Algebra often involves manipulating numbers or other objects using operations like addition and multiplication. Flavio Coelho/Moment via Getty ImagesCourtney Gibbons, Hamilton College You scrambled up a Rubik’s cube, and now you want to put it back in order. What sequence of moves should you make? Surprise: You can answer this question with modern algebra. Most folks who have been through high school mathematics courses will have taken a class called algebra – maybe even a sequence of classes called algebra I and algebra II that asked you to solve for x. The word “algebra” may evoke memories of complicated-looking polynomial equations like ax² + bx + c = 0 or plots of polynomial functions like y = ax² + bx + c. You might remember learning about the quadratic formula to figure out the solutions to these equations and find where the plot crosses the x-axis, too.Graph of a quadratic equation and its roots via the quadratic formula.Jacob Rus, CC BY-SA Equations and plots like these are part of algebra, but they’re not the whole story. What unifies algebra is the practice of studying things – like the moves you can make on a Rubik’s cube or the numbers on a clock face you use to tell time – and the way they behave when you put them together in different ways. What happens when you string together the Rubik’s cube moves or add up numbers on a clock? In my work as a mathematician, I’ve learned that many algebra questions come down to classifying objects by their similarities.
Sets and groups
How did equations like ax² + bx + c = 0 and their solutions lead to abstract algebra? The short version of the story is that mathematicians found formulas that looked a lot like the quadratic formula for polynomial equations where the highest power of x was three or four. But they couldn’t do it for five. It took mathematician Évariste Galois and techniques he developed – now called group theory – to make a convincing argument that no such formula could exist for polynomials with a highest power of five or more. So what is a group, anyway? It starts with a set, which is a collection of things. The fruit bowl in my kitchen is a set, and the collection of things in it are pieces of fruit. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 also form a set. Sets on their own don’t have too many properties – that is, characteristics – but if we start doing things to the numbers 1 through 12, or the fruit in the fruit bowl, it gets more interesting.In clock addition, 3 + 12 = 3.OpenStax, CC BY-SA Let’s call this set of numbers 1 through 12 “clock numbers.” Then, we can define an addition function for the clock numbers using the way we tell time. That is, to say “3 + 11 = 2” is the way we would add 3 and 11. It feels weird, but if you think about it, 11 hours past 3 o’clock is 2 o’clock. Clock addition has some nice properties. It satisfies:
closure, where adding things in the set gives you something else in the set,
identity, where there’s an element that doesn’t change the value of other elements in the set when added – adding 12 to any number will equal that same number,
associativity, where you can add wherever you want in the set,
inverses, where you can undo whatever an element does, and
commutativity, where you can change the order of which clock numbers you add up without changing the outcome: a + b = b + a.
By satisfying all these properties, mathematicians can consider clock numbers with clock addition a group. In short, a group is a set with some way of combining the elements layered on top. The set of fruit in my fruit bowl probably can’t be made into a group easily – what’s a banana plus an apple? But we can make a set of clock numbers into a group by showing that clock addition is a way of taking two clock numbers and getting to a new one that satisfies the rules outlined above.
Rings and fields
Along with groups, the two other fundamental types of algebraic objects you would study in an introduction to modern algebra are rings and fields. We could introduce a second operation for the clock numbers: clock multiplication, where 2 times 7 is 2, because 14 o’clock is the same as 2 o’clock. With clock addition and clock multiplication, the clock numbers meet the criteria for what mathematicians call a ring. This is primarily because clock multiplication and clock addition together satisfy a key component that defines a ring: the distributive property, where a(b + c) = ab + ac. Lastly, fields are rings that satisfy even more conditions. At the turn of the 20th century, mathematicians David Hilbert and Emmy Noether – who were interested in understanding how the principles in Einstein’s relativity worked mathematically – unified algebra and showed the utility of studying groups, rings and fields.
It’s all fun and games until you do the math
Groups, rings and fields are abstract, but they have many useful applications. For example, the symmetries of molecular structures are categorized by different point groups. A point group describes ways to move a molecule in space so that even if you move the individual atoms, the end result is indistinguishable from the molecule you started with.The water molecule H₂O can be flipped horizontally and the end result is indistinguishable from the original position.Courtney Gibbons, CC BY-SA But let’s take a different example that uses rings instead of groups. You can set up a pretty complicated set of equations to describe a Sudoku puzzle: You need 81 variables to represent each place you can put a number in the grid, polynomial expressions to encode the rules of the game, and polynomial expressions that take into account the clues already on the board. To get the spaces on the game board and the 81 variables to correspond nicely, you can use two subscripts to associate the variable with a specific place on the board, like using x₃₅ to represent the cell in the third row and fifth column. The first entry must be one of the numbers 1 through 9, and we represent that relationship with (x₁₁ – 1)(x₁₁ – 2)(x₁₁ – 3) ⋅⋅⋅ (x₁₁ – 9). This expression is equal to zero if and only if you followed the rules of the game. Since every space on the board follows this rule, that’s already 81 equations just to say, “Don’t plug in anything other than 1 through 9.” The rule “1 through 9 each appear exactly once in the top row” can be captured with some sneaky pieces of algebraic thinking. The sum of the top row is going to add up to 45, which is to say x₁₁ + x₁₂ + ⋅⋅⋅ + x₁₉ – 45 will be zero, and the product of the top row is going to be the product of 1 through 9, which is to say x₁₁ x₁₂ ⋅⋅⋅ x₁₉ – 9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 will be zero. If you’re thinking that it takes more time to set up all these rules than it does to solve the puzzle, you’re not wrong.Turning Sudoku into algebra takes a fair bit of work.Courtney Gibbons What do we get by doing this complicated translation into algebra? Well, we get to use late-20th century algorithms to figure out what numbers you can plug into the board that satisfy all the rules and all the clues. These algorithms are based on describing the structure of the special ring – called an ideal – these game board clues make within the larger ring. The algorithms will tell you if there’s no solution to the puzzle. If there are multiple solutions, the algorithms will find them all. This is a small example where setting up the algebra is harder than just doing the puzzle. But the techniques generalize widely. You can use algebra to tackle problems in artificial intelligence, robotics, cryptography, quantum computing and so much more – all with the same bag of tricks you’d use to solve the Sudoku puzzle or Rubik’s cube.Courtney Gibbons, Associate Professor of Mathematics, Hamilton College This article is republished from The Conversation under a Creative Commons license. Read the original article.
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The Unfavorable Semicircle Mystery: The YouTube Channel That Uploaded Tens of Thousands of Cryptic Videos
In 2015, the YouTube channel Unfavorable Semicircle gained attention for its enigmatic and abundant video uploads, totaling over 70,000 before its deletion in 2016. Theories about its purpose vary, from automated content generation to digital art experimentation, leaving its intent unresolved.
In the vast digital landscape of the internet, strange phenomena occasionally emerge that leave investigators, tech enthusiasts, and everyday viewers scratching their heads. One of the most puzzling cases appeared in 2015, when a mysterious YouTube channel called Unfavorable Semicircle began uploading an astonishing number of cryptic videos.
Within months, the channel had published tens of thousands of bizarre clips, many of which seemed random, incomprehensible, and visually chaotic. But as internet detectives began analyzing the content more closely, they discovered that these videos might not have been random at all.
The Sudden Appearance of an Internet Mystery
The Unfavorable Semicircle channel reportedly appeared in March 2015, with its first uploads arriving in early April.
Almost immediately, the channel began publishing videos at an incredible pace. Observers estimated that the account uploaded thousands of videos per week, sometimes multiple videos per minute. By the time the channel disappeared in early 2016, researchers believed it had uploaded well over 70,000 videos, possibly far more.
The scale alone made the project seem impossible for a human to manage manually.
Most of the videos shared similar characteristics:
Extremely short or very long runtime
Abstract visuals such as flashing colors, static, or distorted imagery
Little or no audio, or heavily distorted sounds
Titles made of random characters, symbols, or numbers
To casual viewers, the videos looked like pure digital noise. However, online investigators suspected something more deliberate was happening.
Hidden Images Discovered
The mystery deepened when researchers began extracting individual frames from some videos.
When thousands of frames from certain clips were stitched together, the results sometimes formed coherent images. One of the most famous examples involved a video titled “LOCK.” While the footage appeared chaotic at first, combining the frames revealed a recognizable composite image.
This discovery suggested the videos were carefully constructed rather than random uploads.
Theories About the Channel’s Purpose
Because the creator never explained the project, several theories emerged across Reddit, YouTube, and internet forums.
Automated Experiment Many believe the channel was created using automated software that generated and uploaded content at scale.
Alternate Reality Game (ARG) Some viewers suspected the channel might be part of a hidden puzzle or digital scavenger hunt.
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Encrypted Communication Others compared the channel to Cold War “numbers stations,” suggesting the videos could contain coded messages.
Digital Art Project Another theory suggests the channel was an experimental art project exploring algorithms, data, and visual noise.
Despite years of investigation, no single explanation has been confirmed.
Why the Channel Disappeared
In February 2016, YouTube removed the channel, reportedly due to spam or automated activity violations.
By that time, the channel had already become a minor internet legend. Fortunately, some researchers managed to archive a large portion of the videos before they disappeared.
Even today, archived clips continue to circulate online as investigators attempt to decode them.
Unfavorable Semicircle: The Most Bizarre YouTube Mystery
Other Mysterious YouTube Channels
The Unfavorable Semicircle mystery is not the only strange case on YouTube.
One well-known example is Webdriver Torso, a channel that uploaded hundreds of thousands of videos showing red and blue rectangles with simple beeping sounds. Internet speculation ran wild before Google eventually confirmed it was an internal YouTube testing account.
Another example is AETBX, which posts distorted visuals and unusual audio that some viewers believe contain hidden patterns or encoded information.
These cases highlight how automation, experimentation, and creativity can sometimes blur the line between technology and mystery.
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A Digital Mystery That Remains Unsolved
Nearly a decade later, the true purpose behind Unfavorable Semicircle remains unknown.
Was it a sophisticated experiment? A piece of algorithmic art? Or simply an automated test that accidentally captured the internet’s imagination?
Whatever the explanation, the channel stands as a reminder that even in a world filled with billions of videos and endless information, the internet can still produce mysteries that challenge our understanding of technology.
Why Internet Mysteries Still Fascinate Us
Stories like Unfavorable Semicircle capture attention because they combine technology, creativity, and the unknown. They invite people from around the world to collaborate, analyze patterns, and search for meaning hidden in the noise.
And sometimes, the most intriguing part of the mystery is that the answer may never fully be known.
Dive into “The Knowledge,” where curiosity meets clarity. This playlist, in collaboration with STMDailyNews.com, is designed for viewers who value historical accuracy and insightful learning. Our short videos, ranging from 30 seconds to a minute and a half, make complex subjects easy to grasp in no time. Covering everything from historical events to contemporary processes and entertainment, “The Knowledge” bridges the past with the present. In a world where information is abundant yet often misused, our series aims to guide you through the noise, preserving vital knowledge and truths that shape our lives today. Perfect for curious minds eager to discover the ‘why’ and ‘how’ of everything around us. Subscribe and join in as we explore the facts that matter. https://stmdailynews.com/the-knowledge/
Byron Allen’s Starz Stake Signals Bigger Moves in the Streaming Industry
Byron Allen’s Starz: Byron Allen has acquired a 10.7% stake in Starz Entertainment for approximately $25 million, signaling his long-term media strategy amidst industry consolidation. This investment positions him influentially in the evolving streaming market despite intense competition.
Byron Allen — Founder/Chairman/CEO of Allen Media Group
Byron Allen’s Starz investment
Media entrepreneur Byron Allen has taken another step toward expanding his growing media empire. Through his family office, Allen recently acquired a 10.7% stake in Starz Entertainment, purchasing the shares from a fund managed by former U.S. Treasury Secretary Steven Mnuchin.
The transaction, valued at approximately $25 million, gives Allen a significant minority position in the premium cable and streaming platform. While the investment itself may seem modest compared to the billion-dollar deals common in Hollywood, analysts say the move could signal a larger strategy unfolding in the rapidly evolving streaming industry.
Why the Starz Deal Matters
The shares were sold by Mnuchin’s Liberty 77 Capital fund, which previously invested in the company when Starz was still connected to its former parent, Lionsgate.
In 2025, Lionsgate completed a corporate restructuring that separated its operations into two distinct companies:
Lionsgate Studios – responsible for film and television production
Starz – focused on premium cable and streaming services
Following the spin-off, Starz became an independent publicly traded company. As a result, investors are still determining the platform’s long-term value in an increasingly crowded streaming marketplace.
Despite facing intense competition from larger platforms such as Netflix, Disney+, and Amazon Prime Video, Starz continues to maintain a strong subscriber base and recognizable content franchises.
Outlander – historical drama series
The Power franchise created by Courtney A. Kemp and executive produced by 50 Cent
Byron Allen’s Long-Term Media Strategy
Allen’s investment strategy has long focused on owning media distribution and infrastructure rather than simply producing content.
The Weather Channel
Dozens of local television stations across the United States
Multiple niche cable networks and digital platforms
Over the past several years, Allen has also pursued larger acquisitions, reportedly exploring deals involving companies such as Paramount Global and BET Media Group. While those deals did not materialize, they signaled his ambition to expand Allen Media Group into a major force in global media ownership.
The Bigger Picture: Industry Consolidation
Allen’s investment arrives during a time of significant disruption in the entertainment business. Traditional cable television continues to decline as audiences migrate toward streaming platforms. At the same time, major studios and media companies are struggling to make streaming services consistently profitable.
Industry observers believe these pressures could lead to a new wave of consolidation across Hollywood and the streaming sector. Smaller platforms like Starz could become attractive acquisition targets for larger companies seeking additional subscribers and content libraries.
A Potential Hidden Opportunity
For now, Allen’s 10.7% stake does not give him control of Starz. However, it does provide influence as one of the company’s larger shareholders and leaves open the possibility of increasing his ownership in the future.
If consolidation accelerates and streaming platforms begin merging or forming partnerships, assets like Starz could become significantly more valuable. For Byron Allen—whose career began as a stand-up comedian before evolving into one of the most prominent independent media owners in America—the investment may represent another calculated step in a decades-long strategy built around media ownership and long-term growth.
Dive into “The Knowledge,” where curiosity meets clarity. This playlist, in collaboration with STMDailyNews.com, is designed for viewers who value historical accuracy and insightful learning. Our short videos, ranging from 30 seconds to a minute and a half, make complex subjects easy to grasp in no time. Covering everything from historical events to contemporary processes and entertainment, “The Knowledge” bridges the past with the present. In a world where information is abundant yet often misused, our series aims to guide you through the noise, preserving vital knowledge and truths that shape our lives today. Perfect for curious minds eager to discover the ‘why’ and ‘how’ of everything around us. Subscribe and join in as we explore the facts that matter. https://stmdailynews.com/the-knowledge/
2025 was hotter than it should have been – 5 influences and a dirty surprise offer clues to what’s ahead
The past three years recorded unprecedented global heat, with 2025 being particularly warm. Factors such as greenhouse gas emissions and a decline in solar activity influenced temperatures and extreme weather patterns.
The past three years have been the world’s hottest on record by far, with 2025 almost tied with 2023 for second place. With that energy came extreme weather, from flash flooding to powerful hurricanes and severe droughts. Yet, by most indicators, the planet should have been cooler in 2025 than it was.
So, what happened, and what does that say about the year ahead?
As an earth and environmental scientist, I study influences that affect global temperatures year to year, such as El Niño, wildfires and solar cycles. Some make Earth hotter. Some make it cooler. And one particularly unhealthy influence has been quietly hiding a large amount of global warming – until now.The past three years have been the warmest on record. The chart compiled by the European Union’s Copernicus Climate Change Service shows the comparison to preindustrial-era temperatures in the second half of the 1800s. C3S/ECMWF
Factors that made 2025 cooler than 2024
The Earth’s climate is the result of many factors that change from year to year. Some that helped make 2025 cooler than 2024 include:
La Niña’s arrival: La Niña is part of the El Niño-Southern Oscillation, a natural climate pattern that fluctuates between warm El Niño conditions and cooler La Niña conditions. During El Niño, the Pacific Ocean heats up along the equator, influencing the atmosphere in ways that can cause intense storms, droughts and heat waves around the planet. La Niña does the opposite; it’s like putting an ice pack on the atmosphere.
Both 2023 and 2024 were El Niño years, but in 2025 conditions shifted to neutral and then to La Niña starting in September.
The solar cycle: The Sun reached its solar maximum near the end of 2024, the peak of its energy output in an approximately 11-year cycle, and began declining in 2025. So, while the sun’s output was still stronger than average in 2025, it was less than in 2024.
Fewer wildfires: Despite some destructive blazes, the world also saw fewer wildfires during 2025 than 2024, which put less carbon dioxide – a planet-warming greenhouse gas – into the atmosphere.Major warming and cooling influences from 2016 to 2025. Each graph starts at 2016. Anthropogenic warming, natural carbon sinks and sulfur dioxide (SO2) reductions start from zero in 2016 to illustrate cumulative changes to existing reservoirs; El Niño/La Niña and the solar cycle show real-time influences on the global temperature, relative to mean values. Michael Wysession. Data: Global Carbon Project (Anthropogenic Global Warming, Natural Carbon Sinks); NOAA (El Niño/La Niña, Solar Cycle); SO2 Reductions (FaIR Analysis by Carbon Brief)
Several other factors made 2025 warmer than expected, and some are likely to continue to increase in 2026. They include:
Greenhouse gas emissions: The big driver of global warming is excess greenhouse gas emissions, largely from burning fossil fuels, and 2025 had plenty.
Greenhouse gases trap heat near Earth’s surface like a blanket, raising the temperature. They also linger in the atmosphere for years to centuries, meaning gases released today will continue to warm the planet well into the future. The levels of carbon dioxide, methane and nitrous oxide in the atmosphere all increased in 2025.
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Sources of worldwide carbon dioxide emissions that have grown the most in recent decades. Carbon Brief, CC BY
Earth’s energy imbalance: Other sources can disrupt the natural balance between the amount of sunlight that reaches Earth and the lesser amount radiated back to space. A recent study found that Earth’s energy uptake surged and temperatures rose quickly when a rare three-year La Niña in 2020-2022 shifted to El Niño in 2023-2024.
Declining polar ice, which efficiently reflects sunlight back into space, also affects the energy balance. As sea ice declines, it leaves dark ocean water that absorbs most of the sunlight that reaches it. In a spiraling feedback, warmer water melts sea ice, allowing more sunlight into the ocean, warming it faster; 2025 had the lowest winter peak of Arctic sea ice on record and the third-lowest minimum extent of Antarctic ice.
Air pollution: Sulfate aerosol pollution from coal combustion and burning heavy fuel oil in shipping has also been affecting Earth’s energy balance. It has been masking the full effects of human-caused greenhouse gases for years by reflecting sunlight back into space, creating a cooling effect. But sulfate aerosol pollution is also a serious health hazard, blamed for about 8 million human deaths per year from lung diseases.
Recent reductions in sulfate pollution – now 40% less than 20 years ago – have meant about a 0.2 F (0.13 C) increase in global temperatures. Much of the reduction was from China’s efforts to reduce its notoriously bad air pollution in recent years and international shipping rules in effect since 2020 that have reduced sulfur emissions from large ships by 85%.Sea ice levels were near record lows for both Arctic and Antarctic ice in 2025. Carbon Brief, CC BY
Taking all factors together, humans are now warming the planet at a faster rate than at any point in human history: at about 0.5 F (0.27 C) per decade. That extra heat can fuel extreme weather, including flash floods, heat waves, extended droughts, wildfires and coastal flooding, affecting human lives and economies.
Predictions for 2026
Most climate models predict 2026 will be about as hot as 2025, depending on whether a Pacific El Niño develops, which forecasters give about a 60% chance of happening. The planet is already starting the year out warm, even if it doesn’t feel like that everywhere. While January was very cold in parts of the U.S., globally, Earth saw its fifth-warmest January on record, and much of the western U.S. saw one of its warmest winters on record.
Dive into “The Knowledge,” where curiosity meets clarity. This playlist, in collaboration with STMDailyNews.com, is designed for viewers who value historical accuracy and insightful learning. Our short videos, ranging from 30 seconds to a minute and a half, make complex subjects easy to grasp in no time. Covering everything from historical events to contemporary processes and entertainment, “The Knowledge” bridges the past with the present. In a world where information is abundant yet often misused, our series aims to guide you through the noise, preserving vital knowledge and truths that shape our lives today. Perfect for curious minds eager to discover the ‘why’ and ‘how’ of everything around us. Subscribe and join in as we explore the facts that matter. https://stmdailynews.com/the-knowledge/
Courtney Gibbons, Associate Professor of Mathematics, Hamilton College This article is republished from The Conversation under a Creative Commons license. Read the original article.
