March 21, 2010

Knotted Staircase

It looks like a spiral staircase that had lost its way before finally winding back to its starting point. The looped sculpture, titled Révolutions, stands in a little park near the entrance to the Papineau Métro station in Montreal. Created by Michel de Broin, the artwork pays tribute to the curving iron stairs that serve as entries to many city residences.

Constructed from aluminum, the sculpture stands on several legs, lifted about 10 feet off the ground.

In this case, the elevated, twisty staircase forms a mathematical knot, with no end and no beginning. "Stairs are a symbol of progress, of linear inward and upward," de Broin notes, "but in the knot they become a continuous circuit."

When I saw the sculpture on a recent visit to Montreal, I couldn't quite figure out what type of knot it is. But I could tell right away that the loop was not one-sided or one-edged, like a Möbius strip. The main clue was the railing. It did not appear on all sides of the knotted, spiraling staircase as it would have if the surface were truly one-edged.

At the same time, knots and Möbius strips are not incompatible. Artists have been creating striking Möbius strip trefoil knots, for instance. John Robinson's Immortality is one notable example.

From any angle, Révolutions is an unexpected pleasure and a treat for mind and eye.

Photos by I. Peterson

March 10, 2010

Deciphering Integer Sequences

Regularly compiling the NumberADay blog for the MAA website has made me more aware than ever of how much we are surrounded by numbers, from street addresses and highway signs to serial numbers and other identifiers. Curious sequences of numbers may also pop up unexpectedly.

While visiting Madison, Wisconsin, for example, I happened to notice that each bus shelter lists, in numerical order, the numbered bus routes serving that particular stop. I couldn’t help wondering whether a given list of numbers has any mathematical significance. 

Many bus lines serve this stop near the state capitol in Madison, Wisconsin.

To find out, I turned to one of my favorite resources on the web: Neil Sloane’s On-Line Encyclopedia of Integer Sequences (OEIS). Sloane has spent more than 40 years amassing a vast database of known integer sequences.

Entering the sequence 2, 3, 4, 6, 7, 8, 11, 12 in the search box, for example, yields seventeen results—known sequences that contain this particular string of consecutive integers. One is the sequence defined as follows: If n is in the sequence, then so are 2n and 4n – 1. Sloane’s entry provides information on how that particular sequence arose, along with comments, formulas, references to the literature, links to other websites, related sequences, and computer programs. You can even graph or listen to it!

If I were to extend the sequence by one term, adding 13, the OEIS search produces seven results; six results with the addition of 14, and 2 results with 15 added. However, adding 29 produces no results.

Bus shelters in different parts of town are sure to show different sequences, so I’m now tempted on my next visit to Madison to collect more examples, just to see what sorts of integer sequences come up.

I also use Sloane’s database to find interesting properties of individual numbers, just by entering a single number in the search box. Entering 2658, for instance, generates 95 results. One result reveals that 2658 is the number of 8-digit numbers in base 6 with adjacent digits differing by 1 or less. I’m not sure why anyone would want to know that, but there it is.

Sloane was at the Joint Mathematics Meetings in January 2010, promoting the transition of his indispensable database from essentially a one-man operation to a wiki format at, giving each sequence its own web page. A board of 50 associate editors will moderate submissions, relieving Sloane of much of the burden of checking new sequence submissions, which come in at a rate of more than 50 a day, and making all the necessary changes to web pages.

Sloane has set up the non-profit OEIS Foundation as owner of the database.

See also "The EKG Sequence" and "Sequence Puzzles."

Cipra, B. 2010. What comes next? Science 327(Feb. 19):943.