Friend of Strong Towns, Vince Graham, shared an interesting email conversation with us a few weeks ago involving a quest to find the equation that would prove the value of gridded streets. We share the story with you now in the hopes that you might learn something as you follow along with his adventure.
In the early 1990s, I lived in downtown Beaufort, South Carolina. I did a lot of running back then, and usually chose routes through historic Beaufort for my 3-5 mile runs. Because of my interest in architecture, I often varied my routes in order to view different buildings from different angles. After a while, it occurred to me that Beaufort's grid provided a seemingly endless number of possible routes. Being a numbers guy, I became increasingly curious about the possible number of routes.
During the summer of 1994, my youngest brother Geoff, then a rising junior at Davidson College, visited Beaufort, bringing with him classmate Casey Hawthorne. Casey, a math major, had previously scored a perfect 800 on the math portion of the SAT. While working construction in Beaufort that summer, I asked Casey to help figure out a formula for the number of possible routes through a grid.
He came up with the following factorial equation, which I deemed "the Hawthorne Equation":
Given points "A" and "B" at opposite corners of a grid where "x" is the number of north-south blocks and "y" the number of east-west blocks, the number of possible goal-directed ways to get from A to B = (x+y)!/(x!)(y!).
"Goal-directed" means one can only move in the direction from A to B (up and to the right in the case where A is in the lower left corner of the grid and B in the upper right). One can't deviate from these directions.
In a simple 2-block by 2-block grid, there are thus six possible ways to get from A to B.
(2+2)!/(2!) x (2!) = 4x3x2x1/2x1x2x1 = 24/4 = 6.
In a 3-block by 3-block grid there are 20 possible ways. In a 10-block by 10-block grid, there are 184,756 possible ways.
A better explanation of this can be seen in this Clemson University lecture from two years ago. (Go to the 26-minute mark.)
Over the years, I've tried to come up with various ways to illustrate the power of a grid vs. the conventional using other comparisons. For example:
Casey also came up with an equation that showed the total possible ways without the caveat of goal-directed moves. But this equation involved a lot of sigma notations and I'd long since forgotten my calculus... Because I felt the simplified formula was sufficient for my purposes of explaining the power of networks, I didn't write it down.
But I came to regret this. From time to time over the next two decades, I would reach out to Casey, asking him for the formula. Casey was bouncing around Central and South America, teaching math, eventually earning a doctorate degree in California. He is now a professor at Furman University. I tracked him down there early last year, and again asked him for the formula. He engaged another math professor and we had a conversation about it, but they were not able to come up with a solution.
Part of the problem is in how you define it. I was interested in the number of possible ways, with a single exception from otherwise goal-directed moves. So, imagine you're driving through a grid and come to an impasse (the street is blocked, by an accident, for example, or to fix a pothole). You would typically re-direct a single time before proceeding in a goal-oriented direction.
I was visiting my brother Geoff and his family in Atlanta last Thanksgiving. While there, Geoff introduced me to his friend Nassim Taleb who was also in town visiting family. I shared the Hawthorne Equation with Nassim, who immediately grasped its importance. I also mentioned being on the hunt for the second equation. One of Nassim's hobbies is solving difficult math problems, and he told me he would work on it. But he was off to Moscow the next day, and may have forgotten about it.
Still on my quest, earlier this year, I shared my dilemma with polymath friend Noah Schiffman. Noah has a broad educational background with degrees in psychology, mechanical engineering, and medicine. He currently works in cyber-security. (On a side note, Noah's late father was a professor of psychology at Rutgers, and always impressed upon his sons the importance of being cross-disciplinary and avoiding "silo thinking.")
After showing Noah the "Hawthorne Equation" he told me that it was commonly referred to as "the grid-walking problem" and is among the first things taught in computer science classes. I was somewhat dumbfounded, as I had been sharing the Hawthorne Equation with civil engineers and others for two decades, and they were completely unfamiliar with it.
Noah took on the task and solved the equation! Below is his formula. The caveat to this is that the first two moves from point A, and the last two moves toward point B must be goal-oriented. Otherwise you would be doubling back across the same path.
For a grid with dimensions of "m" blocks by "m" blocks, the number of ways = 2m x ((2m)!/((m-2)! x (m+2)!))
With a 3-block by 3-block grid, using the Hawthorne equation, there are 20 possible ways. Using the Schiffman equation, there are 36. With a 10-block by 10-block grid, the number climbs almost 14-fold from 184,756 to 2,519,400.
This is a mathematical demonstration of what Taleb might call an "antifragile" transportation system. Among other things, I think the resilient nature of a grid might be an important factor when weighing the potential for removing limitations placed upon grids back in the day: for example, when removing large highways that cut through a grid, or reconverting one-way streets to two-way. Another positive attribute of a gridded street network is the number of possible ways enables most streets to be relatively narrow compared to a system based on arterial highways.
So why aren’t we building more street grids, even today, when we have had the better part of a century to learn from our mistakes?
One answer might be that traffic engineers tend to be silo-thinkers, preoccupied with the movement of automobiles. Another thought is that grids, as Rob Steuteville pointed out in a recent article at CNU: Public Space, work so much better than conventional, hierarchically engineered streets systems. They accommodate traffic at a tiny fraction of the cost. This is not so good if you're an engineer or road contractor who earns money by perpetuating the problem of traffic congestion. As Upton Sinclair famously said,"It's hard to get a man to understand something if his salary depends upon him not understanding it."
Meanwhile, my friend Noah, in his "Beautiful Mind" way, continues to work on the equation, thinking he may be able to further simplify it. In the end, though, it’s a simple concept to grasp: Gridded streets create a myriad of clear, potential routes to get from one place to another in a way that no other form of street design can. We should be doing all that we can to make grids a priority in our towns.
(Top photo source: Evan Walsh)
About the Author
Combining modern advances with time-tested urban principles, Vince Graham founded the traditional walking neighborhood of Newpoint in Beaufort, SC in 1991. Since that time he has participated in building eight other neighborhoods: the Village of Port Royal, Broad Street, I'On, Morris Square, Hammonds Ferry, Mixson, and Earl's Court in South Carolina; and East Beach in Virginia.
In addition to garnering numerous design and environmental stewardship awards, these neighborhoods have also been the subject of articles and stories in The Wall Street Journal, Builder, Landscape Architecture, and National Geographic magazines, Home and Garden Television, CNN, the BBC and more. Graham is a passionate advocate for advancing human-scaled urbanism, and has spoken at architectural and planning symposiums in Australia, Europe, and throughout the United States.