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Fun with Venn and Euler Diagrams

A Venn diagram is a mathematical illustration that shows all of the possible mathematical or logical relationships between sets. A Euler diagram resembles a Venn diagram, but does not neccessarily show all possible intersections of the sets. A Euler diagram is often more useful for showing real world data, because not all sets partially overlap with all other sets.

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I used the 3 Circle Venn Diagram Applet to make a diagram of food preferences in my family. I have one child who won't eat much in the way of meat or vegetables, and another who dislikes carbs. The is a classic Venn diagram, which explains why I don't cook as often as I used to.

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This graphic is labeled as a Venn diagram, but it is actually a Euler diagram, because at no point does true happiness intersect with wearing pants. At least for the person who made the diagram.

More diagram fun, after the jump.

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If the circles don't intersect, it's not a Venn diagram. Which can be a very sad thing if you're a circle. This design was found at Threadless T-Shirts. However, Euler diagrams may overlap or not. But Euler is not what the blue circle had in mind.

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I came across this very useful image last week that started my quest to learn the difference between a Venn diagram and a Euler diagram. It explains the geographic terminology used in that area of the eastern Atlantic that confuses Americans. The author labeled the item as The Great British Venn Diagram, then explaned that it was actually a Euler diagram. No doubt people are more familiar with Venn. But no matter how geographically accurate the place names are, Irish commenters predictably objected to being included in anything labeled with the word "British". Another such diagram that includes more islands can be found here.

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These sweet and sour doodles are true Venn diagrams. They were produced by Jessica Hagy of Indexed, a blog full of wonderful diagrams and graphs jotted on index cards.

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Rob Harvilla used Venn and other diagrams to deconstruct the song "This Is Why I'm Hot" by Mims at The Villlage Voice. The diagrams clearly show that the lyrics make no sense at all. Nevertheless, the song was number one at the time this was written.

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Speaking of music, here's a t-shirt for music snobs featuring a two-set Venn diagram. From the product page:

Nothing is any good if other people like it. We've just proven it mathematically. I have a theory that the only thing cartoonists bothered learning in math class was Venn Diagrams.

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Randall Munroe at xkcd has the geekiest love notes ever. This classic Venn diagram is so sweet and simple, until he started to fill in the zones.

Venn diagrams and Euler diagrams are just two more of the many handy learning devices that are used for strange or comedic purposes. See also Periodic Tableware, More Periodic Tableware, and Fun with Flow Charts.

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Essential Science
What Is Infinity?
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iStock

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.

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History
148 Lost Alan Turing Papers Discovered in Filing Cabinet
Courtesy University of Manchester
Courtesy University of Manchester

You never know what you’re going to uncover when you finally get around to combing through that decades-old filing cabinet in the back room. Case in point: The University of Manchester recently unearthed 148 long-lost papers belonging to computer science legend Alan Turing, as ScienceAlert reports.

The forgotten papers mostly cover correspondence between Turing and others between 1949 and his death in 1954. The mathematician worked at the university from 1948 on. The documents include offers to lecture—to one in the U.S., he replied, “I would not like the journey, and I detest America”—a draft of a radio program he was working on about artificial intelligence, a letter from Chess magazine, and handwritten notes. Turing’s vital work during World War II was still classified at the time, and only one document in the file refers to his codebreaking efforts for the British government—a letter from the UK’s security agency GCHQ. The papers had been hidden away for at least three decades.

A typed letter to Alan Turing has a watermark that says 'Chess.'
Courtesy University of Manchester

Computer scientist Jim Miles found the file in May, but it has only now been sorted and catalogued by a university archivist. "I was astonished such a thing had remained hidden out of sight for so long," Miles said in a press statement. "No one who now works in the school or at the university knew they even existed." He says it’s still a mystery why they were filed away in the first place.

The rare discovery represents a literal treasure trove. In 2015, a 56-page handwritten manuscript from Turing’s time as a World War II codebreaker sold for more than $1 million.

[h/t ScienceAlert]

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