constant. It is suspected that π is a normal number, but nobody can say for sure. Does it contain every possible string of digits in base 10? It is hard to say, but you might enjoy this website I made for a particular date a few weeks after π day a while back, along with this accompanying explanatory blog post.

One final use of the word ‘normal’ is in group theory – I have written about groups here before, but essentially they are sets of objects which can be combined in pairs, and the result will be an object from the same set (we say the set is **closed **with respect to the combining operation). For example, numbers form groups, since when you add two numbers together, you get another number.

Groups can be made up of numbers, permutations (which I have also written about here before) or even symmetries. Each element in the group has an **inverse** – if the element is g, its inverse is g^{-1}. For a number, this might be its negative, or for a rotation symmetry it might be a rotation by the same amount in the opposite direction. An element combined with its inverse will cancel out and have no effect.

A subgroup is a collection of objects within a group, which also has to be closed (so, combining any two things in the subgroup, the result has to stay in the subgroup). For example, take the group of symmetries of a square. Any two symmetries can be combined – by applying one followed by the other – to get another symmetry. Within this group, which has eight elements, there is a subgroup consisting of the four rotations (by 0, 90, 180 and 270 degrees) since combining any of these will only ever give another rotation.

A normal subgroup is one which satisfies a particular condition – that it is **closed under conjugation**. **Conjugating** an element, say g, involves taking a second element, h, and combining h with g followed by the inverse of h: hgh^{-1}. (Conjugation is closely related to abelian groups, which I describe in the blog post linked above).

If we take all the elements in a particular subgroup and find their conjugates with h (where h can be any element of the whole group), we will get a new subgroup. If this is the same subgroup as the one you started with, for any h you choose, then we say it is a **normal subgroup**.

This might sound like a very fiddly technical definition, but for a more intuitive way of thinking about it, imagine h is a rotation by 90 degrees. If we rotate something by 90 degrees, perform another transformation to it, then rotate it back again, it is like we have rotated the transformation around. Or, if we are working in a group of permutations, conjugating by a permutation is like relabelling all the objects in the set, performing a shuffle, then relabelling them back again.

Normal subgroups are important, as they help us to understand the structure of the group as a whole, and how elements will behave when combined. Normal subgroups can be thought of as like the prime factors of a number – a **simple group** is one with no normal subgroups, which would be analogous to a prime number. Since group structures can be seen underlying many mathematical ideas, understanding how normal subgroups interact can be an extremely useful tool.

So if you are worried about whether or not something is normal, do not worry – it probably is!

The post Who Wants to Be Normal? originally appeared on the HLFF SciLogs blog.

]]>The post Sci-Me! – A Science-Themed Board Game originally appeared on the HLFF SciLogs blog.

]]>*The Heidelberg Laureate Forum has a single purpose: To provide some of the brightest minds in mathematics and computer science with the space and time to make connections and find inspiration. Some of the connections made at the HLF will echo into collaborations and projects, with some of those efforts leading to concrete developments. The **HLFF Spotlight** series unpacks a few of those examples.*

Michal Jex is an Assistant Professor in Mathematical Physics at Czech Technical University in Prague. He first attended the Heidelberg Laureate Forum in 2014 for the 2nd HLF during his time as a PhD student, and again for the 4th HLF during his postdoc. Finally, he returned as an alumnus at the 10th HLF in 2023. He is one of the most engaged members in AlumNode, the networking community for alumni of the HLF and various programs and institutions funded by the German foundation Klaus Tschira Stiftung. We sat down with Michal to talk about the science-themed board game he has co-developed called “Sci-Me!”, in which players can simulate the everyday life of a researcher.

Part of the challenges of working in more complex fields of research is trying to effectively communicate the essence of that research to a general public – or even explaining to individuals outside of academia what the everyday life of a researcher is like. This can also be essential in interdisciplinary research, where finding common ground can be difficult when approaches, methodology and terminology might all greatly differ from one another, depending on your discipline.

Trying to find common ground between disciplines is something not unfamiliar to Michal Jex, who began his academic career by completing a full Bachelor’s and Master’s in Chemistry, as well as simultaneously earning Bachelor’s and Master’s degrees in Mathematical Physics. He would go on to earn his PhD in Mathematical Physics. This puts him in an interesting role when it comes to how his colleagues see him: “To mathematicians I’m a physicist, to physicists I’m a mathematician and to chemists I’m … who the hell knows?” he tells us during our conversation.

Michal’s research is in the field of quantum physics, where he looks at many-body problems and weakly bounded states. When asked how he would explain his work in layman’s terms, Michal sums it up:

“I’m trying to describe matter from a theoretical point of view … One can think of essentially any object around us. It is composed of a lot of elementary quantum particles – or not really elementary, but protons, electrons, neutrons. They just make this huge mess when they are close and far, and they hit each other, and they interact with each other, and I am trying to make sense out of this soup of particles.”

He is employed at the Faculty of Nuclear Sciences and Physical Engineering of the university, which, he says, “is a really unfortunate name … because everybody just thinks that … you are doing nuclear reactors. So I’m always saying to people, ‘no, you really don’t want a nuclear reactor made by me.’”

While he says his academic background has helped him bridge the gap between various disciplines (“I had more tools in my toolbox”), it did not necessarily always allow him to help both sides to communicate better with each other. Particularly for mathematics, he felt that researchers outside of mathematics would often have difficulties concretely imagining what his day-to-day work might look like.

The same holds true even more so for many outside of academia, for whom it can often be difficult to picture what a mathematician spends his days with. Such a lack of insight into the scientific process can often lead to misunderstanding and misinformation around scientific results and achievements. This became especially evident during the COVID pandemic, where mistrust towards scientists and misinformation regarding the virus were rampant.

At the same time, however, Michal reflected on why scientists, even if they encounter colleagues from other disciplines, tend to trust in each other’s work: “I trust their results, because I’m convinced that they during their work do several steps which are universal … that other scientists start with certain hypotheses, and then we need to go through some series of experiments, or prove authoritative calculations to show that our hypothesis is either correct or wrong.”

He says, “I wanted to convey this message and was thinking, ‘people hate to learn new things in general’ … But if you make it the byproduct of some activity, they are usually fine with that.”

So it came that during a ‘Sciathon’ at the Lindau Nobel Laureate Meetings, which took place online due to the pandemic, Michal – along with Lena Schorr (German Cancer Research Center (DKFZ), Heidelberg), as well as Vindula Shakthi Kumaranayake Magurawalage (University of Münster) and Mark Christian Guinto (Nara Institute of Science and Technology) – came up with the idea for the board game “Sci-Me!”. After the Sciathon, Michal and Lena continued the project and created a working prototype. The goal of the game was simple: to show the public what the daily life of a researcher was like, and have a little fun along the way.

The board game puts players in the role of a researcher whose goal is to publish a set amount of papers before the other players can do so. You begin as an assistant professor with a full lab, hiring PhDs and postdocs to work for you. To pay your employees, however, you need to secure funding – this can be through either grants or patents. Certain random events can give players advantages – receiving a coveted prize, ensuring more funding in the future – or disadvantages – like having to deal with the unexpected flooding of one’s laboratory. There are occasional bits of silliness – often inspired by real life events – such as one event card that has players keep their lab employees motivated and happy by installing an ice cream fridge in the office.

Players go through all of the essential steps needed to create and publish solid research, taking into account the typical checks and balances that serve as guardrails for that process, such as having one’s work be peer-reviewed.

Ironically, Michal points out that developing the board game was a lot like conducting research: You have a concept and form a working prototype (read: ‘hypothesis’), then go out and test it; if you do not achieve the desired results, you adjust accordingly. Michal stresses that the feedback one gets during testing can be essential.

He notes that in playtesting the game, one of the first reactions came as part of the players’ requirement to secure funding: “When we were first at a board game fair, and we were showing the game to the first person, we found out that people don’t understand also the funding of science. ‘What do you mean? Getting money for doing research? You just do research, right? It’s for free.’”

In fact, Michal and Lena came to realize something else as well: The game did not resonate that much with a very general audience. Perhaps the process remained too arcane, too abstract, to players unfamiliar with the academic world. The groups that did respond very well were usually familiar with the scientific process in one way or another, whether that meant they were already researchers themselves or undergraduates at the beginning of their studies: “If you are really in academia, it’s fun. Because in a way, it’s your everyday life, but without the risk of actually affecting your everyday life … You can do crazy stuff without risking your position … I think if you are at the start of your potential academic career, this can really help you. ‘Hmm, do I really want to go into academia? This sounded like fun! So that might be a good idea.’”

For those immersed in an academic environment, Michal says it really resonated with them, since they felt their own experience so well represented in the game:

“What we found really fun when you play with people from academia or research is that … they start to kind of create their own personal stories, based on things they are doing on a board. For example, when you hire people, it’s like in real life. You don’t know how good they are. You might see their CV, but the CV might not tell the full story.”

“It feels like a bit of a Stockholm syndrome that you work in this environment the whole day, and then there is a free evening, and you are like: ‘Hey Let’s do it again just in a board game version.’”

In another example of the parallels between the scientific process and developing a board game, Michal and Lena found the process of finding a publisher quite difficult:

“If you try to pitch it as educational, that’s usually game over,” Michal tells us, because publishers “are really scared of the topic,” especially if it has “limited business potential.”

“During our first pitch we started with ‘educational board game’, and then, essentially, we knew that we could just leave. During the first sentence we knew that we’d lost immediately.”

Michal explains that they very quickly learned they had to adapt and developed what he jokingly called the “’Dictionary of Forbidden Words’, or words which you might mention as a footnote on a second page of a one pager which you don’t turn around.”

So, for now at least, the target audience of “Sci-Me!” seems to be shifting more towards academic insiders. Thankfully, the game is highly adaptable itself, with numerous physical prototypes for different disciplines already developed. These stretch from an arithmetic-based edition, over physics (classical mechanics), all the way to logic and German. There is even a travel-sized version that functions as a card game.

The development of the board game was not without its obstacles. The COVID pandemic meant that Michal and Lena had to work on the project remotely. But after two years of working on it, Michal says it was a real pleasure to finally be able to meet up with Lena in person. The occasion could not have been a more joyous one: to present the finished prototype of “Sci-Me!” to the public for the first time at a board game fair.

You can learn more about the board game here. If you would like to try it out for yourself, get a friend (or several) and try out the online test version!

If you would like to reach out to Michal with any questions or comments about “Sci-Me!”, you can do so at michal.jex@fjfi.cvut.cz.

The post Sci-Me! – A Science-Themed Board Game originally appeared on the HLFF SciLogs blog.

]]>The post June Huh, Combinatorics, and the Strange Allure of Chess Knight Problems originally appeared on the HLFF SciLogs blog.

]]>In high school, Huh wanted to be a poet. He even dropped out of high school to focus on his poetry, feeling exhausted by the relentless studying. His poetry did not do all that well.

His university studies were also fraught with difficulties. He wanted to major in physics and astronomy and become a science journalist. However, his attendance record was poor and he had to repeat several courses after failing them.

If there was anything Huh knew he didn’t want to do, it was mathematics. “I was pretty good at most subjects except math,” he told the New York Times. However, there were glimpses of Huh’s mathematical brilliancy, it just did not seem like math.

Huh recalls one such flash coming from the unlikeliest of settings: a computer horror game. The 11th Hour, a 1995 computer game, had the player investigate a series of grisly events and piece things together. The player is tasked with solving various problems, some of which were notably complex.

Among them, one puzzle haunted Huh. It was a chess puzzle.

The puzzle seems simple enough. It is only a few squares and you only have four knights (the knights move in an “L” shape, like in regular chess). The goal is also straightforward: swap the positions of the black and white knights.

At first glance, the task seems trivial. You just move the knights around until something works. But it is not that simple – try it out.

It took the young Huh a week to solve, but for him, it was not just about solving the problem. It was about understanding what the problem actually meant.

In a mathematical sense, the way the knights move does not matter. The shape and size of the board also does not matter. What matters is the geometry of the blocks and the relationships between them. Essentially, the chess puzzle can be reinterpreted as a graph. Each knight can move to an unoccupied space on this graph, and this makes solving the problem much easier.

The first step is establishing a notation.

With this notation, instead of visualizing the knight moves geometrically, you can see them as a graph of possible moves from one space to another.

The knight can move from “1” to “5” and nowhere else. From “5,” it can go back to “1” or to “7.” In fact, the only box that has three places the knight can go is “2.” Herein lies the key.

When we add the knights, the problem takes this form:

With the above image, it becomes much more apparent what the way forward is. We have the “9” square that we can use as a hideaway for one knight, while we maneuver the other three. If we want to swap the white and black knights, we simply need to tuck the black ones away one at a time and move the white ones to their left (as shown on this graph). Try it!

This approach of visualizing problems in a mathematical fashion, fascinated Huh and drew him to mathematics. This particular type of problem is called a chromatic polynomial.

Imagine you have a map or a network of nodes connected by edges, like cities connected by roads or regions on a map bordered by boundaries. The chromatic polynomial of this network (or graph) tells you the number of ways you can color the nodes (or regions) with a given number of colors, such that no two adjacent nodes (or neighboring regions) share the same color.

This idea is not just about chess. It has practical applications in logistics and optimization problems, statistical physics (particularly in the study of phase transitions), and network analysis.

Translating problems into mathematical expressions became more than just a hobby for Huh. In his last year of college, Huh started to really fall in love with math. He attended a course by Heisuke Hironaka, a Japanese mathematician who had won a Fields Medal in 1970. Huh would become Hironaka’s Master’s student and received a letter of recommendation. It was a strange combination: a student who had failed several mathematics courses but had an enthusiastic recommendation from a leading mathematician. It was almost not enough. All but one of the universities that Huh applied to rejected him. The University of Illinois Urbana-Champaign put him on a waiting list and then, ultimately, accepted him.

Thus began the research journey of June Huh. Following a reluctant path into mathematics, he would go on to revolutionize the field of combinatorics, reinvigorating the field and intertwining it with some of the most esoteric areas of geometry.

However, Huh is far from the only mathematician drawn to chess knight puzzles.

Perhaps the most famous knight problem is the so-called “knight’s tour.” The premise is, once again, very simple. You have a square chessboard of a given dimension and one knight. You have to move the knight in a way that covers all the squares, without going to the same square twice. Something like this:

The problem can be applied to a number of different board sizes.

This too can be generalized as a Hamiltonian path problem in graph theory. But unlike Huh’s problem, this one dates from much earlier.

The earliest known reference to a knight’s tour problem dates back to the 9th century, from a Sanskrit work. Kashmiri poet and literary theorist Rudrata presented the knight’s tour as a verse in four lines of eight syllables each – another surprising mixing of poetry and mathematics.

Famous mathematician Leonhard Euler also looked at the problem, but it was H. C. von Warnsdorf who developed the first traceable algorithm to solve the knight’s tour. Warnsdorf postulated the following rule: “Move the knight to a square from which it will have the fewest available subsequent moves.”

This simple rule is surprisingly effective, but it is not perfect. It produces valid tours in the vast majority of cases on boards smaller than 50×50, but not all the time.

As it turns out, finding a workable, generalizable algorithm for knight’s tours is more difficult than it may appear. This comes from the sheer complexity of larger boards. For instance, the smallest board for which a knight’s tour yields solutions is 5×5. For a board of that size, there are over 1,700 possible tours (tours are considered different if they start from different places or have different trajectories; they can be mirrored or rotated).

That number quickly gets much, much larger.

n | Number of directed tours (open and closed)on an n × n board |

1 | 1 |

2 | 0 |

3 | 0 |

4 | 0 |

5 | 1,728 |

6 | 6,637,920 |

7 | 165,575,218,320 |

8 | 19,591,828,170,979,904 |

Remarkably, the problem even lends itself to a neural network algorithm. This was first discussed in 1992 in a paper spearheaded by Yoshiyasu Takefuji from Keio University in Japan, with a network set up so that each move is essentially a neuron in the network.

The allure of such knight problems and the surprising richness and depth that emerge from them says something about mathematics in itself. There is a generalizable trait of mathematics that is often ignored: the inherent beauty in its patterns and solutions.

For June Huh, this realization came from a place of artistic pursuit and a seemingly unrelated game puzzle, illustrating how the paths to mathematical success are as varied as the individuals who walk them.

The post June Huh, Combinatorics, and the Strange Allure of Chess Knight Problems originally appeared on the HLFF SciLogs blog.

]]>The post A Slice of π originally appeared on the HLFF SciLogs blog.

]]>Last month, mathematicians marked π day, named after the mathematical constant pi, whose value (to two decimal places) is 3.14. Pi Day is observed on March 14th (in the American calendar, 3/14) and gives mathematicians everywhere an excuse to celebrate the circle constant and share their favourite things about it.

One way to celebrate π is to try to calculate it, and with modern computers this can be done to a staggering level of precision. On π day this year, it was announced that a team of computer engineers had calculated 105 trillion digits – beating the previous record of 100 trillion digits, and taking a computer around 75 days to crunch through the calculations.

Calculations of π have been attempted throughout the ages – Wikipedia’s thorough and informative Chronology of Computation of Pi starts way back in 2000 BCE (estimated) with the Ancient Egyptian approximation of \(4 \times \big(\frac{8}{9}\big)^2\), through biblical references to π being equal to 3, all the way through to modern computational records.

These approximations suddenly get significantly more accurate when mechanical and electronic calculation devices join the fight, making it much easier to get accurate values. But the really hard-core mathematicians would much prefer to calculate π the old-fashioned way: by hand. If we check the records for the largest calculation performed prior to the invention of the desk calculator, that prize goes to a mathematician called William Shanks.

Shanks achieved a calculation of 527 digits, which he completed in 1853 – by hand, on his own, and over the course of a few decades. In order to approximate π by hand, Shanks used a formula introduced by John Machin in 1706, which states that:

\[ \frac{\pi}{4} = 4 \tan^{-1}\left(\frac{1}{5}\right)-\tan^{-1}\left(\frac{1}{239}\right) \]

The inverse tan function, sometimes also called arctan, might seem too difficult to compute by hand. But thanks to a formula first discovered in the 14th century by Indian mathematician Mādhava of Sangamagrāma, we can do this:

\[ \arctan(x) = x\ – \frac{x^3}{3} + \frac{x^5}{5}\ – \frac{x^7}{7} + \cdots \]

This means that each of the two arctans in Machin’s formula can be replaced with an infinite series of terms involving increasing powers of \(x\). Since in the formula we are using \(x\) is a fraction with 1 on the top, raising it to higher powers will just make it smaller and smaller.

So as we move along this sequence of fractions, the values will get smaller – at some point, they will be so small that they do not affect the earlier digits of π at all – so if we want to calculate, say, the first ten digits of π, we just need enough terms of this sequence to get us past the point where we are adding terms which affect those digits.

In practice, since each term has a power of the same value, this process largely involves dividing by the same value each time – the step from \(\big(\frac{1}{5}\big)^3\) to \(\big(\frac{1}{5}\big)^5\) is just division by 25, as is the step from \(\big(\frac{1}{5}\big)^5\) to \(\big(\frac{1}{5}\big)^7\). Calculating using this formula involves a lot of long division!

Machin himself used his formula, along with this arctan expansion, to compute π to 100 decimal places, but it was taken further by Shanks, who kept going to 527 digits. (He also later continued his calculations to 707 digits, but unfortunately he made a small error right at the start, meaning all of these new digits were incorrect).

More recently, Stand-up Mathematician Matt Parker has been attempting to calculate π by hand for his YouTube channel, where he posts a video every two years involving an increasingly elaborate method of calculation. Starting 11 years ago with an attempt to calculate π using pies, he has since calculated it using an out-of-control car, a pendulum, and even by counting molecules.

Two years ago, Matt attempted a hand calculation, using the same formula as William Shanks, in the town where Shanks himself performed his own calculation. This year, Matt wanted to take things one step further and attempt to beat Shanks’ record. This time, rather than using the two-term arctan formula of Machin, he employed a different one from the same mathematical family.

The term ‘Machin-like formula’ is used to describe formulae for π which have a similar form – a sum of arctan terms, with coefficients in front, which add to \(\frac{\pi}{4}\). There are hundreds of possible formulae for π in this form, including this one, attributed to Carl Størmer in 1895:

\[ \frac{\pi}{4} = 4\arctan\left(\frac1{5}\right)-\arctan\left(\frac1{70}\right) + \arctan\left(\frac1{100}\right) \]

\[ + \arctan\left(\frac1{5000}\right)-\arctan\left(\frac1{10101}\right) \]

As well as:

\[\frac{\pi}{4} = 322\arctan\left(\frac1{577}\right) +76\arctan\left(\frac1{682}\right) + 139\arctan\left(\frac1{1393}\right) \]

\[+\ 156\arctan\left(\frac1{12943}\right) + 132\arctan\left(\frac1{32807}\right) + 44\arctan\left(\frac1{1049433}\right)\]

And, if you do not mind your fractions having a non-integer on the bottom:

\[\frac{\pi}{4} = 83\arctan\left(\frac1{107}\right) +17\arctan\left(\frac1{1710}\right) +22\arctan\left(\frac1{103697}\right)\]

\[-\ 12\arctan\left(\frac1{1256744.5}\right) -22\arctan\left(\frac1{9140003941.5}\right) \]

Each of these can still be calculated using the arctan expansion, and depending on the number of arctan terms and the size of the denominators, different numbers of individual calculations will be needed. The larger the denominator, the more quickly the sequence will produce π – but the more individual calculation steps will be needed per division (long division by the square of a 10-digit number is quite a headache!)

For his own calculation, Matt chose one with seven arctan terms, which he hoped would allow him to delegate parts of the work, or “parallelise” the computation more efficiently – allowing more of the 400+ volunteers he had collected to join in and help. He picked a formula that would converge to the value of π as quickly as possible, while keeping the calculations as simple as possible. After multiplying through by 4, the formula used was:

\[\pi = 1587\arctan\left(\frac1{2852}\right) + 295\arctan\left(\frac1{4193}\right) + 593\arctan\left(\frac1{4246}\right) \]

\[+\ 359\arctan\left(\frac1{39307}\right) + 481\arctan\left(\frac1{55603}\right) \]

\[+\ 625\arctan\left(\frac1{211050}\right)\ – 708\arctan\left(\frac1{390112}\right) \]

If you would like to see the results of the attempt, Matt’s video about the 2024 calculation tells the whole story (and you may see me pop up a few times, as I was involved in the organisation).

The post A Slice of π originally appeared on the HLFF SciLogs blog.

]]>