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

John Napier Promptuary

John Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

This spring was the first time my mathematical travels took me to Spain. The National Archaeological Museum of Spain, in Madrid, is home to the only known period example of John Napier's Promptuary. I've been focused on John Napier and his calculating devices for quite a few years, but I feared I would never have opportunity to see this one. I'm so glad this visit worked out. 

Drawers in the John Napier Promptuary

The drawers of the chest contain two types of strips for carrying out multiplication through horizontal and vertical placement. The thicker strips (central open drawer above) are covered in digits and form a vertical base when multiplying. The thinner strips (open drawer to the left) are laid horizontally atop the others. These thinner strips contain perforations that reveal digits on the thicker strips below them. I've included a picture of the set up below, but it's blurry due to how far away from me these strips were in the glass case and due to the darkness of the room, but hopefully you can get the idea.

Napier's Promptuary - multiplication using the strips

Results of the multiplication process are read diagonally down from right to left, any necessary carries being made diagonal to diagonal. This device is derived from Napier's earlier calculating device known as "Napier's Bones" or "Napier's Rod." Calculating can be carried out more quickly with this device than with the bones or rods. A couple of drawbacks to this device in comparison to the rods (bones) is that it is more difficult to create, and it cannot be used for operations other than multiplication.

Napier Promptuary - full display - National Archaeological Museum of Spain (Madrid)

Thankfully, this chest includes not only the strips described above but also a set of Napier's Bones (bottom left in the photo above). The bones were a widely used calculating device for hundreds of years and can be used in carrying out multiplication, division, and the taking of square and cube roots.

Napier's Bones - National Archaeological Museum of Spain (Madrid)

Napier's Bones - National Archaeological Museum of Spain (Madrid)

Napier's Bones - National Archaeological Museum of Spain (Madrid)

When I think of this Napier display in Madrid's Archaeological Museum, the phrase that comes to mind for me is "mathematician's toybox." There is so much more to this chest than meets the eye at first glance. 

Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

At the very top of the chest is a lid that's been left askew for easier viewing. This is the spot in which to tuck away Napier's Bones when you are done with your calculations.

Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

At the base of the chest is a large drawer on which calculations can be carried out. To the right we see the multiplication strips of the Promptuary. On the left we see a plate listing powers of the digits starting with squares and going up to the seventh power. Below that reference plate is the square root plate for Napier's Bones. (If you flip that plate over, you'll find what you need for cube roots.)

Exponent Reference Plate and Napier's Bones Root Plate

Even the space on the interiors of the chest doors is not wasted. There are reference charts here too. On the left-hand door there is another reference chart for exponentiation of digits. Sadly, the right-hand door (third picture below) is damaged, and it was too hard for me to decipher. I think the chart on this side has to do with numbers relating to geometric figures.

Detail of left-hand door of Napier's Promptuary

Detail of left-hand door of Napier's Promptuary

Detail of right-hand door of Napier's Promptuary

Below are some different views for context.

Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

Door of Napier's Promptuary - National Archaeological Museum of Spain (Madrid)

As I create this post I have the sense of joy again that I did when I visited Madrid's NAM this spring. Napier's Promptuary really does feel like a toy box for a mathematician. And it's so cool how you can pack it all back up - putting the bones in the top slot, pushing in the drawers, closing the doors, and locking it all up with a key. If you're ever in Madrid, I recommend a trip to the National Archaeological Museum to see this device. One hint I have for you, in case you get there but can't find the Promptuary, is to ask where the "abacus" is. Napier's Promptuary is one of the treasures of this museum and very well known, but it is known there as an "abacus." It is currently on the second floor and to the right, but museum displays can tend to move. :-)

National Archaeological Museum of Spain (Madrid)