How to Build a Cathedral: With a square, a circle and a diagonal, you can generate an entire cathedral

  How To Build A Cathedral



Notes from How to Build a Cathedral: BBC 4

• Gates of Heaven
• Everything in existence has symbolic value
• Man's actions images of Divine Order
• Architects = Master Masons
• 1 : √2

There are a number of algorithms for approximating the square root of 2, which in expressions as a ratio of integers or as a decimal can only be approximated. The most common algorithm for this, one used as a basis in many computers and calculators, is the Babylonian method of computing square roots, which is one of many methods of computing square roots.
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The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1.

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Another proportioning system is the ratio of (Square root of 2) : 1. The simplicity of the derivation (square root of 2 is the diagonal through a square of side length 1) is paralleled by the ease of maintaining the proportion through division or multiplication of the proportioned rectangles. 

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Clearly the golden section proportion is closely connected with the square, the most neutral rectangular proportion (1 : 1) imaginable. (The "Modulor" books are square!) Compared with other proportions, the golden section rectangle is relatively long. That creates a certain tension between golden section and square, which may contribute to the interest that this proportioning scheme can maintain (see Corbu's Modulor), especially when compared to schemes that use the square as only proportioning scheme (see O.M.U.).

Now, does that constitute any understandable reason to connect golden mean proportioning inseparable with beauty? Without doubt: No. Because of the non-linear nature of the golden section, as clearly demonstrated in the Modulor derivations, it is possible to find some base length and some subdivisions close enough to the ratio of the golden section in anything that may be perceived as beautiful. But that may have to do with the underlying structuring into non-equal divisions that establish scale and generate more interest because of the increased amount of detail that is generated or that is cause of the inequal divisions. 

Another proportioning system is the ratio of (Square root of 2) : 1. The simplicity of the derivation (square root of 2 is the diagonal through a square of side length 1) is paralleled by the ease of maintaining the proportion through division or multiplication of the proportioned rectangles. The sum of two rectangles of proportion (Square root of 2) : 1 long side by long side is (Square root of 2) : 2. Divided by the square root of two we arrive at 1 : (Square root of 2), the same ratio as the two rectangles that were added together, only with a change of orientation. [source]

\This is suggested by the Roman architectural forerunner Vitruvius and his discussion of the application of the side and diagonal of a square. He pays great homage to Plato for stating and showing in Meno that the square on the diagonal of another square has twice the area of the smaller square. Vitruvius emphasizes the great utility of this result. He notes that this surmounts an arithmetical impossibility (i.e., writing down the square root of two) with a geometric solution. This ascribes to the ratio of the side of the square to itsdiagonal a special status—it is a profound principle. Its profundity, association with Plato as noted by Vitruvius, and long-standing traditional use may have given a reverence and prestigeto this principle during the medieval period.


The rediscovery of the mathematical schema, including the side of the square and its diagonal, employed at a specific church is a challenging problem within architectural history. As an example, Durham Cathedral, an Anglo - Norman Romanesque church, built 1093–1130/1133, in the northeast of England has many mathematical points of interest. Consider the constructional- geometric procedure for the major lengths of the building and the widths of the transepts. A design motif that was common, though not standard, in the large Anglo-Norman Romanesque churches was basically, in terms of interior lengths, that the west tower/nave (HD in Figure 6) to the west tower/nave/crossing/choir up to the chord of the central east-end apse (HB in Figure 6) is in the same ratio as the side of the square to its diagonal or equivalently, the half-diagonal to the side of the square. A slightly different situation appears at Durham Cathedral. The “cut-point” possibly should be the interior east wall of the transept chapels (C in Figure 6), rather than using the interior west wall of the transept or the transept piers (D in Figure 6). The length of the choir up to the chord of the central east-end apse (BC in Figure 6) to the width of the choir (AB in Figure 6) are also in the ratio of the side to the half-diagonal of the square. One of the other common larger scale relationships, for Anglo - Norman churches with attached monasteries including Durham, is that the length of the cloister’s side adjoining the nave (DG in Figure 7) to the length of the tower and nave (HD in Figure 7) equals the ratio of the side of the square to its diagonal, or equivalently the ‘half-diagonal’ of the square to its side. The thorough application of the square’s side and diagonal also occurred in the ground plan of the south transept and suggests a relationship between the full interior width of the south transept and the interior width of the nave and its north and south aisles [Figure 8]. These relationships are examples of the application of practical or constructive geometry in the design and laying out of Durham Cathedral. [source .pdf]

• Proportion, Ratio and Symmetry = spiritual qualities the reflected the harmony of creation
• With a square, a circle and a diagonal, you can generate an entire cathedral
• Arches = basic building block of cathedrals
• Scared Theater: architecture, sculpture, music all combined in harmony
• Sacred Scenes on facades, in windows
• Skeletons of Stone 

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The Parthenon: an objective basis of beauty that mirrors the proportions of an ideal human body

When the Parthenon was completed in 432 B.C., the gleaming marble temple atop the Acropolis would have been visible from almost anywhere in the ancient city. [source]

Nova: Secrets of the Parthenon

Notes from the transcript (emphasis mine):

•  NARRATOR: It is the Golden Age of Greece, a unique window of time that gives birth to Western ideals of beauty, science, art and a radical new form of government: democracy. To immortalize those ideals, the Greeks build what will become the very symbol of Western Civilization, the Parthenon.
• JEFFREY M. HURWIT (University of Oregon): It was the physical embodiment of their values, their beliefs, of their ideology.
•  The Parthenon is a 20,000 ton, 70,000 piece, three-dimensional jigsaw puzzle. And worse, it's a puzzle that doesn't include instructions. No one has found anything resembling architectural plans.
•  JEFFREY M. HURWIT: The Parthenon was the greatest monument in the greatest sanctuary in the greatest city of classical Greece. It was the central repository of the Athenians' lofty conception of themselves and the physical—marble—embodiment of their values, their beliefs, their myths, their ideologies. It was as much a temple to Athens and the Athenians as it is to their patron goddess, Athena Parthenos.
•  He [Pericles] spearheads an ambitious campaign to rebuild Athens and ushers in the Golden Age of Greece, a unique window of time that establishes Western ideals of beauty, science, art and a radical new form of government: "demos" meaning "people," and "cratos," "power"— people power, or democracy.
• In a powerful statement of their self-confidence, the people of Athens vote to rebuild the Acropolis, and at its center, a building to embody their ideals, the Parthenon.

The Plan of the Parthenon

• CATHY PARASCHI: On the beautiful island of Naxos, we see this temple which is one of the early archaic Greek temples, made of stone.
  NARRATOR: The temple of Demeter was constructed about 100 years before the Parthenon. It, too, was built with few right angles or straight lines.
  
•  MARGARET LIVINGSTONE: This is another classical illusion. If you have two straight lines, if you add converging lines, these two lines seem to bow in the middle. So if the floor of the Parthenon has converging cues as to depth and perspective, you could have an illusory sag in the floor of the Parthenon. NARRATOR: Perhaps to compensate for the illusory sag, the builders left extra marble in the middle. The ancient Greeks realized that to construct a building that appears perfect, they would have to come up with a design that tricks the eye. What they invent is a system of optical refinements.
  CATHY PARASCHI: Their concern was the visual perfection of the building.
  NARRATOR: This small stone temple, on Naxos, provides evidence of the Greeks' keen observation over hundreds of years.
  CATHY PARASCHI: Here we can see the first optical refinements already experimented by the people building the temple. Here lies, literally, the D.N.A. of the Parthenon.
•  Architect Mark Wilson Jones believes the enigmatic Salamis Stone, depicting an arm, hands and feet, may be a conversion table for the different measuring systems, Doric, Ionic and Common. MARK WILSON JONES (University of Bath): This is a tracing I've done that shows the stone, and you can immediately see how the main measures work. We have this foot rule here. That's 327 millimeters, more or less, the Doric foot. And here you have a foot imprint that's roughly a 307-millimeter-long foot, which we tend to call the Common foot. And there are, in fact other feet. For example, this dimension here is one Ionic foot. So there is a, kind of, whole network of different interrelated measurements here.
  NARRATOR: The Salamis Stone represents all the competing ancient Greek measurements: the Doric foot, the Ionic foot, and, for the first time, the Common foot—virtually the same measurement we use today.
  Wilson Jones finds evidence of all three measuring systems in the height of the Parthenon.
  MARK WILSON JONES: That distance is, at one and the same time, 45 Doric feet, that's the ruler on the relief; it's also 48 Common feet, which is the foot imprint; and it's 50 Ionic feet, all at the same time. And these are quite exact correspondences.
  NARRATOR: So the Salamis Stone may have provided a simple way for ancient workers from different places to calibrate their rulers and cross-reference different units of measurement.
  But the Salamis Stone may also be a clue to how the ancient Greeks were using the human body to create what we now regard as ideal proportions.
  MARK WILSON JONES: What's extraordinary about this, is that at the same time as being a practical device, it's also a kind of model of theory, architectural theory, that a perfect, ideal human body, designed by nature, is a kind of paradigm for how architects should design temples.
  NARRATOR: Among the first to record that Greek temples were based on the ideal human body was the Roman architect, Marcus Vitruvius. He studied the proportions of temples like the Parthenon, in the first century B.C.E., 400 years after it was built.
  MANOLIS KORRES: Vitruvius's work gives us the overall frame which is necessary to understand the system of proportions of the Parthenon.
  NARRATOR: According to Vitruvius, Greek architects believed in an objective basis of beauty that mirrors the proportions of an ideal human body. They observed, among many examples, that the span from finger tip to finger tip is a fixed ratio to total height, and height is a fixed ratio to the distance between the navel and the foot.
  Two thousand years after the Parthenon, another artist was also searching for an objective basis of beauty.
  
  
  
  
  
  
  
  MARK WILSON JONES: This is a very famous image. It's drawn by Leonardo da Vinci, in the Renaissance, and it's based on Vitruvius's description of the ideal the human body. And he encapsulates this idea of its theoretical importance. And what's really interesting for us is that when we superimpose the Salamis relief on this drawing, we see that there's a remarkable correspondence. There are differences, but it's the same principle. You have the same interest in the anthropomorphic principle of getting a kind of sacred fundamental justification for these measures.
  NARRATOR: Da Vinci's ideal Renaissance man famously stands in a circle surrounded by a square. Da Vinci named this image "Vitruvian Man" after the Roman architect.
  The ratio of the radius of the circle to a side of the square is 1 to 1.6. That ratio is sometimes attributed to the Greek mathematician, Pythagoras, who lived 100 years before the building of the Parthenon. In the Victorian age, it became known as the "golden ratio." It was a mathematical formula for beauty. For centuries many scholars believed the golden ratio gave the Parthenon its tremendous power and perfect proportions. Most notably, the ratio of height to width on its facades is a golden ratio.
  Today the golden ratio's use in the Parthenon has been largely discredited, but Manolis Korres and most scholars believe another ratio does in fact appear in much of the building.
  MANOLIS KORRES: The width, for instance is 30 meters and 80 centimeters; the length is 69 meters and 51 centimeters, the ratio being 4:9.
  NARRATOR: The 4:9 ratio is also found between the width of the columns and the distance between their centers, and the height of the facade to its width.
  JEFFREY M. HURWIT: The Parthenon, like a statue, exemplifies a certain symmetria, a certain harmony of part to part and of part to the whole. There's no question that the harmony of the building, which is clearly one of its most visible characteristics is dependent upon a certain mathematical system of proportions.
  MARK WILSON JONES: For the Greeks, there was nothing better than a design based on the coming together of measures, of proportions and harmonies and shapes. It's rather like an orchestrated piece of music in which the harmonies of the various instruments are, sort of, fused together in a wonderful, glorious, orchestrated symphony.
  NARRATOR: With something like the Salamis Stone's use of the human body as units of measure, and the idealized human form to define perfect proportions, the Parthenon literally embodies the words of the Greek philosopher Protagoras, who lived in Athens during the construction of the Parthenon, "Man is the measure of all things."
  
•  NARRATOR: One of the subtlest of these curves can be found on the Parthenon's columns. LENA LAMBRINOU: If we pull a string, we can see that from the middle of the column and up, we can see a curve, a very slight curve.
  NARRATOR: The curve is gentle, starting a little less than halfway up and tapering again near the top. It's an optical refinement called "entasis."
  CATHY PARASCHI: Entasis means tension. It gives life to the column visually. It resembles an athlete trying to lift the weight, even the deep breadth of the swelling of its chest. It is no longer dead stone. It has life in it. It has pulse.
  JEFFREY M. HURWIT: These deviations from the straight, from the perfectly vertical, from the perfectly horizontal are analogous to the curvatures and the swellings and the irregularities of the human body. And in that sense the Parthenon strikes me as being a sculptural as well as an architectural achievement.
  NARRATOR: The entasis curve on the side of the column is so subtle and so slight, restorers can only draw it by computer. For the ancients to have drawn it at full scale, they would have had to set their compass at an impossible radius of nearly a mile. How they constructed the curved columns was one of the last great riddles left by the ancient Greek temple builders.
  The answer literally "came to light" at Didyma, 200 miles from Athens, in what is, today, Turkey. Here, a team of German archaeologists was exploring the ruin of the Temple of Apollo.
  Built at the time of Alexander the Great, 150 years after the Parthenon, it was the biggest Greek temple ever conceived: 120 columns, each one more than twice the height of the Parthenon's.
  The German team noted an optical refinement, a curvature, on the base of the temple, similar to that of the Parthenon.
•  NARRATOR: The Parthenon was completed in 432 B.C.E. As the ultimate expression of Athenian ideals, the temple is adorned with mythological battles of victory: justice over injustice, civilization defeating barbarity, order prevailing over chaos. And, perhaps for the first time on a Greek temple, the Athenians, mere mortals, depict themselves alongside the gods. JEFFREY M. HURWIT: And so, if the human beings, the Athenians on the Parthenon frieze, are elevated near the rank of gods, the gods are represented in a way that makes them human. And the difference between gods and mortals, between Athenians and the Olympians is not one so much of kind, as of degree. This is an extremely humanistic way of representing themselves.
  

A full-scale, 12-ton replica of the colossal Athena Parthenos is the centerpiece of a recreation of the Parthenon in Nashville, Tennessee. While the replica is gilded in gold leaf, the fifth-century statue was plated in some 2,400 pounds of solid gold. [source]

Notes from the Interview with Jeffrey Hurwit, a professor of art history at the University of Oregon and expert on the architecture of the Acropolis:

• Hurwit: I think the artists and the architects of any culture strive for what they consider the perfect expression of their ideas of beauty or their beliefs. Clearly the Athenians, and the Greeks in general, had notions of perfection; they had notions of what they called symmetria, the harmonious relationship of part to part and of the part to the whole. And the Parthenon, like certain statues created in the fifth century, is an expression of these ideas.
• So the Parthenon was an attempt on the part of Pericles and Athens to assert the city's cultural, political, and military dominance over the rest of Greece and the Aegean. Pericles called Athens "the school of Hellas," an education unto Greece, and the Parthenon was intended to be the main text in the curriculum.
• Hurwit: The Parthenon was built completely of marble from the base of the temple to its roof tiles. It had two large-scale pediments, each filled with over 20, larger-than-life-sized marble figures in compositions that extolled Athena and her power. It was adorned with 92 exterior sculptured metopes [decorated rectangular panels near the top of the temple]. It also had an Ionic frieze running around the top of the cella walls [the interior walls of the building] representing an idealized and pious Athenian citizenry. It had great roof ornaments, acroteria, in the form of victory figures, Nikai, alighting as if descending from heaven.
  And the great statue of Athena Parthenos inside the Parthenon, made of gold and ivory, held in the palm of her hand another image of Nike, some six feet tall, offering it to the Athenians as if to confirm their military predominance over the rest of Greece.
• The statue had all kinds of sculptural decorations. She not only held in her hand a statue of victory, but the shield on which she leaned with her left hand was adorned on the outside with a battle of the Greeks against the Amazons. And on the inside, there was a representation of the gods fighting the giants. And on the sandals of the statue there was a representation of the Greeks fighting the centaurs. All of these mythological battles represent the struggle between the forces of justice and injustice, of civilization against barbarity, of order versus chaos.
  These three battles (as well as episodes from the Trojan War) were also represented on the outside of the Parthenon, on the metopes of the building. So there is a thematic unity from the exterior to the interior of the Parthenon. This theme of victory, of order over chaos, would have been drilled into any visitor to the Parthenon. These mythological battles between gods and giants, Greeks and centaurs, Greeks and Amazons were regarded as mythical allusions to historical victory, the recent victory of the Athenians over the Persians.
  One interesting thing about the Athena Parthenos, however, is that this glorious expression of Athens' patron goddess stood on a base decorated with a representation of the birth of Pandora. Pandora is best known through Hesiod, the epic poet, as responsible for letting loose evils from her famous box. It's curious that Pandora, whom Hesiod calls a beautiful evil, should adorn the base of a statue that otherwise expresses Athenian might, wealth, and power.
  I think an intellectual member of Athenian society might have wondered whether Pheidias wasn't attempting to perhaps slightly undercut the glorification of Athens, that this great statue hinted that Athens might have to deal with circumstances not of its own choosing.
• Hurwit: There is no question that the Greeks conceived of their architecture in anthropomorphic terms. The most obvious expression of this is found on the Acropolis itself, where in the south porch of the Erechtheum, columns take the form of six maidens, or caryatids. Vitruvius, the Roman architect, talks about how the Doric order is masculine and the more elaborate, thinner Ionic order is feminine. There is no question that in the Greek mind, there was an analogy between the architectural form and the human form.
• Q: Did the ancient Greek architects have a mathematical formula for beauty?
  Hurwit: Well, the idea of symmetria, the harmonious relationship of one part of a body or of a structure to another part of a body or a structure, seems to have been of paramount importance to the Greeks, not just architects but sculptors as well.
  In the middle of the fifth century, while the Parthenon was under construction, Polykleitos of Argos, a great sculptor, actually created a statue, which we call the Doryphoros, or Spear Bearer, to be the embodiment of a geometrical or mathematical system of proportion. It is likely that Greek architects also strove for an architectural symmetria—for some formula, some geometrical or mathematical proportional system that would enable them to achieve a perfect harmony of part to part and of parts to the whole.
  Q: So is it fair to say that the Parthenon's beauty comes from such a formula?
  Hurwit: The Parthenon, like a statue, exemplifies a certain symmetria. Its symmetria largely depends upon the 9:4 ratio, which is present in various dimensions of the building—the length of the stylobate [the platform that forms the base of the building] to the width of the stylobate, the width of the stylobate to the height of the column and entablature [the top section between the columns and roof] together. There's no question that the harmony of the building, which is clearly one of its most visible characteristics, is dependent on a certain mathematical system of proportions.
  Because we assume that the Parthenon was perfectly designed, we almost have a compulsion to focus on the basic mathematical ratio that governs the design. But I think it is the irregularities and variations that are the most interesting things about the Parthenon and that give it an organic feel. 

Nashville, Tennessee is home to a full-scale replica of the Parthenon, including a recreation of its original Athena statue.