Category: Mathematics

Humans are greedy—for knowledge, for power, and quite frankly, anything we can get our hands on. Our puny desire to explain the questions of the universe leaves us struggling to grasp just how unfathomable our world is. Fortunately, mathematics serves as a medium in an attempt to conceptualize such otherworldly phenomena, one being the golden ratio.  

The golden ratio can be seen everywhere and at any given moment. It is a naturally occurring mathematical phenomena that can be seen in nature to optimize space and sunlight. 

Sunflower seed arrangement exhibiting phyllotaxis, a natural pattern 

associated with the golden ratio. 

What’s really behind the golden ratio is a concept most students have been exposed to since elementary school: a sequence. More specifically, the golden ratio is based on the mathematical concept of the Fibonacci sequence. 

While the name may beg to differ, the Fibonacci sequence was in reality founded by ancient Indian mathematicians centuries before explored by western mathematicians. It is named after the Italian mathematician Leonardo Pisano (also known as Fibonacci), who introduced the number sequence to Western Europe in 1202. 

This series is a sequence of numbers where every number is the sum of the previous two numbers. For example, the very first term will begin with 0 and be followed by 1. 0 + 1 = 1, and 1 + 1 = 2. Thus, the sequence continues, 0, 1, 1, 2 … and so on. It can be written in general form as a recursive sequence such that

F_n = F_n-1 + F_n-2

_{where n = the nth term of the sequence}

As the sequence gets larger, if you divide any Fibonacci number by the one immediately before it, the answer gets closer and closer to 1.618, which is the Golden Ratio! 

If you want to find a specific Fibonacci number without adding up all the numbers before it, there’s a formula for that too—Binet’s Formula. 

F_n = [ɸⁿ – ( –ɸ)^–n ] / √5 

where ɸ = 1.618… (the Golden Ratio) or more accurately ɸ = (1 +  √5) / 2. 

This sequence can also be drawn into a spiral. First start with the squares of the numbers in the sequence (0² = 0, 1² = 1, 1² = 1, 2² = 4). These numbers will all make small rectangles side by side. Then, you can link opposite corners with a curved line, making a spiral.

This spiral (and its mathematical counterpart—the logarithmic golden spiral) is used mathematically to study fractal geometry, self-similarity, approximation limits, and polar coordinate systems. While artists value its beauty, mathematicians use it as a foundational tool to understand growth patterns where a shape scales up without changing its core proportions. In example, it can be used to calculate a hawk diving at its prey at a visually appealing angle while keeping it proportional to the lens. 

Another application of the Fibonacci sequence is in the stock market. The Fibonacci sequence is used in technical analysis to identify potential support and resistance levels during price pullbacks. Traders use specific ratios derived from the sequence (most notably 23.6%, 38.2%, 50%, and 61.8%) to determine optimal entry points, stop-loss orders, and profit targets. The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.) has unique mathematical properties, where each number is the sum of the two preceding numbers. In stock trading, these numbers are converted into key percentages representing how much a stock’s price might “retrace” before continuing its overarching trend. Below is a breakdown of these key percentages: 

38.2% & 61.8%: Considered the “golden ratios” of trading. These levels indicate areas where an asset is likely to bounce off its lows or find resistance during a rally.

50%: Though technically not a true Fibonacci number, it is universally accepted in technical analysis (inspired by Dow Theory) as a primary pivot point.

23.6%: Indicates a very short-term or shallow retracement, often watched in highly trending stocks.

In the depressing trenches of AP studying, your eyes fixate on a half-opened bag of chips right across the living room. Naturally, you decide to move towards the chips with purpose. It seems simple—but according to Zeno of Elea, you might never actually reach it. 

Of course, nothing should come between you and your chips. It turns out Zeno isn’t some random, but an incredibly famous Greek philosopher, well-known for proposing various interesting and mind-boggling paradoxes. Zeno was a student of Parmenides and because of that, many tend to believe his paradoxes were meant to defend his teacher’s idea of an unchanging reality. However, this interpretation mostly comes from later speculations, including those from Plato’s dialogues. Some of his paradoxes include The Antinomy of Large and Small, The Antinomy of Limited and Unlimited, and The Paradoxes of Motion—one of which is the Dichotomy Paradox. 

The Dichotomy Paradox is explained by Zeno as follows: let’s say a runner intends to meet a goal. If the goal is one meter away, the runner must cover a distance of ½ meter, then ¼ meter, then ⅛  meter, and so on ad infinitum. Because this process continues indefinitely, Zeno argues that the runner can never reach the final goal. To expand, the regressive version of the Dichotomy Paradox states that the runner can’t even take the first step because any step may be divided conceptually into a first half and a second half. Before taking a full step, the runner must take a ½ step, but before that, he must take a ¼ step, but before that, a ⅛ step, and so forth ad infinitum. Seems convincing, right? 

Well, no, not really. In fact, this paradox has been resolved in both math and physics. 

To begin, let’s envision the runner with an impending goal of one meter. Zeno breaks this one meter into ½ meter, ¼ meter, ⅛ meter, and so on forever. To express the total distance traveled:

Total Distance Traveled = ½ + ¼ + ⅛ + … 

By a convergent geometric series, we see that the total distance equals exactly one meter. Hint: Notice how the sum of all the individual boxes still results in the whole box. See, even though the runner is completing infinitely many subdivisions, the total distance still adds up to a finite amount. 

Unfortunately for us, mathematics alone isn’t enough to provide a full solution. To fully resolve this paradox, we need to realize that this paradox isn’t simply about dividing infinite parts, but the physical concept of a rate. Zeno’s paradox feels convincing because it only takes into account distance, without factoring in time. Motion isn’t limited to how far one moves, but how far one goes in a select amount of time. “The reason objects can move from one location to another (i.e., travel a finite distance) in a finite amount of time is because their velocities are not only always finite, but because they do not change in time unless acted upon by an outside force” (Siegel 2020). 

There is another detail of the Dichotomy that needs resolution. How does Zeno’s runner complete the trip if there is no final step or last member of the infinite sequence of steps (intervals and goals)? During the process of “taking a trip,” can there be an absence of the crucial “last step”?  The Standard Solution answers “no,” while the intuitive answer “yes,” held by Zeno, Aristotle, and the average person today, must be rejected when embracing the Standard Solution. Even if there is no “last step,” the runner can still finish the journey because completion stems from the limit of infinitely many steps, not the final step itself. 

Now, unfortunately for Zeno, you can confidently say you made it across the room and got the chips—no paradox stopping you.

Authored by Chandhana Lingam Muhilan and Katie Huang