# What Is Infinity?

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Albert Einstein famously said: “Two things are infinite: the universe and human stupidity. And I'm not sure about the universe.”

The notion of infinity has been pondered by the greatest minds over the ages, from Aristotle to German mathematician Georg Cantor. To most people today, it is something that is never-ending or has no limit. But if you really start to think about what that means, it might blow your mind. Is infinity just an abstract concept? Or can it exist in the real world?

#### THERE'S MORE THAN ONE KIND

Infinity is firmly rooted in mathematics. But according to Justin Moore, a math researcher at Cornell University in Ithaca, New York, even within the field there are slightly different uses of the word. “It's often referred to as a sort of virtual number at the end of the real number line,” he tells Mental Floss. “Or it can mean something too big to be counted by a whole number.”

There isn't just one type of infinity, either. Counting, for example, represents a type of infinity that is unbounded—what's known as a potential infinity. In theory, you can go on counting forever without ever reaching a largest number. However, infinity can be bounded, too, like the infinity symbol, for example. You can loop around it an unlimited number of times, but you must follow its contour—or boundary.

All infinities may not be equal, either. At the end of the 19th century, Cantor controversially proved that some collections of counting numbers are bigger than the counting numbers themselves. Since the counting numbers are already infinite, it means that some infinities are larger than others. He also showed that some types of infinities may be uncountable, as opposed to collections like the counting numbers.

"At the time, it was shocking—a real surprise," Oystein Linnebo, who researches philosophies of logic and mathematics at the University of Oslo, tells Mental Floss. "But over the course of a few decades, it got absorbed into mathematics."

Without infinity, many mathematical concepts would fall apart. The famous mathematical constant pi, for example, which is essential to many formulas involving the geometry of circles, spheres, and ellipses, is intrinsically linked to infinity. As an irrational number—a number that can't simply be expressed by a fraction—it's made up of an endless string of decimals.

And if infinity didn't exist, it would mean that there is a biggest number. "That would be a complete no-no," says Linnebo. Any number can be used to find an even bigger number, so it just wouldn't work, he says.

#### CAN YOU MEASURE THE IMMEASURABLE?

In the real world, though, infinity has yet to be pinned down. Perhaps you've seen infinite reflections in a pair of parallel mirrors on opposite sides of a room. But that's an optical effect—the objects themselves are not infinite, of course. "It's highly controversial and dubious whether you have infinities in the real world," says Linnebo. "Infinity has never been measured."

Trying to measure infinity to prove it exists might in itself be a futile task. Measurement implies a finite quantity, so the result would be the absence of a concrete amount. "The reading would be off the scale, and that's all you would be able to tell," says Linnebo.

The hunt for infinity in the real world has often turned to the universe—the biggest real thing that we know of. Yet there is no proof as to whether it is infinite or just very large. Einstein proposed that the universe is finite but unbounded—some sort of cross between the two. He described it as a variation of a sphere that is impossible to imagine.

We tend to think of infinity as being large, but some mathematicians have tried to seek out the infinitely small. In theory, if you take a segment between two points on a line, you should be able to divide it in two over and over again indefinitely. (This is the Xeno paradox known as dichotomy.) But if you try to apply the same logic to matter, you hit a roadblock. You can break down real-world objects into smaller and smaller pieces until you reach atoms and their elementary particles, such as electrons and the components of protons and neutrons. According to current knowledge, subatomic particles can't be broken down any further.

#### THE INFINITIES OF THE SINGULARITY

Black holes may be the closest we've come to detecting infinity in the real world. In the center of a black hole, a point called a singularity is a one-dimensional dot that is thought to contain a huge mass. Physicists theorize that at this bizarre location, some of the singularity's properties are infinite, such as density and curvature.

At the singularity, most of the laws of physics no longer work because these infinite quantities "break" many equations. Space and time, for example, are no longer two separate entities, and seem to merge.

According to Linnebo, though, black holes are far from being an example of a tangible infinity. "My impression is that the majority of physicists would say that is where our theory breaks down," he says. "When you get infinite curvature or density, you are beyond the area where the theory applies."

New theories may therefore be needed to describe this location, which seems to transcend what is possible in the physical world.

For now, infinity remains in the realm of the abstract. The human mind seems to have created the concept, yet can we even really picture what it looks like? Perhaps to truly envision it, our minds would need to be infinite as well.

# The Science Behind Why the Earth Isn't Flat

Earth as captured from near the lunar horizon by the Lunar Reconnaissance Orbiter in 2015.
NASA

On March 24, 2018, flat-earther Mike Hughes set out prove that the Earth is shaped like a Frisbee. The plan: Strap himself to a homemade steam-powered rocket and launch 52 miles into sky above California’s Mojave Desert, where he'd see Earth's shape with his own eyes.

It didn't matter that astronauts like John Glenn and Neil Armstrong had been to space and verified that the Earth is round; Hughes didn't believe them. According to The Washington Post, Hughes thought they were "merely paid actors performing in front of a computer-generated image of a round globe."

The attempt, ultimately, was a flop. He fell back to Earth with minor injuries after reaching 1875 feet—not even as high as the tip of One World Trade Center. For the cost of his rocket stunt (\$20,000), Hughes could have easily flown around the world on a commercial airliner at 35,000 feet.

Hughes isn't alone in his misguided belief: Remarkably, thousands of years after the ancient Greeks proved our planet is a sphere, the flat-Earth movement seems to be gaining momentum. "Theories" abound on YouTube, and the flat-Earth Facebook page has some 194,000 followers.

Of course, the Earth isn't flat. It's a sphere. There is zero doubt about this fact in the real, round world. To say the evidence is overwhelming is an understatement.

#### HOT SPINNING BODIES

Not every celestial body is a sphere, but round objects are common in the universe: In addition to Earth and all other known large planets, stars and bigger moons are also ball-shaped. These objects, and billions of others, have the same shape because of gravity, which pulls everything toward everything else. All of that pulling makes an object as compact as it can be, and nothing is more compact than a sphere. Say, for example, you have a sphere of modeling clay that is exactly 10 inches in diameter. No part of the mass is more than 5 inches from the center. That's not the case with any other shape—some part of the material will be more than 5 inches from the center of the mass. A sphere is the smallest option.

Today the Earth is mostly solid with a liquid outer core, but when the planet was forming, some 4.5 billion years ago, it was very hot and behaved like more like a fluid—and was subject to the squishing effects of gravity.

And yet, the Earth isn't a perfect sphere; it bulges slightly at the equator. "Over a long time-scale, the Earth acts like a highly viscous fluid," says Surendra Adhikari, a geophysicist at the Jet Propulsion Laboratory in Pasadena, California. The Earth has been spinning since it was formed, and "if you have a spinning fluid, it will bulge out due to centrifugal forces." You can see evidence for this at the equator, where the Earth's diameter is 7926 miles—27 miles larger than at the poles (7899 miles). The difference is tiny—just one-third of 1 percent.

#### THE SHADOW KNOWS

The ancient Greeks figured out that Earth was a sphere 2300 years ago by observing the planet's curved shadow during a lunar eclipse, when the Earth passes between the Sun and the Moon. Some flat-Earth believers claim the world is shaped like a disk, perhaps with a wall of ice along the outer rim. (Why no one has ever seen this supposed wall, let alone crashed into it, remains unexplained.) Wouldn't a disk-shaped Earth also cast a round shadow? Well, it would depend on the orientation of the disk. If sunlight just happened to hit the disk face-on, it would have a round shadow. But if light hit the disk edge-on, the shadow would be a thin, straight line. And if the light fell at an oblique angle, the shadow would be a football–shaped ellipse. We know the Earth is spinning, so it can't present one side toward the Sun time after time. What we observe during lunar eclipses is that the planet's shadow is always round, so its shape has to be spherical.

The ancient Greeks also knew Earth's size, which they determined using the Earth's shape. In the 2nd century BCE, a thinker named Eratosthenes read that on a certain day, the people of Syene, in southern Egypt, reported seeing the Sun directly overhead at noon. But in Alexandria, in northern Egypt, on that same day at the same time, Eratosthenes had observed the Sun being several degrees away from overhead. If the Earth were flat, that would be impossible: The Sun would have to be the same height in the sky for observers everywhere, at each moment in time. By measuring the size of this angle, and knowing the distance between the two cities, Eratosthenes was able to calculate the Earth's diameter, coming up with a value within about 15 percent of the modern figure.

And when Columbus set sail from Spain in 1492, the question wasn't "Would he fall off the edge of the world?"—educated people knew the Earth was round—but rather, how long a westward voyage from Europe to Asia would take, and whether any new continents might be found along the way. During the Age of Exploration, European sailors noticed that, as they sailed south, "new" constellations came into view—stars that could never be seen from their home latitudes. If the world were flat, the same constellations would be visible from everywhere on the Earth's surface.

Finally, in 1522, Ferdinand Magellan's crew became the first people to circle the globe. Like Columbus, Magellan also set off from Spain, in 1519, heading west—and kept generally going west for the next three years. The expedition wound up back at the starting point (though without Magellan, who was killed during a battle in the Philippines). And speaking of ships and seafaring: One only needs to watch a tall ship sailing away from port to see that its hull disappears before the top of its mast. That happens because the ship is traveling along a curved surface; if the Earth were flat, the ship would just appear smaller and smaller, without any part of it slipping below the horizon.

#### THE EVIDENCE IS ALL AROUND (AND ALL ROUND)

But you don't need a ship to verify the Earth's shape. When the Sun is rising in, say, Moscow, it's setting in Los Angeles; when it's the middle of the night in New Delhi, the Sun is shining high in the sky in Chicago. These differences occur because the globe is constantly spinning, completing one revolution per day. If the Earth were flat, it would be daytime everywhere at once, followed by nighttime everywhere at once.

You also experience the Earth's roundness every time you take a long-distance flight. Jetliners fly along the shortest path between any two cities. "We use flight paths that are calculated on the basis of the Earth being round," Adhikari says. Imagine a flight from New York to Sydney: It would typically head northwest, toward Alaska, then southwest toward Australia. On the map provided in your airline's in-flight magazine, that might look like a peculiar path. But wrap a piece of string around a globe, and you'll see that it’s the shortest possible route.

"If the Earth were flat," Adhikari says, "the trajectory would be completely different." How different depends on which way the globe is sliced into a flattened map, but if it looked like it does on a Mercator-projection map, it might head east and pass over Africa.

Engineers and architects also take the Earth's curvature into account when building large structures. A good example is the towers that support long suspension bridges such as the Verrazano Narrows bridge in New York City. Its towers are slightly out of parallel with each other, the tops being more than 1.5 inches further apart than their bases. If the Earth were flat, the bottom of the towers would be separated by the exact same distance as the top of the towers; the planet's curvature forces the tops of the towers apart.

And for the last half-century, we've had eyewitness and photographic proof of the Earth's shape. In December 1968, the crew of Apollo 8 left Earth for the Moon. When they looked out of the Command Module windows, they saw a blue-and-white marble suspended against the blackness of space. On Christmas Eve, lunar module pilot William Anders snapped the famous "Earthrise" photograph. It gave us an awe-inspiring perspective of our round planet that was unprecedented in human history—but it wasn't a surprise to anyone.

# How Long to Steep Your Tea, According to Science

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The tea in your cabinet likely has vague instructions about how long to steep the leaves. Bigelow, for instance, suggests two to four minutes for black tea, and one to three minutes for green tea. According to Lipton, you should "try singing the National Anthem" while waiting for black tea leaves to infuse.

But while it's true that tea brewed for 30 seconds is technically just as drinkable as a forgotten mug of tea that's been steeping for 30 minutes, drinkable shouldn't be your goal. Taste and—depending on the tea you're drinking—antioxidant and caffeine levels all depend on the amount of time the leaves are in contact with the water. So how early is too early to pluck out a tea bag, and how long can you leave it in before passing the point of no return?

#### THE SCIENCE OF STEEPING

To achieve the perfect timing, you first need to understand the chemical process at work when you pour hot water over tea leaves. Black, green, white, and oolong tea all come from the leaves and buds of the same plant, Camellia sinensis. (Herbal teas aren't considered "true teas" because they don't come from C. sinensis.) The teas are processed differently: Green and white tea leaves are heated to dry them, limiting the amount of oxidation they get, while black and oolong tea leaves are exposed to oxygen before they're dried, creating the chemical reactions that give the tea its distinct color and flavor. Damaging the tea leaves—by macerating them, rolling them gently, or something in between—helps expose the chemicals inside their cells to varying levels of oxygen.

Both green and black teas contain a lot of the same chemical compounds that contribute to their flavor profiles and nutritional content. When the leaves are submerged in hot water, these compounds leach into the liquid through a process called osmotic diffusion, which occurs when there's fluid on both sides of a selectively permeable membrane—in this case, the tea leaf. Compounds on the surface of the leaf and in the interior cells damaged by processing will diffuse into the surrounding liquid until the compounds in both the leaf and the water reach equilibrium. In other words, if given enough time to steep, the liquid in your mug will become just as concentrated with tea compounds as the liquid in your tea leaves, and the ratio will stay that way.

Osmotic diffusion doesn't happen all at once—different compounds enter the water at different rates based on their molecular weight. The light, volatile chemicals that contribute to tea's aroma and flavor profile dissolve the fastest, which is why the smell from a bag of tea leaves becomes more potent the moment you dunk it in water. The next group of compounds to infuse with the water includes the micronutrients flavanols and polyphenols, which are antioxidants, and caffeine. They're followed by heavier flavanols and polyphenols such as tannins, which are the compounds responsible for tea's bitter flavor. (They're also what make your mouth feel dry after drinking a glass of wine.) Tea also has amino acids like theanine, which can offset the sharpness of tannins.

Water temperature is another factor to take into consideration when steeping your tea. High water temperature creates more kinetic energy, which encourages the compounds to dissolve. "The heat helps you to extract the compounds out of the tea leaves," Shengmin Sang, a North Carolina A&T State University researcher who studies the chemistry of tea, tells Mental Floss. "If you put it into cold water or low-temperature water, the efficiency to extract these compounds out of the leaves will be much lower." But not all water is equal: Bigelow Tea recommends using water at a rolling boil for black tea, and barely boiling water for green tea.

#### LOOSE LEAF VS. TEA BAGS

Osmotic diffusion takes place whether you use loose leaves or tea bags, but there are some notable differences between the two. When given room to expand, loose tea leaves swell to their full capacity, creating more room for water to flow in and extract all those desirable compounds. Tea that comes prepackaged in a bag, on the other hand, only has so much room to grow, and the quality suffers as a result. This is why some tea companies have started selling tea in roomier, pyramid-shaped bags, though the size matters more than the shape.

But even before the tea touches the water, there's a difference in quality. Loose leaf tea usually consists of whole leaves, while most teabags are filled with broken pieces of tea leaves called dust or fannings, which have less-nuanced flavors and infuse fewer antioxidants than whole leaves, no matter how long you let them steep.

So if you have a choice, go with loose leaf. But if tea bags are all you have on hand, don't bother adjusting your brewing method: The difference in taste and antioxidants isn't something that can be fixed with a few extra minutes, and according to Sang, you should follow the same steeping times for both tea bags and loose leaf.

To calculate the perfect brew times for what's in your mug, first consider what you want most out of your drink.

#### IF YOU DRINK TEA TO BE HEALTHY

Suggested steeping time: 2 minutes, 30 seconds to 5 minutes

Tea leaves are packed with beneficial compounds. Research indicates that flavanols such as catechins and epicatechins, found in both green and black teas, help suppress inflammation and curb plaque build-up in arteries. Drinking tea may improve vascular reactivity, which dictates how well blood vessels adjust to stress. According an analysis of multiple tea-related studies published in the European Journal of Epidemiology in 2015, drinking three cups of tea a day reduces your risk of coronary heart disease by 27 percent, cardiac death by 26 percent, and total mortality by 24 percent. Polyphenolic antioxidants in tea may also protect against diabetes, depression, and liver disease.

Past research has shown that it takes 100 to 150 seconds to extract half the polyphenol content from green and black tea leaves. According to a study published in 2016 in the journal Beverages, you can get more polyphenols into your drink if you allow the leaves more time to steep. However, the returns may not be worth the extra effort: Most of the compounds the researchers measured after 10 minutes of steeping were extracted in the first 5 minutes.

Sang makes another argument for not waiting too long to drink your tea. Antioxidants are slightly unstable, which means they will eventually break down and lose their healthy properties after infusing with water. “After you extract the compounds from the tea bag, you can not keep the solution for too long,” he says. “Because these compounds are not stable, they will be oxidized. So if you brew it in the morning, then you drink it in the afternoon, that's not good.” This oxidation can occur even after the tea leaves are removed from the cup, so if your tea has been sitting out for a few hours, it's better to brew a new batch than to pop it in the microwave.

#### IF YOU DRINK TEA FOR THE CAFFEINE BOOST

Suggested steeping time: 3 to 5 minutes

Though less potent than its rival coffee, a properly brewed cup of tea packs a caffeine punch. According to a 2008 study published in the Journal of Analytical Toxicology [PDF], letting your tea brew for at least a few minutes has a big impact on the caffeine content. The study found that after brewing for one minute, a cup of regular Lipton black tea had 17 milligrams of caffeine per 6 ounces of water, 38 milligrams per 6 ounces after three minutes, and 47 milligrams per 6 ounces after five. (The nutritional information for Lipton black tea says a serving contains 55 milligrams of caffeine per 8 ounces, so it's pretty accurate.)

Some people may use those numbers as an excuse to steep their tea past the five-minute mark in an attempt to reach 100 percent dissolution. But a longer brewing time doesn't necessarily equal a stronger caffeine kick. Yes, more caffeine molecules will enter the tea, but so will other compounds like thearubigins. Caffeine works because it's perfectly shaped to bind to certain neuroreceptors in your brain, thus blocking the chemicals that tell you to feel tired. But caffeine is the right shape to bind to thearubigins as well, and if that happens first, less caffeine will get to those neuroreceptors. So if you're looking for a highly caffeinated cup of tea, you should remove the leaves after most of the caffeine has been extracted—after about three to five minutes—rather than waiting for every last milligram of caffeine to dissolve.

#### IF YOU DRINK TEA BECAUSE IT TASTES GOOD

Suggested steeping time: 1 to 3 minutes

There's nothing wrong with enjoying a cup of tea for taste alone. Flavor is the most subjective factor influenced by steeping times, but for the sake of simplicity, let's assume you prefer a pronounced tea taste that's not overshadowed by bitterness. To extract those more delicate flavors, you don't need to steep your tea leaves for very long at all. Some of the first volatile organic compounds to break down in tea are geraniol and phenylacetaldehyde, tied to a tea's floral aroma, and linalool and linalool oxide, which give tea its sweetness.

The other compounds we associate with tea's distinctive taste are tannins. They're the difference between an aromatic, fruity cup of tea and a bitter cup that needs to be diluted with milk before it's palatable. But tannins aren't all bad: Some people prefer their tea to have a bracing astringency. Because tannins are some of the last molecules to dissolve into tea, if you want to add some bitter complexity to your drink, steep your tea for a minute or two longer than you normally would. A good way to keep track of the strength of your tea is to look at the color: Like tannins, pigments are heavy compounds, so if you see your tea getting darker, that means it's getting stronger as well.

And what about herbal teas? Feel free to leave the leaves in as long as you like. Because herbal teas are high in aromatic compounds and low in tannins, drinkers can be more liberal with their steep times without worrying about getting that astringent taste. Some teas, like rooibos and chamomile, also contain antioxidants, which is another reason to take your time.

And if you're new to the world of tea and aren't sure what your preferences are, put a kettle on the stove and start experimenting.

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