An Elegant LED Illustration of a Mathematical Identity

As a full-stack developer and lifelong math enthusiast, I‘m fascinated by the deep connections between mathematics and computer science. From the foundational logic of Boolean algebra that underlies all digital circuits to the high-level abstractions of category theory that inform modern functional programming paradigms, math is the lifeblood of our trade.

One area of mathematics that I find especially relevant to my work as a programmer is combinatorics, which studies the fundamental principles of counting, arrangement, and combination. Combinatorial identities and techniques arise frequently in tasks like analyzing algorithms, optimizing data structures, and solving probability problems. A solid grasp of these concepts is an essential tool in any developer‘s mental toolbox.

To deepen my own understanding and appreciation for one particularly elegant combinatorial identity, I recently challenged myself to create a dynamic LED light display that would illustrate the identity in action. Specifically, I wanted to build a physical model of the following equation:

This identity states that the sum of the first n-1 positive integers (the left side of the equation) is equal to the number of unique ways to choose 2 items from a set of n items (the right side). In mathematical notation, this quantity is often written as $\binom{n}{2}$ or "n choose 2", and is known as a binomial coefficient.

Binomial coefficients are fundamental objects in algebra and combinatorics, with deep connections to problems like counting subsets, expanding powers of sums (the Binomial Theorem), and finding paths through Pascal‘s triangle. They are defined more generally as:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

where n! denotes the factorial of n.

As a concrete example, let‘s consider the case of n=4. The identity claims that:

$1 + 2 + 3 = \binom{4}{2}$

And indeed, we can easily verify that both sides evaluate to 6. There are 6 ways to choose 2 items from {1,2,3,4}:

{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}

Combinatorially, we can think of this as the number of edges in a complete graph on 4 vertices, or the number of handshakes that would occur if 4 people each shook hands with everyone else exactly once.

To illustrate this identity in a more visceral, interactive way, I decided to construct a triangular array of LEDs, with n green LEDs along the base and red LEDs at each vertex of the "n choose 2" triangular numbers, like so:

The key insight is that each red LED corresponds uniquely to a pair of green LEDs, visually demonstrating the two-to-one relationship between the binomial coefficient and its combinatorial interpretation. More precisely, the red LED at the vertex of a triangle formed by the i-th and j-th green LEDs (counting from 0) represents the selection of that pair of items from the base set.

Turning this visual model into a dynamic LED display required translating the mathematical abstraction into concrete electronics and coding up the control logic to animate the lights in an illustrative sequence.

On the hardware side, I used an Arduino Uno microcontroller to drive the LEDs, with the standard setup of an current-limiting resistor on each LED to prevent overloading the pins. The real fun began in sketching out the software to bring the idea to life.

As a programmer, my first instinct was to consider how to efficiently map the indices of the green "selector" LEDs to the corresponding red "indicator" LED. In other words, given a pair (i,j), how do we determine which red LED should light up? My first thought was to use the combinatorial formula directly, effectively calculating $\binom{i+j-1}{1}$ to locate the red LED in row i+j-1, column 1 of a Pascal‘s triangle indexing scheme.

However, I quickly realized that there‘s a much simpler way to achieve the same mapping. Since the red LEDs are arranged in a triangular grid, we can simply start numbering them from the bottom row, going from left to right. With this indexing scheme, the red LED corresponding to green pair (i,j) is located at position j-i+(n-2) within the red LED array, as derived visually here:

Implementing this logic in code is now a straightforward matter of nested loops and clever array indexing. Here‘s the complete Arduino sketch in all its glory:

const int n = 4;  
const int greenPins[] = {2, 3, 4, 5};
const int redPins[] = {6, 7, 8, 9, 10, 11};

void setup() {
  for (int i = 0; i < n; i++) {
    pinMode(greenPins[i], OUTPUT);
  }
  for (int i = 0; i < (n * (n-1)) / 2; i++) {
    pinMode(redPins[i], OUTPUT);
  }
}

void loop() {
  for (int i = 0; i < n; i++) {
    for (int j = i+1; j < n; j++) {
      int redIndex = j - i + (n - 2); 
      digitalWrite(greenPins[i], HIGH);
      digitalWrite(greenPins[j], HIGH);
      digitalWrite(redPins[redIndex], HIGH);
      delay(500);
      digitalWrite(greenPins[i], LOW);
      digitalWrite(greenPins[j], LOW); 
      digitalWrite(redPins[redIndex], LOW);
      delay(500);
    }
  }
}

Let‘s break this down section by section. First, we define some constants: n for the size of our base set, and two arrays to store the pin numbers for the green and red LEDs respectively. Note that there are n green LEDs and $\binom{n}{2}$ = n(n-1)/2 red LEDs in total.

The setup routine is responsible for initializing each of the LED pins to output mode, enabling us to programmatically turn them on and off. We use two simple for loops to iterate over the green and red pin arrays.

The loop function is the heart of the program, containing the logic to animate the LEDs in a meaningful way. We use a pair of nested for loops to generate all $\binom{n}{2}$ combinations of green LEDs, using the outer loop variable i to track the "first" LED and the inner loop variable j for the "second" (where j always starts one position ahead of i to avoid repetition).

For each (i,j) pair, we calculate the corresponding red LED index using the formula derived above, then light up that red LED along with the two green ones. After a brief delay to make the pattern visible to human eyes, we turn all three LEDs off again and continue to the next pair.

The result is a captivating light show that cycles through all possible pairings of the green LEDs, flashing the red LED that uniquely represents each selection. It‘s mesmerizing to watch the intricate, symmetrical patterns emerge and grow more complex as n increases. For instance, here is the display in action with n=8:

https://gfycat.com/ifr/FarflungWellgroomedHammerheadbird

This whole project was a fantastic opportunity to combine my passions for math, programming, and hands-on tinkering. It challenged me to think deeply about mathematical abstractions and how to convey their meaning in concrete, interactive terms. I especially enjoyed the iterative process of prototyping and refining the LED mappings, both in my head and in code.

From a software engineering perspective, this project also reinforced some valuable lessons about the power of choosing the right data representation. By shifting from a 2D "Pascal‘s triangle" indexing scheme to a 1D "flattened" red LED array, I was able to greatly simplify the index calculation and make the code more intuitive. As a professional programmer, I‘m always on the lookout for these kinds of simplifying abstractions.

On a deeper level, this little Arduino sketch is a great example of how mathematical identities often have multiple valid interpretations that shed light on the underlying concepts from different angles. The algebraic proof of the identity offers a formal verification, while the combinatorial counting argument reveals the identity‘s meaning in terms of discrete structures. And finally, the LED display provides a tangible, dynamic illustration that ties it all together viscerally.

This multifaceted understanding is what makes mathematics so beautiful to me as a programmer. By studying these abstract relationships from many angles, we gain powerful mental models and tools that can be applied to a wide range of practical coding problems. Combinatorial identities like the one illustrated here come up all the time in areas like algorithm analysis, probability theory, and data science.

For instance, consider the task of optimizing a function that computes the Fibonacci numbers recursively. A naive implementation might use the direct recursive definition:

def fib(n):
    if n <= 1:
        return n
    else:
        return fib(n-1) + fib(n-2)

However, this approach is grossly inefficient, with a time complexity of O(2^n) due to redundant recursive calls. A much better solution is to use dynamic programming to store previously computed values. And it turns out that the number of unique subproblems that need to be solved is precisely $\binom{n}{2}$, as each pair of adjacent Fibonacci numbers forms a new subproblem. Understanding this combinatorial identity gives us a way to analyze and optimize the algorithm.

Another classic example is the binomial heap data structure, which is often used to implement priority queues efficiently. Binomial heaps are based on a clever recursive structure that maintains a collection of binomial trees, each of which is defined by the binomial coefficients. Specifically, a binomial tree of order k has $\binom{k}{i}$ nodes at depth i, and the number of distinct binomial trees of order k is the k-th Fibonacci number. Once again, combinatorial identities provide a key insight into the design and analysis of this important data structure.

As a final example, consider the famous "hat check" problem from probability theory:

n men check their hats in a restaurant, and the hats are returned randomly. What is the probability that no man receives his own hat?

It turns out that this probability is given by the beautiful formula:

$P(n) = 1 – 1 + \frac{1}{2!} – \frac{1}{3!} + \cdots + (-1)^n \frac{1}{n!}$

which is known as the derangement number Dn. And astoundingly, the derangement numbers are related to the binomial coefficients by the following identity:

$\binom{n}{0}D_0 + \binom{n}{1}D_1 + \binom{n}{2}D_2 + \cdots + \binom{n}{n}D_n = n!$

This identity can be proven algebraically using the inclusion-exclusion principle, but it also has a nice combinatorial interpretation in terms of counting permutations with fixed points. Once again, we see how combinatorial identities often tie together seemingly disparate mathematical concepts in surprising and elegant ways.

As programmers, we are uniquely positioned to appreciate and exploit these deep connections between mathematics and computation. By staying curious and playful in our explorations, we can discover new ways to use abstract mathematical structures to solve concrete coding challenges. I hope this LED illustration of the "n choose 2" identity has sparked your imagination and renewed your sense of wonder at the beautiful patterns that underlie our craft.

I encourage you to try building your own interactive visualizations of mathematical concepts, whether with LEDs, graphics, sounds, or any other medium that inspires you. The process of translating between different modes of understanding is incredibly enriching, both intellectually and creatively. And by sharing our experiments and insights with each other, we can collectively deepen our appreciation for the timeless ideas that unite mathematics and computer science. Happy tinkering!