How can maths help us design the perfect car park?

A car park in Singapore. Image: Getty.

There can be few things as frustrating as being stuck in a car park for four hours on a scorching Sunday afternoon; yet this was the unhappy fate of shoppers at a new multi-storey grid car park at an IKEA store in Reading, UK.

The delays were caused by a few factors, including a particularly busy adjacent road, the popularity of the store’s opening weekend and the convergence of vehicles on a single exit ramp. Despite the retailer’s major £4m investment in constructing good access routes, the infrastructure simply could not cope with the volume of traffic leaving the car park.

As well as prompting a full-scale investigation by IKEA, such chaos invites us to consider how further gridlock can be prevented. Fortunately, mathematics can provide some basic guiding principles, as we consider how to design the perfect car park.

A numbers game

First, we need to decide how many parking spaces there should be. This is a perennial problem, to which there is no prescriptive answer. Too many spaces are costly and look ugly, while too few spaces result in distraught and dissatisfied customers. Yet some relatively simple maths can help us avoid the worst congestion.

Suppose that daily peak demand averages, say, m cars. We can get a sense of how much the number of cars is likely to vary from the average using standard deviation – let’s call this value s. If s is small, it means that the daily peak demand is quite consistent. If s is larger, it means that there’s a bit more variation; perhaps attendance spikes on Sundays, or over long weekends, or during sale periods.

Once we know these values, we can use the normal distribution to evaluate the probability that a car park with a given number of spaces will overflow. For example, if m=750 and s=100, then a car park with 800 spaces will overflow on 31 per cent of days, whereas a car park with 1,000 spaces will overflow on only 1 per cent of days. Greater accuracy can be achieved by modifying this simple model to focus on the more extreme events.

While avoiding excess capacity is an important consideration for some organisations, such as airlines, it’s better for car parks to err on the side of generosity. While some travellers might be willing to catch later flights in return for compensation, drivers are likely to find the prospect of being redirected to a multi-storey car park two blocks away somewhat less appealing.

The perfect angle

It’s also better to err on the side of generosity when it comes to the size of individual parking bays: my own modestly sized vehicle is pitted with craters caused by other doors banging into my side panels.

Bays should leave ample space around the cars to enable pedestrian access and to allow for the axle tracks of turning circles, so that vehicles can enter and exit smartly without cutting the corners of adjacent bays. This can also be achieved with sufficiently wide access lanes, so that cars are parallel to the lines when entering their bays.

Now consider the layout of parking spaces. Assuming that the building has a rectangular plan, there are some simple rules that ensure a convenient and dense population of bays. Rather than having access lanes around the perimeter of each storey, moving the lanes inwards allows us to place bays around the edges and increases the number of spaces.

Use your space well. Image: The Conversation.

Dead ends are undesirable, as they require drivers to reverse against the flow of traffic, so ramps should be located to avoid these. One-way flow systems throughout the car park also helps to avoid congestion and confusion, while allowing access lanes to be narrower than for a two-way flow of traffic.

A diagonal layout of car parking spaces offers significant advantages over a rectangular layout. Imagine proceeding along an access lane and finding an empty bay. With a rectangular layout you need to change your direction of travel by 90 degrees, which requires a substantial lane width to accommodate your turning circle.

But for a diagonal layout, the bays on both sides are inclined towards you. These require less course adjustment and the access lane can be narrower, so we can fit more parking bays into the same space. For a large car park, a 45 degree bay angle leads to an efficiency saving of 23 per cent. You also need to change your direction of travel much less, so manoeuvring is easier and safer when later reversing out of the bay.

The perfect parking lot

Clearly, there should be at least as many exits as there are entrances. Ramps should be not so steep that they are dangerous for drivers who queue or stall on them. Nor must they be too flat, or they will occupy too much precious space. Other factors are important, including the car wheelbase, ramp length and curvature, and possible transitional ramps.

Isaac Newton’s second law of motion – which tells us that that the rate of an object’s acceleration depends on the mass of the object, and the magnitude of the force which acts upon it – together with a simple vector diagram, help to balance the car’s weight with the slope’s normal and frictional forces, to determine acceptable limits for the gradient. One in ten is a generally accepted maximum slope, with lesser gradients often preferred.

A helix form car park in Newcastle city centre. Image: Trevor Littlewood/Wikimedia Commons.

If one were to design a new car park from scratch, one of the best of all systems is epitomised by the helical car park design. With one entrance, simple traffic flow and one exit, it is safe for pedestrians and uses the available space efficiently. Crucially, it is also reasonably pretty. Perhaps IKEA should ditch its grid design, and give the helix a whirl.The Conversation

David Percy is professor of mathematics at the University of Salford.

This article was originally published on The Conversation. Read the original article.


How the rise of anti-crime politics caused lasting harm to Black Americans

"I see an awareness that has developed in the Black community in the last 10 years or so about how deeply racist the criminal justice system has become," James Forman Jr. says. (David McNew/Getty Images)

The police killing of George Floyd, and the protest movement that emerged from it, has reinvigorated a national conversation around reinventing criminal justice policy in the United States.

At the same time, reports that violent crime is rising in many US cities have resurrected talk of the much-disputed “Ferguson effect,” a theory put forward by law enforcement professionals, and some researchers, who argued that police slowdowns in the wake of the first wave of Black Lives Matter protests resulted in elevated rates of violent crime. President Donald Trump is trying to weaponise this narrative, paired with images of federal officers clashing with protesters in the streets of Portland, to wage a 1968-style backlash election campaign.

“People who want to mobilise a lock-them-up style of either policing or prosecution have tried to weaponise those short-term increases,” says James Forman Jr., professor of law at Yale Law School. “Criminologists will say you have to be very, very cautious about short-term movement [in crime statistics]. We don't know whether or not what we're seeing right now [with violent crime increasing] is going to sustain itself. But the fact is, it's here and people are talking about it.”

In 2018, Forman won the Pulitzer Prize in nonfiction for his book Locking Up Our Own: Crime and Punishment in Black America. Drawing on his experience as a public defender in Washington, DC, he traced the emergence of anti-crime politics in late 20th century Black communities. Forman showed how newly empowered Black politicians fought for policies they believed would protect and uplift Black Americans, but inadvertently contributed to mass incarceration. 

CityMetric recently caught up with Forman to discuss crime trends, where he sees reason for hope in this moment and how the Black political class’s attitude toward crime and punishment has shifted since the latter part of the 20th century. 

This interview has been edited and condensed. 

There is talk right now about a resurgence of crime and violence in American cities. We saw similar, more localised concerns after the initial 2015 Black Lives Matter protests in Ferguson and Baltimore. Do you fear this could reinvigorate the kind of politics you describe in your book among segments of the Black community and political class?

I fear that it could be reinvigorated nationally and also in the Black political class. Look at the political conversations that are happening in Atlanta right now, for example, a city that also has seen a short-term uptick in crime as it is a site of a lot of protests about George Floyd and Breonna Taylor on the national level, as well as Rayshard Brooks and Ahmaud Arbery more locally in Georgia.

I think that you can already see in some of the language of the local elected officials this idea that we have to be very careful about pulling back. [They are saying] “while the protesters may make some valid points, we can't risk returning to the ‘80s and ‘90s.” Those decades really traumatised the United States, and particularly traumatised Black communities. There's a deep fear about returning to the levels of the violence that we saw in the crack years.

You write a lot about class divides among Black Americans, where middle income and elite Black people don't suffer as much from extremely punitive policies. They also have closer ties to the politicians who are creating these policies. There are very specific groups of people, even in marginalised communities, whose voices are heard.  As a result of these dynamics, you write about Black politicians fighting for things like mandatory minimum prison sentences or against decriminalising marijuana. Is there still that disconnect between those who suffer the most from criminal justice policies and those who are actually heard in political discourse?  

Let me just say a caveat, that when we talk about class divisions in the Black community it's important to hold two truths in our head at the same time. Bruce Western and others have shown the way in which class, educational status, income can dramatically reduce the likelihood of being hardest hit by the criminal system – namely incarcerated. Middle class and upper middle class Black people get some measure of protection. It's also true at the same time that Black people of all classes are worse off relative to their class counterparts in the white community. 

One area where class is least protective is policing and police stops. The police do not know how many degrees you have. They don't know how much money you have in your bank account. I want to be very clear that in making this point about class, I'm not making the argument that race or racism don't matter in this context. 

In terms of how it plays out now, I see an awareness that has developed in the Black community in the last 10 years or so about how deeply racist the criminal justice system has become. Twenty or 30 years ago they had a consciousness, but there's levels of understanding. Many of the people I write about in the book wanted to promote the interests of the Black community. They weren't motivated by indifference or callousness. When presented with mounting evidence of how awful this system has been in Black lives, they're reconsidering and recalibrating. 

Lots of former elected officials have said to me some version of “I didn't know at the time and I appreciate that you showed us in our full complexity. I appreciate that you showed the pressures we were under. If I had known then what I know now, maybe I would have been less quick to go along with some of these harsh measures.” 

The second thing that has affected the Black political class has been the emerging movements, led by Black people in particular and led by young people. They not only educated leaders, but pressured them and made them understand that there is a political cost. If you're not moved by the moral argument, then you'll be moved by the political argument. You'll be moved by the people protesting outside the office of District Attorney Jackie Lacey in Los Angeles, for example, where Black Lives Matter LA has held, I believe, a year of consecutive protests against a Black district attorney who has had really some of the worst practices.

From what I can tell, she's been pressured by the movement to change some of her positions on important issues like prosecution of low-level drug offenders, for example, and the aggressiveness with which she prosecutes police officers for acts of violence.

What do you make of the calls to defend or even abolish the police?

What I find so compelling about abolition, initially in the prison context and extended to the police as well, is that it shifts the conversation and forces us to go through experiments in which we imagine what it would take to build that world. I think that exercise is very important, because it pushes us further than we are naturally inclined to go. Cultivating a broader imagination is an incredibly important part of this work, because as you know from my book, often it was lack of imagination that caused people to fall back on [punitive policies]. 

That's what caused D.C. Councilmember David Clarke to call the police rather than public health experts when he was overwhelmed with letters about heroin addicts in public space. He was anti-drug war, but he couldn't imagine responding to a call for help with heroin addicts with anything other than police. That's a very common move from even really good and progressive people. 

People who are for defunding, for abolition, are absolutely right about reinvesting that money into alternative structures that support communities. But the reinvestment part doesn't follow naturally from the terms. We might want to come up with a term that captures the new stuff we want to do. I think that's particularly important because one of the reasons Black communities have ended up supporting more police is that Black communities have always wanted their fair share of the resources.

Then, the evidence suggests the United States has too many police officers doing prophylactic, preventative, or stop-and-frisk style policing. The style of policing that leads to district level harassment, pulling people over for no reason. But we have too little investment in the parts of police departments that investigate unsolved crimes. I'm talking about the investigator or the detective who comes to your house after there's been a robbery, an assault, a rape, or homicide. 

As compared to European countries, in the United States we actually underinvest in those parts of our police departments. Jill Leovy’s book Ghettoside shows this in dramatic detail. She describes an LAPD that's stopping and frisking Black drivers wantonly and yet the homicide detectives are still relying on a fax machine and the fax machine is broken. They have to go with their own money to Staples to buy a printer. Meanwhile, other aspects of the department are kitted out in this ridiculous riot gear that makes them look like they're in Fallujah. 

That under investment is particularly damaging to Black communities because we're disproportionately victimised by crime. Because of racism and this allocation of resources, the police are less likely to respond in Black communities. The kids I used to work with in the charter schools in DC, we talk about no snitching, but one of the reasons they would never call the police after they'd been victimised by crime is they would say, “They're not even going to come. You're wasting time.” 

I did a Q&A with Jill Leovy too and her argument is one I've struggled to articulate in our present moment. She argues the state doesn't have a monopoly on violence in low-income Black neighbourhoods, because investigations of violence are deemphasised and crime victims or their loved ones often take retribution into their own hands.  But right now, establishing or preserving the state's monopoly on violence isn't an appealing talking point. 

Yes, this is another thing nobody's talking about. Whatever we're going to do instead of the police has to be accountable to the public. The best, most direct way to have accountability is to have the individuals be public employees. As long as we have 300 million guns in this country at least some of those state employees are going to themselves be armed. It's unreasonable to ask them to do the job without it. Not as many need to be armed as are armed now, but some of them need to be. But they can't be hiding behind union contracts or civil service protections which make it impossible to remove even the worst performing, most abusive officers. 

We can not call them police if we want to. That's semantic, but maybe symbolism matters. But those people have to be state employees. They can work with community-based nonprofits, but there are also communities that don't have as robust of a nonprofit network, and they deserve protection too. These [community] groups have to be accountable to the state and, when they don't exist, the state has to be there. 

Progressives get all the points I just made when it's applied to education. The notion that things be public and accountable to the state is understood when it comes to schools. It's exactly why so many people on the left are opposed to charter schools, because they say they don't have public accountability. They want these things to be a state function. But this point about the difficulty in removing this entirely from the hands of the state is, I think, one that liberals and progressives understand from other contexts.

Jake Blumgart is a staff writer at CityMetric.