Numbers Beyond Belief

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Numbers Beyond Belief

What is the biggest number you can think of? Or better yet, what is the biggest number you can’t think of? Graham’s number is a quantity so mind-bogglingly large that if you tried to think of it, your head would quite literally turn into a black hole. The maximum amount of entropy you can store in your brain is related to a black hole with the same radius as your brain, and the entropy of this black hole carries less information than it would take to store Graham’s number in your head. The number is so large that the entire observable universe would not be able to store it, even if each digit was the size of a planck volume, the smallest measurable space. Graham’s number is a truly godly value, but where does it come from and why do we need to know about it? Come with me as we journey to the fringes of infinity as we explore one of the biggest number ever used constructively, Graham’s number.

Before we can consider Graham’s number, let us take a look at this math problem:

Let N be the smallest dimension n of a hypercube such that if the lines joining all pairs of corners are two-colored for any n≥N, a complete graph K4 of one color with coplanar vertices will be forced.

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If you are like most people who are not well versed in combinatorics, this question probably makes very little sense. Luckily, Hoffman proposed an equivalent analogy problem that is likely more accessible to the common person. The analogy problem is stated like this:

 

Consider every possible committee from some number of people n and enumerating every pair of committees. Now assign each pair of committees to one of two groups, and find N*, the smallest n that will guarantee that there are four committees in which all pairs fall in the same group and all the people belong to an even number of committees.

 

 

In a rather complex proof, Ronald Graham, an American mathematician, proved that the answer to this question is somewhere between 6 and Graham’s number.

To get an appreciation for how large Graham’s number is, we need to turn to “arrow notation”, proposed by the legendary computer scientist Don Knuth. First, let us begin with just one arrow:

 

 

3↑3=33= 27

So far, we are dealing with numbers we know and love. However, the numbers start to get really big, really fast. Let us explore two arrows now:

 

3↑↑3=3↑(3↑3)=327=7.6 trillion

As you can see, adding just one arrow escalates things dramatically. However, 7.6 trillion is a number we can still fathom. It’s about equal to the number of bacteria on eight human bodies. When you add just one more arrow, the numbers become quite literally out of this world.

 

3↑↑↑3=3↑↑(3↑↑3)=33333333....333 where there are 7.6 trillion 3’s in the stack of 3’s

We aren’t even close to Graham’s number yet. However, we now have the tools to start making sense of Graham's number. Let us first define the first pivotal quantity, g1:

 

g1=3↑↑↑3

As you know by now, g 1 is absolutely gargantuan. We can now define g 2 :

 

g2 =3↑↑↑↑........↑↑↑↑3, where there are g 1 number of arrows

Naturally, g3 has g2 number of arrows, and so on and so forth. Onwards we go until we hit g64, which has g63 number of arrows. Finally, you’re done! Graham’s number is g64.

For a long time, Graham’s number was the largest number ever used in a mathematical proof. Nowadays, tree algorithms have produced bigger numbers, including the titanic TREE(3), but Graham’s number will always have a place in mathematical lore. For most of us, numbers this big will have no impact on our lives, but in our most philosophical moments, as we ponder the universe and what is beyond, we can remember that everything in existence cannot hold such a big value, and this colossal number is infinitely smaller than an infinite amount of numbers. Eternity is quite a lot bigger than you might think.

 

References

  • Gardner, Martin (November 1977). "Mathematical Games"
  • Padilla, Tony; Parker, Matt. "Graham's Number". Numberphile. Brady Haran.
  • Ron Graham. "What is Graham's Number? (feat Ron Graham)" Numberphile. Brady Haran

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Six Degrees of Separation

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Six Degrees of Separation

In the early 2000’s Samy Kamkar wrote JS.Spacehero, a bit of computer code that displayed the words “Samy is my Hero” on your screen when you clicked on his MySpace page. The code did not stop there however; when you clicked on his profile, JS.Spacehero also copied onto your MySpace page, so anyone who clicked on your page also saw the words “Samy is my Hero” on their screen. The code would also copy onto their computer, and whoever clicked on their page would see the message, and so on and so forth. In just 20 hours, the virus hit over a million different accounts, making JS.Spacehero the fastest spreading virus ever. When Samy deleted his page in a panic, he took down the entirety of MySpace with him. Of course, the US government frowns upon this kind of technological chicanery, and Samy Kamkar was hit with a felony charge. The importance of this story is not about computer hacking or technological laws; it's about something much bigger. Just how connected is everybody on earth?

You may have heard of the idea of Six Degrees of Separation, the theory that everyone in the world is interconnected in a path of 6 or fewer intermediaries. Before we can explore this idea however, we must introduce the mathematical notion of a network, or more properly, a graph. A graph is a structure made of nodes and connections between nodes, called edges. 6 examples of graphs are shown below. The red dots are nodes and the black lines between nodes are edges.

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Graphs can be used to represent many different things, but they are very helpful in modeling social networks. A graph can theoretically be used to model the connection between every person on earth, but for the sake of simplicity, we must first explore a small subset of the population: popular musicians. As we do this, let us also play a game. The object of the game is to link any artist to Kanye West in the shortest number of steps. Artists will be represented by nodes and the songs that the artists worked together on will be represented by edges. Let’s start with an easy example: We will link rapper Drake to Kanye. This is a simple path; they were both on the song “Forever.”

Simple enough, right? Now let us try someone potentially harder: Shakira.

It turns out we need to go only two steps to reach Kanye West from Shakira. Shakira sang “Can’t Remember to Forget You” with Rihanna, who featured on the Kanye West song “Famous.” As it turns out, Rihanna is a very important node in this graph, as she connects many disparate artists to Kanye West. In graph theory, nodes like Rihanna are known as hubs because their number of connections far exceeds the average.

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One important feature we see in most graphs is clustering. The idea of clustering is that related nodes are highly interconnected with each other. In this case, musicians in Kanye West’s genre, rappers, have many connections to each other.

The final concept we need to explore is the strong law of weak ties. Simply stated, the path between completely unrelated nodes are made shorter by seemingly random or surprising connections between two nodes. In this case, let’s take the path between two very different musicians: Kanye West to the Turkish singer Tarkan. It turns out that Tarkan made a song with Wyclef Jean, called “Aman Aman.” Wyclef Jean made a song with Kanye West called “Sweetest Girl.” The importance of this connection in the structure of the graph cannot be understated, because it connects a whole new group of artists to Kanye West in a relatively short amount of steps!

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Now let’s look at the big picture. Can any two random people in the world really be connected in 6 or fewer steps? The evidence suggests this is possible. Facebook data suggest that 92% of all Facebook users are connected in 5 steps. The connections on Twitter are even more close than that. Studies have shown that the average path length between two Twitter users is between 3 and 4 steps. Our friend groups can also give us insight into how interconnected we are. You can think of your tight knit group of friends like the rappers in our musician graph since most of them are friends with each other. If these were our only connections however, the world wouldn’t be very interconnected at all. Our more popular friends, the big names on campus that function much like Rihanna on our musicians graph, connect us to large amounts of people. Even more important are our random acquaintances. Our friends across campus, across towns, or across oceans are the people that really connect us to the rest of the world. So go out there and meet people! You never know who you might connect with.

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Eating Healthy

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Eating Healthy

In the past few years, I’ve developed an interest in cooking. I binge-watched the Great British Baking Show and practically worshipped Ina Garten. But after months of churning out pastas and pies, I began to miss the foods that were most familiar to me. I reminisced about my grandmother’s cheese filled paratha and fresh poori we had during our Indian Thanksgiving. So one day, I called my grandmother over and asked her to teach me how to make chana batura. She pulled out her infamous silver spices tin native to Indian cooking and threw the spices in without any measuring cups. After an indeterminable amount of time, the smell of turmeric and onions filled my home, and we dug into the fruits of our labor.

Although many Indians love the food of their homeland, many of them have also suffered the consequences of consuming large amounts of ghee and fried foods in the form of cardiac disease and diabetes, two of the biggest killers in the United States. Many of the diets focused on mitigating these diseases are based on Western foods. If you look through the recommended recipes on the American Diabetes Association website, the closest they get to a culturally diverse recipe is a burrito bowl, which isn’t even an authentic Hispanic dish.1 On the American Heart Association website, none of the foods labeled “Asian” are at best Americanized imitations of Asian dishes.2 Although there are some less well-known websites that list out diabetes-friendly recipes based on culture, the majority of health diets do not take into account the cultural diversity of hospital populations in this country.3

I called my grandmother a few weeks later to ask her whether there were dishes in Indian cuisine that were considered healthy. She said that most Indian dishes were highly caloric because people there were more concerned with having enough to eat rather than eating too much. This could explain why in times of excess, people who are eating large amounts of fattening foods are suffering the consequences. Still, there were some dishes she knew about that seemed promising. She mentioned that when there was someone sick in the household, they were fed a mixture of yogurt, rice, and lentils, a high protein dish that used minimal butter.

Could it be possible to balance our food culture with maintaining a healthy diet? Can we make changes that preserve flavor while sacrificing the less savory, fatty ingredients? Throughout this blog series, I will interview people from various cultural backgrounds to determine their “health foods’ and analyze the historical background and nutritional value behind popular recipes. At the end of this, I hope to better understand what it means to “eat healthy” from both a cultural and nutritional perspective.

 

References:

  1. American Diabetes Association. www.diabetes.org/food-and-fitness/food/cookbook-recipes. (accessed Nov. 20, 2017)
  2. American Heart Association. www.recipes.heart.org. (accessed Nov. 28, 2017)
  3. Eating Right Ontario. www.eatrightontario.ca (accessed Nov. 23, 2017)
  4. Image: https://petitworldcitizen.files.wordpress.com/2014/02/img_1678.jpg

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How to Stop Unwanted Thoughts

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How to Stop Unwanted Thoughts

We’ve all had our share of unpleasant experiences -- flunking a test, going through a breakup, failing an interview. Most of the time, memories of these experiences linger for a few days or weeks but eventually fade away, and we can continue with our lives stress and worry free.

However, sometimes, these unwanted thoughts constantly come back to haunt us no matter how much we try to distract ourselves or make ourselves forget them. Recently, a group of researchers at the University of Cambridge have discovered the mechanism behind why this phenomenon occurs.1

Participants of the study memorized pairs of words containing a cue and a memory. If the cue was shown in green, the participants were told to recall the memory. If the cue was shown in red, the participants were told to block all thoughts of the memory or try to “push it out of mind.” For example, if the pair was BEACH-AFRICA and BEACH was shown in red, the participant would try to suppress thoughts about AFRICA.

During the trials, researchers imaged the participants’ brains using functional magnetic resonance imagining (fMRI) and 1H magnetic resonance spectroscopy (MRS). These techniques allowed researchers to track the concentration of the GABA neurotransmitter in the hippocampus, a region of the brain responsible for memory.

GABA, or gamma-aminobutyric acid, is a chemical that relays signals between nerve cells in the brain by acting as an inhibitory transmitter, meaning that it suppresses the activity of cells receiving the signal. The researchers found that higher concentrations of GABA in the hippocampus were associated with a better ability to suppress thoughts, both positive and negative. They hypothesize that GABA prevents retrieval of memories, which in turn suppresses unwanted thoughts.

According to the researchers, the ability to control unwanted thoughts is essential for mental health.2 Most of us aren’t affected too severely by negative thoughts. However, having constant intrusive memories, thoughts, or hallucinations is a symptom characteristic of mental illnesses like anxiety, major depression, PTSD, and schizophrenia. For people struggling with mental health issues, this research presents a new approach to develop treatments that could improve their wellbeing and happiness.

 

References:

  1. Schmitz, T. W.; Correia, M. M.; Ferreira, C. S.; Prescot, A. P.; Anderson, M. C. Nature Communications 2017, 8 (1).
  2. http://www.cam.ac.uk/research/news/scientists-identify-mechanism-that-helps-us-inhibit-unwanted-thoughts
  3. Image: https://ww2.kqed.org/mindshift/2016/03/04/how-to-turn-on-the-part-of-your-brain-that-controls-motivation/

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The Science of Beauty

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The Science of Beauty

Henry David Thoreau, in his book Walden, details his almost two-year excursion of simple living in the woods of Massachusetts. He viewed nature as a way to achieve a higher understanding of the universe, and enjoyed being one with the solitude and beauty it has to offer. Nature, thus, has a way of connecting humans to our emotions and eliciting positive thoughts and feelings. For example, it is a universal truth that a rainbow after a rainy day brings a smile to anyone’s face. The aurora borealis, or Northern Lights, are regarded by many as breathtaking, a must-see on planet Earth. But how does nature capture our attention and scintillate our senses? What are the long-term effects of spending time in the outdoors?

Beauty in the natural world affects humans subconsciously: spending time in the outdoors is connected to overall mental well-being. A simple stroll through a forest, for example, can allow us to distance ourselves from our otherwise chaotic thoughts. We are forced to regard every stimuli around us, from the sun shining down upon us to the tall trees shrouding us to the the small squirrels and insects we are careful not to harm. Compared to the contemporary world, which forces humans to live life in the fast lane through the influence of technology and commerce, nature is Earth at its most basic level. It allows humans to take a step back and a breath in, and entices us with its many facets of simplicity and serenity. Thus, the environment melts stress and releases endorphins that can decrease feelings of depression and fatigue.

Nature’s ability to distract us from the present also increases creativity and intelligence. David Strayer of the University of Utah showed that hikers were able to solve more complex puzzles after a four-day backpacking trip compared to a control group. The prefrontal cortex, which controls decision-making and social behavior, undergoes much strain from daily usage of technology and multi-tasking. This area of the brain can take a break when we respond to purely nature-driven stimulus. Nature allows the brain to reset so that it can perform tasks with renewed energy.

A change of environment can also makes humans kinder and more generous. There is an out-of-body feeling associated with viewing an awe-inspiring landscape that makes one feel that one is part of something bigger than the present. It can make day-to-day inconveniences seem inconsequential and remind us that there is more to the world than what goes on in our lives. Humans are also more likely to be more ethical when faced with moral dilemmas after spending time in nature. Experiments conducted at the University of California, Berkeley, found that participants playing the Dictator Game (which measures the degree to which individuals will act out of self-interest) were more likely to be generous to their peers after being exposed to alluring nature scenes.

Planet Earth’s most primitive offerings actually present us with complex and diverse benefits. A quick breath of fresh air can melt away feelings of stress and anxiety, while increasing cognitive focus and creativity. Perhaps we can create our own “Walden” and take a break from studying or working to simply enjoy the outdoors and spend time appreciating the many sides of our ever-changing world.

 

 

References:

  1. “How Nature Can Make You Kinder, Happier, and More Creative.” Greater Good, greatergood.berkeley.edu/article/item/how_nature_makes_you_kinder_happier_more_creative.
  2. Louv, Richard. “Ten Reasons Why We Need More Contact with Nature | Richard Louv.” The Guardian, Guardian News and Media, 12 Feb. 2014, www.theguardian.com/commentisfree/2014/feb/13/10-reasons-why-we-need-more-contact-with-nature.

  3.  

    “Henry David Thoreau.” Henry David Thoreau, transcendentalism-legacy.tamu.edu/authors/thoreau/.

     

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