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.

A Queen's Visit

"John Dee Performing an Experiment before Elizabeth I" Henry Gillard Glindoni (1852-1913), Public Domain

John Dee, advisor to Queen Elizabeth I, is often remembered as "The Queen’s Conjurer." Dee chose the astrologically favorable date of her coronation, advised her on calendar reform, tutored her in alchemy, trained her seafarers in navigation, and was summoned to Windsor Castle in 1577 to explain the ominous appearance of a comet.

From the Archive of the Bodleian Library MS Ashmole 1789

On September 17, 1580, Elizabeth did something unusual—she visited Dee at his own home. Despite being her trusted advisor, Dee was not a nobleman, and monarchs did not visit the homes of their subjects unless, perhaps, being formally hosted by wealthy nobles while on progress.

Detail from what remains of Richmond Palace

Dee’s house at Mortlake lay just two and a half miles from her palace at Richmond. She visited him more than once, often pausing by the wall that bordered his property and the neighboring church. Usually, she traveled by barge along the Thames, but on this day she arrived by coach.

The Thames River, Mortlake, front John Dee's former property

We know of the visit through Dee’s own hand. In the margins of his almanacs—ephemerides—he recorded the moment in detail. Elizabeth, he wrote, came by coach across the fields, turned toward his house, and when she saw him, beckoned with her hand. He approached the coach, whereupon she removed her glove for him to kiss her hand and asked that he “resort to her court.” Why not just send a messenger?

The wall between Dee's property and St. Mary the Virgin, Mortlake (note church tower in background)

That a queen should ride to the home of a commoner, no matter how scholarly, and stand at his garden wall is both surprising and telling of the value she placed on Dee.

J. E. Littlewood

 

Nevile's Court, Trinity College, Cambridge

On this day, September 6, in 1977, Cambridge mathematician J. E. Littlewood passed away. He is perhaps best remembered for his 35-year collaboration with G. H. Hardy, one of the most famous collaborations in all of mathematics. 

They created four rules governing their work, one of which was that every paper would carry both names, credit always being shared equally—whether or not both had contributed.

Their influence on British mathematics was so great that, in 1947, Harald Bohr quoted a colleague as saying: “Nowadays there are only three really great English mathematicians: Hardy, Littlewood, and Hardy–Littlewood.”

Though they lived only steps apart at Trinity College, they typically chose to collaborate by letter or through notes sent by messenger, rather than in person. From the time of his arrival until his death 65 years later, Littlewood kept the same rooms in Nevile’s quad—rooms overlooking the Wren Library on the River Cam. Given the view, who can blame him?

Nevile's Court, Trinity College, Cambridge

Staircase D, Nevile's Court, Trinity College, Cambridge

Wren Library as seen from Nevile's Court, Trinity College, Cambridge

Wren Library, Trinity College, Cambridge as seen from the River Cam

If you’d like to explore more about Littlewood, Hardy, and their sometime collaborator—the brilliant and tragic Srinivasa Ramanujan—the 2015 film The Man Who Knew Infinity offers a moving glimpse into that world.

Memorials in Trinity College, Cambridge, Chapel (Littlewood & Ramanujan)

Memorials in Trinity College, Cambridge, Chapel (Hardy)

Fazio Cardano

University of Pavia

On this day, 501 years ago, Fazio Cardano crossed the threshold into the unseen world he had courted in life. His name lingers in the shadow of his son, the great Girolamo Cardano, yet Fazio himself was no less a figure of wonder. True to the Renaissance spirit, Fazio was a man of many parts: jurist, physician, and mathematician. He taught geometry at the Piatti Foundation of Milan and lectured at the University of Pavia, within sight of that city’s towers that still rise as pictured below.

Fazio was a close friend and mathematical consultant to Leonardo da Vinci, and he translated a 13th-century book on optics by Archbishop John Peckham of Canterbury, which Leonardo used in setting up his own optical experiments. But Fazio’s story does not rest in academia alone. From his youth he was haunted—or perhaps chosen—by spirits. He once drove away a folletto, a mischievous sprite, but later kept a familiar spirit, seeking its aid in trials, and turning only in direst need to the saints. In August of 1491 he recorded a visitation by seven beings at once: demons, angels, or travelers from some other realm—who can say?

Half a millennium later, we remember Fazio Cardano—Renaissance scholar, mathematician, occultist, and companion of spirits.

Pavia, Italy

Pavia, Italy

Base of the Towers of Pavia, Italy

University of Pavia

Classroom in the University of Pavia

Another view of the classroom at the University of Pavia

The Towers of Pavia, viewed while leaving the University