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 shows up 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, 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)