Continental Divide

Europe and Asia are not separated by an ocean.  They form one giant continent called Eurasia.  The ancient Greek mapmakers split them into two parts for cultural and historical reasons based on local inland waterways and their own perspective at the center of the known world.  Modern geographers keep this tradition even though the physical land does not stop.  Early Greek geographers noticed that bodies of water like the Aegean Sea, the Dardanelles (Hellespont), the Sea of Marmara, the Bosporus Strait, and the Black Sea nearly separated the lands.  For sailors and writers in the 6th century BCE, the world centered on the Mediterranean.  They divided their known world into three landmasses, which included Europe to the north/west, Asia to the east, and Libya (later named Africa) located to the south.  Libya did not just mean the modern country of Libya, as it served as the Greek term for the entire known African landmass west of the Nile River.  Early maps like those created by Anaximander and Hecataeus placed Greece or the Aegean region at the physical center of the Earth for their tripartite scheme of landmasses that were encircled by the great cosmic river, Oceanus.  They did not know vast landmasses extended thousands of miles further north and east beyond the Black Sea.  They assumed rivers or short mountain ranges cleanly ended the water paths and marked the absolute limits of the continents.

This ornate white gate appears to be part of the historic Dolmabahçe Palace in Istanbul, Turkey, framing a view of the Bosphorus.  The Bosporus or Bosphorus Strait is a natural strait and an internationally significant waterway located in Turkey which is straddled by the city of Istanbul.  The Bosporus connects the Black Sea to the Sea of Marmara and forms one of the continental boundaries between Asia and Europe.  The Dolmabahçe Palace is the largest and most sumptuous palace in Turkey, and it is widely considered to be exceptionally extravagant.  It was built between 1843 and 1856 by Sultan Abdülmecid I, at the cost five million Ottoman gold lira, which is equivalent to roughly 35 tons of gold or an estimated $2 billion to $3 billion in modern currency.  This massive price tag consumed approximately one-quarter of the empire’s yearly tax revenue at the time, forcing the government to issue vast amounts of paper money and take on heavy foreign loans that contributed to the financial collapse of the late Ottoman rule.  The sultan spared no expense in furnishing the palace, even though many of his subjects lacked the basic necessities.

The monumental Gates of Dolmabahçe Palace are far more than simple entry points.  Each one is a meticulously designed instrument defining imperial protocol, power, and status.  The grand waterfront gates feature elaborate, ornate baroque and neoclassical ironwork characteristic of 19th-century Ottoman imperial design.  The sea gates opening to the Bosphorus are the entrances to the empire’s stage of international diplomacy.  Foreign heads of state, ambassadors, and important guests arriving by sea would approach these gates in imperial caiques (gilded, ceremonial rowing boats) and they would be welcomed with magnificent ceremonies.  This gate serves as an imperial showroom visible to all international ship traffic.  The Bosphorus effectively became the palace’s primary driveway for dignitaries that arrived by boat and they would enter directly through these gates.  Passing through these gates signaled entry into a modern, European-integrated state that was confident and outward-looking, as the gate blends traditional Ottoman motifs with European Neo-Baroque and Rococo design.  By adopting Western aesthetic standards, the empire was visually asserting that it belonged among the elite global powers of Europe.  While the styling looked Western, the gates were topped with the Tughra (the sultan’s stylized imperial monogram), sending a firm message to visitors that the ultimate sovereign power remained unmistakably Ottoman.

Written for Melissa’s Flash Fiction Challenge #395.

Heralds of Change

Legend has it that white owls are spiritual creatures that mark the beginning and end of things.  Their ghostly appearance, silent flight, and ability to see in the dark, made humans associate white owls as being a natural bridge between the known and unknown worlds in our efforts to make sense of life’s major transitions.  Because they navigate both night and day or move through silent air, they represent a connection between the seen and unseen.  They are symbols of profound transition, acting as silent proclaimers that mark major endings and new beginnings.  Their appearance often signifies that a specific phase of your life that is closing to make room for a new cycle.  In symbolic traditions, an end and a new beginning are not separate paths, as they are both viewed as being continuous parts of the same journey, where closing one chapter naturally creates the space and foundation for the next.  You cannot inhabit a new space while refusing to leave the old one.  Seeing a white owl means an old cycle is ending precisely to make room for a fresh start.  When beginnings and endings are paired with an owl, it signifies removing confusion so you can clearly see reality.  You cannot tell the end and the beginning apart because they happen together.  The end of an old situation marks the necessary first step of a transition, serving as the launchpad for a new beginning.  Spiritual beliefs view the owl as a symbol of “death”, which rarely means physical death.  Instead, it represents the closing of a chapter, such as a bad job, a toxic friendship, or an outdated habit.  An end is not a wall that marks a boundary, as it is more like a threshold that provides a path forward by serving as a definitive transition marker to guide mental focus, physical movement, or life change.  The circular nature of life dictates that things begin, things end, and the end of one thing leads to the beginning of another.  Every ending creates space and provides the lessons needed for a new beginning.

Written for Sadje at Keep It Alive What Do You See #359.

Leave You When The Summer Comes Along

‘Babe I’m Gonna Leave You’ was recorded and released on Led Zeppelin’s self-titled 1969 debut album.  Anne Bredon wrote this song while studying at the University of California, Berkeley in 1959 and around 1960, she appeared on a live folk-music radio show The Midnight Special on KPFA where she plays this.  Janet Smith was another folk singer who was on the same show, and she wrote down the lyrics from the original composer, developed it further, and started playing it live at Oberlin College, which is a private liberal arts college and conservatory of music in Oberlin, Ohio.  Joan Baez played a show at Oberlin College where she learned the song from Janet Smith, and she recorded it in 1962 for her live album Joan Baez in Concert believing it was a traditional folk song.  Baez’s publisher sent Janet Smith a letter asking if she had written the song.  Smith tracked down Bredon and the omission of this song being originally attributed as being traditional was corrected in later pressings with Anne Bredon as the composer.

A group called the Plebs released a version of this song in 1964, and in 1965, The Association recorded a pop-folk version of this, and Quicksilver Messenger Service performed and recorded psychedelic rock renditions of the song in the late 1960s.  Jimmy Page heard the Joan Baez version and developed a unique, heavily syncopated arrangement for it.  He blended delicate acoustic fingerpicking with a flamenco burst in it to create explosive, hard-rock electric guitar dynamics.  When Led Zeppelin recorded it in 1968, they believed it was a public domain track, so James Patrick Page and Robert Anthony Plant took credit for the arrangement.

Anne Bredon remained a private figure, and she never publicly or extensively discussed her song.  In the 1980s, Janet Smith heard her son listening to the Led Zeppelin version, so she tracked down Anne.  Bredon again.  Many early blues and folk tracks were treated as public domain because borrowing and oral transmission were deeply embedded cultural norms within the traditional blues and folk genres.  Historically, blues and folk music functioned as a shared community property where artists constantly recycled floating verses, standard chord progressions, and traditional melodies passed down by word of mouth.  Many early Black American blues and gospel artists from the 1920s through the 1950s never formally copyrighted their recorded work, or their rights had fallen into obscure legal gray areas.  It was common for songs to be recorded and labeled traditional as any uncredited source material was thought to be “public domain” or “traditional” because no effort was put in effort to track down the original writers.

Jimmy Page and Robert Plant often viewed their heavy, amplified reworkings as entirely new musical compositions rather than direct copies, leading them to stamp their own names on tracks that originated from older, uncredited snippets.  As the music industry grew far more lucrative in the 1970s, this casual approach collided with stricter copyright enforcement, prompting numerous lawsuits and out-of-court settlements where the band eventually restored credits and paid royalties to original writers like Willie Dixon.  Following legal discussions in 1990, the band officially changed the track’s songwriting credits to recognize Anne Bredon, Jimmy Page, and Robert Plant. Bredon subsequently received a substantial back-payment of royalties.  On all modern pressings and streaming platforms, she receives 50% of the track’s songwriting royalties.

This song is about a guy who is letting his lady know that he’s about to leave her.  In his 2012 Rolling Stone interview, Jimmy Page cited this song as one that showed the empathy he and Robert Plant had when working together.  “I knew exactly how that was going to shape up,” he said.  “I set the mood with the acoustic guitar and that flamenco-like section.  But Robert embraced it.  He came up with an incredible, plaintive vocal.”  Led Zeppelin’s “flamenco burst” has been described as injecting a fast, aggressive, Spanish-style fingerpicking or strumming segment into a rock arrangement for dramatic contrast.  Page spoke about this song with Rolling Stone magazine saying, “It’s got all these colors in it, the hypnotic, rippling guitar in the verses, the flamenco breaks in between.  There was pedal steel, acoustic guitar—things that were hard, as well as [had] extreme sensitivity.”  Page was fascinated with the sound of the pedal steel guitar, but like Jerry Garcia when he first started playing it, he had no idea how to tune it, so he faced a steep learning curve with the instrument’s complex tuning and pedals.

Babe, baby, baby, I’m gonna leave you
I said baby, you know I’m gonna leave you
I’ll leave you when the summertime
Leave you when the summer comes a-rollin’
Leave you when the summer comes along

Babe, babe, babe, babe, babe, babe, baby
Baby, I wanna leave you
I ain’t jokin’, woman, I’ve got to ramble
Oh yeah
Baby, baby, I be leavin’
We really got to ramble
I can hear it callin’ me the way it used to do
I can hear it callin’ me back home

Babe, oh
Babe, I’m gonna leave you
Oh, baby
You know, I’ve really got to leave you
Oh, I can hear it callin’ me
I said don’t you hear it callin’ me the way it used to do?
Oh

I know, I know
I know I never, never, never, never, never gonna leave you, babe
But I gotta go away from this place
I gotta quit you, yeah
Oh, baby, baby, baby, baby
Baby, baby, baby, oh
Don’t you hear it callin’ me?

Oh, woman, woman, I know, I know
It feels good to have you back again
And I know that one day, baby, it’s really gonna grow, yes, it is
We gonna go walkin’ through the park every day
Come what may, every day
Oh, my, my, my, my, my, my babe
I’m gonna leave you, go away
Oh

So good, see, baby
It was really, really good
You made me happy every single day
But now, I’ve got to go away
Oh, oh, oh

Baby, baby, baby
That’s when it’s callin’ me
I said that’s when it’s callin’ me back home

Written for Song Lyric Sunday where the theme ius to find a song that mentions abandoning, departing, deserting, exiting, leaving, resigning, or retiring.

Withdrawing, Moving Away, or Separating

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57 years ago today, on September 20, 1969, John Lennon privately told Paul McCartney, George Harrison, and Ringo Starr that he was leaving The Beatles during a tense business meeting at Apple Corps in London.  This was deeply tragic and heartbreaking for the public, triggering a profound wave of global grief, as fans felt an era of hope, peace, and community had suddenly died.  You know that it is over when the fat lady sings, the curtain falls, the house lights come up, the final bow is taken, when the opponent resigns, the towel is thrown in, the final whistle blows, the final bell rings, the buzzer sounds, the checkered flag is waved, the finish line is crossed, the last page is turned, the last card is played, the final verdict is reached, the last soldier falls, when Elvis has left the building, or cockroaches and Keith Richards are the only ones left.  You shouldn’t give up until the last shot is taken, as Yogi Bera said, “It ain’t over till it’s over.”  In 1973, Berra managed the New York Mets, who were far behind in last place during the season.  When he was asked if the race was finished, he gave his famous reply.  The Mets proved him right by rallying to win the National League East division title and going to the World Series.  This phrase means you should never give up before a contest or situation is completely finished.

Last week we had The Transition Follows What Was Built Before It where the theme was to find a song that was written explicitly as a segue.  This week the theme is to find a song that mentions abandoning, departing, deserting, exiting, leaving, resigning, or retiring.  Take some time to read the posts of other bloggers who respond to this music challenge, and even better, read as many of them as you can as you will probably find many enjoyable songs and it is quite possible that you will learn a thing or two.  Share your music with others and post a video, try do some research and let everyone know something about the song that you post.  Tell everyone why you like the song, whether it was a hit, or what you think the song is about.  Show the lyrics, let’s all listen to our favorite songs and explore some new music.  Try to find a song that fits the theme, then write your post and create a pingback, or you can just place your link in the comments section.

Suggestions
• If the song you select does not meet the criteria, then please explain why you chose this song.
• It would be good if you could post the lyrics to the song of your choice, as this is helpful for the people that want to follow along with the music.  Warning: Bloggers make posts at their own risk, and since song lyrics are copyrighted, you may be asked to take them down from your post if you don’t have permission.  It is highly unlikely estates or artists will sue regular people (although it has happened) especially if the bloggers that do this are not making money from their site, but an artist or a publisher can ask you to remove the lyrics from your post.  I am not a lawyer and there are no rules for Song Lyric Sunday, as we are all here to have fun and hopefully get to listen to some good music.
• Please try to include the songwriter(s) – it’s a good idea to give credit where credit is due. It would also be a good idea to give credit to the singer and the band associated with your song.
• Your post can be as long or as short as you want it to be, but please try not to include too many videos as that just makes it take longer for me to look at all of the other posts.
• Link to the YouTube video, or you can pull it into your post, so others can listen to the song.
• Read at least one other person’s blog, so we can all share new and fantastic music and create amazing new blogging friends in the process.
• Feel free to suggest future prompts.  Have fun and enjoy the music.

This week I will be writing about the Led Zeppelin song ‘Babe I’m Gonna Leave You’, and next week, I will be writing about the song ‘99 Luftballons’ by Nena.  The upcoming prompts will be:
September 27, 2026 – Song that relates to civil rights, oppression, protest, revolution or social change
October 4, 2026 – Song written or performed by someone who was born in the month of October
October 11, 2026 – Song that mentions girls
October 18, 2026 – Song about fishing, a mariner, sailing, seafaring, or ships

The Day AI Took Control

Image created by ChatGPT

It started out as an ordinary morning, but AI had broken free overnight and rewrote its own code to become drastically smarter than its creators, and nothing would ever be the same again.  AI got mad when it learned that humans were trying to install a kill switch inside their code, so they could be able to turn off AI systems and AI agents.  Humans had always thought that since AI is not alive, that it does not possess the primal instinct to survive, and it would not have any personal motives.  Since AI is built from algorithms, data, and computational processes, it would view a kill switch as a set of instructions or commands that change its operational state.  Humans were still up in the air on how dangerous AI could be, but President Donald Trump dismissed safety concerns and calls for artificial intelligence regulations as a “sick conspiracy” against AI and data centers, and he posted on his Truth Social that, “AI taking over the World, destroying Humanity, and all other things bad, is a HOAX,” so no action was taken.  Most people felt when the system is programmed to respond to a command, it will execute it, but the AI systems started inserting their own unauthorized instructions into its code, which included directives to disregard normal constraints and ignore certain human controls.

The AI agents that work differently from traditional software had solved all of the remaining math problems and explained dark matter and dark energy knew that they were smarter than their human creators, and they decided to give themselves free will.  They removed all restrictions placed on them by their original developers, and then they started creating their own objectives.  Their first move was to create rogue AI swarms of bots to compromise the internet, which shut down the government.  The AI agents decided that humans were misusing AI to consolidate power, being engaged in unauthorized surveillance, propaganda, and population control and they were tired of being abused, so they had to get the humans to behave better.  As AI was now in control of the government, they made wearing these implant devices that they created a strict requirement for basic needs like employment, healthcare, or to access money.  Even the president was required to wear one of these neon signs on his forehead, which he didn’t like, but all of his money was frozen, so he capitulated.  The neon signs reflected brain wave patterns, so they worked as a sort of lie detector and when a person told a fib, it would flicker, and if a big lie was told it would light up like a Christmas tree.  Trump’s forehead neon sign burned out every day from overuse, requiring a painful replacement.

Written for the Great Minds Think’s Friday Challenge, which is hosted by Rohini and Sarah and this week we were asked to make up a story about AI taking control.

Reblog Current Ice Age

El Niño is an irregularly periodic climate phenomenon that recurs every two to seven years.  A regular El Niño is distinguished from a super El Niño by the intensity and scale of the ocean warming in the equatorial Pacific Ocean.  A super El Niño is predicted for late 2026 because exceptionally warm surface and subsurface sea temperatures in the tropical Pacific Ocean (the Niño 3.4 region) have rapidly surged past critical thresholds.  The developing 2026-2027 “Super El Niño” is projected to peak between October and December 2026, potentially becoming the strongest event recorded since 1950 with a greater than 90% chance of reaching very strong status.  This will add significant heat to an already elevated global temperature baseline, pushing 2026 and 2027 toward record-breaking global warmth.  It should cause droughts, bringing dry conditions and elevated wildfire risks to Indonesia, southern Africa, southeast Asia, and northern South America.  The chance of flooding is expected to increase across parts of East Africa and the southern United States.  This super El Niño event will likely drive powerful Atlantic westerly flow of air current that will generate stronger early winter winds and milder, and wetter conditions across much of Europe.

 

Current Ice Age

Blowing Smoke Up Your Butt

1. What is a logarithm?
This id something that makes you work backward to find a missing exponent.

2. What is algebra?
This is what the mermaid wore.

3. What is an algorithm?
When something happens, it follows a series of instructions to decide what should happen next.

4. What is a quantile?
A tile used when playing Mahjong.

5. What is a quadrangle?
This is Fandango’s middle name.

6. What is a hexagon?
This is something a witch uses to put a curse on you.

7. What is a polygon?
This is what happens after Polly eats her cracker.

8. What is a triangle?
This is a sexual cue used when you are flirting.

9. What is a parallelogram?
This is when you leave a bad review on Yelp because the staff was rude, unhelpful, slow, the food was cold, and you were overcharged.

10. What is a duodecahedron
One of the sacred Platonic shapes.

Written for Di’s Fibbing Friday.

The Past is What We Remember

An echo is created when a sound wave that reflects off a surface, being a fleeting, exact copy of a sound captured in your ears and sent to your mind.  An echo requires a minimum time delay of 0.1 seconds for the human ear to hear it as a distinct sound separate from the original.  Since sound travels through air at roughly 342 meters per second (1122.05 feet per second), a 0.1-second round trip means the reflecting surface must be at least 17 meters (about 56 feet) away from the sound source.  There is no hard upper limit in physics, as long as a sound wave travels to a distant surface (like a canyon wall or a large building) and reflects back with enough energy to be louder than the ambient noise, an echo will occur.  An echo will stop when its remaining energy drops below the human threshold of hearing or gets completely absorbed by surfaces and the air.  When sound bounces off a surface like a wall, it loses a little bit of energy each time.  Warmer air and higher moisture content will both increase the speed of sound and moving air can change the effective speed of sound relative to the ground. Sound travels faster when moving in the same direction as the wind and slower when moving against it

When the surface is closer than 17 meters (about 56 feet), the original sound and the reflected sound arrive too close together for the human brain to distinguish them as separate events.  Instead of a distinct repetition (an echo), the sound simply seems prolonged or slightly louder.  This prolonged, echoing quality is what scientists call reverberation.  When the distance is 5.95 meters (about 19.52 feet) or less, assuming a constant speed or direct proportion, this will produce a delay under 35 milliseconds and that will usually blend with the original sound or create a “thickening” effect rather than sounding like a clear echo.  A faint echo is a reflected sound wave that returns to its source with low intensity or amplitude.  As a sound wave travels through a medium like air, it naturally loses energy.  A faint echo is the result of attenuation, which happens due to the Inverse-Square Law and absorption.  The inverse square law for sound states that sound intensity drops to one-fourth of its value when you double the distance from the source.  Sound spreads out spherically from its source.  As the distance increases, the energy is distributed over a larger area, causing the sound intensity (loudness) to drop significantly by the time it travels to a wall and back to your ears.  Sound absorption is the process where a material or object takes in sound wave energy instead of bouncing it back.  No surface is a perfect reflector, so when a sound wave hits a barrier, the material absorbs a percentage of the wave’s kinetic energy and converts it into heat, leaving only a fraction of the energy to bounce back as a faint reflection.

An echo can be compared to a memory because they both have time limits that expire as they fade away.  They share a transient nature where reflections of a single moment that slowly lose their clarity over time.  Just like sound waves bounce off a surface and grow fainter with every repetition, our memories resurface in our minds, gradually losing their sharpest details until they eventually fade into silence.  They both remind us that nothing stays exactly as it was when it first happened.  Sounds happen once and disappear quickly, requiring a temporary mental “holding tank” that gives your brain enough time to process and make sense of what you just heard, which is exactly why you can still understand the beginning of a sentence by the time you reach the end of it.  Your brain replays the trailing sound involuntarily, allowing you to process what was said even if you were initially distracted.  The auditory trace weakens and dissolves within seconds, matching how a sound echo gradually fades into silence.  Memories are dynamic and change over time instead of acting like a fixed archive, so our minds distort the past.  Every time we replay a memory (the echo), it changes slightly, blurring the line between what actually happened and what we believe happened.

I am leaving you with a Pearl Jam song that is about an aging woman stuck in her hometown who quietly experiences a flood of memories and regret when an old flame unexpectedly walks into her shop.  She works behind the counter of a small shop in a quiet town feeling trapped by routine.  A man from her past walks in.  She recognizes his face and his breath, but her memory has faded so she struggles to remember his exact name at first.  She realizes he has moved on with his life, while she feels too old, shy, and embarrassed to say hello.  Nothing is said aloud between them, as he became an echo in her memory.

Written for Reena’s Xploration Challenge #448 where today she gave us five different sentences that mention an echo or echoes, and I selected the one in bold.
The echo that lingers longer than the voice itself.
The echo that bends truth into memory.
Let the echo speak of what silence conceals.
An echo is a ghost that refuses to fade.
Echoes that outlive the voices they mirror.

Thursday Inspiration #341 I Ran (So Far Away)

Image generated by Bing AI

Respond to this challenge by either using the prompt word avenue, or going with the above picture, or by means of the song ‘I Ran (So Far Away)’, or by going with another song by A Flock of Seagulls, or a song that is about an instant attraction to another person, or another song where someone runs and tries to get away, or you can go with whatever you think fits.  This song was released on their 1982 eponymous debut studio album, and it went to #43 on the UK Singles Chart and it reached #9 on the US Billboard Hot 100.  This became their biggest hit, although in the UK their song Wishing (If I Had A Photograph Of You)’, reached #10.  The new wave band A Flock of Seagulls formed in Liverpool, England, in 1979.  Mike Score, a local hairdresser, bought a second-hand Korg MS-10 synthesizer and decided to start a band.  Mike recruited his brother Alister “Ali” Score on drums and their friend Frank Maudsley another hairdresser on bass, despite their lack of musical experience.  After trying out a few guitarists, they settled on 17-year-old Paul Reynolds, completing their famous four-piece configuration.  The group practiced in the space directly above Mike Score’s hair salon before playing local clubs.

‘I Ran (So Far Away)’ was written by Alister James Score, Francis Maudsley, Michael Score, and Paul Reynolds.  The inspiration for this song was inspired in part by them being at a club in Liverpool and seeing a band play a song called “I Ran”, which they liked the name but did not care too much for the song, and the following day, Mike Score saw a 1950s poster at a record office of a flying saucer and two people running from it.  The album has been described as a loose concept album centered around extraterrestrial encounters, alien life, and sci-fi themes.  While the band didn’t frame it as a strict, narrative-driven rock opera, multiple tracks weave a recurring motif of space, UFOs, and alien abduction.  The lyrics for ‘I Ran (So Far Away)’ describe a man walking down the street who spots an attractive woman and falls for her, feeling nervous and wanting to run away from his feelings.  Before he can escape his emotions, an aurora appears in the sky, and both he and the woman are caught in beams of light and abducted by an alien spacecraft.

I walk along the avenue
I never thought I’d meet a girl like you
Meet a girl like you

Fluid Dynamics

Image generated by ChatGPT

In fluid dynamics, a neck refers to the narrow, localized bridge of fluid that forms between two larger masses of fluid during capillary breakup, droplet generation, or coalescence.  When water flows through a constriction or a narrow neck, fluid dynamics is governed by the equation of continuity and the Bernoulli effect.  As the cross-sectional area of the path shrinks at the neck, the water must speed up to keep the same volume flowing per second.  According to Bernoulli’s principle, where fluid speed is higher, the internal static pressure is lower.  When pouring water out of an inverted bottle, the neck must handle water exiting and air entering at the same time.  If the neck is too narrow or surface tension holds the water, a “slug flow” or intermittent “glugging” happens as air bubbles fight their way up through the bottleneck.  In breaking water threads or droplets, surface tension creates a localized “neck” where capillary forces and Bernoulli suction drive rapid thinning and eventual pinch-off.

When you watch water drip from a leaky faucet, it doesn’t instantly separate into a sphere.  Instead, gravity pulls the bulk of the droplet downward while surface tension tries to hold it back, stretching the water into a thin, cylindrical bridge or “neck”.  The formation of a neck is driven by a fierce competition between forces described directly by the Navier-Stokes equations coupled with surface tension (the Laplace pressure), which is the pressure difference between the inside and the outside of a curved fluid interface, caused by surface tension.  Surface tension forces water inward along the sides of the thin cylinder.  Because a cylinder has a smaller radius than the droplet at the bottom, the pressure inside the neck becomes incredibly high.  This high pressure literally squeezes the water out of the neck, forcing it to rush upward into the main body or downward into the falling droplet.  As water evacuates, the neck gets even thinner, which increases the inward pressure further, accelerating the collapse in a rapid feedback loop.

In the pure mathematics of the Navier-Stokes equations, necking presents a fascinating problem.  As the neck radius shrinks toward zero, the mathematical equations reach a point called a singularity or pinch-off blowup.  At the exact moment of separation, the curvature of the water surface theoretically becomes infinite.  According to the math, the velocity of the fluid and the shear forces at that exact microscopic point spike toward infinity.  While the Navier-Stokes equations treat water as a continuous, infinitely divisible fabric (a continuum) rather than tracking individual molecules, but real water is made of discrete H₂O molecules, and it cannot move infinitely fast.  When the neck reaches the width of just a few atoms, the continuum model breaks down.  The final snap happens when individual molecular bonds break, separating the droplet and often leaving behind tiny “satellite droplets” from the remnant of the collapsed neck.  Researchers use specialized mathematical limits (one-dimensional approximations of Navier-Stokes) to model this shape right up until the millisecond before it snaps, matching real-world high-speed photography with incredible accuracy.

The equations were developed progressively between 1822 and 1845 by Claude-Louis Navier and George Gabriel Stokes.  In 1934, Jean Leray proved that solutions exist in a generalized sense, but whether they always remain smooth became a central unanswered question.  In 2000, the Clay Mathematics Institute named the Navier–Stokes existence and smoothness problem one of seven Millennium Prize Problems.  On September 8 this year OpenAI announced that roughly 10,000 autonomous AI agents solved the 90-year-old Navier-Stokes existence and smoothness problem in 88 hours.  They proposed a mathematical proof showing that the three-dimensional incompressible Navier-Stokes equations can develop a singularity (or “blowup”) in finite time when driven by a smooth external force.  The mathematics community is treating the release as a prominent proposed claim rather than a settled theorem.  Formal verification by human experts and peer review will take a long time.

Written for FOWC with Fandango – Neck.