A Spire for Greatness

Over the past few months, I’ve been thinking about the connections between symbols, information and the progress of human kind, scientific and otherwise. I’d like to share an essay I wrote a while back about skyscrapers.  In a very naive sense, the heights of our tallest buildings reflect the progress of civilization. In reality however, our advancement is more closely linked to the  depths of our abstractions than the heights of our concrete constructs.

Without further ado, here is my (abridged) essay, entitled “A Spire for Greatness”

A Spire for Greatness

A few weeks ago, I found myself in John F. Kennedy International airport on my way to meet my grandparents for the Thanksgiving Holiday. The flight attendant at the gate scanned my ticket and wished me a safe flight as I walked through the gate. Just past the gate was a short declining passageway connecting the plane to the terminal. The passageway only had one full plaster wall, and on the center of that wall was a single poster. The poster, which featured a slew of skyscrapers peaking through the clouds, read: “This is the story of human ambition”. At the time I didn’t know why I did it, but I snapped a picture of the poster before boarding the plane. It was only later that I realized why the poster had such a strong effect on me: it was precisely because I was in an airport – one of the largest and most advanced in the world. I was about to soar above the clouds in a giant metal cylinder with wings.

Fig 1. Poster at JFK that reads “this is the story of human ambition”.

Airplanes and skyscrapers are alike in a superficial, as well as in a fundamental way – they are both metal boxes, which take humans up into the sky. But airplanes fly higher than even the most imaginative designs for skyscrapers. As a child I remember learning that the Wright brothers epitomized dreaming and innovation. Amelia Earhart, who fearlessly flew solo across the Atlantic Ocean, seemed to embody the intrepid. And yet, in this air-transportation hub, where it would be most apt to revere plane and pilot, it is the skyscraper that symbolizes human ambition.

What is it about the skyscraper that screams to us of human achievement? And why, instead of using a word accepted as positive like progress, did the poster highlight ambition, a word riddled with connotations both good and bad? On the one hand, ambition is often viewed as a negative quality, an inimical mixture of Greed and Pride, two of the seven deadly sins. On the other hand, ambition is in our modern society somewhat synonymous with a desire and a will to succeed. The ambitious are vigorous, and take their own initiative – these are the type of mates parents plead with their children to find and latch on to. As I later found out, the poster was just one in a series of such posters in airports all over the world. The posters were created and financed by the bank HSBC to promote human ambition – a thinly veiled ploy to improve their global image. By shining a positive light on ambition, the company could shift its associations from the avarice of banks to the aspiration to contribute positively to society.

Although we usually think of skyscrapers as epitomizing negative ambition, they can, and should represent the positive type. For skyscrapers to add to their urban environments, and the world as a whole, they must embrace the horizontal along with the vertical.In such a world, skyscrapers no longer ‘scrape’ or ‘pierce’ the skyline. Instead, they create it.

The word “skyscraper” traces back to the late 1700s, when it was introduced as the name of a racehorse, and subsequently used to describe a light sail at the top of a mast. “Skyscraper” was first applied to tall buildings in 1888, four years after the construction of the Home Insurance Building in Chicago, which, at ten stories tall and metal-framed, is regarded as having been the first modern skyscraper. Architect Leroy Buffington proposed building a “stratosphere-scraper” with the same metal frame and almost three times as many floors. Perhaps “stratosphere-scraper” was too long or unwieldy, or “stratosphere” sounded too scientific for the common vernacular, but “skyscraper” caught on. A century later and “skyscraper” has maintained its context.

A compound word, “skyscraper” conjures images of literally “scraping the sky”, clawing at some impenetrable boundary above. This paints a picture of human curiosity, wherein the skyscraper enables man to pull back the curtains of the clouds and view the space beyond. But this tale of exploration and inquiry might be better suited to a “stratosphere-scraper” than a “skyscraper”. Unlike the physicality of “stratosphere”, “sky” carries with it a spiritual significance, pointing toward the divinity and immortality of Heaven above. “Scrape” implies a level of violence and disrespect toward the object of its scraping – in this case the sky. However, “scrape” seems unfit to describe the action of the spired, towering buildings in modern metropoles. A better verb might be “puncture”, conveying the violence of a spire shooting up into the sky.

Nonetheless, the precise definition of “skyscraper” differs from source to source. Some impose cutoffs based on height or number of floors; some count spires while others don’t; and some require that the building rise above the rest of its urban environment. There are even designations of “supertall” and “megatall” reserved for the exceptionally tall skyscrapers. For the purposes of this paper, we do not need such measures. Instead, we satisfy ourselves with the following:

Def: A skyscraper is a tall structure built by humans 
     and in which humans live or conduct daily 

We don’t need to specify exactly how tall. What is important is that it is tall in the sense of our experience. Moreover, the second clause excludes structures such as the Washington Monument, the Statue of Liberty, and the Pyramids of Giza. This definition does apply, however, to tall buildings that were constructed before the word “skyscraper” came into being.

In this general sense, the tower of Babel can be considered a skyscraper. In the Book of Genesis, the people say, “come, let us build ourselves a city, with a tower that reaches to the Heavens, so that we may make a name for ourselves; otherwise, we will be scattered over the face of the whole earth”. Taken literally, the people hope to avoid being “scattered”, or spread outward over the face of the earth. To these people, the tower represents the vertical, whereas, relative to their body matrix – their natural coordinate system  – this scattering threatens to spread them horizontally. The vertical then, is supposed to fend off the scattering, instead bringing them together. The tower of Babel can be seen as an axis mundi. But the “Heavens” here refers to more than just that which is physically above them; it implicates God. “Reaching to the Heavens” is paramount to equaling God, and through the tower of Babel, their skyscraper, the people hope to equal the Divine Creator. “So that we may make a name for ourselves” only further illustrates this human pride. Moreover, it acknowledges that the tower of Babel is a product of self-serving ambition.

As the infamous tale concludes, God reasons that if the people build the tower, “nothing they plan to do will be impossible for them”. As a result, he “confuses their language”, scattering them over the earth. God’s rationale further situates the primitive skyscraper as the ultimate symbol of ambition.

With the tower of Babel in mind, it seems fitting that the language that we use to describe tall buildings perpetuates this association of the skyscraper with self-serving ambition. In English, a set of stairs, taking you from one floor to another is commonly referred to as a ‘flight’. Obviously, people don’t fly up stairs; they walk on two feet. But “flight” makes us ever conscious of our ascending and descending. In particular, it reminds us of the myth of Daedalus and Icarus, in which Icarus flew too close to the Sun, melting the wax that held his wings together, and falling precipitously to his death. In Greek mythology, the Titan Helios personifies the Sun, imbuing it with divine significance. Just like the Heavens in the story of Babel, the Sun represents something that humans should not attempt to equal, and Icarus’ flight is often interpreted as a story of personal ambition.

In a skyscraper, the alternative to a flight of stairs is an elevator. Indeed, the elevator was technologically a precondition for the modern skyscraper. Being tall, these buildings often have tens of floors, and it is inconvenient and even impractical for people to climb all of those stairs. The name “elevator” is apt, as it does elevate people. However, in its earliest use, the verb “to elevate” means “to raise in rank or status”. This etymology suggests a literal hierarchy of height; the higher up one is physically, the higher their status in society.

The skyscraper embraces this hierarchical structure in two ways: within a single skyscraper, and between competing skyscrapers. Inside high-rise apartments, the more luxurious, expensive rooms tend to be on the upper floors, with quality (and cost) increasing with height. Some buildings even have separate elevators reserved for occupants of the upper floors. The highest rank one can have is to own the penthouse – the most grand apartment in the entire building. The penthouse suite is almost always on the top floor of the apartment complex, often taking up the entire floor. A similar hierarchy ensues in office buildings, where penthouse is replaced by oh-so-coveted corner office with a view on the top floor. The skyscraper thus fosters an atmosphere of competitiveness.

The hierarchy between skyscrapers is perhaps even more evident. With the completion of the Home Insurance Building and the arrival of the modern skyscraper has come the quest to construct the tallest building. This competition was on display in the volatile skyline of New York City in the mid 20th century. In early 1928, the Woolworth Building stood as the tallest building in New York after eclipsing Trinity Church in 1913. But that summer saw Manhattan host the “race for the sky” between the Chrysler Building and the Bank of Manhattan Trust Building. During the construction of these two buildings, the height advantage shifted back and forth before topping out at similar levels. After the Chrysler’s groundbreaking ceremony, the architect of its competitor at 40 Wall Street “increased the height of his project by two feet and claimed the title of the world’s tallest building”. In response to this, William Van Allen, the architect of the Chrysler Building added a 125-foot spire to the top of the building, reclaiming the title.

Today, there is even an “official” title of the “world’s tallest building”, bestowed by the Council on Tall Buildings and Urban Habitat (CTBUH). Every year or so, it seems, this title is handed down (or up if you will) to the next champion of the sky. The CTBUH’s database, The Skyscraper Center, confirms that the five tallest buildings in the world were all completed after 2010.

What about skyscrapers in particular fosters this competition? Among skyscrapers, one simple answer seems evident: taller buildings command more attention. A tall building brings its architect into the public eye. The company asserts its presence in the community, and the architect, as a creator, “makes a name for himself” as attempted those constructing the tower of Babel. It’s no surprise that many skyscrapers of this period in the late 19th to mid 20th century were named after the companies and wealthy individuals who purchased them: “Woolworth”, “Chrysler”, “Home Insurance” and “Manhattan Company” are just a few examples.

What is surprising is that many of these structures emphasize their “building-ness”. The fact that the Chrysler Building is a building – that it contains a roof and walls and stands of its own accord – is self-evident, and the word “Building” is unnecessary in conveying that. Taking “building” as part of a building’s name, and consequently its written and spoken identity, however, gives us pause to think. In the English language, the suffix “ing” is predominantly used in gerunds – verbs constructed from nouns. For example, in the sentence “Computing is fun”, “ing” transforms the verb “to compute” into the noun “computing”. Yet we don’t speak of this new noun as an entity of its own – “A computing” is unnatural. “Building” is similarly a gerund, in this case formed from the verb “to build”. Nonetheless, we speak of “A building” when we speak of the “Chrysler Building”. The presence of the word “building” thus implicates the verb “to build”, insinuating that the process of constructing and building is ongoing. By so doing, these skyscrapers implicitly claim to be building on top of the other members of their urban environment, asserting their place above the city. A similar story applies to the word “tower”, which appears in “Trump Tower”, the “Eiffel Tower” and “Burj Khalifa” (Burj meaning tower in Arabic). Although “tower” can function as a noun, its original usage was as a verb. The presence of “tower” in the names of these skyscrapers then speaks to their “towering over” their built environments.

Inside the skyscraper, it is the ambition of the occupants, not its architect and sponsor that is promoted. As extensions of these buildings, balconies provide a window through which to understand intra-skyscraper self-assertion. Balconies have a long and rich history that begins centuries before the construction of modern skyscrapers. A quick online search of “balcony” returns images of exquisitely decorated balustrades with artful railings. A search for “city balcony”, however, reveals images not of the structure and artistry of the balcony itself, but of the view from that balcony. Some of these pictures capture the edge of a railing in the frame, while others ignore the balcony entirely, with camera tilted slightly downward, giving an air of superiority in which the on-looker is literally “looking down on” the rest of the city. The higher one is, the more removed from the turmoil below. He who stands out on the balcony peering out upon the city rises above the city and its mortality. After all, he is literally in rarefied air, and rarefied company.

The complementary “looking up” of people below at the balconies adds to this sentiment. According to Schopenhauer, the balance of load and support is fundamental to architecture. In his treatise On the Aesthetics of Architecture, Schopenhauer writes that because balconies protrude from a building without any obvious support, “they appear suspended, and disturb the mind”. The higher up one is, the less apparent lack of support plagues their eyes and minds. He who rests on the highest perch is immune to these worries. In contrast, those on the street below see the undersides of every balcony, and must fear that such unsupported load will come crashing down upon them.

And while for the most part skyscrapers do not physically crash down upon us, dystopian science fiction films cultivate the association of skyscrapers with our fear of the collapse of humanity. Skyscrapers are conflated with human pride through the frequent use of darkened cityscape and skyline as a setting for artificial intelligence. By trying to create something intelligent – and in their own image, humans raise themselves to the level of the Divine Creator, God, “reaching toward the Heavens”. One example of this that I find particularly poignant – and which was received with great success internationally, speaking to its thematic universal appeal – is the Wachowski siblings’ Matrix. In the Matrix, humans create thinking machines in an attempt to take away the AIs’ source of energy. This backfires, as the machines realize that they can use humans as batteries, and they begin “harvesting” human beings in gigantic sky-scraping towers referred to as “power plants”. The machines hold the humans in these towers in an unconscious state by hooking their brains up to “the Matrix”, effectively feeding them a false reality. While the few humans outside the Matrix live underground, the skyscrapers are the domain of the machines. They are portrayed as dark, formulaic, and inhuman, and we see the “humans” in the tower as somehow less human. This speaks to our fear that skyscrapers will somehow detract from that which makes us human.

Fig 2. Image of two “skyscrapers” used to house humans
as batteries in the Matrix.

But this doesn’t have to be the case. There is nothing inherent to tall buildings that necessitates that they promote greed or pride. Nor is it necessary that they figure as symbols of negative ambition in our society. In her famed novel The Fountainhead, philosopher Ayn Rand espouses her view through character Gail Wynand that skyscrapers are “the will of man made visible”. Of course, this will can, and has been turned toward personal gain. But “will”extends beyond the will to succeed; human resolve and determination also manifest themselves as the will to live and the will to love. By the same token, skyscrapers can outgrow the literal connotations associated with scraping the sky and come to reflect the positive side of human ambition.

In fact, there are instances of skyscrapers contributing to their communities going back all the way to the dawn of the modern skyscraper. In 1889, the Eiffel Tower became the first such structure to weave its way into the fabric of an entire nation. While the Eiffel Tower is not generally considered to be a skyscraper in the conventional sense, it satisfies the definition given in this paper. The height of an 81-story building, the tower was the tallest structure in the world for more than four decades. It is a monument, but unlike the Washington Monument, the Eiffel Tower – at least initially – satisfied the final constraint: its architect Gustav Eiffel lived in the penthouse apartment at the top of the tower. Now the apartment is open to public tours, so the tower’s designation as a skyscraper no longer holds.

As a skyscraper, the Eiffel Tower did succumb to many of the obvious signs of self-promotion. Not only does the tower bear Eiffel’s eponym, the architect signed the contract for the tower in his own name instead of that of his firm, ensuring himself all profits from the deal. Moreover, the so-called “tower” was an entrance to the 1889 World’s Fair, making the implications of the word “tower” even more significant. However, the tower was able to transcend these ambitions as a symbol of France’s 100th anniversary celebration of its Revolution. Eiffel said that the tower would reflect “a century of Industry and Science… prepared for by the great scientific movement of the eighteenth century and the Revolution of 1789”. Indeed, the tower certainly paid its respects to science by memorializing 72 French scientists and engineers. In addition, it aided in the contribution to scientific discovery: the Eiffel Tower was instrumental in the discovery of cosmic rays, and to experiments in meteorology and air resistance. Just as importantly, the tower came to be a symbol of French patriotism – of national pride.

Over the past few years, America has been similarly united by skyscraper-induced patriotism. On September 11th 2001, coordinated terrorist attacks sent hijacked airplanes crashing into the North and South Twin Towers, collapsing the structures and taking the lives of thousands of people working inside. Debris from the towers rained down upon and destroyed the World Trade Center. In addition to causing the United States an estimated loss of three trillion dollars, the attacks sent a wave of fright and uncertainty across the nation, wiping away the feeling of American security.

Shortly after the catastrophic events of 9-11, plans were proposed for the construction of a new building at Ground Zero, and by 2004, like a phoenix rising from the ashes of its predecessor, One World Trade Center was born. Even the building’s name speaks to its purpose: While “one” technically refers to its address, it combines with “world” to suggest togetherness and advocate for one universal community. The building’s designer Daniel Libeskind sought to create “a monolithic glass structure which reflects the sky”. As simple as it seems, this statement speaks to a truly different mindset toward the construction of skyscrapers. Instead of piercing or even reaching the sky, the architects hoped to reflect it. To “reflect”, or “to cast back without absorbing”, suggests an acknowledgement and acceptance of human’s inability to capture and hold onto the divine. For Icarus, tragedy took the form of melting wax unfastening his wings. Implicit in this melting is the absorption of heat from the Sun. Reflection stands in stark contrast to this demise, demonstrates a divergence from prideful ambition. In addition to its physical interpretation, “reflection” admits a second definition – “serious thought or contemplation”. This meaning is synonymous with “meditation”, and conveys deep introspection and concern for how one’s actions affect others. In the case of One World Trade Center, reflecting the sky becomes a kind of meditation on the sky, in which the building is conscious of its relation to the Heavens – that it is below the Heavens.

These initial hints of thoughtfulness are substantiated by the building’s form, which embraces other New York skyscrapers rather than eclipsing them. In his original design proposal, Libeskind intended to complement the Statue of Liberty, and to integrate One World Trade Center with the surrounding buildings. The revised plans however, do even more – paying tribute to many of New York’s most iconic structures. One World Trade Center’s façade draws inspiration from the original North Tower, and its spire closely resembles those of the Empire State Building and the Chrysler Building.

Fig 3. Spire at the top of One World Trade Center,
paying homage to the spire on the Chrysler Building.

Just as this spire helped join together the staples of the New York skyline, it helped unite the American people by transforming One World Trade Center into a symbol of resilience and renewed national faith. In the “race for the sky”, Van Allen threaded a spire through the top of the Chrysler building to edge out the Manhattan Company Building. This “race” was fueled by competition, and the spire was added for the purpose of claiming the title of tallest building in New York (and in the world). Upon its completion, One World Trade Center did reach higher than had any previous building in New York. However, its spire was introduced to raise the building to a height of precisely 1776 feet to commemorate the signing of the Declaration of Independence. The purpose was to “restore the skyline with a tall and dramatic new symbol for New York”; it needed to be tall to make a statement, but the resulting designation as tallest in New York was an afterthought. In other words, while it did reach up into the sky, its ambition was to reach out toward the rest of the skyline, and to the people of America. In this ambition it has certainly succeeded.

And One World Trade Center is not alone. Over the past few years, architects across the world have begun to integrate skyscrapers into their urban communities. In London, this movement from domination toward contribution has manifested itself in high-tech buildings that pay respect to the history surrounding them. One such building, the Cheesegrater is the product of massive commercial backing, yet is not named after the financing company. Rather, along with other newly constructed London skyscrapers like the Gherkin, the Walkie-Talkie, and the Scalpel, it is endearingly known for the object it resembles. This catchy nickname and association with an everyday object make it easier for people to grow accustomed to the building, facilitating its assimilation into the city’s architectural framework. While the Cheesegrater is recognizable by its distinct wedge shape, it is for the most part unobtrusive. In fact, the building’s architects chose the wedge-shaped form precisely so as to not inhibit views of other London landmarks such as St. Paul’s Cathedral and Westminster Palace. According to architect and art historian Bethan Morgan, for the Cheesegrater, “the structure…determined the aesthetics of the building, and the form was deciphered from the needs of London as a historical context”. The building’s designers, Rogers Stirk Harbourn and Partners, applied technological advances and state of the art architectural techniques in order to add to London’s architectural story while preserving its history. If architecture is, as Ayn Rand asserts, “the will of man made visible”, then the Cheesegrater is a testament to that will. The Cheesegrater illustrates ambition to punctuate instead of puncture the skyline.

Fig 4. The “Cheese grater”, one of
several iconic skyscrapers in London.

Indeed, the idea that a skyscraper can assert itself through accenting and adding to the other buildings in its urban habitat has made its way into the common architectural discourse. In 2012, the CTBUH hosted a design competition on the topic of “Reimagining Tall”. The competition’s chair William Penderson motivated the event: “There has been a major transition in the sense of the value of the tall building and what it can contribute to the urban realm, and society in general”. Hundreds of students across the globe submitted original designs for skyscrapers that added to their environments. One of the winning solutions, entitled “Urban Forest”, proposes a new model for the skyscraper, in which one tall building can “grow” out of another, extending it in any direction. In this way, a single building cannot exist as a whole on its own; each building is a part of those around it. Like trees in one connected forest, these buildings form a cohesive urban environment. Any one building not only adds to the urban environment, it takes part in creating it. As the proposal’s authors describe the forest, “it looks not just to fit within the city, but to extend new fits to nearby buildings and public spaces”. These extensions connect one tall building to another, opening the skyscraper up to the horizontal. Just as a tree must spread its roots laterally to grow upward, so must the skyscraper connect to those around it as it rises vertically.

Fig 5. Artistic rendering of “Urban Forest”

Of course, “Urban Forest” is only one possible model for skyscrapers. But it does point to a deeper truth about tall buildings: If tall buildings are to represent the positive type of ambition – the aspiration to add to their urban habitats in a meaningful way – then they must embrace the horizontal as well as the vertical. Like One World Trade Center’s spire, skyscrapers should reach outward as well as upward.

Identity and Equivalence

One of the conditions of being human – in some ways an advantage, in others a limitation – is that we formulate our thoughts in terms of entities – people, places and things. It’s the best way we know to relate to our surroundings. Simply put, we like to think in terms of distinct, separable, independent entities. And we love to create distinctions – to identify and delineate one thing from another. Scientists especially love to categorize, and to identify fundamental properties of entities. From phylogeny, to the periodic table of elements to the standard model of particle physics, it is a large part of how we make sense of the world.

Sometimes, these divisions are reasonable. One particularly sensible instance is the delineation of places. Within America, there are 50 states. When I’m in California, there’s no sense in which I’m in Alaska or Wisconsin, or even in bordering Nevada. If I were to walk eastward, I would continue to be in California until one particular, well-defined moment when I cross over into a neighboring state. In other words, there are agreed upon, sharp boundaries between one state and the next. And while geographic location may not be the only unique characteristic of a state, it certainly can be used to unambiguously discriminate one from the next.

Other entities are harder to define. Take colors for instance: on its face, color seems fairly tame. Early on we learn about the three primary colors, red, yellow and blue. We are taught that by combining these three, we can arrive at any color we would like. We learn that the rainbow is composed of 7 colors, and we are given a handy acronym, ROYGBIV, for remembering these colors. In most everyday situations, these naive definitions hold up, and there is no cause for concern. Armed with our experience and rules of thumb, we typically agree on what color we see reflecting off of an object.

However, the problem is just that – we almost always agree. Lexicographer Kory Stamper of Merriam Webster wrote a fascinating article in Slate about the exceedingly unhelpful definitions of certain colors that can be found in the historical literature.

One example Stamper gives is Vermillion: “a variable color averaging a vivid reddish orange that is redder, darker, and slightly stronger than the chrome orange, redder and darker than golden poppy, and redder and lighter than international orange.”

If you were given this definition, could you go to a color wheel and pick out a precise location that realizes vermillion? Sure. And so could I. And so could everyone and their grandmother. But there is absolutely no guarantee the “colors” that we pick out would be the same. In other words, while the quote provides a few reference points of colors that are redder, lighter and darker etc., it does nothing to quantify how much so. It tells us nothing about where golden poppy ends and vermillion begins.

And if it’s this difficult to pin down colors, how difficult must it be – or is it even possible at all – to develop consistent and complete definitions for more complicated concepts? And how are we to think about our own identity, personhood, and sense of self?

This is a question that has consumed philosophers for millennia. In fact, it’s really not just a question but an entire sub-discipline of philosophy. Some of the sharpest minds in history have taken a crack at this problem, among them Descartes, Locke, Hume and Kant. The goal here is not to replicate their arguments or try to come up with some tidy all-encompassing definition for what it means to be something rather than something else. Rather, I’m going to draw on a few examples from physics and mathematics that may provide some insight into this problem. It is left to the reader to decide how convincing these arguments are, and to what extent they bear consequences in our lives.

Mathematics, in its ultimate generality, is the study of structure. There are many different branches of pure math, from algebra and calculus, to topology, logic, and number theory. The methods of proof vary by topic, but the common thread is the imposition of a certain “structure” on a set, and the importance of “maps” between sets, which preserve that structure. Forgive the complete lack of rigor in the paragraphs that follow.

Let’s bring this back down to Earth with an example: In the study of logic, the “structure” is informally called “interpretation”. Two statements are “equivalent” in the logical sense if they are true under the exact same models, or interpretations. Take the following two statements:

If Bill Gates is in Palo Alto, then he is in California.
If Bill Gates is not in California, then he is not in Palo Alto.

I and II are not exactly the same in the sense that the words in each statement are arranged in different orders. Yet for any possible situation, the truth or falsity of I and II will always be the same. (II is the contrapositive of I, so I can be derived from II and II can be derived from I. These derivations are related to the map between the statements.) They are logically equivalent. In the context of Logic, this equivalence is what matters. In other words, their logical content is their essence.

How about something a bit more abstract. In graph theory, a graph is defined by its vertices (a set of points), and edges (a set of pairs of vertices), the latter of which describe the connections between vertices.

Take for example the following simple graph:






In this graph, which we can call G, the vertices are a, b, c, d, and e, or V = {a, b, c, d, e}. The edges are the lines drawn between these vertices, E = {(a, b), (b, c), (c, d), (d, a), and (a, e)}. We can write G as G = (V, E) – this contains all of the defining information about the graph.

For graphs, the manifestation of equivalence is in mappings called isomorphisms.






At first glance the two graphs above – a pentagon and a star, seem quite different. On the page (or computer screen), the lines in the star cross each other, while those of the pentagon clearly do not. However, this difference is inconsequential from a graph-theoretic perspective. The two graphs are actually isomorphic – i.e. they have the same underlying structure.

One way to see this is to label the vertices of the pentagon a through e, and then write down all of the edges in the graph in terms of these vertices:

E1 = {(a, b), (b, c), (c, d), (d, e), and (e, a)}.

If we do the same for the star graph, labeling the vertices 1 through 5, we find that the edges are

E2 = {(1, 3), (3, 5), (5, 2), (2, 4), and (4, 1)}.

Looking at these two edge sets, it is obvious that if we make the substitution

a → 1, b → 3, c → 5, d → 2, and e → 4,

then the graphs have the same edge sets. Despite different vertex labels and a different appearance, the graphs behave in the same way – just as in the Logic example, these two have the same essence.

Sure, mathematics may exist in its own world, where it is not confined to the laws of physical reality. Obviously humans are far more complex and nuanced than can be mathematized. But these types of equivalences, which pervade every branch of pure math, beg deep philosophical questions.

What gives an entity its essence?

What make it it rather than something else?

What is essential to my own identity?

In some sense, I am different than the Jacob I was 18 years ago singing the ABC’s and playing with Sesame Street action figures. And I am different from the Jacob that I will be when I have grown bald and hard of hearing. I have matured. I’ve gained life experience. Even the cells in my body, comprising my physical self, die and are replaced every few weeks. Yet through it all I have – and will continue to – retain the fundamental quality of Jacob-ness.

Just like relabeling vertices on a graph, I could legally change my name without changing who I am. I could shave my head, or put on a top hat to disguise myself, but I will just be an equivalent Jacob, as the second logical statement above was merely the first statement in different guise.

John Locke famously believed that one’s material body had no bearing on their personal identity. In his Essay Concerning Human Understanding, he presents an example of a prince who wakes up in the body of a cobbler. Locke claims that after the transfer of consciousness, the person now corresponding to the cobbler is one and the same with the person who previously corresponded to the prince.

Okay so physical presence is not essential to one’s self identity. Then what is? What of opinions, thoughts, beliefs, and even actions? Perhaps quantum mechanics can provide some insight.

In quantum physics, a particle is represented by a mathematical object called a wave-function. One of the simplest quantum systems is the two-level atom. Like a classical analog, there is a ground state and an excited state. The quantum-ness comes into play in that the atom does not have to be in just the ground or the excited state at any given moment – instead it can be in a superposition of the two. The mathematical object used to represent such a system is a two-dimensional complex valued Hilbert space, – the state of the system being any normalized vector in that space.

Things get more interesting when multiple quantum systems interact: such systems can become entangled. When this happens, the state of the combined systems cannot be described individually. For example, when two two-level atoms are entangled, the state of the combined system cannot be described by two vectors in . Rather, they must be described by a vector in . When one atom is measured, the state of the second atom changes instantaneously.

In some sense, the state of a person is their worldview – their thoughts and tendencies. And if only figuratively, we are all entangled with those around us. It is impossible to disentangle our beliefs from the influence of society. Stark changes in the world around us, which we may call formative events, impact our internal state in undeniable ways. Our state – feelings, beliefs and desires – are distinct from our fundamental nature in the same way as the state of a particle is distinct from the mathematical object – the structure – which defines it.

What then is our fundamental essence? Is it an immaterial soul? A naked, unencumbered nature? Or something else entirely? Perhaps it’s a combination of qualities or, as Locke believed, continuity of consciousness. Maybe it is impossible to define precisely, and we must embrace the vagueness, as in our discussion of color. As a scientist, I would like to believe that there is something. But as a human being, I’m torn. Whatever it is, it’s not my place to say what that essence is. It is up to each of us to decide what makes us who we are.

The Question of Science

When I was 12, a friend gave me The Ultimate Hitchhiker’s Guide as a birthday present – 5 books bound together with an ornately engraved gold and black cover. It was a behemoth of a book, chronicling the adventures of Englishman Arthur Dent and his extraterrestrial friend Ford Prefect as they traveled through the galaxy. It seemed like the Bible of Science Fiction, and I absolutely couldn’t put it down. I think I read the entire series over a Spring Break one year, opting out of swimming, instead immersing myself in science fiction lore.

Through Arthur and Ford, I vicariously ventured into all corners of Douglas Adams’ universe. I encountered a cow that wanted to be eaten, a Paranoid Android older than the Universe, and a babel fish, which, when placed in your ear, allows you to understand any language.

In the first installment, The Hitchhiker’s Guide to the Galaxy, a group of hyper-intelligent beings builds a supercomputer called Deep Thought, in order to help them solve the mysteries of the universe. They ask Deep Thought the ‘Answer to the Ultimate Question of Life, the Universe, and Everything’, which turns out to be 42. But, as the computer aptly points out, the Beings know the answer but not the question. In the sequels, armed with an answer, the extraterrestrials backtrack through possible equations and formulas, and build an even more powerful supercomputer to discover the question.

One of the reasons I love the series is that every instance of science or adventure is so incredibly fantastical. But recently I’ve found that some of the ideas in the Hitchhiker’s Guide hit close to home.

When I first thought about writing this blog post, I wanted to write about physical intuition, the ability to qualitatively explain complex processes by thinking about them in terms of everyday experiences. I’ve since come to realize that “everyday experiences” doesn’t quite do justice to the way most people perceive science. Science is about asking questions and, hopefully, getting answers. However, in many ways, we don’t even know which questions we should be asking.

Science asks how, not why. It’s a subtle difference, one that is easy to ignore or overlook. But it lies at the heart of the way we see the world. How looks for an explanation, a sequence of events that brings a system from point A to point B. It is the search for processes that accurately describe the world around us. Even cause and effect lie in the realm of how. Why, on the other hand, begs a purpose, an intention.

These two questions seem so genuinely similar that we group them together, using them interchangeably. The lack of distinction is so ingrained in our society that it has become a part of our language. My roommate here at CERN, Emil Öhman is originally from

Sweden, so English is his second language. Every night over dinner we talk about philosophy and physics, and even though he is fluent in English, he always asks me to clarify what I mean when I use the word reason.

Emil brought to my attention the fact that reason has two completely separate meanings. Sure enough, as thesaurus.com attests, we use the word as a synonym for ‘motive’, the why, and ‘cause’, the how. The former connotes purpose, which is inherently human, while the latter seeks to explain natural occurrences. Consciously or not, we often use reason ambiguously, hinting at both meanings simultaneously.

Syntactical quirks aside, this inability to distinguish between how and why is an integral part of the way we interact with science. Through high school and my freshman year of college, I’ve heard the phrase ‘physical intuition’ thrown around in physics and math classes in the likes of “from our physical intuition, we can see that…” and “it’s obvious if you use your physical intuition, that this problem can be distilled down to…”

In many cases it is the most powerful problem solving technique that we have. Sometimes the math required to solve a problem is incredibly involved, and you have to think back on all of the physical processes you’ve observed in your life, pick out a few that share a semblance of similarity, and compare and contrast them in the hopes of developing a hunch. It allows us to run thought experiments in our heads, taking as an axiom that the physical laws that govern all processes are the same.

The classic examples of using physical intuition tend to fit this mold fairly well. Richard Feynman famously asked his students to find the weakest point on an infinitely round table with four evenly spaced legs. In the absence of physical intuition, this problem would require applied physics and intensive mathematical calculation. Using personal experiences, it is easy to guess that the table is weakest between any two of its legs.

Albert Einstein used physical intuition to formulate his theory of relativity. He realized that if he were in a metal box isolated from the outside world, and he felt a downward acceleration, he wouldn’t be able to tell if the box was a rocket accelerating upward, or if a gravitational body was pulling him closer. Einstein didn’t need to run experiments or create a perfectly isolated system out in space in order to come to his conclusion. Instead, the Equivalence Principle, as it became known (for equating inertial and gravitational acceleration) required physical intuition and a bit of creativity.

But it seems like the analogies are now in large part based on social structures. Physical intuition has grown to view particles as people, and subatomic phenomena as social interactions. I remember my Chemistry teacher in high school explaining the force between protons and electrons as “opposites attract”, as if it were an obvious corollary to a similar phenomenon in dating. We colloquially referred to certain elements as “wanting” to have a complete valence shell of electrons, when we really meant that atoms of that element lose or gain electrons because they are unstable.

We even personify Natural Selection, the process by which life evolves to handle environmental and social conditions. I’ve heard over and over again that Evolution “favors” those that are fit. Evolution itself has no wants or desires. It just happens to be true that on average organisms that are more fit to survive, survive.

Whereas physical intuition used to ask how, our personification of science has added an element of why. Instead of asking, “how did such biodiversity come to exist?” we ask, “why were these organisms chosen to survive?” By bringing inanimate objects and abstract processes to life, we give them thoughts, feelings, and even emotions! We impose upon them the human constructs of intent and purpose.

Indeed, it’s such a satisfying way to view the world. Emil thinks of particles as people that push each other away when they get too close and invade personal space, and pull each other closer when they grow too distant.

I only recently realized that my own conception of particles and elements relies heavily on social interaction. On the whole, the more fundamental a particle, the more stable it is, and the larger and more complex it becomes, the more prone it is to decay. Some, like the proton, have lifetimes longer than the existence of the Universe, while others live for fleeting fractions of a second. The same logic applies to elements. Those with more atoms tend to be less stable than elements containing only a few. I think about these particles and elements in terms of social groups; the larger the group, the greater the likelihood that members will disagree, and someone will want to leave. Complexity spurs on collapse.

There’s something poetic in our relationship with Science. We are composed of particles, but we view particles as if they were innately human. We personify elements and assign rationale to mechanisms that have no inherent purpose. Our social, as well as physical experiences, point us to the answer…

Which brings us back to the ‘Answer to the Ultimate Question of Life, the Universe, and Everything’. Is it wrong to incorporate why into our understanding of Science? Does why complement the empiricism of how, or refute it? Clearly viewing particle physics through the lens of social structures has limitations. It could, however, lend new approaches to old problems. After all, why appeals to something deep down, which how alone can never hope to satisfy. Could there be two fundamental questions, two ‘Ultimate Questions of Life, the Universe, and Everything’?

Inside the Heart of a Glacier

Right now it’s 3:30 in the afternoon and I’m sitting on the side of the road with my friend Nick. Along with Manjari, who is out exploring, Nick and I are in the small French town of Chamonix, situated in a valley in the Alps, about two hours outside of Geneva.

Chamonix is a beautifully odd mixture of idyllic European facades and touristy trinket shops. Cafes and chapels are interspersed with ATMs and hotels. The Tourist Center has a line dedicated to Japanese visitors, although far more come from China. And the music in restaurants is entirely in English, even when none of the waiters speaks the language: The Beatles are especially popular.

The unmistakable golden arches of the fast-food empire are easily visible from the bus stop – which is where we are right now. But the McDonalds logo is dwarfed by Mont Blanc, rising regally in the background.

Nick and I are waiting for the 5:00 bus back the Geneva after a weekend full of sightseeing. Since yesterday morning, we’ve hiked along the Alps, walked on a glacier, and ridden cable cars up to Aiguille de Midi, “Needle of the Noon”.

Today we woke up early and took the first train to the glacier, Mer de Glace. In French, Mer de Glace translates as “Sea of ice”, but the name is a vestige of the past. In the 1800s, the glacier was clear white, and easily visible from Chamonix. It was regarded as one of the world’s foremost natural wonders, and was the subject of many paintings and photographs. Since then, the Sea of Ice has receded hundreds of meters and lost a third of its thickness. Deposits of stone and earth, called moraines, have also diluted the crystal color of the glacier.

The first thing you see when you arrive is a vast cavern; previously filled with ice; now almost empty. As you walk down into the depths of the trench, you pass signs marking the levels of the glacier in past years -1820,  1890,        1920,                   1937… the distances between signs grow longer while the years grow closer together.

At the bottom lies the main attraction – an ice cave dug into what is left of the glacier. In the past, Mer De Glace inspired paintings. Now the glacier itself is the canvas. The artwork is framed only by the Earth’s crust. It moves with the glacier at a rate of a centimeter per hour, so every summer it must be entirely re-sculpted.

Sadly, when we got to Mer de Glace, the ice cave wasn’t open. We couldn’t bear to miss such an opportunity, so the three of us snuck in! Descending the last few flights of stairs, we walked along a blue carpet into the mouth of the cave.

The cave itself is an incredible sight! It is about 10 feet across, and penetrates deep into the glacier. The cavern is roughly cylindrical, with smoothed ridges along the walls, like the inside of the tunnel of a wave. Lights line the walls, each one striking the ice slightly differently, creating its own shades of translucency.

More surreal even than the cave are the sculptures inside it. The cave is composed of multiple cavities, each representing a room in an archetypal house. One cavity contains ice sculpted to look like beds. Another mimics a living room, with chairs, couch, and coffee table. A third contains a giant faucet, representing a bathroom.

Walking through the cave, I was struck by the strangeness of the juxtaposition. Why was such a beautiful natural canvas covered with the quintessentially human? Why did the artists model their creation after a house? Did the sculptures change each year? Out of all that is left of the glacier, the cave is the closest we can come to experiencing and appreciating the Sea of Glass. It seems like adding artificial sculptures would detract from that experience.

A part of me thinks that the sculptures are meant to symbolize the age of industrialism overtaking Earth. Maybe the sculptors are saying that humanity has clawed out the heart of Mother Nature, the same way they carved out the core of the glacier. The rest of me feels like it is just a product of proximity to the often incongruent Chamonix.

At its current rate of shrinkage, Mer de Glace could soon be gone. Living in Saint Genis for the summer, which is in the same country, I had to take two buses and a train, and sneak past tour guides to gain entry into the ice cave. In a few weeks, I’ll be back in America, a continent away from Chamonix and the Alps.

I may never again get to walk in the tunnel of translucent ice under France’s largest glacier. I hope to enter the cave once more before it vanishes. But in the meantime, I’ll be imagining a true sanctuary, an ice cave cleared of sculptures.

So Much for Symmetry

One of the first ideas that drew me to science was symmetry – the concept that everything has inherent balance, proportionality, and temporal harmony. I know it’s romantic to think that everything, from electrons to elephants, obeys the same universal laws; but for such a simple idea, it holds surprisingly true.

In Biology, plants and animals adhere to many different types of symmetry. Sunflowers and Nautilus shells spiral outward according to the golden ratio; trees and leaves display fractal geometry, in which the same pattern appears on many scales; and animals show bilateral symmetry, where left mirrors right.

Chemical bonding, too, is governed by symmetry – elements that have the same number of electrons in their outer (valence) shell, bond in the same way. The whole periodic table is based on this structure!

Much of Physics revolves around symmetry in the form of invariance, or lack of change based on coordinate system. It only seems right that in space up is no different than down! In fact, when drawing the diagrams that bear his name, Richard Feynman was so taken by symmetry, he concluded that a particle traveling forward in time is the same as its antiparticle traveling backward in time. In 1972, the Nobel Laureate PW Anderson went so far as to declare, “It is only slightly overstating the case to say that physics is the study of symmetry”!

But every once in a while nature throws a curveball our way. The narwal’s tusk is on its left jaw. And the wrybill is the only bird with its beak always bent to the right.

It’s almost frustrating that out of the millions of species on Earth, all but a few fit symmetries so nicely. What makes them different? What allows left-handed fermions to interact with the weak force when right-handed fermions cannot?

Chaos theory (pictured below), is the study of sensitivity to initial conditions. It says that simple systems, like the double rod pendulum, can have extremely complex results. With just two rods connected at a joint and hung as a pendulum, the double rod pendulum is one of the simplest mechanical systems. Yet it exhibits extremely complex behavior. It never retraces its earlier path, and never repeats a pattern. What’s more, changing the initial conditions even slightly leads to entirely unrecognizable results.

Maybe this sensitivity extends to other systems. In his book, A New Kind of Science, Stephen Wolfram studies cellular automata, or small grids of squares colored either black or white. He identifies 256 simple rules for deciding what color each square would be based on the squares around it, and runs each simulation on his computer. While the vast majority of the simulations produce simple, easily identifiable patterns, and fractals, a small fraction of the tests give different results.

Rule 110, although close to many of the other rules, produces results so complicated that it takes thousands of iterations, and millions of cells, for a pattern to emerge. Until then, it is essentially impossible to predict the behavior of a cell from the prior results.

The mathematician Horace Lamb once said of the chaotic nature of turbulence, “When I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic”.

Maybe chaotic systems can’t be predicted. Maybe their rules can’t be solved for with mathematics or computational power. But I’d like to think that our Universe is governed by simple, elementary laws; and that usually these laws produce simple results, but every once in a while produce an unrecognizable pattern which we perceive as disorder. Narwals and wrybirds break bilateral symmetry. But I’d like to believe that there’s some deeper, truer symmetry that we’re missing. As childish an ideal as symmetry is, it’s soothing to think that there is some underlying order.

Maybe nature isn’t throwing us a curveball; we might just be looking at turbulence the wrong way!