Later on Daring thefts at the Louvre, thieves managed to escape with priceless crown jewels in broad daylight - here's how a decades-old geometry problem can help museums improve security.
It only took them eight minutes.
In those 480 seconds, the thieves climbed a mechanical platform to reach a balcony on the first floor of the Louvre Museum in Paris, then broke in through a cut-out window in broad daylight.
Once inside, they broke into two glass display cases and then fled with eight pieces of priceless crown jewels from the Napoleonic era.
It was a "daring theft" that shook France to its core.
So far it is seven suspects arrested regarding this theft.
Look HERE a graphic depiction of the great robbery of the Louvre that shocked France.
One of the frequently unanswered questions plaguing investigators was why the thieves were not discovered earlier.
At a hearing before the French Senate immediately after the robbery, Laurence de Cars, director of the world-famous museum, admitted that they had "failed to protect" the crown jewels.
The only camera covering the balcony where the thieves entered, she continued, was pointed in the wrong direction, and a preliminary report found that one in three rooms in the Denon wing that the thieves hit did not have security cameras.
More generally, De Kars admitted that the reductions in the number of security personnel have made the museum more vulnerable and insisted that the Louvre's security system must be strengthened so that every point can be "monitored".
Alarms at the museum reportedly went off when they should have, according to the French Ministry of Culture.
See how thieves escaped from the Louvre after the robbery
Again, this is the third theft at French museums in two months, prompting the Ministry to implement new security plans across France.
While there's no doubt that modern museum security is a complex and expensive undertaking, there's an intriguing 50-year-old mathematical problem that addresses this very issue.
He wonders what the minimum number of guards - or, equivalently, 360-degree security cameras - is necessary to keep an entire museum under surveillance?
This is known as the museum problem, or the art gallery problem.
The solution is very elegant.
We'll assume that all the walls of our imaginary museum are straight lines so that the floor plan is what mathematicians call a polygon, a shape with sharp edges and corners.
The cameras must be in fixed positions, but they can see in all directions.
To make sure the entire museum was covered, we would need to be able to draw a straight line from any point on the floor plan to at least one of the cameras.
Take for example the hexagonal-shaped gallery on the left in the diagram below.
No matter where you place the camera, you will be able to see the entire floor and walls of the entire space.
When in this way every position can be seen from every other, we call the shape of the gallery a convex polygon.
The L-shaped gallery in the middle is non-convex, which means you are limited in your camera placement, but we can still find points from which one camera can see the entire gallery.
An AZ-shaped gallery needs two cameras to cover it - there are always spots that only one camera will miss.
For more interesting floor plans (see the unusual 15-page floor plan below) it's much harder to know how many cameras are needed or where they should be placed.
Fortunately for museum directors of limited museums, Czech graph theorist Vaclav Hvatal solved the general museum problem after it was installed in 1973.
The answer, it turns out, depends on the number of corners (or, as mathematicians call them, "vertices"), because there will be as many walls as there are corners in the room.
A little simple division helps us calculate how many cameras are needed.
By dividing the number of corners in a room by three, we will find out how many cameras are needed to completely cover it, assuming that the cameras have a full 360-degree view.
This works even for complex shapes like the unusual 15-page gallery in the image below.
In this case, there are 15 angles, so 15 divided by three is five.
This works even if the number of corners is not divisible by three.
For a 20-page gallery, for example, the solution is six and two-thirds.
In these cases, you can only take a whole number - so we wouldn't need more than six cameras in a room with 20 sides.
Steve Fisk, a mathematics professor at Bowdoin College in Maine, USA, found a proof in 1978, considered one of the most elegant in all of mathematics, for this lower bound on the number of cameras needed.
His strategy was to divide the gallery into triangles (see the left image of the figure below).
He then proved that you could choose just three colors - for example, red, yellow, and blue - and assign a different color to the corners of each triangle.
This would mean that each triangle in your gallery has a different color in its three corners (see the right image of the figure below, for an example).
This is known as "tricoloring" the corners.
Triangles are one of those "convex" polygons we mentioned earlier, so a camera placed at any corner (or really anywhere in a triangle) can see every point in that shape.
Each triangle has corners with each of the three colors.
This means that you can choose just one of the colors and place the camera in those positions.
Those cameras will be able to see every part of every triangle and therefore every part of the gallery.
But here's the best part.
The beauty of Fisk's proof is that you can choose the color with the fewest dots and still cover the entire gallery.
On the 15-sided shape above, by selecting the red dots we can get away with just four cameras.
Moreover, the red dot in the upper left corner is not necessary, because the next red camera can cover its entire surveillance area.
So, we could even get away with three cameras for this gallery.
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This is especially true if we install modern multi-directional cameras, instead of old-school wide-angle security cameras that would have to constantly scan the area to get a full view, temporarily creating blind spots.
But we should not forget that many traditional museums such as the Louvre have mostly rectangular rooms.
Fortunately, a variation on the art gallery problem shows that when the walls meet at appropriate angles, we only need one camera to cover the entire room.
In her testimony, De Cars also admitted that the Louvre's exterior cameras do not cover all the walls.
"We didn't notice the thieves' arrival early enough... the weakness of our perimeter security is well known," she said.
Fortunately, there are versions of this problem, known as the "fortress problem" or "prison problem," that solve the problem of camera coverage for the exterior of a building as well.
Louvre officials admit security cameras did not cover the entire exterior of the building
What both variants reveal, however, is that finding the right vantage point is crucial.
But it's just important to accept that thieves entering through public galleries are not the only threat facing museums.
The British Museum in London, for example, happened to Cartier ring worth $950.000 goes missing 2011 from a collection that was not on public display.
The museum's jewels were found in an eBay listing in 2020 after they were allegedly taken by one of the museum's curators.
In addition to theft, museums also have to protect collections from vandalism, fire, and other forms of destruction.
Despite this, the art gallery's problem is worthy of attention even to those outside the hallowed spaces of museums.
It has applications in a wide range of fields where visibility and coverage are of key importance.
In robotics, for example, it helps autonomous systems improve efficiency and prevent collisions.
In urban planning, it influences the positioning of radio antennas, mobile phone signal transmission towers, or pollution detectors to ensure comprehensive coverage of public spaces.
Disaster management strategies use similar principles to deploy drones to monitor large-scale disaster sites from the air or to set up field medical stations.
In motion picture and computer image editing, the art gallery problem can help identify visible regions within a scene.
It can help ensure that actors are always lit on stage and even help museums themselves ensure that their galleries are properly lit.
The Louvre did not respond to BBC questions about whether it was aware of the solution offered by the museum's mathematical problem.
The museum undoubtedly has bigger problems to solve at the moment.
But as museums and art galleries around the world reexamine their security after the Louvre theft, it can't hurt to remember the lessons this 50-year-old math problem offers.
Watch the video: Some of the jewels stolen from the Louvre
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Bonus video:
