Mathematical Analysis of the Pantheon Columns: Vitruvius & Didyma Turkey Plans

Josie Noone 2022

I. Introduction

The Pantheon is one of many ancient marvels of Rome. The current and final Pantheon that we see today is its third reconstruction, completed in 125 A.D. by Emperor Hadrian during his reign (Brewminate). Originally dedicated as a temple for all Gods, the Pantheon now serves as a Catholic church as well as a major tourist destination (Brewminate). The Pantheon is one of the best-preserved structures of ancient Rome and its massive portico, columns and dome have been studied and analyzed throughout decades by engineers, architects and mathematicians alike. 

Figures 1 & 2: The Pantheon and surrounding area, including the Macuteo Obelisk; map of Rome showing where the Pantheon is located. 

Hadrian was greatly inspired by both Roman and Greek culture and the Pantheon is a product of those architectural influences, as can be seen specifically in its columns (Artincontext). The massive 39 foot tall columns are a particularly impressive and intriguing feature of the Pantheon. Each column is carved from a single granite stone from one of two different Egyptian quarries and brought to Rome. Each column also has a special shape – a slight vertical curvature, called entasis, with a tapering diameter size from the base to the top (Grabhoff).

This paper examines the measurements of the Pantheon columns and compares those measurements to specifications for what was considered in ancient Roman times to be the ideal shape of architectural columns, as described in 27 BCE – over a hundred years before the final Pantheon was completed – by the Roman architect Marcus Vitruvius Pollio in his treatise De architectura libri decem (Ten Books on Architecture) (Grabhoff). We then analyze the possible mathematical methods used by the Romans to achieve the Pantheon columns’ dimensions and enstasis by examining ancient Greek plans for similar columns found at the Temple of Apollo in Didyma, Turkey. Analysis of the Pantheon columns shows their architectural complexity and the mathematical knowledge of the Romans at that time. 

II. Basic Structure & Layout of the Pantheon

The Pantheon is constructed of a rectangular portico fronting to a circular building topped with a dome, as shown in the floor plan set forth in Figure 4. The concrete dome of the circular building at the back is a near perfect sphere, with the height from the top of the dome matching its diameter, and with a hole in the center to allow light into the building (Brewminate). In front of the dome, the portico juts out in a rectangular shape and is lined with 16 single-stone granite columns, each weighing upwards of 50 metric tons (Grabhoff). The columns are arranged in three rows, the first row with eight columns and the next two rows with four columns each. The massive columns in the portico support a triangular pediment.  

III. Analysis 

To analyze the Pantheon’s columns and their dimensions, a study conducted in 2005 (MPIWG), the Bern Digital Pantheon Project, created a digital model of three-dimensional points of the columns.  Although the Bern Digital Pantheon Project website has not been maintained since 2010, its data, including specific measurements of each column, remains available for analysis thanks to related writings by the project’s participants (MPIWG; Grabhoff). The columns’ measurements demonstrate that the architects of the Pantheon were following approximately the same mathematical proportionality as had been laid down by Vitruvius over a century earlier.   

A.  Pantheon Column Measurements Fit Vitruvius’ Ideal

1) Measurements and Parameters

To analyze the columns of the Pantheon, it is helpful to number each one and consider its lay out in relation to the others, as in the image below:  

To obtain the measurements of the columns, the Bern Digital Pantheon Project digitally scanned each column and obtained a data set consisting of 1-2 million measuring points of horizontal slices (Grabhoff). The radius of each (original) column shaft was measured in meters, at upper and lower horizontal cross sections. The results of those measurements are set forth in the “Lower radius” and “Upper radius” columns in Table 1 below.  The proportional relationship between the upper and lower radii is calculated and recorded in the “Proportion: UR/LR” column in the following Table 1:  

PantheonColumn #	Lower radius,LR	Upper radius,UR	Proportion:UR/LR
2	.738	.648	.878
3	.730	.632	.866
4	.733	.640	.873
5	.741	.655	.884
6	.741	.654	.883
7	.735	.643	.875
8	.760	.672	.884
12	.753	.663	.880
14	.725	.642	.886
16	.736	.637	.865
11	.729	.637	.874
13	.718	.627	.873
15	.739	.649	.878

As seen in the table, the values of all of the lower radii fall closely within a range of .718 to .753 meters while the upper radii fall closely in the range of .627 to .663 meters, with the exception of column #8 which has a noticeably larger radius throughout. The proportional values of the radii measurements for each column are within an exceptionally small range as well (including column #8), and the average proportional value is approximately .8768 meters. These measurements suggest that the columns were carefully designed and intentionally made to follow the same tapering design pattern. 

A visual representation of each column’s data points showing the continuous radii with respect to the column’s height is shown below in Figure 8:

Figure 8: Graph of the radius vs height (in meters) of each of the 13 original columns. Graph from Grabhoff. 

We can see from the graph the taper that the column shafts follow as their heights increase. We also see that the decrease in the columns’ radii as their heights increase does not follow a straight line but rather follows a slight curve. The overall shape of each column line therefore follows a similar tapering trend curvature as the others with surprising precision. The slight variations and jagged lines in the graph may be attributed to the limited tools available at the time of construction and also possibly the variation between different column builders over time. 

2) Vitruvius: Ten Books on Architecture

Vitruvius was a famous Roman architect and engineer during the 1st century BC. Most of what we know today about Classical Roman architecture, design and mathematical principles comes from his widely-recognized Ten Books on Architecture written around 27 BCE (Britannica). While the Pantheon was constructed after Vitruvius’ treatise was written, his treatise on architecture nevertheless provides valuable insight into the knowledge and design principles of the time. Analysis of Vitruvius’ writings as compared to the Pantheon’s measurements suggests that there was a known methodology of constructing curved columns in ancient times.

According to Vitruvius, column proportions should be dictated by the distance between each column, and the columns’ respective diameters. Similarly, the relative thickness of a column should be based on its position with respect to the rest of the columns in the structure. For example, for a corner column, Vitruvius stated that the relative thickness of that column should be 1/50 greater than the diameter of its neighbors (Vitruvius, III.3.xi).  

For aesthetic purposes, Vitruvuis also mentions the tapering curve of columns as an important feature of design. For example, for columns between 40-50 feet (as are the Pantheon’s columns), Vitruvuis notes that the upper diameter of the shafts should have a 7:8 ratio (.875) with respect to the lower parts of the shaft (Vitruvius, III.3.xii.). Using mathematical and proportional analysis in this way, one would end up with an aesthetically pleasing and sound structure and not one of awkward height with respect to the thickness and placement of its columns.  

Comparing the Pantheon’s columns to these articulated general rules for column design reveals that the Pantheon architects were likely aware of the specifications and ideal design for columns articulated by Vitruvius in his treatise. For example, Pantheon column 7 has an lower diameter of .735 and an upper diameter of .643, with the proportionality of .875, which is precisely equal to the 7:8 proportionality laid out by Vitruvius. (7:8 ratio = .875). 

Figures 10 & 11: lower and upper proportions of column #7. 

The other Pantheon columns also fall within a remarkably close range, with a joint average proportionality rate of approximately .8768. Further, the unusual thickness of column #8 can be explained by the fact that it is a corner column, which according to Vitruvius’ teachings, should be thicker than its neighbors. 

While all the Pantheon column dimensions are not exactly identical to each other, given the technology available at the time, the columns are surprisingly accurate, uniform, and consistent with the ideal column standards that Vitruvius envisioned and set forth in his architectural writings. 

2. Column Construction Plans: Didyam, Turkey 

Until recently, we could only guess about how the ancient Romans actually implemented their construction and design ideas. We had bits and pieces of knowledge from Vitruvius’ treatise but it was pure speculation how those ideas were actually brought to life through the construction of the massive buildings from the Roman era. Recently, however, archeologists made an important discovery that sheds light on possible construction and planning methods that were known as far back as 300 BC when the Greeks planned and constructed their columned temples. Those construction and planning methods also likely would have been known and similar plans may have been used centuries later, when the ancient Romans built the Pantheon (around 125 CE).

Specifically, in 1979, archeologist Lothar Haselberger discovered ancient Greek ‘blueprint’ plans at the Temple of Apollo at Didyma, Turkey etched into the marble walls and columns (Haselberger). Although the plans were not specifically Roman and were not found in or around the Pantheon, they nevertheless give us an idea of the knowledge of the ancients at the time and how they were implementing architectural design.  One of the many drawing plans found was a visual depiction of a column construction plan, shown below in Figure 13:

Figure 13 shows the profile of a half column (the other half would be a mirror image). The horizontal direction is drawn to full scale while the vertical direction is compressed by a scale of 1:16 (Haselberger). The horizontal hatch marks correspond to the (full scale) radii of the column as the height changes. This means that the difference between the lower radii and upper radii, ie, the taper of the column, can be directly measured from the image and translated to the construction. The curved line on the left of the radii hatches represents the entasis line of the entire column. This curve can be identified as a circle segment in the drawing’s compressed form. As the construction would be implemented using a (non-compressed) 1:1 ratio, the circle segment depicted would stretch into an ellipse segment and the plan could be directly translated into the physical column design. 

If we can reasonably assume that the Pantheon columns were also following a similar 1:16 compressed circle segment construction model, we can devise two separate (but similar) possible construction models for their column design (Grabhoff).  The first model is described with a fixed construction radius that fully defines the circle segment entasis. A geometrical model of this is shown below, with features labeled as follows: rD is the construction radius; h is the height of a single column; rU and rL are the upper radius & lower radius of column shaft; and a is the difference between upper & lower radius. 

Solving for the construction radius rD, we come up with:

This equation is actually the well known Pythagorean Theorem which directly relates the dimensions of a right triangle (link here for more details). From this model, we can deduce that the construction circle radius (and therefore the entasis curve) is based only on the difference between the upper and lower shaft radii (a) and the height of the column (h). This makes sense considering the measurement tools and knowledge available to the stonemasons carving and constructing the columns who did not have access to the more advanced technology used today. 

A second plausible construction model for the Pantheon columns would use a smaller radius construction circle which, like the previous model, runs through the upper and lower shaft radii. The center point of the circle segment in this model (point M in the diagram below), however, has been moved diagonally closer to the upper and lower radii points, and therefore the construction radius is smaller: 

In this construction plan, the column shafts do not start tapering until well above the base level, just above the dashed curve on the left. (The dashed curve shows where the taper would be if the straight segment of the columns did not exist.) As previously seen in Table 1, all of the Pantheon’s column shafts, regardless of their construction method, have approximately the same 7:8 (.875) proportion for the lower and upper radii. Therefore, to achieve this proportion and the measured tapering, this second model must compensate for the vertical section at the base by having a slightly more pronounced entasis curve towards the top. The smaller construction radius in this model gives the enasis a steeper curve and can thus ‘catch up’ and match the proportions of the first construction method (Figure 14) as well as Vitruvius’ proportions. In other words, because a circle with a smaller radius has a more pronounced “steeper” curvature (in relation to things around it), this second model uses the smaller circle radius to make up for the straight vertical bottom section. 

If we analyze each column individually, we can see that each column fits one of these two different construction models very specifically. Columns #3, 4, 12, 13 & 15 (see figure 7) fit this second model described above. Each of these columns has a straight portion towards the bottom and a more pronounced curve at the top (Grabhoff). The rest of the columns more closely follow the first method described because the entasis curve is present all throughout the columns. This difference in the column construction was most likely due simply to having (at least) two different groups in charge of shaping the columns (Grabhoff).

It can be inferred that the columns were very roughly constructed to shape in the quarries before shipment. This would have made shipping the easiest as the columns were as light as possible. However, because each column’s construction model is not directly aligned with the two separate origin quarries, it can be concluded that the precision shaping of the columns was done on site after the columns were delivered (Grabhoff). In other words, the columns that came from the same quarry did not all follow the same construction method meaning final column precision shaping likely occurred on site.

V. Conclusion

The massive scale, consistency, and near-perfection of the curved Pantheon columns are met by the surprisingly simple yet ingenious design plans. This was achieved using proportionality and ratios laid down by the famous Roman architect Vitruvius. Trigonometry and an understanding of circles, angles and radii were all tools that the ancients used to construct temples, and the builders of the Pantheon likely used these same concepts to develop a complex and sophisticated construction plans that resulted in Pantheon’s ancient tapering columns. These most fundamental mathematical principles were put to use to create one of the most mathematically sound and generally impressive temples in the world. 

Sources: 

Alibarti, L. and Alonso-Rodriguez, M.A. (2016). Shape and Layout of the Coffers of the Pantheon’s Dome in Rome: Measuring the Ideal Model

Artincontext. “Pantheon Rome – a Look at the Roman Pantheon’s Architecture.” Artincontext.org, 23 Aug. 2022, https://artincontext.org/pantheon-rome/.

“Engineering the Pantheon – Architectural, Construction, & Structural Analysis.” Brewminate, 29 May 2022, https://brewminate.com/engineering-the-pantheon-architectural-construction-structural-analysis/.

Grabhoff, G. and Berndt, C., Nexus 2016 Architecture and Mathematics, Ed. Duvernoy, S.  pp. 167-174 (2018), Decoding the Pantheon Columns

Haselberger, Lothar. “The Construction Plans for the Temple of Apollo at Didyma.” Scientific American, vol. 253, no. 6, 1985, pp. 126–33. JSTOR, http://www.jstor.org/stable/24967878. Accessed 15 Sep. 2022.

“The Pantheon Project.” MPIWG, https://www.mpiwg-berlin.mpg.de/project/pantheon-project.

Vitruvius Pollio, The Ten Books on Architecture, BOOK III, CHAPTER V: PROPORTIONS OF THE BASE, CAPITALS, AND ENTABLATURE IN THE IONIC ORDER. (n.d.). Retrieved September 4, 2022, from http://www.perseus.tufts.edu/hopper/text?doc=Vitr.+3.5&lang=original

“Vitruvius.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., https://www.britannica.com/biography/Vitruvius.