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Introduction
Architectural firms all over the world are tacitly competing to see which of them can achieve the most outstanding results in urban development. Every year, people in developed countries see a large number of new urban developments that combine the latest in architectural aesthetics with environmental sustainability within the city’s ecosystem. While many of them do stir the human imagination and delight with genius, it seems that it is not just the geometric shape of the building that matters. Turning to the textbook example of the Louvre Museum of Art clearly demonstrates that even simple spatial shapes can be attractive from an urban and architectural point of view.
It must be said that the Louvre Museum is a world-renowned art museum located in the center of Paris. Before the pandemic, the museum was visited annually by an average of ten million visitors, with almost five times less than the permanent population of Paris (SRD). So, the Louvre Museum is an important tourist attraction visited by a large number of local and foreign tourists interested in art. With regard to the Louvre, it is impossible not to note its unique architectural form, which gives the structure a particular attraction (Fig. 1). The museum has a form of a quadrangular pyramid, at the base of which lies a square. It is important to emphasize that the shape of the museum is not tetrahedral, as it may seem because the central composition of the visible part of the Louvre is directly a square, not a triangle. The pyramid first appeared in the center of Paris in 1989 and has since become an integral part of the city’s topography (PT). It should be said that the actual museum is not inside this pyramid, but it is more of a portal of entry. The pyramid lets visitors into an underground vestibule that connects the three galleries of choice, Sali, Richelieu, and Denon.
Descriptive Geometry
A mathematical analysis of the Louvre is helpful to begin with elements of descriptive geometry, which will give a more accurate idea of the pyramid’s forms. According to Pei, the maximum height of the pyramid at its peak is 20.6 meters, with each of the sides of the Louvre’s square base equal to 35 meters (Figure 2). The measurements of the other sides are not reported, so it seems possible to calculate them using the tool of analytical geometry. To begin with, it must be asserted that the base of the figure is a square: this is indeed so, since all its sides, and consequently, its interior angles, appear to be equal. Proceeding from this assumption, it becomes possible to determine the measurements of diagonals of a regular quadrilateral by the formula [1]. This formula is based on the Pythagoras theorem, which allows to determine the size of the hypotenuse of a blue triangle (Fig. 3) based on the size of two cathetuses (Russick). Again, one can be convinced of the value of the interior angle of 90° because the quadrilateral has four equal sides.
In such a pyramid, the height dropped from the center to the base divides the diagonals strictly in half. Consequently, each of the diagonals is divided into segments of 24.8 m. Now again, we can use the Pythagorean theorem, but now the action is aimed at determining the size of an edge of such a pyramid. Using the known values, it is possible to calculate this value, as shown in [2]. Thus, all the unknown dimensions have been calculated, and there are no more unknown sides left.
Regarding the space used, it would be interesting to trace what proportion of the total volume can be occupied by people if we take into account that a considerable part of the Louvre pyramid is hollow and its space is not used. Let us turn to Figure 6. It shows a quarter of the pyramid (for convenience), and the black area marks the area used routinely. The height of this area is uniform in all directions and is precisely three meters – this includes the height of the average person and the remainder for the ceiling so that even the tallest people would feel comfortable in such a room. Remarkably, there is no such three-meter ceiling in the Louvre Museum, so for now, it is hypothetical. In fact, we are dealing with a truncated rectangular pyramid, for which we can still calculate the volume. For this purpose, let us turn to the formula shown in [10]. In this formula, known quantities are presented, but the letters S1 and S2 denote the areas of the bases of the actual trapezoid. The bases contain squares, so it is not difficult to calculate their areas. For the bottom base, this will be equal to the doubled side of 24.8; that is, the area of the bottom base is 615.04. To calculate the area of the top square, we must first calculate some of its characteristics. In particular, we use the law of similarity of two triangles.
In other words, people usually occupy a volume not exceeding 6,360.96 cubic meters from the Louvre pyramid. It is worth saying that this is only 75.6% of the total volume of the entrance gate of the museum, so a critically small part of the Louvre is really used with benefit. The reason for such a decision is the purely aesthetic appeal of the pyramid, as it has no civil-defense purpose and moreover cannot be used for shelter, maximizing the number of visitors. The value obtained earlier of 3,062 individuals is the maximum possible occupancy of the Louvre if only one layer is used. However, if the museum management was faced with Your Last Name 10 the question of maximizing capacity, they might consider the idea of three-meter-high tiers inside the pyramid. Given the height of each tier and the overall height of the pyramid, there would have been no more than six tiers, with the last one being the lowest and occupying only 2.6 meters. This would have significantly increased the guest capacity of the pyramid but would have created a number of serious problems with ventilation, evacuation, and flow management.
Trigonometry
Not only the tools of analytical geometry, however, but also trigonometry can be used for practical calculations. By now, all the sides of a quadrilateral pyramid are known, which means there is no problem finding its angles. Let us turn to the green triangle from Figure 3; use the law of sines to construct the following equality.
Calculated Analytics
In a strategically important museum like the Louvre, an attendance study is critical. Because of COVID-19, a large number of visitors could not get into the museum due to physical and social constraints – as a result, the Louvre was losing revenue, and exhibits were stagnating. Therefore, it is the task of analytics to examine visitor dynamics. According to “The Next Generation of Digital Content,” the total number of visitors between 2012 and 2017 at the Louvre is given by the function.
It is not known whether this function is valid, but its study may be of interest for museum applications. In this function, the argument x shows the year starting from 2012 (Figure 11). In fact, not all areas of the graph shown make sense since the values of x are limited to two bounds. Figure 12 shows that the number of visitors to the Louvre has risen smoothly over five years, reaching 654% by 2017 of what it was in 2012. It is of interest to clarify the total number of tourists who visited the Louvre over these five years. To do this, we need to refer to a certain integral.
Conclusion
It is worth noting that mathematical calculations are routinely used in a large number of real-life examples, even if it does not seem obvious at first glance. The pyramid in the Louvre Museum in the center of Paris illustrated the power of the synthesis of analytic geometry, trigonometry, and functional analysis to investigate different aspects of the pyramid. In particular, all sides and angles within the pyramid were calculated, and areas and volumes were measured. A calculation was made of the usable area occupied by people and suggested a way to expand the number of tourists who could visit the Louvre pyramid at the same time.
Among the mathematic tools used are the following:
- Geometric formulas for calculating areas
- Geometric formulas for calculating volumes
- Formulas for sines and cosines
- The law of sines and cosines
- The similarity of triangles theorem
- The law of section of three-dimensional figures
- Constructing the equation of a function by two points
- Basic arithmetic calculations
- Functional analysis
- Boundary Integrations
- Rounding of results
This has been a useful practice to improve my own skills and to learn more deeply, not superficially, about the pyramid phenomenon. However, it is worth saying that any deviations in the calculations from those norms that the reference knows can be caused by rounding and by errors in the calculations. Thus, the general conclusion of the whole work is that a large number of mathematical tools are applicable to the Louvre pyramid, which has been demonstrated.
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