Splines
Front View of a #Peripteral (#Sexastyle) #Colonnade with #IonicColumns arranged in #Eustyle #intercolumniation.

Ever since prehistoric architects at #Stonehenge designed rock columns and labored to lift the heavy rocks atop them, humanity has been fascinated with columns and entablatures, whether they were known by that name or not, and the designs have continued to evolve.

#Vitruvius described five classes of temples, designated as follows: "#pycnostyle, with the columns close together; #systyle, with the intercolumniations a little wider; #diastyle, more open still; #araeostyle, farther apart than they ought to be; #eustyle, with the intervals apportioned just right." So, what does it mean to have "intervals apportioned just right?"

Aside from the subjective aesthetic criteria mentioned in https://pixelfed.social/p/Splines/802974815166948953, such as avoiding columns that "look thin and mean" and shafts that "look swollen and ungraceful," there were practical considerations, such as the gap being too wide to support heavy stone entablatures.

There was also the practical matter with intercolumniation that was noo narrow. "When the [temple] matrons mount the steps for public prayer…, they cannot pass through [narrow] intercolumniations with their arms about one another, but must form single file; then again, the effect of the folding doors is thrust out of sight by the crowding of the columns, and likewise the statues are thrown into shadow; the narrow space interferes also with walks round the temple."

So, intercolumniations of 2 column diameters (4µ) or less, as in #pycnostyle and #systyle, were considered too narrow. Likewise 3 column diameters (6µ) or more, as in #diastyle and #araeostyle, were too wide. The consensus sweet spot was 2.25 diameters (4.5µ) between column shafts at the bottom (6.5µ axis-to-axis), except for the two middle columns where the spacing was 3 column diameters (8µ from axis-to-axis).

The image shows this variable intercolumniation.
Splines (@[email protected])

Classical #Intercolumniation is a complex topic with myriad rules that were developed after lots of experimentation by Greek as well as Roman architects. The primary purpose of #colonnades or multiple columns was both practical (to support heavy weight) and aesthetic ("imposing effect of high relief" as #Vitruvious wrote in https://www.gutenberg.org/files/20239/20239-h/20239-h.htm#Page_78). The architects paid keen attention to the thickness of columns as well as the spacing between them. They experimented with "columns close together, … with intercolumniations a little wider, … more open still, … and farther apart than they ought to be," until they settled on column spacing "with the intervals apportioned just right." With variable spacing came the need for adjusting thickness of shafts which had to be "enlarged in proportion to the increase of the distance between the columns," without which, "the column will look thin and mean, because the width of the intercolumniations is such that the air seems to eat away and diminish the thickness of such shafts." They also wanted to avoid proportions that would make the "shaft look swollen and ungraceful, because the intercolumniations are so close to each other and so narrow." So, it wasn't just the design of an individual column that was parameterized with the single parameter µ, first mentioned in https://pixelfed.social/p/Splines/790357912719769731 and further described in https://pixelfed.social/p/Splines/790417950261292263. Intercolumniation was also codified in terms of column width (effectively parameterized by µ). Beyond the appearance, there were practical considerations. While walls (which came before columns) were made of bricks, the dominant material for columns and entablatures were stone. As such, regardless of the width of individual columns, the gap could not be increased beyond certain limits, for the spans above the column could break. Materials such as timber for beams allowed more experimentation.

Pixelfed
Mar 6, 2025 at 1:38amWeb
Show threadSplinesMar 6, 2025
@charlesrkiss "there is an ideal range of values… the largest span… would determine the remaining openings. And this would depend on the material strength." Exactly. Now we can model #elasticity using #HookesLaw (deformation is directly proportional to the deforming force or load — small displacements of a material's constituent molecules, atoms, or ions from normal positions is also proportional to the force that causes the displacement) and #YoungsModulus (measure of the ability of a material to withstand deformation when under tension or compression). Architects back then didn't know those, and had to rely on costly and sometimes catastrophic experimentation lasting years or decades.

#Vitruvius wrote, "With regard to burnt brick, nobody can tell offhand whether it is of the best or unfit to use in a wall, because its strength can be tested only after it has been used on a roof and exposed to bad weather and time—then, if it is good it is accepted. If not made of good clay or if not baked sufficiently, it shows itself defective there when exposed to frosts and rime. Brick that will not stand exposure on roofs can never be strong enough to carry its load in a wall. Hence the strongest burnt brick walls are those which are constructed out of old roofing tiles." It is implicit that roofs sometimes had to collapse for architects to learn that the material or the design dimensions had to be rejected.
Show threadSplinesMar 7, 2025
I responded to this earlier, but the comment was somehow mangled and deleted :-(
Show threadferrousironicMar 7, 2025
Thank you again for this excellent documentation! I have not previously seen or imagined any stories on stone "deformation," or any reporting on #creep analysis across a period of centuries.

Here is a question: Extend this idea of stone deformation not over hundreds of years, but over THOUSANDS of years, on the pyramids of Eygpt, and other historic monoliths.

Given all the purported "refined shaping" of stone to incomprehensible tolerances, could you imagine the super fine spacing between these stones were actually due to creep, and not craftsmanship — creep on the order of 1-2 mm per 1000 years?

What would be a good approximation on such stones, in your opinion? It's probably not zero.
Show threadSplinesMar 6, 2025
@charlesrkiss There is some perverse reasoning in "Brick that will not stand exposure on roofs can never be strong enough to carry its load in a wall."
Show threadSplinesMar 7, 2025
@charlesrkiss Any material can be deformed under the right conditions. Rubber is flexible. Metals are both malleable and ductile. Rocks can be liquefied under intense heat and manifest themselves as molten lava. Stones that resist deformation under force will still break when overwhelming force is applied.

Orthogonal to deformation is erosion, and in my opinion, the grandest spectacle of erosion is in Grand Canyon and other canyons in Colorado and Utah. I'm sure there are others, but I haven't seen them. Yosemite Valley carved out of hard granite by glaciers is another example of nature as both designer and sculptor. River water running over rock also gives it refined shaping to smooth, incomprehensible tolerances.

I have seen human artifacts and #architecture preserved in Rome, Florence, and Paris — and they are awe-inspiring, until you realize that their foundations were laid a millennium and a half earlier. Not sure if the Greek and Roman architects of 2500 years ago had seen the rock arrangements at #Stonehenge, which is more than 5000 years old. If they had, how did they react? Did they scoff, or were they bemused, or did they marvel at the sheer human will, we will never know.
Show threadSplinesMar 7, 2025
Show threadSplinesMar 7, 2025
PixelFed commenting does not behave the way I want it to behave, or even as I believe commenting should work :-) So responses get hidden unless you repeatedly hit "Load More Comments" at the bottom. Also, I cannot add both text and images in the same comment using the web UI. So, @charlesrkiss, I posted a screenshot of #Funicular Forms that cited Robert #Hooke's work in the 17th century after whom #HookesLaw is known. The whole PDF is available at https://www.routledge.com/rsc/downloads/Chapter_3_-_Fabric_Formwork.pdf. Also, @Xover0, with your work on chains, you might find this interesting too
Show threadX Over ZeroMar 8, 2025
Yes! We work with funicular forms pretty often in our work. In teaching and researching architecture we also use them, sometimes even in multiple directions. We were inspired by Gaudi's use of them in many projects. Not to mention Otto, Eisler, and Dieste more recently.
Show threadSplinesMar 8, 2025
Sorry, the #entablature is missing the #dentils.
Show threadX Over ZeroMar 7, 2025
A friend who is an architect trained in India has said that the ideal proportions for buildings of many types were put down in the Vedas, the holy texts of Hinduism. He explained that - especially in a time before our modern split between the role of law and code and that of religion - no one would have followed it were it just "rules." But by elevating to a religious dictate people took seriously what were actually best practices driven by the performance of the materials, etc.

This is even more profound in the need for and thus codification of these rules in religious architecture whether Greek or Hindu. The direct descent of the required spatial and material performance criteria from the liturgical demands and the human performance of it is really amazing.