Bringing an XKCD Comic to Life: My Fascinating Journey

Bringing an XKCD Comic to Life: My Fascinating Journey

Table of Contents:

1. Introduction
2. The Ancient Egyptian Way of Looking at Fractions
3. Fibonacci and Egyptian Fractions
4. Greedy Approach to Egyptian Fractions
5. Is Greed Always Good?
6. Paul Erdesh and Ron Graham
7. Collaborating on Egyptian Fractions
8. Solving the Problem Decades Later
9. Heritage Numbers and Mathematicians
10. The Story of the Rising Paul Erdesh
11. Conclusion

The Story of Ancient Egyptian Fractions and the Rising Paul Erdesh

Introduction

Ancient Egyptian Way of Looking at Fractions

Fibonacci and Egyptian Fractions

Greedy Approach to Egyptian Fractions

Is Greed Always Good?

Paul Erdesh and Ron Graham: Two Mathematician Friends

Collaborating on Egyptian Fractions

Solving the Problem Decades Later

Heritage Numbers and Mathematicians

The Story of the Rising Paul Erdesh

Conclusion

Introduction

Today, I want to talk about how problems from thousands of years ago still echo in our time. This is a story of how I partially brought an XKCD comic to life. XKCD comics are known for their STEM-focused humor, created by a former NASA engineer. In 2009, XKCD posted a strip titled "Apocalypse," where the skies burned, the seas turned to blood, and the dead walked the earth. Although I did not cause any of those things, nor did mathematicians, I found myself fitting into the strip in unexpected ways. To understand how it all connects, we need to go back to ancient Egypt and explore the fascinating world of fractions.

Ancient Egyptian Way of Looking at Fractions

In ancient Egypt, mathematics held immense significance. The Rhine Papyrus, the oldest existing mathematical document, provides insights into how they approached mathematics and the problems they encountered. One aspect that stands out is their unique way of looking at fractions. They were particularly fond of fractions that had a "1" as the numerator, such as 1/7, 1/137, and 1/19. These were considered great fractions. On the other hand, fractions without a "1" as the numerator, like 4/7, 9/29, and 5/111, were frowned upon. Ancient Egyptians sought to work with only their preferred fractions and found creative ways to handle the fractions they disliked.

Fibonacci and Egyptian Fractions

Leap forward a few thousand years to the time of Leonardo de Pisa, also known as Fibonacci. Fibonacci's book, "Liber Abaci," revolutionized mathematics by making it accessible to all. One of his notable contributions is the Fibonacci numbers sequence. Fibonacci discovered that any fraction could be rewritten as a sum of Egyptian fractions with a "1" as the numerator. The key to achieving this was to be greedy in the selection process. By starting with the largest suitable fraction and repeatedly replacing it with smaller ones, any fraction could be expressed in terms of Egyptian fractions.

Greedy Approach to Egyptian Fractions

The greedy approach to Egyptian fractions involves finding the largest fraction with a "1" as the numerator and then iteratively replacing it with smaller fractions until the desired fraction is obtained. For example, if we take the fraction 9/29, the largest suitable fraction below it is 1/4. Thus, we have 9/29 = 1/4 + 7/116. Continuing this process, we replace 7/116 with 1/17 and 3/1972. Finally, after another iteration, we obtain 9/29 = 1/4 + 1/17 + 3/1972 + 1/648,788.

While this approach may become unwieldy for complex fractions, it allows us to express any fraction as a sum of Egyptian fractions. However, it raises questions about the convenience and optimality of the Egyptian fraction system.

Is Greed Always Good?

Although the greedy approach offers a method to represent any fraction as a sum of Egyptian fractions, it challenges the notion that greed is always good. The Egyptian fraction system, while unique, may not be the most efficient or straightforward way to express fractions. Modern mathematicians have evaluated the system and debated the advantages and disadvantages it presents. The conveniences of the system, such as the guaranteed representation of any fraction, coexist with complexities and challenges.

Paul Erdesh and Ron Graham: Two Mathematician Friends

Now, let's shift our focus to two key individuals in our story: Paul Erdesh and Ron Graham. Paul Erdesh, born in 1913, was a renowned mathematician from Hungary. He possessed exceptional mathematical talent and eccentricity that made him stand out. With a burning passion for mathematics, Erdesh lived an itinerant life, collaborating with numerous mathematicians while constantly on the move. On the other hand, Ron Graham, born in 1935, displayed a different personality. He was athletic, charismatic, and well-versed in various fields, including mathematics.

Despite their differences, Erdesh and Graham shared a deep interest in problems related to Egyptian fractions. Their collaboration, which spanned several decades, resulted in numerous papers and an unbreakable friendship. Graham even took over the accounting responsibilities for Erdesh, as they worked tirelessly on mathematical problems together.

Collaborating on Egyptian Fractions

Erdesh and Graham's collaboration extended beyond publishing papers. They engaged in a vast array of unsolved problems, some of which remained unfinished due to Erdesh's passing. However, Graham carried their shared ambition forward and sought to complete those unfinished problems.

Decades after the inception of their collaboration, I had the opportunity to meet Ron Graham and work alongside him. Drawn by our mutual passion for mathematics, we became friends and embarked on a journey to finish a problem that Erdesh and Graham had started together.

Solving the Problem Decades Later

With the aid of modern tools and techniques, we delved into the problem left unfinished by Erdesh and Graham. The focus was on Egyptian fractions with a specific form of denominators composed of three prime numbers multiplied together. The question at hand revolved around whether it was possible to express any whole number using these fractions. After extensive research and calculations, we finally arrived at the solution—an affirmative answer. It is indeed possible to represent any whole number as a combination of these specially crafted Egyptian fractions.

The completion of this problem marked a significant milestone in our mathematical journey. It demonstrated the endurance of unsolved problems over time and the potential for future discoveries in the field of mathematics.

Heritage Numbers and Mathematicians

In the world of mathematics, heritage numbers serve as a measure of the distance between mathematicians. The concept is comparable to the famous "Six Degrees of Kevin Bacon" in the entertainment industry. For mathematicians, the smaller the heritage number, the closer they are to Erdesh in terms of collaborations and connections.

Heritage numbers are determined by the number of steps required to connect one mathematician to Erdesh through joint papers. For instance, Albert Einstein, who published a paper with Ernst Strauss, has a heritage number of two. However, only 512 individuals, including myself, hold the esteemed heritage number of one—the closest possible connection to Erdesh.

The Story of the Rising Paul Erdesh

Returning to the XKCD comic about the apocalypse, where the dead rise from their graves, we find a mathematician responding in a rather logical manner: by doing more math. The comic humorously depicts mathematicians seeking Erdesh's signature, even after his passing, to secure that coveted heritage number of one. Although I achieved this distinction nearly 20 years after Erdesh's passing, it symbolizes the enduring influence and legacy of this remarkable mathematician.

Conclusion

The intertwining stories of ancient Egyptian fractions and the collaboration between Paul Erdesh and Ron Graham impart valuable lessons about the progress and mysteries of mathematics. From the unconventional methods of the ancient Egyptians to the modern-day pursuit of unsolved problems, the exploration of fractions and their representations continues to fascinate mathematicians. The rising Paul Erdesh, even after death, serves as a testament to the enduring impact mathematicians leave on the field. As one problem is solved, numerous others emerge, beckoning future mathematicians to unravel their secrets.