For the measurements used in the Parthenon (plan and facades) refer to:

http://athang1504.blogspot.com/2011/05/parthenon-more-measurements.html

Regards

Thanks Aineias Oikonomou. Angelopoulos' proposal includes fractions, powers and square roots of both integer numbers and classic mathematical constants (π and ф) and modern ones (he use Euler's e-number). Are you sure it's so difficult?

For my part, Francisco, I’m skeptical.

Those who see golden sections in the end elevations of the Parthenon never mention the side elevation. To harmonize part of a building and not all of it sounds like an oxymoron. Who would bother? The extravagance and diversity of the various overlaid explanations might also tell us something. Especially as these overlays don’t seem to acknowledge the well-known optical corrections in this building.

Unfortunately, there’s no documentary evidence and the question can’t be settled empirically. Phi advocates allow themselves a bit of a fudge factor in fitting the Golden section to actual rectangles and especially to non-rectangular things like a gable end. So, for example, one would never know whether a designer had used Alberti’s “sesquitertia doubled” (9:16) ratio or the irrational Golden ratio, or neither. And if the conceptual differences don’t matter, then surely phi doesn’t matter.

Ultimately, however, the doctrine of the phi rectangle is an element of sacred geometry. As such it is an article of faith, outside the realm of critical thinking. So if a phi rectangle has been revealed to someone in a fish or an assemblage of marble blocks or a crumpled newspaper, then this cannot be denied.

In this question you have to be skeptical. Many things have been said. However, some system of proportions was used in its construction.

Fortunately, we have the Parthenon archeology as unquestionable documentary evidence, and the question can be solved empirically. All you need is a digital record of the geometry ...

By the way, do you know digital planimetry of the Parthenon?

Article Parthenon Peristasis, Fibonacci Analysis

This may be a start. Five dimensions of the Parthenon's peristalsis precisely coincide with the integers 5,13 and 21, according to Penrose's measurements and Ugur Gulcugil 's verifications.

Are five measures enough to propose the divine proportion as a design criterion of the Parthenon? What happens with the rest of the measures? What system of proportions relates them? Do you think it is the golden number?

Dear Colleagues,

As far as the Parthenon's plan is concerned, the basic rectangle of the sides, measuring 30.88: 69.50 with an analogy of 4: 9, was formed by three rectangles with sides 3-4-5. Even the rectangle of the nave (sekos) -without the pilasters- has the same analogy of its sides 4: 9.

Berger determines the height-to-width ratio equal to 4: 9, and finally the height-to-width-to-length ratio equal to 16: 36: 81 (square of 4: 6: 9) (Berger 1980). In my analysis, it is clearly shown that the grid that defines the stylobate is based on a square measuring 25x25 Attic feet of 30.85 cm.

Based also on my present analysis of the plan of Prof. Korres, it appears firstly that the sekos has the specific proportions (5: 12) that Anne Bulckens (1999) notes, but a very different module (70 by 168 feet of 30.85 cm) and secondly that the ratio of the outer width to the sekos width is extremely close to the square root of 2, as it is in fact 10: 7.

The length of the inner temple (cella or naos) is 100 feet (ekatompedon) or 40 paces, thus equal to the outer width and its width is 70 feet (5x14). The grid of 14 Attic feet (30.85 cm) seems to be the grid which most accurately determines the axial distances of the columns and the overall design of the Parthenon’s plan.

https://dx.doi.org/10.1080/15583058.2021.1899339

With Kind Regards,

Aineias Oikonomou,

Dr. Architect Engineer

Dear Aineias Oikonomou,

Your proposal is very different from Angelopoulos's and simpler. So if Pythagorean triads (3-4-5) and Pythagorean quasi-triads (7-7-10) were used, isn't the golden number or another irrational value present?

I suggest to start from a different point of view. I think it is totally wrong to take just the facade in order to make fit some sort of hidden geometry in general, not just GR. Let's start with an integrated approach to design analysis that is from the accurate survey of the building made. With photogrammetry and laser scanner, then with the help of archaeologists and restorers it is necessary to get rid of "false friends". Once. You obtain three main drawings (plan sections, elevations) you have to reconstruct the whole design process. Areas, requirements, lengths with some functional relation. Find modularity, achieve the whole graphic pattern used by the architect. Then we may or not verify our hypothesis within a certain tollerance

Dear colleagues,

You have to bear in mind that the construction of the Parthenon, lasted from 448 to 438 BC, while the sculptural decoration was completed approx. in 432 BC.

It is more than physical that, the architect’s foot (30.85) and modular system that were used for the plan’s tracing differ from that of the façade. This is something well known also for the medieval great cathedrals.

You can always have simple Pythagorean geometry on the plan, and harmonic ‘‘golden’’ proportions on the façade. So, according to my point of view, if Phi was ever used in the Parthenon, it was definitely used only for the façade, and not for the plan.

I agree that all measurements have to be considered, not just the facade. We have long awaited the publication of a photogrammetric or a laser scanner survey of the Parthenon. In any case, it must be the most measured building in history (F. Penrose and N. Balanos). There are many dimensions to analyze. For this reason, theories based on few dimensions are not rigorous. Honestly, I believe that it is not based on the Pythagorean triads (regular grids) or on the Fibonacci sequence of the golden number (regular grids too), but it is based on another simple geometric proportion.

I think that when considering a module or a fixed proportion between sides of rectangles present in the Parthenon must be taking in account if they were really usseful for the laying out. The first and most important step should be the setting out of the plan on the stylobat. I found that the Berger's system of 81 x 36 in plan modules, not only satisfy a mathematical aim (9x9 and 9x4 and so the proportion 9:4) but it permits a very accurate and simple laying out of the columns. Its necessary to draw the resultant grid (which as far as I know It has not ever made) and to see if it matchs with them. According to the measures of Atanasius Orlandos and Athanasius Angelopoulos the diferences among interaxes resultant of Berger's hypotesis are as average lower than 5 mm. Besides Athanasius Angelopoulos' drawing shows that the real columns are not distributed with a preciselly regularity. So at least the system of Berger should permit the setting of the columns with a more than reasonable precision, including the corner ones.

Dear Dr. Rafael García García,

It is true that the axis to axis distance of the Parthenon’s columns is approximately equal to 5 modules of 85.7 cm. In that way, the proposed grid arrangement of 36x81 modules could have been used for the laying of the stylobate and also for the placement of the columns.

The only problem is that the module of 85.7 cm is an artificial one (as far as we know) and derives from the division of 100 attic feet of 30.85 by 36 (the overall width of the stylobate: 36) - or the division of 225 attic feet by 81 (the overall length of the stylobate: 81).

In fact, the mean axis to axis distance of the columns (431.9 cm) is exactly equal to 14 attic feet of 30.85 cm.

In that way, the most simple and logical explanation is that:

a) for the geometrical tracing of the stylobate, a 4x9 grid of 25 attic feet was used and also

b) for the placement of the columns, a 7x16 grid of 14 attic feet was used.

Please refer to https://dx.doi.org/10.1080/15583058.2021.1899339

With Kind Regards,

Aineias Oikonomou,

Dr. Architect Engineer

Dear colleagues,

The system of proportions must justify all dimensions. So far we have divided few measurements into equal parts, checking that their proportions have different valid fractional solutions. What if the measures were not always made up of equal parts?

It is an element from the Greek civilization. The Greek mathematician and sculptor Phidias used the golden ratio when designing the Parthenon, which still stands on the Athenian Acropolis in Greece [source: Horn]. For centuries, it was widely believed the Parthenon, with its appearance of balanced, straight lines, was also built according to the golden ratio.