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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.
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.
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.
- YouTubewww.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.
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.
Many middle class families are sharing that they have nothing in savings right now.
According to an April 2024 Gallup poll, 54% of Americans identify as part of the middle class, with 39% identifying as "middle class" and 15% identifying as "upper-middle class." That percentage has held fairly steady for years, but what it feels like to be a middle-class American has shifted for many.
Notably, inflation caused by the pandemic has hit middle-class families hard, with incomes not keeping up with cost-of-living increases. Housing costs have skyrocketed in many areas of the country, mortgage interest rates have risen to levels not seen since the pre-Obama era, and grocery bills have increased significantly. One government study found that the cost of living has increased between around $800 and $1,300 a month, depending on the state, since 2021, putting a squeeze on everyone, including the middle class.
One woman shared that her family is just getting by and asked other middle-class people to "chime in" with what they have in their savings accounts.
@abbyy..rosee somethings gotta give #savings #middleclass #relatable
"I swear, every paycheck I am putting money into my savings, but needing to transfer it back within a few days," shared @abbyy..rosee on TikTok. "My registration is due. My husband's registration is due. He needed two new tires, even though they had a warranty. That's $300. My oldest needs braces, he needs a palate expander, that's $120 a month. Not to mention groceries are $200 more a week. Forget about feeding your family great ingredients because who has $500 a week to spend on perfect ingredients to feed your family?"
A depressed couple doing their bills.via Canva/Photos
She explained that her husband makes enough money that they should be able to live comfortably, and that she quit her job because the cost of daycare was more than she was making.
"At some point, something has to give," she said. "What is going on? How do I save money?"
People in the comments chimed in with their savings account totals and it was quite eye-opening. Many people shared that they have $0 saved.
"We make the most money we ever have and have zero savings. We live paycheck to paycheck and every month I don’t know how we get by."
"I think the middle class is 1 personal disaster away from bankruptcy."
"Y’all got savings accounts?!?! 😂"
"I used to freak out if I had under $10k in savings, now I’m happy when I have over $150. 😫"
"We make almost 100,000 a year with no savings!!!! It's always something!!"
"I'm lucky if we have $500-$1K for an emergency. Every single time we start saving, something happens: the vet, the cars, the kids... something."
"Savings account? I transfer money each paycheck but always end up needing to transfer it back. My husband makes great money too but we are scraping by."
"$803 but we have to pay a $750 deductible this week b/c my Husband hit a deer soooo… back at it 😭 It’s exhausting. Constantly draining it, refilling it, transferring."
An upset couple doing their bills.via Canva/Photos
Some people shared that they do have some savings, but several said it was because they'd had an inheritance or other chunk of money come their way. Many people shared that their savings has dwindled as increased costs have taken their toll. Some people gave lifestyle advice to save money, but most agreed that just the basics have gotten so expensive it's harder to make ends meet much less put extra into savings.
Thankfully, the inflation issue appears to be waning, but even just plateauing at their current financial reality isn't ideal for many American families. Middle class is supposed to be a comfortable place to be—not rich, but well enough off to feel secure. That's not how many middle class folks feel, though. Most Americans don't have anything close to the amount of money saved that is recommended across the age spectrum, but at least hearing that others are in the same boat is somewhat comforting.
An upset couple doing their bills.via Canva/Photos
Further, a 2024 study found that 37% of Americans can't afford an unexpected expense over $400, and nearly a quarter of them don't have any emergency savings at all. “Not all surprises are good, and people know it. The study suggests financial precarity at a time when household finances may be stretched due to rising prices and inflation,” says Rebecca Rickert, head of communications at Empower. “Life happens, and people are stressed about the surprise expenses that could tip them off-balance.”
It can be vulnerable to share your financial reality, but it's helpful to hear what other people are doing and dealing with so we all feel less alone when we're struggling. Perhaps if people were more open about money, we'd all be able to help one another find ways to improve our financial situations rather than lamenting our empty savings accounts and wondering how to change them.
This article originally appeared last year.
Elder wisdom can come in handy.
With age comes wisdom, or at least we hope it does. As we get older, we collect life lessons that we can pass along to younger generations, sometimes with lengthy stories, sometimes with quippy sayings.
Adages like "A penny saved is a penny earned," or "Early to bed, early to rise makes a man healthy, wealthy, and wise," have been part of our collective treasure chest of life advice for generations, but the aphorisms that spring from the experience of our loved ones and mentors are often the most meaningful.
Someone shared that they'd read and appreciated this old man's advice: “My grandpa once told me 'if you get on the wrong train, get off at the nearest station, the longer it takes you to get off, the more expensive the return trip will be.' He wasn’t talking about trains.” The person asked for more tidbits of wisdom from old folks and people started sharing sayings they heard from their elders growing up.
from AskOldPeopleAdvice
Here are some of the best "old man advice" sayings and how they can be applied in a person's life.
Another commenter put it another way: "Don't look behind you, you aren't going that way." This adage is about not dwelling on the past. Many of us have a tendency to spend more time mentally in the past, rehashing old memories or being nostalgic for what once was, than we do in the present or looking forward to the future. There's nothing wrong with a little reflection, but if most of our focus is in the past, we miss out on the present. Rear view mirrors are for quick glances, not where our focus should be.
This saying is about good character and true integrity. If you notice someone drop a $20 bill and no one is around to see but you, do you give it back to them or do you pick it up and keep it? If you realize that a store didn't charge you for an item, do you point it out and make it right or do you allow the mistake to remain? There are opportunities each day for us to choose between right and wrong, and those choices really speak to who we are if they are made without anyone else knowing.
bart simpson episode 21 GIFGiphy
Don't get cocky and don't burn bridges you think you won't need to return to. There's value in being kind for its own sake, but there's also the reality that being kind also makes people like you. When people like you, they're more likely to lend you a helping hand, and you never know when you're going to be in a position to need one. It's also a good reminder that you're not inherently better than anyone else just because of where you are in life. We're all constantly in flux, so it's important to stay humble and kind.
In other words, "Make all your words sweet because tomorrow you may have to eat them."
The power of silence in general is often underrated, but it can be an especially useful tool in a negotiation. Some people are so uncomfortable with silence that they will make concessions simply to avoid it. And sometimes the best response to an unreasonable demand is to just say nothing and stare, letting the other party come to the realization themselves. It takes calm confidence to simply be quiet and let the silence fill the room, which can feel surprisingly intimidating.
Eddie Murphy Shut Up GIF by BounceGiphy
Ah, the strange and mysterious sense of intuition that we can't really describe but know when we feel it. Whether it's getting a creepy vibe about a person or a little voice telling you to do or not do something, those "gut instincts" can serve us well. Of course, if we are prone to anxiety, our instincts can sometimes be confused with anxious thoughts, but "go with your gut" is solid advice anyway.
Sometimes people trying to find their way end up hitting roadblock after roadblock, which may mean they just haven't found the right path yet or might mean they need an entire overhaul of their life. That might look like switching career paths entirely, rather than trying to find a job in your field that fits. It might mean changing majors in the middle of your studies when you find yourself not enjoying any of your classes. It might mean finding a new community or reevaluating your relationships.
Several sayings line up with this one, like "Luck is the intersection of preparation and opportunity," and "Luck favors the prepared mind." There's a lot to be said for fortune and hard work going hand in hand. If we expect good things to just land in our lap, we will likely be disappointed, but if we move in the direction of things we want to happen and do the work of preparing for good things to come our way, "luck" frequently seems to follow.
Finding a saying that resonates can be really helpful when we're facing a specific challenge in life, especially when we commit it to memory and repeat it often.
This article originally appeared in March.
Alexis Smith was crowned Miss Kansas in June 2024.
Domestic violence survivors cheered on the winner of the 2024 Miss Kansas competition after she stood on stage and called out her abuser who showed up to the pageant. Before Alexis Smith was chosen from 26 participants in the state competition on June 8, 2024 to represent Kansas in the 2025 Miss America contest, she was asked to speak on stage about her Reclaimed Respect initiative.
“My vision as the next Miss Kansas is to eliminate unhealthy and abusive relationships,” Smith said. “Matter of fact, some of you out in this audience saw me very emotional because my abuser is here today. But that’s not going to stop me from being on this Miss Kansas stage and from representing as the next Miss Kansas. Because I, and my community, deserve healthy relationships. We deserve a domestic [violence] free life.”
Watch:
@lexlex_smith Respect Reclaimed is about reclaiming your power and standing firmly in it. On the night of Miss Kansas, my journey took an unexpected turn when someone I have been healing from tried to disrupt my peace. Instead of falling into silence, I chose to live out my vision for a better world. I took back my power—not just for myself, but for my dreams and everyone watching and listening. This isn't about shunning others; it's about turning our pain into purpose and channeling it in a way that unifies and uplifts. I'm ready to use my story, tools, and resources to end unhealthy relationships in all forms. My voice and advocacy will empower everyone to reclaim their own power in their own unique way. I might be small in stature, but I stand tall in strength, purpose, and power with hopes of inspiring others to do the same. #fyp #abuse #miss #misskansas #missamerica #pageant #awareness #me #relationship #respect #tiktok
In the video shared on TikTok, the new Miss Kansas wrote, "Respect Reclaimed is about reclaiming your power and standing firmly in it. On the night of Miss Kansas, my journey took an unexpected turn when someone I have been healing from tried to disrupt my peace. Instead of falling into silence, I chose to live out my vision for a better world. I took back my power—not just for myself, but for my dreams and everyone watching and listening."
She wrote that it wasn't about shunning anyone, but about "turning our pain into purpose and channeling it in a way that unifies and uplifts."
"I’m ready to use my story, tools, and resources to end unhealthy relationships in all forms," she wrote. "My voice and advocacy will empower everyone to reclaim their own power in their own unique way.
I might be small in stature, but I stand tall in strength, purpose, and power with hopes of inspiring others to do the same."
@lexlex_smith Bye bye #DV #RespectReclaimed #FYP #MissKansas #Shoot #photography #photoshoot #TikTok
People who have experienced abuse themselves applauded her advocacy.
"As a victim of domestic violence I applaud you for speaking out!! I watched this 10 times!! I’m still getting bullied by his parents even with a no contact order. I plan on helping women like us as well."
"WHAT A WOMAN. This is absolutely incredible. From one survivor to another, I am SO SO proud of you for reclaiming this moment for yourself. You will do amazing things🫶🏽🫶🏽"
"Incredibly brave of you. You just made a statement for all women. I appreciate you so much."
Love shouldn't hurt.Photo credit: Canva
"As an old survivor…I’m so damn PROUD OF YOU!! Love, A Stranger💚"
"We got to see you ACTIVELY showcasing your platform LIVE IN PERSON! My utmost respect to you Miss Alexis. This is beyond any crown, I cannot wait to watch your journey. As someone who grew up around domestic violence, I have chills watching you. You will always have a supporter in me.💖 You absolutely ate that. 👑"
"YAS GIRL! As a fellow survivor, you are an inspiration and I’m so proud of you for using your voice and showing your strength."
""I experienced emotional and psychological abuse for a very long time," Smith shared with KMUW, "and it was recognizing that I was losing control over my own personal emotions, trying to save the emotions of someone else, and so to be able to save myself. I recognize that we don't want to both go down together. I need to be able to leave this relationship. That way, I'm able to pursue a future, because you just never know what can happen to your partner or happen to yourself when you're in those relationships."
According to The Wichita Eagle, Smith uses her 19 years of experience as a ventriloquist to teach kids about healthy relationships with puppets as part of her Reclaimed Respect initiative. She also works full-time as a cardiothoracic ICU nurse.
This article originally appeared last year.
Greetings in Japanese sound quite odd when literally translated to English.
Studying a language other than your own native tongue is always a trip. Wrapping your head around completely different grammar and syntax rules, trying to create sounds your mouth isn't used to, sometimes learning entirely new alphabets (or characters when there is no alphabet), all just to be able to communicate with more of your fellow human beings. (Seriously, when are we going to decide on an actual universal language?)
Linguistics is wild, as evidenced by certified Japanese teacher Hikari's video demonstrating what Japanese introductions would sound like if they were translated literally into English.
One of the first greetings you learn in Japanese is "hajimemashite," which is generally interpreted as "Nice to meet you," or "How do you do"—something you say when you meet someone. But the literal meaning of the phrase is "at the beginning" or "first time," which of course sounds odd in English. (Similarly to if you were to literally translate "How's it going?" from English to another language, the understood meaning of "How are you feeling right now?" wouldn't come through, since the words "it' and "going" have nothing to do with how you're feeling.)
Then there are name introductions, which seem like they should just be straight up names, but aren't because of what they mean. "I'm Under the Forest." "I'm Inside the Field." Huh? And wait til you see how they share their ages. Watch:
- YouTubeyoutu.be
To make things a little clearer, the name "Under the Forest" is almost certainly Morishita, a common last name in Japan (Japanese people generally introduce themselves by last name). The translation "under the forest" comes from Chinese characters used in Japanese, Kanji, that most Japanese last names are written in. Morishita is 森下 in Kanji, with 森 (mori) meaning "forest" and 下 (shita) meaning "below" or "under." "Inside the Field" would be the name Tanaka, with a similar explanation, but with different characters.
As far as ages go, that's a whole other cultural quirk. In Japan, time is separated into imperial eras based on whoever the emperor is, and each era has a name. "Shining harmony 63" means she was born in 1989, or the 63rd year of the Shōwa ("shining harmony") era. Moons are months, and days are, well, days.
Imperial eras of time. Canva Photos.
Japan is the only country where Japanese is an official language, but thanks to the tech boom there in the 80s and 90s—and the rise of the popularity of anime worldwide—the Japanese language has seen continually growing interest outside the archipelago nation. According to Brittanica, there are around 127 million Japanese speakers worldwide, with some 1.5 million Japanese immigrants (and their descendants) living in parts of North and South America with "varying degrees of proficiency in Japanese."
sailor moon walking GIFGiphy
(Side note: Having studied Japanese myself, I can attest that it's a very fun language to learn. The alphabets and Kanji are the hardest parts—the phonetics are consistent and the grammar is quite logical, with far fewer exceptions to the rules than English.)
In addition to videos like this one, Hikari offers Japanese lessons on her YouTube channel, Japanese with Hikari. You can also sign up for her newsletter, chock full of motivational posts, encouragement, and resources to keep subscribers motivated to learn Japanese.
This article originally appeared last year. It has been updated.