A Mathematical History Tour: Milan

Sunday, November 23, 2025

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 an octave on a keyboard:

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

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*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.

Monday, July 15, 2024

Da Vinci and his Math Consultants: MILAN

Sforza Castle, Milan, Italy

Leonardo da Vinci worked for various patrons in various cities over the course of his life. One of these patrons was Ludovico Sforza, and one of these cities was Milan, where Sforza ruled as duke. The pictures above and below are of Sforza Castle.

Sforza Castle, Milan, Italy

Leonardo (1452-1519) first worked in Milan from 1482 to 1499 and then again from 1508 to 1513. And though his patron was Ludovico Sforza, he was free to take commissions from others while there. One of these was the painting The Virgin of the Rocks (the one that is now in the Louvre).

Madonna of the Rocks, Leonardo da Vinci (Louvre)

Another commission was that of The Last Supper, which he painted for the monastery of Santa Maria della Grazie.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

Santa Maria della Grazie, Milan, Italy

Before moving on to Leonardo's mathematical consultants, I want to share one more of his works that is in Milan - one I had not heard of before traveling here. This one is found inside the Sforza Castle. It's in a room, or I should say, "IS the room" of Sala della Asse.

Salla della Asse, Castello Sforzezco, Milan, Italy

It's hard to make out, but this fresco on the walls and ceiling are of intertwining plants and trees and is intended to make the room seem to be an outdoor setting rather than an interior room. The intent was for it to look like a grove of mulberry trees. It doesn't help that my pictures were taken over and around barriers blocking the opening to the room which is undergoing renovations. The painting dates to about 1498.

Salla della Asse, Castello Sforzezco, Milan, Italy

While in Milan, Leonardo met mathematician Luca Pacioli. Pacioli had been invited to work in Milan by Duke Ludovico Sforza in 1497, but Leonard already know of Pacioli and had purchased his book Summa arithmetica in 1495 - one of the most expensive books in Leonardo's library. Pacioli became a geometry tutor and collaborator to Leonardo. The collaboration was extensive, and though they were forced to leave Milan in 1499 due to war with France, their paths did not separate until 1506.

Portrait of Luca Pacioli (attributed to Jacopo de' Barbara - source: Wikipedia - Public Domain)

Pacioli was born in Sansepolcro in the Republic of Florence in about 1447 (making him 5 years Leonardo's senior). As well as being a mathematician, he was also a Franciscan friar and is considered the father of accounting and bookkeeping. In the mid-1460s, Pacioli lived in Venice where he worked as a mathematics tutor. It was during this time that he wrote his first book, a book of arithmetic for the boys he was tutoring. About 10 years later, he moved to Perugia. He continued to work as a private tutor there until become the first chair of mathematics at the university there. One of Pacioli's books, Divina proportione (Divine proportion) was completed during his time in Milan (1498), and it was illustrated by Leonardo da Vinci. I don't know of any other mathematicians who can claim the honor of having one of their books illustrated by da Vinci!

Illustrations by Leonardo da Vinci, from Luca Pacioli's Divina proportione, 1509 edition (Public Domain - Wikipedia)

First printed illustration of a rhombicuboctahedron, Leonardo da Vinci, published in Divinina proportione (Public Domain - Wikipedia)

Another of Leonardo's mathematical consultants was Fazio Cardano, father of the more famous Girolamo Cardano. Fazio and Leonardo met early in Leonardo's time in Milan, and Fazio may have been Leonardo's first tutor in the sciences. We know that in 1490 Leonardo traveled from Milan to nearby Pavia where Fazio was professor of mathematics.

University of Pavia

Towers just outside the historic University of Pavia buildings

This trip resulted in Leonardo's famous Vitruvian Man, which he completed later back in Milan.

Vitruvian Man by Leonardo da Vinci (Public Domain - retrieved from Wikipedia)

One topic of conversation between Leonardo and Fazio was that of optics and perspective. Fazio had edited the first printed edition of a book the topic of optics, Perspectivo communis by John Peckham (1230-1292), which was the first printed book that included diagrams (beating out even Euclid's Elements for that distinction). Leonardo mentions Fazio Cardano in his notebooks many times, especially in the form of reminders to himself to ask for one or another book on proportions.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

Circling back around to Leonardo's Last Supper, we see his use of perspective as the room in which the supper is being held recedes into the distance. The painting is on the wall of the refectory where the monks of the day would have eaten. The perspective of the painting would make it look as if the monks were joining the disciples in the Last Supper.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

Young Girolamo Cardano, who would grow up to be one of the most prominent mathematicians of the 16th century, was used by his father Fazio as a bit of pack mule - having him carry his bags as Fazio went about his business. This included visits to Leonardo. As a teenager, Girolamo Cardano met Leonardo da Vinci and saw the Last Supper.

Leonardo da Vinci's Last Supper, Santa Maria della Grazie, Milan

In one of my readings on Girolamo Cardano (and sadly, I can't remember which one) he comments that when he visited the Last Supper somewhat later in life that Leonardo's masterpiece was already disintegrating mere decades after he painted it. It is on Girolamo Cardano that my travels in northern Italy are focused, but it's pretty cool that I get to "meet" other folks along the way - other folks such as Pacioli, Fazio, and, not least, Leonardo da Vinci.

Oh, and before concluding, I should mention that a significant portion of Leonardo da Vinci's notes - the Codex Atlanticus - reside in Milan's Biblioteca Ambrosiana. Here you can find a room dedicated to this codex and with pages from it on display.

Biblioteca Ambrosiana, Milan

Detail of da Vinci notes from Codex Atlanticus, Biblioteca Ambrosiana, Milan

Just before getting to the room containin the codex, there is a room containing paintings by da Vinci and early copies of da Vinci works. Below are a couple of examples.

Biblioteca Ambrosiana, Milan