A Million Dollar Question

Math seems like such a well-defined subject that it is difficult to believe that there are still any big questions left unanswered. After all, for a subject that’s been studied extensively for at least 3000 years and one that builds on itself so easily, how can there be anything left unknown? However the reality is there are so many unproven conjectures and areas of research that it would be impossible for me to list them all. 

At the turn of the millennium, the Clay Mathematics Institute chose 7 hugely important unanswered problems still plaguing mathematics today. Whoever solves one of the “Millenium Problems” is entitled to receive a whopping $1,000,000 for their work. 

So what kinds of solutions are worth a million dollars?  The following is the statement of the Poincaré Conjecture, the only Millenium Prize Problem to have been solved so far:

“Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere” [1].

So, let’s break down exactly what this statement is saying. First, what is a 3-manifold? In mathematics, a manifold is essentially a surface with the property that at any given point, the surface resembles our regular intuition of a space with some number of dimensions [2]. We actually see a manifold in action every day by living on the Earth. Although the Earth is a sphere, when we walk on it, it’s easy to believe we lie on a flat 2-dimensional surface. Thus, the earth, and any 3-dimensional sphere, is a 2-manifold since when we walk on it, we effectively walk in a 2-dimensional space. 

More generally, a n-manifold is a higher dimensional (for simplicity's sake we can just assume the dimension is n+1) surface, but when an observer zooms in on a particular point on the manifold, the observer will see that an n-manifold will behave like n-dimensional Euclidean space. Thus, the 3-manifold is a 4-dimensional surface that behaves in 3-dimensions when we zero in on it.

A 3-sphere is a specific example of a 3-manifold, but what exactly is it? To understand what a 3-sphere is, it is useful to examine a 2-sphere. To begin, a 2-sphere is what we normally think of when we see the word “sphere”: a 3-dimensional surface with each point equidistant from a given center. Likewise, a 3-sphere is just the 4-dimensional analog; it is a 4-dimensional surface with each point equidistant from a given center. 

The conjecture proposes that any 3-manifold that is continuous (without any holes or other weird features) can be molded in a way that transforms the 3-manifold into the 3-sphere. For a slightly more rigorous definition, each point on a 3-manifold can be mapped to a unique point on the 3-sphere and vice-versa. 

Though we had to go through some complex definitions to understand the conjecture, the statement itself is simple enough and seems rather intuitive. However, it was actually proposed by Henry Poincaré in 1904, before being solved over one hundred years later by Grigori Perelman [3]. That’s not to say no one else had attempted to solve the question. By some metrics, the Poincaré Conjecture has had more false proofs than any other statement in recent history [3]. Rather, the fact that it took over a century to solve it shows how difficult proving this seemingly-simple statement is and there are six more just like it! 

There are still a lot of unanswered questions in mathematics, but with each solution, we get closer to a better understanding of the universe we live in.

References

1. [Image] Veisdal, J. The Poincaré conjecture - cantor’s paradise - medium https://medium.com/cantors-paradise/the-poincar%C3%A9-conjecture-cb4ca7014cc5 (accessed Mar 1, 2021).

2. http://www.owlnet.rice.edu/~fjones/chap1.pdf (accessed Mar 1, 2021).

3. Prize, P. The Hundred-Year Quest to Solve One of Math’s Greatest Puzzles by George G. Szpiro.

4. Poincaré Conjecture https://www.claymath.org/millennium-problems/poincar%C3%A9-conjecture (accessed Mar 1, 2021).

Mathematical Patterns in Plants

Ever wondered how your middle school algebra lessons could possibly apply to real life? It turns

out that math is ever-present in the world around us, and your garden may be the first place to start

looking. Studies of phyllotaxis, the arrangement of leaves on plants, have led to discoveries that traits

like divergence angles of leaves from stems can form fascinatingly exact patterns [1]. One of the most

prevalent patterns in plants is the well-known Fibonacci sequence (1, 1, 2, 3, 5, 8, …), which is also found

in many other iterations within the natural world [1]. The resulting spirals align with primordium plant

growth and are evident in the center of a sunflower, the aerial view of a radial succulent, and the

polygons of Brassica oleracea in the image above [2]. Aside from the Fibonacci spiral, common formulas

that plant branches follow include distichous (alternating), decussate (paired at right angles), and

tricussate (whorled trios of leaves) patterns [3].

But why would plants need to follow this math? There is substantial evolutionary justification

for this visually stunning geometry; plants are largely immobile creatures who adapt to their

environments primarily through their own growth (or lack thereof). As such, a plant’s best chance of

survival requires using growth forms to maximize a leaf’s exposure to sunlight and protection from the

elements – and then doing the same for every other leaf [4]. Leaves must grow in close proximity to

conserve space and resources on the plant, but current hypotheses state that they also biochemically

repel other leaves from growing too near them in order to fully utilize their environment [3]. This level

of complexity requires the absolute precision that is found in the mathematical formulas we use to

describe these natural phenomena.

While it’s true that certain patterns are far more recognizable than others – the golden ratio and

Fibonacci sequence are rarely inconspicuous – that doesn’t mean that plants with seemingly irregular or

random branching patterns don’t follow discernable equations. For instance, the species Orixa japonica

has the eponymous orixate arrangement, which was long thought to be mathematically inexplicable [3].

Its leaves grow asymmetrically rather than in a spiral or radial form and occur in a repeating series of

four different angles from consecutive leaves. Surprisingly, this highly specific pattern occurs in other

plant species across the evolutionary tree, indicating that there must be some mathematically driven

mechanism causing it. Researchers only came up with an appropriate model for its manifestation in the

past few years using computer simulations, and in doing so revised a previous phyllotaxis formula to

better describe several other plants [3].

Despite the longstanding fascination of the relationship between mathematical concepts and

the botanical world, concepts of it being leaf-inhibitory based remain in speculation and there is no

unifying theory for determining formulas or the mechanisms behind this [2]. There is undoubtedly a

great deal of complexity in this phenomenon, and yet its amazing visual effects can be appreciated by

even the most untrained eye in the simplest plant in a garden.

References:

1. Shipman, P.D.; Newell, A.C. Phyllotactic Patterns on Plants. Physical Review Letters 2004, 92, 1-4.

2. Newell, A.C.; Shipman, P.D. Plants and Fibonacci. Journal of Statistical Physics 2005, 121 937-968.

3. Yonekura, T.; Iwamoto, A.; Fujita, H.; Sugiyama, M. Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis. PLoS Computational Biology 2019, 15, e1007044.

4. Burakoff, M. Decoding the Mathematical Secrets of Plants’ Stunning Leaf Patterns. Smithsonian Magazine 2019. https://getpocket.com/explore/item/decoding-the-mathematical-secrets-of-plants-stunning-leaf-patterns?utm_source=pocket-newtab (accessed September 1, 2020).

5. Image source: Hesselink, A. romanesco; 2005.

G-Forces and Greyouts: The Science Behind Roller Coasters

There is something enticing about roller coasters: the drop in your stomach as you plummet down a hill, the weightless feeling as you hover at the top of a loop, or the pressing sensation on your head as you fight to maintain consciousness...no, not that last one.

On certain coasters, it is common for riders to experience a momentary loss of vision and/or consciousness. Commonly called “greying out,” this phenomenon is a result of high amounts of G-force acting on your body during high-intensity points of the ride. G-force is a measure of acceleration divided by the gravitational constant, g (acceleration due to gravity), and can induce serious physiological effects on the human body [1]. In particular, a strong force in the +z direction (from your head downwards, as shown in Figure 1) will push blood towards your feet and away from your brain [2]. This is often caused by powerful upward movement. The sudden decrease in brain oxygen can cause a wide variety of symptoms, ranging from the loss of peripheral vision, to the loss of color vision (greying out), to temporary blindness (blackout), to G-Induced Loss of Consciousness (G-LOC) [1].

G-force directions and resulting responses. –Gz acts upwards (weightlessness), and can cause redouts (blood rushes to the brain and causes vision to redden). +Gz acts downwards (pressure), and can cause greyouts, blackouts and G-LOC. Created with Biorender.com.

Greyouts commonly occur in airplane maneuvers or certain roller coasters with high levels of G-forces. The human body is equipped to handle a max +Gz-force of up to 4.7 G, based on a +Gz-time tolerance curve predicting the point at which G-force exceeds human tolerance [3]. To put this number in perspective, the force of gravity we experience daily has a magnitude of 1 G; however, there are several Six Flags roller coasters that have reported much higher G-forces, such as Taz’s Texas Tornado in Six Flags AstroWorld (6.5 Gs) and Batman and Robin in Six Flags Great Adventure (5.0 Gs) [1]. 

However, although prolonged exposure to high G-forces can cause serious brain damage, the vast majority of roller coasters are engineered to keep the rider under safe force thresholds [2]. The negative physiological effects of G-forces are time-dependent: a study has shown that although exposure to a force of any G level for more than 4.2 seconds can induce G-LOC, the human body can withstand up to 10 Gs for a very brief period of time [1]. In the cases of rides with extreme G-force measurements, as long as these values are only maintained for a fraction of a second, there is not enough time for blood to pool in the extremities and induce G-LOC [2]. In fact, it is very common to experience high levels of G-force––for safe amounts of time––in everyday activities such as sneezing (2.9 Gs) or sitting down quickly (10.1 Gs) [1]. In the case of amusement parks, greyouts are likely exacerbated by a variety of outside factors: hypoxia (low blood oxygen), heat stress, dehydration, fatigue, and consecutive rides can all increase the risk of greying out [4].

So the next time you buckle yourself into a scary-looking coaster, make sure you’ve had plenty to drink and adequate rest time between rides; this way, you can enjoy every loop or corkscrew, and keep plenty of blood in your brain!

References

1. Braksiek, R. J.; Roberts, D. J. Amusement Park Injuries and Deaths. Ann. Emerg. Med. 2002, 39 (1), 65–72.

2. Smith, D. H.; Meaney, D. F. Roller Coasters, g Forces, and Brain Trauma: On the Wrong Track? J. Neurotrauma 2002, 19 (10), 1117–1120.

3.  Whinnery, T.; Forster, E. M. The +Gz-Induced Loss of Consciousness Curve. Extrem. Physiol. Med. 2013, 2 (1), 19.

4. McMahon, T. W.; Newman, D. G. G-Induced Visual Symptoms in a Military Helicopter Pilot. Mil. Med. 2016, 181 (11), e1696–e1699.

5. Batman and Robin [Image]. https://www.screammachine.net/rideinfo.php?ridecode=11139&active= (accessed Oct 15, 2020).

Hello Quantum Worlds!

Quantum computing has been projected as a sort of messiah to the technological plateau that humanity is experiencing. The general prevailing belief is that with the advent of QC, we will be able to do the impossible — break cryptographic codes, solve problems that have eluded computer scientists for years, and even occupy interstellar space.

This is what everyone has to offer when asked what quantum computing is:  “Well, these computers can tackle multiple problems at once. It’s because classical computers can only be in one state at a time — 0 or 1 — while quantum computers can be in both states at the same instant of time!”

The only problem with this explanation is that it is wrong, and misleading to say the very least.

No, quantum computers are NOT in the same state at the same time. At least not technically, and although we might be able to break cryptographic codes faster, we won’t be able to solve puzzles miraculously.

Justin Trudeau was asked to summarize[1] quantum computing and he summed it up better than any layman explanation would:

“Very simply…normal computers work, either there’s power going through a wire or not— a one, or a zero. They’re binary systems … What quantum states allow for is much more complex information to be encoded into a single bit…a quantum state can be much more complex than that, because as we know, things can be both particle and wave at the same time.”

He was careful not to use the term “can be in both states at the same time,” which is commendable because that is where the problem lies. It turns out that staying in two states is physically impossible. Rather, quantum computers fundamentally take advantage of the superposition principle in quantum mechanics which states that any particle can be assumed to be in all states until and unless observed. Upon observation, the particle randomly takes one of the states. In other words, quantum computers form entangled states[2] of 0 or 1, and stay in those states, which is vastly different from staying in two states at the same time. A geometric way to think about it is to think of 0 and 1 as only the poles of a sphere, and a “qubit” as any point on that sphere. Due to a multitude of these points, data of many orders of magnitude[3] more can be stored using the same number of qubits and classical bits.

While we may still be able to crack conventional cryptographic techniques much faster, it is because of this enormous capability to store more data and not because of duality of states. The class of problems known as non deterministic polynomial complexity or NP — problems that don’t seem to have polynomial time solution-getting algorithms — will unfortunately still remain unsolved,  because quantum computers don’t so much mathematically model a problem as physically model it; in theory, we let nature do the math for us, and just watch where the final state ends up. The newfound capability to break crypto sequences wouldn’t be a problem in the long run either, because there are ways to make it even more secure using quantum cryptography. In fact, Google has already begun testing[4] such techniques. Even current state of the art research cannot guarantee that the “speed-up” on computational problems we expect from quantum computing will happen for all problems. Recently, Ewin Tang from the University of Texas at Austin proved that one of the major advances in quantum computing was redundant[5] and can be achieved by classical computing, which set back the quantum industry by decades. Add to that to the fact that we are at least a decade away from the world’s first meaningful quantum computer, and we’ve been that way for more than a decade, the picture is not so rosy anymore.

But there’s more reason to be optimistic than dismal. Intel has already created 49 and 17 qubit processor chips[6] that offer a glimpse into the enormous potential of quantum computing. They demonstrably prove that most traditional solvable problems will be solved in milliseconds, compared to minutes in the traditional way. The only major hurdle for stable quantum computing remains to achieve absolute zero-like temperatures. Qubits require temperatures 250 times colder than outer space to sustain their wave-like behavior. Attempts to recreate those environments in today’s laptops have yielded little fruit, however, the news that major companies have already started preparing for a quantum future is reason enough to be optimistic. And while we may not have quantum computers in our pocket anytime soon, watch out for each quantum of progress they make.

References

1. Morris, David Z (April 17, 2016). “Justin Trudeau Explains Quantum Computing, And the Crowd Goes Wild”. Fortune Magazine. http://fortune.com/2016/04/17/justin-trudeau-quantum-computing/

2. Beall, Abigail and Reynolds, Matt (February 16, 2018). “What are quantum computers and how do they work?”. Wired.  https://www.wired.co.uk/article/quantum-computing-explained

3. Aaronson, Scott (2008). “The Limits of Quantum” https://www.cs.virginia.edu/~robins/The_Limits_of_Quantum_Computers.pdf

4. Greenberg, Andy (July 7, 2016). “Google Tests New Crypto to Fend Off Quantum Attacks”. Wired. https://www.wired.com/2016/07/google-tests-new-crypto-chrome-fend-off-quantum-attacks/

5. Hartnett, Kevin (July 31, 2018).“Major Quantum Computing Advance Made Obsolete by Teenager”. Scientific American. https://www.quantamagazine.org/teenager-finds-classical-alternative-to-quantum-recommendation-algorithm-20180731/

6. Greenemeier, Larry (May 30, 2018). “How Close Are We—Really—to Building a Quantum Computer?” https://www.scientificamerican.com/article/how-close-are-we-really-to-building-a-quantum-computer/

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.

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)=3^{3333333....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