A Retrospective

Me

A Retrospective

The 14^th of February, 2013

As I look back over the past century, I cannot help but be proud. It has been exactly seventy years since my death, but so much has happened. I had lost faith towards the end of my life. Upon my death, the Nazis had restaffed the university and removed many of my Jewish colleagues, many of whom were never seen again. But we shall not cry because they are gone, but rather celebrate their accomplishments. I myself am agnostic. However I believe that mathematical truth is independent of the existence of God. Being at the center of the axis in World War II, I had lost faith in an end to the terror caused by the Nazis. Luckily two years after my death, the terror ended. Over the next fifty or so years the world was revolutionized by advancements in mathematics and technology. Moreover, much of my life’s work is still used today, a very humbling fact.

One of my proudest moments was the presentation of my twenty-three problems, and looking back on it, I believe they had a very large impact. Many mathematicians have pondered my problems, and a good lot still are unsolved. I’m hopeful that these problems will guide the next one hundred years as well. So far, only ten of the problems have been solved, and the solutions are nothing short of genius. As I watched the solutions unfold, I struggled to follow along. I fear the newer generations of mathematicians may be smarter than I. A few of my problems (most notably the first, which regards infinite sets and the continuum hypothesis) have been partially solved. In regards to the continuum hypothesis, two brilliant gentlemen by the names of Ernst Zermelo and Abraham Fraenkel proved that this problem is impossible to prove or disprove! What a conundrum!

I was also amazed by the advances made in science over the past seventy years. In this short amount of time, humans have traveled into space, and even been to the moon! Speaking of space, Hilbert Space (a geometrical concept which I pioneered) is still used frequently today. Mathematicians find it an advantage to use Hilbert Space because it allows for infinite dimensions. This lack of parameters acts as a creative wonderland for mathematicians and makes it a more favorable theoretical space for mathematicians to work. My lasting impact on the mathematic community, and my inspiration of future mathematics, is a treasure.

Upon reflection, for I have had a lot of time to think, I wish I had led a more humble life. Although my problems and concepts are still prevalent, the name Hilbert is seldom uttered outside of symposiums and graduate level lectures. In hindsight, possibly being more humble would’ve awarded me with timelessness, like my colleague Albert Einstein. I will end this with the same words inscribed on my tombstone, words which I think should govern discovery and creativity for as long as the sun burns. Wir müssen wissen. Wir werden wissen.

We must know.

We will know.

My Tomb

Letter #3

Fibonacci

September 3rd, 3000

Dear Leonardo,

I have laid this letter upon your grave. I know the Gods will deliver it to the underworld safely. I have written to thank you for your many contributions to the world as it is today. You may not know who I am, so I shall tell you. You may remember Abdul, a Muslim man from Syria, who taught you how to use the decimal system, which you then introduced to the greater Europe. I am his grandson, Farhad. My grandfather was incredibly fond of you.

 As a young child, he would sit me on his lap and tell me stories of your so called “math adventures” together. He told me that one-day, as you two were discussing roman numerals, Abdul asked if you, a most intelligent scholar, had heard of the decimal system. He was surprised to learn that you had not. Abdul always laughed about the inquisitive look you gave him after hearing the word, “decimal.” However, you were quick to learn the math system, which you obviously know was based on ten digits with a decimal point and a symbol for zero. Abdul says you soon fell in “mathematical” love with this method, realizing how hard it was to perform operations with roman numerals. You soon brought this system to Europe in your famed book, “Liber Abaci”, convincing many European mathematicians to adopt this system. You definitely simplified things for us merchants, Leonardo!

Another most famous contribution you made of the Fibonacci sequence also made a huge impact on the world, as we know it today. My grandfather told me you discovered it when studying the number of rabbits in subsequent generations and the number of ancestors in consecutive generations of bees. The sequence also appears in plants, such as the daisy head, pineapple, and pinecone.

You provided a valuable link between math and the natural world. Leonardo, you proved that math is an instrument for understanding rather than something that is removed from the natural world. For this I thank you. Before the Fibonacci sequence, I knew many people who scoffed at math’s relation to nature; you proved them wrong. Math is incredibly significant to daily life, not just trade, and helps us understand how the world functions.  

            I know the underworld has placed you in a specific mathematical sect in which you solve mathematic equations everyday J. We all miss you. Within this letter I have enclosed a page of a new copy of your book “Liber Abaci” which is fresh off the print. Hope you like the new graphic. It’s a sunflower whose petal pattern occurs by adding the previous number of petals to the current one and repeating the cycle, thus producing a spiral.

Rest in Peace, Leonardo.

Sincerely,

Farhad Rafael.

Best Wishes,

(3)

Abdul Rafael

A Melancholy Day for Art and Mathematics

Dear George Arnold Escher,             March 30, 1972

It pains me greatly to report the passing of one of our century's most well liked artist, respectable mathematician, loving husband, and for you, your humble son. I hope that you are well, but am saddened to inform you of his death, for he fell ill and couldn't conquer this sickness that his doctors called cancer. I must say that even after decades of marriage, at the closure to his life, he secluded himself greatly from his friends and even our family. This aspect saddens me more, but from the mumbles in his sleep, I could tell that he was attempting to grapple with the pain of his illness. I know that you, or Sarah Escher, have not seen your son for some time, but I am so proud to have called Maurits my husband. Can you believe the career your little Mauk had, I knew he cherished that nickname, ranging from impossible perspective pieces, to incorporating geometry, and fiddling with the two dimensional plane? His art truly stands on it's own, but I feel so pleased that I went on this

Drawing Hands, my 2nd favorite piece.

artistic journey with him; I saw him marvel at Alhambra, and frown at the landscapes in the Italian countryside, thinking about what he could do to blow the viewer's mind with a seemingly weird viewpoint. From his tiny charcoal sketches, to his intricate wood works, and everything in-between, I have watched Escher grow, and intensify his technique and care for his work. I wonder if he has always been so pessimistic about the quality of his art, perhaps it started when he was a young boy, but until his death, he continued to fix and re-make his last piece of art. It is called Snakes and he printed it tirelessly and continually for the last years of his life. I have to say that it is my favorite piece, with my

My husbands last piece, "Snakes".

second favorite being Drawing Hands. In addition to his art, I find it crazy that we have lived through two heartbreaking world wars. We moved as a family multiple times due to Escher's desire to keep our family safe and in a war-free zone, and he did try to stay out of all of the politics and opinions about the wars. It seemed as though all of the chaos from the wars forced him to look inward, and find the strength to finish his art pieces. I feel so lucky to have been married to this wonderful man, and am so proud that in the end, he published his notebook. He swore years ago that the book was solely for his own reference, and not to be shown to others. Escher had a change in heart when he realized that his math could help others, and it would further the connection between mathematics and art. He has given me an abundance of things to be proud of, and although I am very sad currently, I am so appreciative for the life he gave me. In addition, I am sure that within the coming years, the true amazement and appreciation for my love's work will become known. I hope that his techniques will soon be marveled at not only from an artistic point of view, but also from a highly technical mathematical perspective. Our sons and I have decided to feature some of his works in the Hague museum, and are considering selling some to Cordon Art, but I will leave my children with that decision.

What your son has done with his life is truly remarkable, and although he didn't follow your desired career path, I hope that you are just as proud of him as I am. I send my condolences for the loss of your son as well.

Sincerely his loving wife,

Jetta Umiker

Journal Entry 351
February 13, 1943

    What a full life I have lived. It seems like almost yesterday when my mother enrolled me in gymnasium, only to find that I hated it. I recall convincing her to let me transfer to one more oriented towards math and sciences. It was immediately after graduating that my true mathematics career began. I attended the local university and was finally surrounded by people who shared my interests. I thrived at the University of Königsberg. I befriended a man named Hermann Minkowski, a gentleman who I had no idea would become my lifelong colleague. I recall college only became better when Adolf Hurwitz, an associate professor, arrived from Göttingen in 1884. The three of us would conduct scientific conversations on the highest level, and it was thrilling.

    I also am pleased with how my contributions have affected other fields, such as physics. Up until about 1912, my focus was solely on mathematics. You could say I was somewhat of a purist, with no regard for any other field. However on a trip to Bonn, I began to take a liking to physics. Actually for as long as I can remember, all of my work with physics is because of my friend Hermann. Our work pertaining to physics before 1912 was because of him, and it was he who first got me interested in it. After he passed away in 1909 due to an appendicitis, my focus shifted entirely to physics. Having lost my best friend made me reassess my life, and I decided it was time for a change, in honor of Hermann.

    I decided to get myself a personal physics touter, and after learning a great deal of the foundations of physics, I began to experiment with general relativity. My goal was to create the axiomatic derivation of field equations (the equations which govern general relativity). I actually had the privilege of working alongside Albert Einstein (or rather he had the honor of working with me). I had invited him to come and speak at the college when Einstein first found out that I too was working on field equations for general relativity. I could tell his effort increased significantly after this discovery. I could sense his apprehension as I shook his hand prior to his departure after the lecture. That November, we both published papers on the field equations of gravity, but I generously gave Einstein all of the credit. My work also influenced the field of quantum mechanics. My work helped support the work of well known scientists like Werner Heisenberg and Erwin Schrödinger. I tried to make physics less “sloppy” and more rigidly governed, to help prolong the legacy of my good friend Hermann.

A Picture of My Great Friend Hermann Minkowski

Math & Art: The Unbeatable Duo

Dear H.S.M. Coxeter,                            June 2, 1959                                                                                  

This is my tile artwork Circle Limit III

I cannot fully express the delight and honor I am experiencing right now, after reading your paper about my sketches. I am so very excited that you see the mathematical perfection within my pieces of art. I am finding myself involving myself more into this realm of mathematics, even though as a child I took no interest in the subject. My brother, Berend sparked my interest in mathematics when he sent me one of Póyla's, the hungarian mathematician, papers from 1924, and now I am thoroughly engrossed in this topic. Even though I have heard that your paper was published four years prior, it has taken due time to reach me. Only fourteen years ago I began to record my thoughts in my notebook and now, wow, I have entered into a whole new phase in my career where I am working with my notes, and translating them into my artwork. In my notebook, that I have decided to name The Regular Division of the Plane with Asymmetric Congruent Polygons, I am attempting to re-create a perspective of infinity on a two dimensional plane. I have also designed a system to categorize shapes so that I can use them in my wood cuttings and sketches. I was so inspired by your paper that depicted how hyperbolic tessellations can be formed using only the diagrams on the paper, and have decided to create my own rules and laws around mathematics in art. The system is quite simple, and I believe it would interest you greatly, so I'll briefly describe it here to you. 
I have split up shapes into two different categories, quadrilaterals, and equilateral triangles, on regular tessellations. From quadrilaterals it branches into variables that define that individual quadrilateral, the type of polygon, and the symmetries in the tessellation. I have assigned a letter to each of the five polygons as seen below.

        A - Parallelogram
B - Rhombus
        C - Rectangle
D - Square
        E - Isosceles Right Triangle

        In my notebook I have created a graph that shows all of the possible symmetries, describing whether the shapes make translations, rotations, glide reflections or a combination of all three. I plan on creating art that represents each shape with the symmetries. Although I have viewed and studied many Moorish sketches, I will make sure that my art doesn't resemble their style, because they refuse to include symbols and shapes from real life. I find it ridiculous, for, how should an artist expect their viewer to relate, if the shapes are not identifiable? I have been, and will continue to work from 8-4 throughout the day until I complete the pieces that include the work from my notebook. I look forward to conversing with you at a future date, and hearing your thoughts on my new findings. In addition, in a few years I strive to have art to show you that represents all of my notes in the book. I neither hope, or wish, to publish my notebook, but will allow you the honor to read it due to our growing friendship. In the future, I hope that our life paths lead to one another.
My best thoughts,
M.C. Escher

Letter #2

Leonardo Fibonacci

July 24, 1204

Dearest Frederick,

I hope you are doing well.  I know the Roman Empire is a bit chaotic these days, mostly fighting on the outskirts of Turkey. Yet, I know you possess the skills to bring peace to a realm renowned for its advancement in technology, education, and organized government. My friend Abdul just told me of your most recent experiment. How did you decide to lock ten convicts in a room so small they suffocated, and then open the door to see if their souls came out? You have always been so creative, my dearest Frederick. I know the life of an emperor is most very busy, but I have written to you to share a new discovery. I last saw you at court in Florence in about 1203, where we were involved in a competition of algebraic equations, which I quickly won. I have continued to develop my mathematic theories ever since. As you know I wrote the book, Liber Abaci, two years ago. However, I do not think you have read this particular problem, which I have begun to dissect and answer completely.

           
 “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive?”

The sequence is

1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

After much study and deliberation of the rabbits, I now know that each group of rabbits’ born is the sum of the two preceding groups of rabbits’ born. I find this sequence very interesting, and have begun to apply it to nature. For example, did you know that most daisies have 34, 55, or 89 petals? I’m sure you quickly realized that 34+55 is 89. I plan to do much work exploring this sequence, and will share my thoughts with you as soon as possible. However, dearest Freddy, I knew you would appreciate an update from your closest math scholar.

            Please send my regards to your beloved wife Penelope and two children. Henry and Jack were always so athletic and skilled at archery; what fine boys they must be by now. I would also love it if you would make a few copies of the text Liber Abaci, and deliver it around the empire. Many mathematicians across the empire are sending me requests of shipment at least once a fortnight. Within this letter, I have enclosed a painting of me, since you have always wished for one to hang in the section of your library, which houses the math texts. Hope all is well.

Many Thanks.

Sincerely,

Fibonacci

(2)

Letter # 1

Leonardo Fibonacci
May 2nd, 1200

Dear Bonaccio,

                        I am currently in Syria, having just left Algeria by camel. It is so hot here. I wear my toga everywhere; the white-stained fabric can barely conceal my skin against the burning orange sun. Thankfully I have left the nickname of Bigollone behind, I am no longer known as the “dunce” or “blockhead.”  My new friends just call me Leonardo, but I miss the joking way in which you used to yell at me “Bigollone, finish your systems of equations and sell these items!”

                        Father, I have something to confess. I know you have always dreamt of the day I become a merchant like you, the day I take over the position of the official trader in Algeria, but I have decided on a different profession. I know this will not disappoint you, as you have stuck by me faithfully all my life. After spending time in the bazaars here, I have become fascinated by the way the skilled merchants, almost as skilled as you, calculate prizes and averages of all items. The process of selling does not interest me but the process of calculation never ceases to. Father, I have decided to become a mathematician. It is my passion in life, and you know that numbers have always fascinated me. I hope you are not disappointed in the path I have chosen.  

                    Where I reside in Damascus, there are mostly Muslims, which is perfectly fine with me. I find Moorish math fascinating, and have already begun to study some of their systems. The more time I spend in this part of the Byzantine Empire, the more gratitude I feel for you.  If you had not encouraged me to travel so freely around the empire, a privilege of being a merchant, I would not have been able to visit all the centers of trade; for this, I thank you. Within these centers I have learned the mathematics of scholars. A friend of mine named Mohammed is a vendor and calculates the prices and value of all items east of the Nile, and I have begun to learn the calculating schemes in popular use here. A calculation technique I have learned from my other friend Abdul has proved vital to studies here. This new system contains ten digits, a zero, and fraction units. It will soon replace the European style of Roman numerals, which do not possess the property of zero, or a non-existent amount, yet can be used as a place holder in value systems. Abdul calls it the “decimal system”. I plan to write a book on the system called “Liber abaci” which will explain the rules of adding, subtracting, and multiplying different numbers. I know it sounds difficult, but you have taught me to persevere and I know it will be completed by at least 1202. 

Hope all is well in Algeria. Much love.

1

Sincerely,

Your son Leonardo

P.S. I have included a map of Syria and all the major trading routes, which I know you will find interesting.  Damascus needs your skills J