Complete #ModernIonicCapital in-situ with #column and #entablature.

#Arch with #Ionic #Entablature and #Keystone Detail

The #dentils arrangement we saw in https://pixelfed.social/p/Splines/791013152244518907 goes well with the classic entablature #profile we saw in https://pixelfed.social/p/Splines/790888454384861893, and they both go well with #simpleIntercolumniation, also known as #architravato.

However, with arches, the entablature profile has to be adjusted a bit so that the dentils arrangement is as shown here. The shape, size, and gap between individual dentils remains the same, but a crucial difference is that the dentils at the #outer corners touch each other.

As I mentioned in https://pixelfed.social/p/Splines/803615973439041638, in #arcadeIntercolumniation, the entablature is repeated on the wall behind the half-column. It doesn't end at the columns and has two "outside" corners and one "inside" corner. While the dentils at the outer corners touch each other, there is a single dentil in the inside corner that is shared by both walls.

A bedrock principle of dentils (like that with #flutes and with eggs in the #EggsAndDarts motif) is that when viewed directly from the front or the sides, a dentil must be centered on the column axis. It is this principle that forces us to adjust the profile of the entablature in arcade intercolumniation giving us the arrangement shown here.

The image also shows the detail of the decoration in front of the #keystone. The most easily recognizable component of that is the large #volute, which is the exact same size as the ones on the #capital. The smaller volute is exactly half the size of the larger one. It is mirrored, rotated and put within a bounding rectangle whose height is exactly 2µ (288 units). The channels of both volutes are bridged with #sinusoids derived from half turn of #helix curves that have been flattened.

This motif in the keystone, where volutes of different sizes are combined with sinusoids is very common. It will be seen in the #modillions of the #modernEntablature.

#Arcade #Intercolumniation without #Pedestal

In https://pixelfed.social/p/Splines/803089629244302486, we saw #simpleIntercolumniation, also known as #Architravato.

Roman architects combined columns with walls thick enough to bury half of the column width inside the walls and added arches to them for better load distribution. An arcade (multiple arches) can be run in series along a single wall, or in parallel to form a walkway. They can also be combined in both series and parallel configurations, perhaps the most famous of which is the #Colosseum in Rome.

In the Colosseum, the outer walls follow an elliptical curve (even though it looks circular from the outside), and it has multiple tiers of arches in series. The interior has arches in concentric passageways in the lower tiers giving it a lattice-like design.

Because arches distribute the load from above, they allow for wider intercolumniation. The rules for #ArcadeIntercolumniation differ depending on whether the columns have pedestals or not.

Besides the arch itself, which is part of the wall, the figure shows some new architectural elements.

The narrow part of the wall immediately behind a column is known as a #pier. The visible face of a pier between a column and the opening under the arch is known as #alette. The base of the pier has a molding, the flat part of which has the same height as the column base (µ) while the rest follows the #fillet and #cavetto or #conge of the #shaft.

As we move up the pier, there is a horizontal molding known as #impost just below where the arc of the arch starts. The impost wraps around on the sides of the pier.

Around the arc is a circular molding known as #archivolt, the bottom portion of which has a #fascia that is aligned with the face of the wall.

The wall itself extends all the way to the top of the #entablature. It is worth noting that the entablature is repeated on the wall. It doesn't end at the columns and has two "outside" corners and one "inside" corner.

Classic #IonicCapital #Tectonic Surfaces Plan

We already made the 8 unit tall #fillet at the bottom of the #capital a part of the #shaft in https://pixelfed.social/p/Splines/791794072490907090. So, excluding that, the remainder of the capital is 14 parts or 112 units tall, for the bottom half of which we use the #revolve operation (like the #columnBase and #columnShaft), and for the top half we use the #extrude operation (like the #pedestal, #entablature, and #plinth).

Starting at the bottom, we have an #astragal that is 2 parts or 16 units tall and has the same profile as a #reed and #torus, falling in between the two in terms of size. The arc AD is shown in gray because it is an invisible #virtualSurface that envelops the decorations like #eggsAndDarts on the #ovolo. This is the measurement that is given in #Scarlata's #PracticalArchitecture, but it makes no mention of the #decorative and #tectonic surfaces. Arc BC with a radius of 4 parts or 32 units is the tectonic surface on which the Ovolo decorations rest. Such decorations have a variable or uneven surface which may not exceed 1 part or 8 units.

Points E and F mark the horizontal tangent or maxima of the second spiral and the first (outermost) spiral, respectively. The gap between them is exactly 4 parts or 32 units. GH is the profile for the vertical side surface on which part of the #ribbon and #braid lie flat, protruding exactly 6 units to coincide with the invisible virtual flat surface through EF.

The #cymaReversa is 2 parts or 16 units tall and 1.5 parts or 12 units wide. It starts 4 units to the right of F and stops 4 units short of the top fillet, which is one part or 8 units tall and 20 parts or 160 units from the #columnAxis.

Of the 4 parts or 32 units between G and H, the lower 3 parts or 24 units are part of the #voluteChannel groove and the top 1 part or 8 units is a fillet that follows the curve of the #volute and progressively gets narrower until it converges with the #eye of the volute.

This sketch shows the arrangement of #dentils in the classic variation of the #IonicEntablature. It shows the full layout, but most of the top is obscured by the top portion of the #cornice. Only the outside square shapes are actually visible.

Each #dentil has a square "footprint" that is 4 parts by 4 parts (32*32 units) and is 6 parts (48 units) tall. The spacing between each dentil is 2 parts (16 units).

Dentils project 4 parts (or 32 units) from the face of the #fascia on which they rest.

Each face of the fascia has 7 dentils with the middle dentil laterally centered and directly in front of the column axis. The 2 side dentils are on side faces, and that is apparent in the darker shading in the sketch at https://pixelfed.social/i/web/post/790782316675150160. Take the time to reconcile this with the numbers listed in #Scarlata's #PracticalArchitecture.

The 3D reconstruction from the #primaryProfileCurves is very similar to that of the #IonicPedestal, with #extrusion, #mitering, #joining, and #capping planar holes as described in https://pixelfed.social/i/web/post/790645054230337543 — just set the dentils aside, for now.

Once you have capped the #planarHoles to get a solid, analyze the edges of the solid in the #CAD program for #nakedEdges and #nonManifoldEdges.

Then, extrude the dentils outline (in the top view) to a height of 48 units (in the front view).

Now perform a #booleanUnion of the two solid shapes to get the complete #entablature.

Finally, check the edges of the solid in the #CAD program AGAIN for #nakedEdges and #nonManifoldEdges.

With this, we have finished two of the three main components of the #IonicOrder. There's a modern version of the Ionic entablature with #modillions, which I will describe later.

Next, we move on to the biggest, most conspicuous part of the order — the #IonicColumn.

There are two variations of the #IonicEntablature. The classic variation has #dentils, which are teeth-like structures shown here above the #frieze. The modern version has #modillions, which are projecting brackets under the #corona of the #cornice. Well, "modern" is a relative term. For designs that are more than 2000 years old, even an alteration 1000 years ago would qualify as modern.

Although the sketch shows the #entablature with a square footprint, in practice, it runs the entire length of a #colonnade (multiple columns) or an #arcade (multiple arches).

#CAD construction of the entablature is very similar to that of a #pedestal.

The first step is to consult #Vignola's #RegolaArchitettura for the visual appearance, and then consult #Scarlata's #PracticalArchitecture for #VignolaProportions in tabular form.

It is convenient to create a spreadsheet to convert the measurements given in Scarlata's book from module "parts" to your own model units based on your choice of value for the module parameter µ.

Armed with these measurements, it is time to plot the points and draw the #primaryProfileCurves on our standard 2D grid with minor grid lines 8 units apart and major grid lines 32 units apart.

In the first pass, skip the dentils and draw the profile curves for the rest of the moldings. Just as with the pedestal, I will show the macro-level plan as well as the detail plan. So, you don't have to go to Scarlata's book, but you know it's there if you want to.

I will show the dentil arrangement in a subsequent post.

Based on µ = 144, the classic Ionic entablature is 648 units (36 parts, or 4.5*µ) tall. Of this, the #architrave at the bottom is 180 units (10 parts, or 1.25*µ) tall, the frieze in the middle is 216 units (12 parts, or 1.5*µ) tall, and the cornice at the top is 252 units (14 parts, or 1.75*µ) tall.

Here is a #textureMapped #IonicPedestal using a photo of a marble slab I took at #TheLouvre in Paris a few years ago.

Also, notice the refined bottom and top of the pedestal with #cymaRecta and #cymaReversa based on flattened half-#helix curves rather than two elliptical arcs.

I am going to focus on #modeling rather than #textureMapping or use of #shaders. I am just putting this up here to show how far we have come in just 10 posts.

Next up is the #entablature of the #IonicOrder

#Vitruvius defined the single parameter called #module or µ, which is the radius of a column at its base [https://pixelfed.social/p/Splines/790357912719769731].

#Vignola divided µ into 12 parts for the #Tuscan and #Doric orders, and into 18 parts for the #lonic, #Corinthian and #Composite orders. To make it easier to move between the orders, it is helpful to choose a value for µ that is a multiple of both 12 and 18. We could start with module µ = 36, which is the least common multiple of 12 and 18.

However, in classical design, it's a lot easier if minor grid lines are multiples of 8 (as in music 'octaves' and poetry 'octets'). So, I use a grid with minor grid marks at 8 and major grid marks at 32, which leads us to choose µ = 144 as that is the least common multiple of 12, 18, 8, and 32.

This value of µ = 144 is still abstract because we haven't chosen a physical unit like mm or inch, but it does not matter at this point. Once the model is constructed with the chosen value of µ, it can be scaled up or down in software to get the desired column width or order height in physical units.

The total height of columns for the same µ are in the ratio 7:8:9:10 for the #Tuscan, #Doric, #Ionic, and #Corinthian, with #Composite the same as Corinthian. When µ = 144, the diameter is 288. So the height of the Ionic column is 288*9 = 2592 units.

The #pedestal, #column, and #entablature are always in 4:12:3 ratio across all orders. So the height of the Ionic pedestal is 864 and height of the entablature is 648, and the total height of the order is 4104 units.

Remember that in the Ionic order, µ is divided into 18 "parts". So each part is 144/18 = 8 units. With this we can convert between µ and parts in either direction.

Total order height of 4104 units is then 4104/8 = 513 parts. If you are trained in computer science, you will grow despondent that it is 513, not 512. Let go, and everything will be all right.

The pedestal is the easiest component, and we start with that.

This is a sketch of the complete #IonicOrder, excluding #intercolumniation and #arches, which came later.

Different people have different abilities and different levels of mathematical knowledge. I make few assumptions about the minimum knowledge one must possess to follow my posts. At a minimum, one must understand ratio, proportion, similar, congruent triangles, Pythagoras, and basic properties of circles, including radius, diameter, circumference, tangents, secants, and chords.

No trigonometry or calculus is assumed, but people who have a knowledge of differentiable continuity, maxima, minima, and inflection points will have increased appreciation of the nuances of some designs featuring smooth curves and surfaces.

I start with first principles, even if it might be a little boring for people with advanced skills. The most basic requirement is that one must be able to mark points on a 3D grid, draw a straight line between two points, and draw a circle or arc from the center. The CAD tools should help with the rest, for example, to find a point of tangency, draw a circle through three arbitrary points, or tangential to three curves (if possible).

There are three components in the #Ionic order. Starting at the bottom is the #pedestal (which is optional), the #column, and the #entablature. Each of these three components has three subcomponents:
— Pedestal has #basement, #dado, and #cap.
— Column has #base, #shaft, and #capital.
— Entablature has #architrave, #frieze, and #cornice.

The pedestal, column, and entablature are always in 4:12:3 ratio. If all components are present, the total order height is divisible by 19. If there's no pedestal, the total height is divisible by 15.

The entire order is parameterized by a SINGLE parameter — the radius of the column at its base. #Vitruvius called the radius a "module" (µ) — an abstract unit of measure independent of physical units.

Components of Ionic column and entablature also have classic and modern variations.