Fibonacci Day

 

Biblioteca Ambrosiana, Milan*

November 23, which is 11/23 in month/day notation, always brings to mind the Fibonacci Numbers. I think some people even celebrate Fibonacci Day, though I haven't seen it get as much traction as Pi Day, which is March 14 or 3.14.

The Fibonacci Numbers are 1, 1, 2, 3, 5, 8, 13, 21 and so on, a sequence begun with 1, 1 and in which each succeding term is found by adding the previous two numbers. For example, the next number above would be 34, since 13+21=34.

As we'll see in a moment, this number sequence shows up in the natural world and in other places around us, which has made them popular, but first I want to spent a moment on their name and origin.

Fibonacci Memorial - Camposanto - Pisa, Italy

Because the number sequence 1, 1, 2, 3, 5, 8 . . . is called the Fibonacci Sequence, it is often thought that Fibonacci developed it. This is not true; it was known long before his time, going back well over a thousand years before his birth. Fibonacci did include a problem in his work Liber abbaci (Book of Calculating) that resulted in this number sequence. It was here that it became popularized and is why it bears his name.

Fibonacci Memorial (left-most sculpture) - Camposanto - Pisa, Italy

Despite knowing about this sequence for decades, what I didn't realize until recently is that Fibonacci's name was nearly lost in the mists of time. In fact, though he lived from about 1170AD to about 1250AD, he wasn't known as Fibonacci until 1838. His name was Leonardo, and he was from Pisa, so he was called Leonardo of Pisa (sort of like Leonardo da Vinci) or Leonardo Pisano. Fibonacci is a contraction of the term filius Bonacci ("son of Bonacci" or "of the house of Bonacci") that Leonardo refers to himself as in his book. Professor Keith Devlin, of Stanford University, went in search of Fibonacci's life and legacy beginning in about 2010, a journey that lasted about a decade and resulted in three very iteresting books: The Man of Numbers, Finding Fibonacci, and Leonardo and Steve - each of which I highly recommend if you want to learn more about about Fibonacci and the profound and ongoing impact of his work. For purposes of this post, we will not turn our attention to Fibonacci Numbers around us.

This number sequence often (though not always) shows up in the number of flower petals (or tepals).

The cala lily has 1 spathe.

The day lily had 3 petals and 3 sepals. (The three tepals with purple and yellow are petals; the other white tepals are sepals. Little did you know there would be botany terminology in this post!)

Many flowers display the "Fibonacci Five," as I like to call it. Here are a few examples:

Here we have 8 petals..

Daisies often display 13, 21, 34, or 55 petails; the daisy below has 21:

It's not just numbers of tepals that display Fibonacci Numbers but also numbers of spirals in natural objects. When I first learned this many years ago, I wasn't sure exactly what it was that I was supposed to be counting in order to find these numbers, so I'm including two explanatory photos below this pinecone, which has 8 spirals if you count the going clockwise and 13 spirals if you count them going counter-clockwise. Note that 8 and 13 are consecutive Fibonacci Numbers.

Pineapples too generally have a number of spirals that are Fibonacci. There are three spirals on a pineapple, and each of the three Fibonacci Numbers is consecutive.

And it's not just the amount of spirals that turn out to be Fiboacci. There are alsp spiral shapes that have the Fibonacci Numbers hidden in them.

The shape of the Chambered Nautilaus shell above can be modeled well by using the Fibonacci Numbers to create a grid - staritng with a square of side-length 1, then another square side-length one, and then a square next to those with side-length 2 and so on as shown below - and then spiraling out from the original square to the outermost square:

And, here are the Fibonacci Numbers 5, 8, and 13 showing up in a C-major scale:

Whatever November 23 holds in store for you, I hope it's a happy Fibonacci Day!

_______
*The picture at the top of this post is of Milan's Biblioteca Ambrosiana. In their collection is a manuscript copy of the Liber abbaci. During a visit to Italy in autumn 2024 I consulted their copy, and I found it to be a deeply moving experience to handle an 800-year-old manuscript, a book that introduced the number system we used today into Europe in a way that merchants and others could make use of it. I would rank the importance of this book near the importance of the printing press for the impact it has had in ushering in the modern age.

George Boole's 210th Birthday

Lincoln Castle - as viewed from a rooftop tour of Lincoln Cathedral

 On this day, November 2, in 1815 George Boole was born in Lincoln, England.

The appropriately named "Steep Street" in Lincoln, England

Lincoln, England

As George grew up, he attended primary school, but when his father's shoemaking business declined, there was no longer the means to provide George with a formal education. Undeterred, he began to educate himself. Early on, he had a strong interest in language. He may have had help with Latin from a local bookseller. In his early teens, he submitted to a local newspaper a poem he'd translated from Latin. His work was of such high quality that he was accused of plagiarism, such an accomplishment seeming impossible in one so young and uneducated Not long after this, he received math texts from someone who knew his father - and a calculus text from a local minister. Mathematics did not come as quickly to him as languages, and without a teacher, it took him many years to master calculus.

Boole's Academy - Lincoln, England

Finances continued to be a problem for the Boole family, and George began teaching at the age of 16 to support his family. By the age of 25, he opened a boarding school at 3 Pottergate in Lincoln (pictured above), and his family moved in with him there.

Plaque on George Boole's boarding school - Lincoln, England

Boole's Academy (in red brick) - Lincoln, England

The house in which he ran his boarding school was originally part of the cathedral enclave. It is on Pottergate (Street) between the cathedral and one of its arched medieval gates, the Pottergate, which can be seen at the right in the photo above - just down the street from his school. Below is a picture of the other side of the Pottergate.

The Pottergate - Lincoln, England

Pottergate Plaque - Lincoln, England

Lincoln Cathedral Tower as seen from 3 Pottergate

Passing Boole's house again on my way back up from the Pottergate to the cathedral, I decided to turn around and check out the view from the house. Sure enough, a tower of the cathedral is visible. Certainly, the Boole family and students, living nearly in the shadow of the cathedral, would have heard the bells throughout the day.

Lincoln Cathedral

Lincoln Cathedral Cloisters

Lincoln Cathedral

Lincoln Cathedral

The cathedral is such a central part of Boole's city of birth, and the boarding school he established was so close to the cathedral, that I spent a lot of time in and around the cathedral to get a sense of what he would have seen and heard. In his mid-thirties, Boole moved to Dublin and became the first professor of mathematics at Queen's College Cork, where he developed groundbreaking new areas of mathematics - not bad for a self-educated young man whose learning of mathematics came about slowly - but that's another story for another post.

For now:

HAPPY 210th BIRTHDAY, GEORGE BOOLE!!

Hamilton Walk

 

Me at Brougham (aka Broom) Bridge, Dublin

In ancient times, Archimedes had an "AHA moment" so astonishing that it caused him to leap out of his bath and run through the streets shouting, "EUREKA!"

In 1843, William Rowan Hamilton had a similarly amazing "AHA moment," but instead of running naked, he carved his realization into the stones of Dublin's Broom Bridge.

Hamilton Memorials at Broom Bridge, Dublin

Hamilton had long sought a means of extending the complex numbers to represent rotations in 3 dimensions. Even his young sons knew of his quest and would ask him when he came down to breakfast, "Well, Papa, can you multiply triplets?"

Hamilton Memorials at Broom Bridge, Dublin

On October 16, 1843, while walking into Dublin with his wife from their home at Dunsink Observatory, the answer suddenly came to him. In his excitement, he took out his pen knife and carved the equations into the stone of the bridge: i² = j² = k² = ijk = -1    

William Rowan Hamilton Plaque at Broom Bridge, Dublin

Hamilton's discovery supported early work on light and quantum mechanics – and today relates also to virtual reality, spacecraft navigation, and even how your phone knows which way is up.

Since 1990, there has been an annual walk where people retrace Hamilton’s steps. The group sometimes numbers 100 or more and includes everyone from school children to world-famous mathematicians, all here to honor Hamilton and his discovery of Quaternions.

Dunsink Observatory, Dublin, former home of W. R. Hamilton

Dunsink Observatory, Dublin, former home of W. R. Hamilton

Dunsink Observatory, Dublin, former home of W. R. Hamilton

Looking west from Broom Bridge, Dublin

Looking east under Broom Bridge, Dublin

Broom Bridge, Dublin

Sadly, I missed this year's event by a mere 25 days, but I was glad for the opportunity to make my own pilgrimage to the bridge and see the plaque commemorating the discovery.

Me honoring the discovery of Quaternions at Broom Bridge, Dublin

Möbius

 

Royal Observatory Göttingen - Home to Möbius's Teacher Carl Freidrich Gauss

On this day, September 26, in 1868, Augustus Ferdinand Möbius passed away.  He was a mathematician and theoretical astronomer who studied under the legendary Carl Friedrich Gauss in Göttingen, Germany.

Display at Göttingen's Museum of Mathematical Models (Möbius Band at top left)

If you’re familiar with the name Möbius, it is probably due to an object known as the Möbius Strip or Möbius Band. Here is the item as displayed at the Museum of Mathematical Models at the Mathematical Institute of Göttingen.

A Möbius Band at Göttingen's Museum of Mathematical Models

And if you’ve heard of the Möbius Band before, you probably know of its special properties. If not, you may want to make one and explore for yourself. To make a Möbius Band, follow the directions below.

STEP 1: Take a strip of paper and give it a single twist.

Möbius Band - Step 1

STEP 2: Tape the ends together. (This can be a bit difficult depending on how long or short your strip is; be patient with it.)

Möbius Band - Step 2a

Möbius Band - Step 2b

You can stop here and admire your museum-worthy work of art, OR you can move on to:

STEP 3: Explore a property of this band by cutting it lengthwise down its middle, all the way around until you come back to where you started. If you’ve never done this before, I think you’ll be surprised with the outcome.

Möbius Band - Step 3a

Möbius Band - Step 3b (Cut all the way to where you started!)

The Möbius strip appears in many places outside mathematics. In literature, we find it in Howard Nemerov's poem Creation Myth on a Möbius Band; we also find it in short stories such as Martin Gardner’s No-Sided Professor, Armin Deutsch’s A Subway Named Möbius, Mark Clifton's Star, Bright, and Arthur C. Clarke’s The Wall of Darkness. Many pieces by artist M. C. Escher make use of this shape, and the symmetry in the score of Bach’s Goldberg Variations can be thought of as having been written on a Möbius Strip.

Thanks to the mathematical mind of Augustus Ferdinand Möbius for developing this interesting object whose influence still twists through culture today.