The Math Of The Louvre (A Modern Pyramid)

The Louvre Museum is one of the best known landmarks in Paris. There are over 35,000 objects on view at the Louvre, dating from prehistoric times to the 19th century. The works of art are displayed over an area of 652,300 square feet. 

Originally built as a fortress in the 12th century under King Phillip II’s reign, the building has undergone lots of transformations through the ages, eventually becoming the museum that we know today.

In the 14th century, the fortress was transformed into a castle. It was home to several generations of kings. Eventually, kings chose to live in other castles prompting the demolition of the palace by King Francis I in the early 16th century. Francis I then had the palace rebuilt in the Renaissance style. (Source: Louvre Museum)

After the French Revolution, the palace was converted to a public museum. The museum is one of the most famous tourist destinations in the world. The museum has over 9 million visitors per year, making it the most visited museum in the world.

It contains some of the most famous masterpieces in the world including the Venus de Milo statue, Leonardo DaVinci’s Mona Lisa, the Regent Diamond, the Law Code of Hammurabi, the portrait of King Louis XIV, and The Raft of the Medusa. (Source: museums.eu)

The Louvre Pyramid

The Louvre Pyramid is a glass and metal structure that marks the center of the museum; visitors to the museum enter through the pyramid. The pyramid took about a decade to complete.

In 1981, then president Francois Mitterand announced that the entire Louvre would be made into a museum. At that point, only part of the building was used as a museum.

Some of this was motivated by the fact that the Louvre was attracting so many visitors – the existing building couldn’t accommodate the large numbers. His hope was that the project would be completed by 1989, the bicentennial of the French Revolution and the 100th anniversary of the Eiffel Tower’s completion.

The Chinese American architect Leoh Ming Pei was selected for the task. The project was called “The Grand Louvre Project.” The Pyramid sparked lots of debate and controversy – Parisians were concerned that the modern style of the pyramid would not complement the Renaissance style of the palace.

Pei felt that the pyramid shape would be the least obtrusive shape and made sure that the top of the pyramid would be lower than the surrounding palace. He used glass panels so the pyramid wouldn’t block the view of the rest of the museum – the older, Renaissance style museum can still be seen through the pyramid. (Source: rostarchitects.com)

There are actually five pyramids that were part of the project. The main pyramid has the same proportions as the Great Pyramid of Giza. Alongside the main pyramid are three smaller pyramids.

And, a fifth inverted pyramid known as the Louvre Inverted Pyramid is visible from underground. The inverted pyramid is a skylight that is inside the Carrousel du Louvre, an underground shopping mall that’s in front of the museum. 

Just for fun

While the Louvre Pyramid certainly isn’t the tallest or largest pyramid-shaped building or structure in the world, it’s interesting to compare the various pyramids or pyramid-like structures from around the world. 

The math of the pyramid

Pei settled on the shape of a pyramid as the optimal design for the Louvre’s needs. The straight lines of the pyramid complement the lines of the Renaissance style palace building. The pyramid is made of 603 glass rhombus panels and 70 glass triangle panels.  

Problem 

Given that the glass pyramid at the Louvre reaches a height of 71 feet and each side of its square base is 112 feet, what is the volume of the pyramid?

Solution

To solve this, we need to use the formula for the volume of a pyramid: 

V_pyramid  = Bh/3

where B is the area of the base, and h is the height of the pyramid. The height is measured from the center of the base to the vertex of the pyramid.

In this case, the area of the base is just the area of a square, which is simple enough. The area of a square is the length of a side squared, so we have:

B = 112*112

B = 12,544

Plug in the values B = 12,544 and h = 71 to obtain:

V_{Louvre pyramid}  = (12,544)(71)/3

V_{Louvre pyramid}  = 296,874.667

Rounded to the nearest whole number, the Louvre pyramid has a volume of 296,875 ft³! Wow!

(Source: rostarchitects.com)

The Louvre Colonnade

The eastern facade of the Louvre, known as the Louvre Colonnade, is considered one of the most important French architectural achievements of the 17th century. As mentioned, the Louvre was constantly transformed throughout the ages.

King Louis XIV directed the project to build the eastern facade. A council of 3 members was appointed to design the eastern facade of the Louvre. This council was made up of Louis Le Vau, Charles Le Brun, and Claude Perrault.

The east facade is beautifully symmetric – it has two pavilions, one on each end with a central entrance. In between the pavilions and the entrance are the wings, made up of double columns. 

The design of the long facade uses a five part pattern that was a trait of French palaces – two pavilions, two recessed wings and a central entrance with a gabled roof that’s higher than the rest of the eastern facade, but not too much higher!

The roof is mostly horizontal with just the triangular form of the center’s roof breaking up the horizontal lines. The overall result is stunning and is quite an architectural feat! The geometry used in the design is evident at a glance. 

Notice, the columns of the facade are in pairs. Prior to this, it was customary in architecture to space columns evenly apart so this was quite a departure and had a stunning effect.

At the time of the facade’s construction, the double columns were considered one of the most startling aspects to its design! (Source: smarthistory.org)

Exit strategy

With an increase of terrorism in Europe and elsewhere, large facilities are constantly trying to figure out the fastest way to help people exit a building to get them to safety.

COMAP, the Consortium for Math and Its Applications, sponsors an international contest every year in which undergraduate students from all over compete on teams to research, analyze, and solve real world problems using mathematical modeling.

In 2019, the Interdisciplinary Contest in Modeling contest challenged the teams to come up with a comprehensive plan to help Louvre visitors exit quickly and safely. The Louvre gets about 26,000 visitors per day, so that’s a lot of people to move!

Competing against 11,000 teams, a group from Duke University finished in the top 1% of contest participants. Their team spent 100 hours devising a computer simulation that would aid in the safe egress of museum visitors.

Duke undergraduate students Vinit Ranjan, Junmo Ryang, and Albert Xue figured out how long it would take for the museum to be completely evacuated in the event of an emergency. The team used a simulation of the Louvre to safely evacuate the thousands of visitors.

The Duke team split each floor of the Louvre into five sections, with the instructions given to the visitors that they should find their way to the nearest exits unless instructed otherwise.

The program accounted for flow rates, which is the number of people that can “flow” through an exit per second. They used average walking rates to help figure out how long it would take to clear out the museum.

The simulation also took into account things like panic among some of the visitors and disabled individuals who may not be able to get through a particular exit. The team concluded that with their plan, the Louvre could be evacuated in about 24  minutes! That’s pretty good! And, that’s some pretty great math.

The team also made recommendations that could improve the exit time – for example, they suggested widening some exits or adding another exit. (Source: researchlog.duke.edu)

For more information about the 2019 contest rules, check out the challenge: COMAP’s Louvre contest. 

Next time you’re in France, be sure to take a day to visit the Louvre. Take special note of the Pyramid entrance and be sure to locate the nearest exits as you stroll through the galleries. Enjoy!

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