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Fun with Venn and Euler Diagrams

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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.


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.


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.


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.


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.


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.


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.


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.


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.

Can You Figure Out How Many Triangles Are in This Picture?

Time for another brain teaser. How many triangles do you see here? A Quora user posted the image above (which we spotted on MSN) for fellow brainiacs to chew on. See if you can figure it out. We’ll wait.


So, as you can see, all the smaller triangles can combine to become bigger triangles, which is where the trick lies. If you count up every different triangle formed by the lines, you should get 24. (Don’t forget the big triangle!)

Some pedantic Quora users thought it through and realized there are even more triangles, if you really want to go there. There’s a triangle formed by the “A” in the signature in the right-hand corner, and if we’re counting the concept of triangles, the word “triangle” counts, too.

As math expert Martin Silvertant writes on Quora, “A triangle is a mathematical idea rather than something real; physical triangles are by definition not geometrically perfect, but approximations of triangles. In other words, both the pictorial triangles and the words referring to triangles are referents to the concept of a triangle.” So yes, you could technically count the word “triangle.”  (Silvertant also includes a useful graphic explaining how to find all the pictorial triangles.)

Check out the whole Quora discussion for in-depth explainers from users about their methods of figuring it out.

[h/t MSN]

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7 Discoveries That Started as School Assignments
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Some schoolwork is worth more than a good grade. The young students behind these assignments earned recognition from scientists, paleontologists, and mathematicians in the real world. Here are seven class projects where the effects were felt far beyond the classroom.


For a middle school science project, Simon Kaschock-Marenda discovered that a sweetener found in most grocery stores doubles as a nontoxic insecticide. With help from his dad, Drexel University biology professor Daniel Marenda, Simon gave artificial sweeteners to fruit flies raised in their home. Of all the groups they studied, the flies fed Truvia had the shortest lifespan (5.8 days compared to 38 to 51 days). Daniel realized the significance of their findings and conducted further experiments at his work lab. Along with his colleagues, he identified the ingredient responsible—erythritol—and published the work in the journal PLOS ONE. The chemical compound won’t be replacing all synthetic pesticides anytime soon, but for small-scale applications it could be a safer alternative.


Minnesota teenager Ethan Manuell didn’t expect to make a major medical breakthrough when he was asked to make a science fair project. As he told USA Today, his main concern was getting a good grade. After receiving the assignment from his eighth grade science teacher, Mrs. Omland, Manuell started tinkering. Using materials he had at home—including old battery-operated toy bugs and the batteries from his hearing aid—he discovered a way to extend the lifespan of zinc hearing aid batteries by 85 percent.

Some hearing aid battery packages instruct users to remove the plastic tab on the back of the battery and let it sit for one minute. Manuell further tested this suggestion, allowing the tab-less batteries to sit for various amounts of time before testing their longevity in his vibrating toy bugs. He found that five minutes was the golden length of time needed to achieve maximum results.

The information is shared in doctor’s offices today. By using Manuell’s trick, hearing aid wearers can save an average of $70 a year on batteries.


A group of 8- to 10-year-olds from Blackawton Primary School in Blackawton, England may qualify as the youngest authors of a published scientific journal. The students conducted their research on the flower selection habits of bees in a local churchyard. The results, which they wrote up themselves, appeared in the respected Royal Society journal Biology Letters in 2010.

With their teacher, Dave Strudwick, and neuroscientist Dr. Beau Lotto as their mentors, the kids set up a color-coded, plexiglass box for bees to navigate. The puzzle contained two types of artificial flowers: one filled with sugar water and one with salt water. They discovered that “bumblebees can use a combination of color and spatial relationships in deciding which color of flower to forage from.” The findings suggest that bumblebees possess a more advanced awareness of their surroundings than some scientists give them credit for. Another takeaway from the study? “We also discovered that science is cool and fun because you get to do stuff that no one has ever done before,” the authors wrote.


Kevin Terris couldn’t have asked for better luck during a field trip he took as a 17-year-old. While scanning the ground for fossils at Grand Staircase-Escalante National Monument in Utah, the student spotted a baby dinosaur skull poking out of the dirt. Once the rest of the remains were uncovered, paleontologists concluded they belonged to the smallest and youngest duck-billed Parasaurolophus dinosaur ever recorded. They nicknamed the specimen “Joe.”

Terris and his classmates visited the dig site as part of a paleontology program at their California high school. The field had already been surveyed by experts when the students arrived, which makes the discovery even more impressive. After receiving his high school diploma, Terris went on to study geology in college. Joe, meanwhile, is on display at the Raymond M. Alf Museum of Paleontology in Claremont, California, after providing important insight into the development of duck-billed dinosaurs.


Getting out of the classroom and into nature led to a major discovery by middle school students at the Dedham School in Maine. Under the guidance of Vanderbilt University scientists via Skype, the students conducted an experiment on metridium (a type of sea anemone) they collected from the harbor. Their research revealed the presence of sulfilimine bonds in the creature’s structure, an observation that had never been made before.

While the bond is found in complex animals like mammals, it’s less common in simpler organisms. The students hypothesized that the quality is what prevents some creatures from regrowing limbs, while other animals like starfish are able to. “You see lots of war heroes that come back and they have a severed arm or a severed leg and so we can’t grow it back and we think maybe this has to do with that our cells are too bonded together: Maybe we need to loosen it up,” Dedham student Braedan Ward told WABI. Whether or not their hunch is proven correct, it's clear the budding scientists are asking the right questions.


When solving binomial expansions for his high school math class in 2013, sophomore Brock Brown could have used Pascal’s triangle like the rest of his classmates. Instead, he invented a brand-new theorem that allowed him to get homework done faster. His method eventually caught the attention of Ben Moulton, the math professor teaching Brock’s mother at Utah Valley University at the time. Moulton described the formula, now known as “Brock’s Theorem,” as an “elegant and simple” alternative to more common binomial theorems. The professor offered to develop a proof for Brock and later submitted it to the American Mathematical Association of Two-Year Colleges. Moulton and Brown presented their work at an Anaheim, California math conference in 2013.


In Dr. Robert Pergolizzi’s class at Bergen's Science and Technology Magnet School in New Jersey, students are given a lot of freedom. "There are no tests in this class, no homework," Pergolizzi told New Jersey News 12. Their only assignment is developing an original research project. Freshman student Joshua Meier took that prompt and ended up discovering a possible treatment for cancer.

Meier began his research by looking into the causes of rapid aging in artificially-generated stem cells. He discovered that synthetic stem cells are missing a third of their DNA, which makes them age faster. By controlling mitochondrial DNA deletion levels, he was able to slow the aging process.

As a junior, Meier used his findings to come up with a potential cancer treatment. Instead of slowing aging in stem cells, he realized he could reverse the process to expedite aging in cancer cells and stop them from growing. The research earned him second place at the Intel International Science and Engineering Fair.


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