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45 years ago, Carl Sagan beautifully explained the fourth dimension with just an apple

Sagan was the Mister Rogers of complex scientific concepts.

via Carl Sagan Planetary Society/Wikimedia Commons and John Finkelstein/Pexels

Carl Sagan used a sliced apple to perfectly explain the fourth dimension.

The concept of the fourth dimension seems beyond human comprehension. As three-dimensional beings, we are unable to see beyond a physical object's height, width and depth. What else could there be? Even if you understand the concept, it is almost impossible to picture it in your mind, which is bound by the limits and realities of the physical world around us.

Enter Carl Sagan, revered as one of the greatest science communicators of his time. Perhaps best known for his research into extraterrestrial life, he was one of the first people to demonstrate that life could have existed on Mars. Sagan possessed a unique gift for demystifying complex scientific concepts, making them accessible and thrilling for the general public. If you never had the pleasure of watching him on television, you could imagine him as something of a Scientific Mister Rogers. Friendly, a wonderful storyteller, and always able to distill difficult lessons into their simplest form.

In 1980, on Episode 10 of the groundbreaking PBS show “Cosmos,” Sagan embarked on a mission to explain the seemingly impossible fourth dimension.


carl sagan, cosmos ,4th dimension, 3-D, 4-D, 2-D, physics, theoretical physics, math, science, space, spacetime, einstein A great communicator and handsome, to boot. Giphy

Many of us have commonly heard of time being considered the fourth dimension. That's not so hard to understand — in order to locate an object in the universe, you'd need to know three dimensions of its spatial location and also the time during which it exists.

But there is also a more theoretical and harder to understand place, where all four dimensions are spatial. It is nearly impossible for any of us to comprehend... without the help of a gifted teacher.

What’s excellent about Sagan’s explanation is that he uses simple and relatable objects: an apple and a Tesseract, or a hypercube.

carl sagan, cosmos ,4th dimension, 3-D, 4-D, 2-D, physics, theoretical physics, math, science, space, spacetime, einstein Sagan explains that if an apple existed in a 2-dimensional space, anyone living in this "flatland" would only see a cross-section of it at a time. Giphy

"In discussing the large scale structure of the cosmos, astronomers sometimes say that space is curved. Or that the universe is finite but unbound," Sagan begins. "Whatever are they talking about?"

Yeah, this guy gets it.

Sagan then goes on to explain how a two-dimensional being living in a flat world would perceive a three-dimensional object like an apple.

Watch his full explanation here. It's hypnotic and entertaining and incredibly enlightening.

- YouTube www.youtube.com


“Imagine we live in this ‘Fllatland’/2-D plane with no concept of ‘up’ or ‘down.’ Then along comes a 3-D object like an apple. We do not even notice it until it crosses our plane of existence — and even then, we have no idea what the apple is,” Sagan explains. “We see only a fragment as it passes through our plane. There is no way we can comprehend the 3-D quality/dimension of the apple, because it is more than we can understand. We only have the evidence of what has passed through our plane.”

To further demonstrate, Sagan stamps the apple into an inkpad and then onto the surface in front of him, which represents Flatland and all of its inhabitants. Inside Flatland, the apple exists only as its points of contact on the paper; or four small dots. He adds that as the apple passes through the 2-dimensional Flatland, its cross-section changes. So someone living in that plane of existence would experience the apple as an ever-shifting and rearranging set of shapes or objects. Wild!

Sagan then related this two-dimensional experience of the third dimension to how we might try to understand the fourth. To do so, he used the Tesseract, a four-dimensional cube, to demonstrate how difficult it is for us to perceive or visualize dimensions beyond our own three.

carl sagan, cosmos ,4th dimension, 3-D, 4-D, 2-D, physics, theoretical physics, math, science, space, spacetime, einstein A tesseract can not exist in 3-dimensional space, but it can be approximated the same way a cube can be drawn on paper. Giphy

Sagan explains that the tesseract is a cube expanded into a 4th dimension, but "I cannot show you a tesseract because I, and you, are trapped in three dimensions." But what he can do is show us a 3-dimensional rendering of one. Just like a cube can be drawn and approximated (or cast a shadow) onto a piece of paper, a 4-dimensional tesseract can be imperfectly represented in 3-dimensional space. Still following?

At this point, Sagan is asking the viewer to expand their minds to understand the fourth dimension metaphorically. Though we cannot see it or even properly visualize it, that doesn't mean that the things we can see can't offer clues and lessons about the fourth dimension.

Studying 4-dimensional space can help in our understanding of the universe around us. Just because we see and experience only three dimensions doesn't mean that's all that exists. It's critical for physicists and mathematicians to be able to understand and map these theoretical spaces to better comprehend things we otherwise can not explain. Remember the ever-changing, rearranging set of shapes as the apple passes through Flatland?

Sagan’s demonstration of the fourth dimension isn’t just a wonderful explanation of a scientific idea that many of us find difficult to comprehend; it’s also a great example of how to teach complex ideas by combining clear explanations, everyday concepts everyone can understand, and brilliant storytelling.

This article originally appeared last year. It has been updated.

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Doctor's unique 4-7-8 technique could fix your sleep problems instantly

Try this next time you're staring at the clock wondering why you can't sleep.

via Andrea Piacquadio/Pexels
The 4-7-8 technique can help you fall asleep.

Are you having a hard time falling asleep? If so, you're not the only one. Falling and staying asleep typically becomes more difficult as adults get older. We spend less time in deep sleep and REM sleep, and we struggle to fight off our worries and anxieties while lying in silence. Did you know there's actually a name for the phenomenon where you keep yourself awake by lying there and wondering why you're still awake? It's called "spectatoring" and it's incredibly frustrating.

Worse yet, the older we get the more likely we are to wake in the night and have trouble falling back asleep. This is why more and more Americans are turning to white noise, melatonin, meditation apps, special pillows, and anything they think can help them get most rest.

Fortunately, a doctor has shared the “most powerful” relaxation technique he knows, and it doesn’t require any equipment or cost a dime.


sleep, dr. andrew weil, fall asleep fast, relaxation techniques, breathing techniques, calm, anxiety, science, meditation Ever stare at the clock and repeatedly wonder why you can't fall asleep? It's called "spectatoring." Photo by Mpho Mojapelo on Unsplash

Dr. Andrew Weil has dubbed it the 4-7-8 method and it’s backed up by science.

Dr. Weil is an expert in integrative medicine and the founder and director of the Andrew Weil Center for Integrative Medicine at the University of Arizona.

The technique is simple:

  1. Breathe in through your nose while you count to 4.
  2. Hold your breath while you count to 7.
  3. Exhale while you count to 8.

It's a unique protocol, especially when compared to another popular technique like Box Breathing. Box Breathing calls for an inhale, hold, exhale, and another hold of equal time (4 seconds). The extended exhale, in particular, is what makes Dr. Weil's 4-7-8 so original.

Here's Dr. Weil explaining his method:

- YouTube www.youtube.com

Dr. Weil says the method creates a "very pleasant, altered state of consciousness" that you may not experience the first time but will come as a "reward" of regular practice. Dr. Weil insists that the 4-7-8 technique is a practice, and you must do four breath cycles at least twice a day to get the benefits. "After a month, you can increase to 8 breath cycles if you're comfortable with it," adding that's the "absolute maximum."

Dr. Weil says that 4 to 6 weeks of doing the practice can lower heart rate, improve blood pressure, digestion and circulation and can promote sleep. A study published in Physiological Reports agrees, saying that practicing the 4-7-8 technique reduces heart rate and blood pressure for several minutes. It's important to note, though, that research is limited on the longterm benefits of 4-7-8 and experts urge us not to overhype it as more than it is.

4-7-8 is also an easy, fast, and effective way to help you fall asleep.


sleep, dr. andrew weil, fall asleep fast, relaxation techniques, breathing techniques, calm, anxiety, science, meditation Deep, intentional breathing may jumpstart melatonin production, helping us sleep. Photo by Alexander Grey on Unsplash

"If you get up in the middle of the night for any reason, it is the most effective anti-anxiety technique that I've found,” Dr. Weil says.

Deep breathing, interestingly enough, has been shown to increase melatonin production in our bodies; that's the hormone that signals to our body that it's time to sleep. Combined with its ability to calm our mind and body, it's no wonder that intentional breathing exercises can put us to sleep in no time.

Why does deep breathing help us calm down and relax so effectively? There are a few reasons. First, when we're stressed or anxious, our breathing naturally becomes more shallow and irregular. Breaking that anxious breath pattern signals to our body that things are OK, that we're in control. Counting and being mindful of our breath also gives our mind something neutral to focus on instead of the usual chaotic images, intrusive thoughts, or worries. Third, deep breath settles down the part of our nervous system that controls our "fight-or-flight" response — and helps with elevated heart rate and muscle tension.

Remember again that 4-7-8 breathing is a practice. It can be used situationally to great effect, but for the best benefits its founder urges you to try it every single day for a cycle of 2-8 cycles.

This article originally appeared last year. It has been updated.

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Woman's art contest 'victory' dance is interrupted when she realizes boyfriend is proposing

The crowd was cheering, but not for the reason she thought.

Images via Canva

A woman didn't realize she was being proposed to

We take the wins where we can get them. And sometimes that win is showing off your amateur painting to a "paint and sip" group, only to have the audience erupt in cheers.

- YouTubewww.youtube.com

That's what happened when a woman and a man stood before a group and were prompted to reveal their renditions of a red and black "model" painting. "Three, two, one!" we hear on the short clip posted on Australian YouTube user @pintoandpicasso's channel. They reveal their work simultaneously, and the crowd goes wild.

They both light up with smiles, but she's especially excited that her art is so appreciated. She first shakes her hips back and forth and then begins a victory lap, reminiscent of Beyoncé's "Single Ladies (Put a Ring on It)." The man turns his painting toward her, perhaps thinking she'll see it then or that the crowd's supportive outburst might make her look. But she's deep in her "moment," even when she circles back to him.

Beyonce, strut, proud, dancingBeyonce struts in concert. Giphy

Finally, she looks and sees what the crowd has seen all along. On his canvas, he has painted the words, "Will You Marry Me?" When she spots it, she seems in shock at first. We hear an outcry of "Oh my God" from the crowd. And it's that moment that he gently puts the sign down and gets on bended knee, popping out a ring. She covers her mouth and crouches down with him as she tries to regain her composure.

Eventually, they both stand up, as she seems to nod a yes over and over — overwhelmed with joyous tears.

marriage, proposal, diamond ring, loveA man proposes with a ring. Giphy Broad City Season 5

On the subreddit r/MadeMeSmile, a commenter posted the video with the caption "Men in Love." There are over 77 thousand upvotes and counting, with nearly 600 comments. One writes, "The best part is when she THINKS they are shouting for her painting skills and models it out~ and then '... Oh.'" To which someone replies, "What a good day for her TBH. First she wins. Then she gets proposed."

Another jokes that perhaps he only chose to propose because he was impressed by her art skills. "Probably proposed because she won."

Others compare it to other similar, public proposals. "It's almost like that proposal at a game on the kiss cam, where the lady just vibed and danced. And when people pointed at the screen, she just kept dancing like 'yeah yeah I know,' then she finally realized."

kiss cam, hockey game, love, marriageA man kisses a woman at the Carolina Hurricanes hockey game.Giphy

A few dared to ask a more cynical question: what was her answer? "Did she say yes or is this a plot-twist: too embarrassed to say no?" A few offer answers. "Could be nodding or hyperventilating… let's see how this develops."

The crux of the comments, however, remains their love of her unabashed confidence. "The fact she thought everyone just loved the s--t out of her painting haha. I want the confidence." With another exclaiming, "That cracked me up. No hesitation, just 'wow, they must really love my painting.'"

And best of all, Redditors took note of just how patient he was with her oblivion. "And the fact he was just laughing at her and waiting for her to notice tells you a lot about how he understands her."

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A professor reveals the secret of how to make 'gratitude lists' really work

"Those adults are regularly moved to tears to learn of the positive impact they had."

Photo credit: Canva

A woman crosses her arms in gratitude. A person writes a letter.

It's true that "gratitude" has become a buzzword and the very idea of it seems trite and empty to some. For others, the world might feel too chaotic at the moment to focus on the tiny, good things we're told to hang onto. But a professor of psychology at Gonzaga University gave some insight on how to make the concept of gratitude truly enhance our lives.

It's easy to take things for granted sometimes. Keeping a list of stuff we're grateful for, big and small, can certainly help put life into perspective—and not just items at the top of Maslow's hierarchy of needs (like air and water), but all along the pyramid. It can change daily from "I got good sleep last night" to "I'm glad the TV show I like is back for a new season."

- YouTubewww.youtube.com

There's a guy on TikTok called David the Grateful Guy (@today.iam.grateful), and as you might have guessed, he posts clips of different things for which he's grateful. One day, he expressed his appreciation for salt. That’s it. Just salt.

@today.iam.grateful

Today I am grateful for salt! Anyone else add salt to almost everything?! #grateful #gratitude #gratitudejournal #todayiamgrateful #thankfulfor #salt #gratitudepractice

What was extra special is this short video garnered nearly 5,000 comments, binding a community together with jokes and "salt-recognition." It was the sharing of his gratitude that spread joy, more than the salt itself. Kind of like a "Oh yeah, salt IS nice." (Note: a cardiologist might not share this sentiment.)

Professor Monica Y. Bartlett, who in her own words teaches "courses on resilience and human flourishing," shares how important it is to do this. She writes in The Conversation that aside from being aware of your gratitude, "a second method for practicing is expressing that gratitude to others. This can look like writing a letter of gratitude and delivering it to someone who has made a positive impact in your life."

- YouTubewww.youtube.com

This can even be a letter in retrospect for someone who has passed on. The act of expressing one's appreciation is just as important as the appreciation itself. Now, this can't always be implemented, but when it can, it might be extremely impactful. Bartlett continues, "When my students do this exercise, it often results in touching interactions. For instance, my college students often write to high school mentors, and those adults are regularly moved to tears to learn of the positive impact they had. Expressing gratitude in work settings can boost employees’ sense of social worth."

A thread, "What's something you're most grateful for?" yielded many comments on Reddit. Lots of people answered food, shelter, air, family. But a few got specific. One person vulnerably shared, "Being gay in a country that doesn't illegalize it. It's still hard, but at least I don’t have to fear for my life."

Another person answered, "Coca-Cola. I'm a simple gal," and another got more specific, "Great red wine—spend a little more than £15 a bottle—but no more than £80—and pick good company (this bit is more important than the absolute price of the wine) and you have a fantastic evening."

red wine, gratitude, friendship, clinking glasses, relaxingPeople clinking their glasses of red wine for a cheers. Photo by Kelsey Knight on Unsplash

And one commenter simply wrote, "I'm grateful to see another day." Perhaps implementing the idea of writing gratitude letters to all the people who help make them feel that way would even triple the impact of happiness. To end with Bartlett's insightful words, "In a world that may currently feel bleak, a letter of gratitude may not only help the writer recognize the good of others, but also let others know that they are making a beautiful difference in the world."

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