Roman Temple of Évora: a golden double-parallelepiped

LOSSIAN BARBOSA BACELAR MIRANDA

Federal Institute of Education Science and Technology of Piaui

lossianm@gmail.com

In this brief communication we prove that the Roman Temple of Évora possesses, in relation to its external measurements, the form of a golden parallelepiped with constant of proportionality equal to (1+√5)/2. We have also proved that this temple, in relation to its internal measurements, has the form of a generalized golden parallelepiped with constant proportionality equal to the real root of the cubic equation x³=x²+x+1.

This work is together with Liuhan Oliveira de Miranda and Lohans de Oliveira Miranda.

1. Definitions

We call a golden parallelepiped to any parallelepiped whose sides measure z, φz,  φ²z, being φ=(1+√5)/2 and z∊𝐑₊*. If the sides are equal z, tz,  t²z  and, furthermore t³=t²+t+1, we say that the parallelepiped is golden generalized with a constant of proportionality equal to t. In this case t≅1.8393  is the real root of the cubic equation x³=x²+x+1. These definitions can be generalized naturally to any dimension n. In this case the equation is 1 + x + ... + xⁿ⁻¹ = xⁿ.

2. Main dimensions of the Roman Temple of Évora

Theodor Hauschild attributes 24 meters long and 15 meters wide to the temple (Theodor Hauschild, 1991, p.107). In figure 7 on page 113 of this same work, the author makes it clear that the distance from the base of the top layer of stones from the podium to the top of the architrave is approximately equal to 9.3 meters. These dimensions make it a golden parallelepiped because, 24/15≅1.6  and 15/9.3≅1.613. These two proportions are very close to (1+√5)/2≅1.618.

If in figure 3 of page 109 of the aforementioned scientific work, if we draw four tangent lines to the column bases (passing through the inner part of the temple) we will obtain an ABCD trapezoid with sides approximately equal to: AB=20.977m; BC=10.986m; CD=20.977m and DA=11.433m. We have AB/BC≅1.909  and AB/DA≅1.834. These ratios are close to the real root of x³=x²+x+1, which is approximately equal to 1.839. The ratios between the interior widths of the temple and the length of the shaft, which is approximately 6.2m, are:  11.433/6.2≅1.844  and 10.986/6.2≅1.772.

The dimensions of the podium mentioned by Theodor Hauschild, namely 3m, 15m and 24m, together with the 9.3m between the base of the highest stone layer of the podium and the top of the architrave, make it clear that the structure of the temple, seen by outside, is that of a golden parallelepiped. Internally, the structure is that of a generalized golden parallelepiped with constant proportionality equal to the real root of x³=x²+x+1. The Maison Carrée temple in Nîmes, which bears a great resemblance to that of Évora, has a rectangular floor and has an interior width of 10,986m. The arrangement of the outer columns, thirty in number, is the same as suggested by Vitruvius (VITRUVIUS, 1914. Book IV, p.115, Chap. IV).

3. Geometric characteristics of golden parallelepipeds

When z=4  the numerical values of the total areas of the parallelepipeds are equal to the numerical values of the volumes. The areas of the faces are in the same proportion as the edges of the parallelepipeds. The total areas are equal to 4φ³z² and 4t³z², and the volumes, equal to (φz)³ and (tz)³, indicating that the volumes are cubable with ruler and compass, the areas of the faces are all squareable with ruler and compass and the parallelepipeds are constructibles with ruler and compass from the previous construction of φ and t. The latter, however, is not constructible with ruler and compass.

References

Theodor Hauschild. EL TEMPLO ROMANO DE EVORA. TEMPLOS ROMANOS DE HISPANIA CUADERNOS DE ARQUITECTURA ROMANA, VOL. 1, 1991, PÁGINAS 107-117. Available on revistas.um.es/car/article/download/68101/65561  

VITRUVIUS. THE TEN BOOKS ON ARCHITECTURE. HARVARD UNIVERSITY PRESS, 1914. Translated by Morris Hicky Morgan, Book IV, p.115, Chap. IV. Available on  http://academics.triton.edu/faculty/fheitzman/Vitruvius__the_Ten_Books_on_Architecture.pdf