LOSSIAN BARBOSA BACELAR MIRANDA
Federal
Institute of Education Science and Technology of Piaui
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
