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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, einsteinA 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, einsteinSagan 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.

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“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, einsteinA 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.

Public Domain

Albert Einstein and Charlie Chaplin at the premiere of "City Lights" in 1931

Albert Einstein and Charlie Chaplin are two of the most famous figures of the 20th century for completely different reasons, and an exchange early in their friendship sums up those differences perfectly—and hilariously.

According to the Nobel Prize committee, Einstein had only been keenly interested in meeting one person in Hollywood: Charlie Chaplin. He got the chance while visiting the U.S. when the scientist and silent film star were introduced during a tour of Universal Studios. The two hit it off, and in 1931, Einstein attended the premiere of Chaplin's 1931 film, "City Lights."

Reportedly, the men exchanged an expression of mutual admiration that demonstrated the actor's quick wit.


"What I most admire about your art, is your universality," Einstein said to Chaplin. "You don’t say a word, yet the world understands you!"

Chaplin replied with perfect comedic timing.

"True. But your glory is even greater!" he told the physicist. "The whole world admires you, even though they don’t understand a word of what you say."

The anecdote was shared by the Nobel Prize organization, but there is some question as to the accuracy of the quotes—though not the sentiment of the joke.

According to Quote Investigator, there are a few published accounts of similar quotes referring to Chaplin being beloved because everyone understands him and Einstein being admired because no one does. For instance, in October 1933, an article written by Charlie Chaplin for “Woman’s Home Companion” included the following anecdote attributing a similar quote to Einstein's son:

“We sat down to delicious home-baked tarts made by Mrs. Einstein. During the course of conversation, his son remarked on the psychology of the popularity of Einstein and myself.

"'You are popular,' he said, 'because you are understood by the masses. On the other hand, the professor’s popularity with the masses is because he is not understood.'"

In another instance, the joke was attributed to Chaplin. Einstein's physician and friend János Plesch shared in a memoir published in 1947:

"Once when Einstein was in Hollywood on a visit Chaplin drove him through the town. As the people on the sidewalks recognized two of their greatest, if very different, contemporaries, they gave them a tremendous reception which greatly astonished Einstein. 'They’re cheering us both,” said Chaplin: “you because nobody understands you, and me because everybody understands me.' There was a good-humoured pride in his remark, and at the same time a certain humility as at a recognition of the difference between ready popularity and lasting greatness."

​Different biographies of Chaplin and Einstein include different versions of the story, including the one shared by the Nobel Prize organization on Instagram. Though we don't know for sure exactly what was said or by whom, it's clear that the two men shared a joke about their unique paths to fame and popularity.

How delightful that the mutual admiration of these two legends not only led to them meeting but forging a genuine, if unlikely, friendship.