Ivars Peterson's MathTrek

June 6, 2005

Knot an Interchange

Switching from one busy highway to another can be a logic-defying, heart-stopping endeavor—one that a carefully engineered interchange is meant to ease.

In the United States and in many other countries, drivers can make right turns easily without crossing any lanes of traffic. The common "cloverleaf" interchange takes advantage of this fact—in effect, replacing a 90-degree left turn with a 270-degree right turn. The opposite would be true in England and other countries where drivers stay to the left side of the road.

One of the most intriguing of all highway crossings is the interchange that links I-95 and I-695 northeast of Baltimore. Here, the interchange has both left and right exits.

According to Dror Bar-Natan of the University of Toronto, who features this interchange in his image gallery of knotted objects, the crossing has an outer layer of right turns, followed by a braiding of the lanes, then an inner layer of left turns.

The interchange of I-95 and I-695, northeast of Baltimore.
Credit: EROS Data Center, U.S. Geological Survey

In the current issue of the Mathematical Intelligencer, Michael Kleber follows up Bar-Natan's observation with additional information about this elegant, knotted interchange. Such interchanges are rare, he notes, mainly because they take up a lot of land. From a human-factors viewpoint, the combination of left and right exits is also a somewhat undesirable feature. Drivers can get confused about which lane to be in to make the desired turn.

Nonetheless, the interchange does appear to work well. Kleber quotes a highway expert who notes that the curves of the roadway and the ramps are gentle enough that cars have to slow down only slightly from full highway speeds.

"It's also a win topologically," Kleber writes. "In a cloverleaf, two lanes of traffic must cross [each other], as cars coming off one ramp need to switch places with ones entering another ramp. This interchange avoids that problem: Drivers entering from any direction can turn either left or right without crossing any other lanes of traffic."

Unlike with a cloverleaf interchange, however, you can't use a combination of exit ramps to make a U-turn.

Interestingly, there's an interchange similar to the one near Baltimore where I-20/I-59 crosses I-65 west of Birmingham, Alabama.

The twists and turns of highway interchanges can certainly lead you in interesting directions!

References:

Kleber, M. 2005. Cartographiana. Mathematical Intelligencer 27(No. 2):35-40.

Mathematician Dror Bar-Natan of the University of Toronto has an image gallery that includes the interchange of I-95 and I-695 just northeast of Baltimore as an example of a knotted object. See http://www.math.toronto.edu/~drorbn/Gallery/KnottedObjects/Interchange/index.html.

Michael G. Koerner's favorite highway features, including interchanges, can be found at http://www.gribblenation.com/hfotw/index.html. See the entries for Jan. 9, 1999, and Nov. 21, 1998, to view the Baltimore and Birmingham knotted interchanges.

A mathematical problem related to interchange overpasses and underpasses, as suggested by Bill Thurston of the University of California, Davis, can be found at http://www.math.toronto.edu/~drorbn/Gallery/KnottedObjects/Interchange/ThurstonW.txt.