Left side of this diagram shows the #profileCurves for the cap of #ModernIonicCapital from the front. The right side shows a perspective view of the cap surfaces obtained by revolving the profile curves about their respective axes and after some of those have been trimmed away

The measurements for the floor plan of the modern ionic capital are given in https://pixelfed.social/p/Splines/807782440025967685 with further links to relevant pages in #Scarlata's book at the bottom.

I won't bore you with the bottom portion of the modern #capital because it is very similar to that of the classic capital shown in https://pixelfed.social/p/Splines/792124787573855518. A significant difference is that the bottom #ovolo is shorter, with a total height of 32 units instead of 40

For the cap, we need two identical copies of a single profile curve that is 30 units wide and 48 units tall. The curves marked by A and B in the diagram are oriented in the same direction and are spaced 100 units from each other.

The bottom of profile curve A lines up with the neck of the #columnShaft at 120 units from the column axis. The revolution axis for this curve is located at 416 units from the column axis at the center of the largest circle in the floor plan.

We #revolve profile curve A full circle about its revolution axis. Then, we #rotate the resulting surface about the column axis to get 4 identical copies.

We revolve profile curve B full circle about the column axis. Then, we trim the resulting surface along with the 4 others at each intersection to get the side and corner surfaces for the cap of the capital.

We #join the trimmed surfaces, cap #planarHoles to convert them into a closed solid, and verify that the resulting solid is #airtight with no #nakedEdges and no #nonManifoldEdges.

The cap is in the correct final orientation. The volutes will be at 45° angles, but when we construct them, it will be easier to rotate the whole plan 45° so that the #volute #spiral is on the XZ plane.

Plan for the #ModernIonicCapital

If the design in https://pixelfed.social/p/Splines/807569519962747338 looks daunting, let me assure you it is far simpler than the work that went into the reconstruction of just the #scroll for the #classicIonicCapital. Be sure to check out #MileStone4 at https://pixelfed.social/p/Splines/795361973789834465.

With the modern #IonicCapital, the designers went back to the basics of using just straight lines and circular arcs to define the geometry of the essential elements of the capital. No #braids, #keystones, or #modillions, and no #helix curves or #sinusoids.

We start the floorplan for the modern ionic capital with a circle of radius 5/6 of µ (120 when µ = 144) which marks the neck of the #columnShaft.

Tangent to this circle is a large circle of radius 296 units centered on the X axis exactly 416 units from the column axis. This is the circle that marks the curve of the #abacus, which is always tangential to the column shaft at the neck. This circle also marks the curved faces of the interior portion of the #volute wedge. Without the raised volute spirals, the interior wedge appears flush with the abacus as they follow the same circular arc.

Concentric to this large circle is another circle with a radius of 280 units to mark the extent of the raised volute spirals which are 16 units thick. Another concentric circle of radius 266 units marks the outer edge of the top of the capital.

The gap between the outermost large circle and the innermost concentric circle is 30 units, and that is reflected in another pair of circles centered on the column axis with radius of 250 units and 220 units to define the four corners.

The capital footprint fits in a square 396 units wide — or 24.75 parts horizontally from axis, per #Scarlata in https://babel.hathitrust.org/cgi/pt?id=mdp.39015031201190&view=1up&seq=45.

Use this with the sketch in https://babel.hathitrust.org/cgi/pt?id=mdp.39015031201190&view=1up&seq=142

In https://pixelfed.social/p/Splines/803615973439041638, we saw #Arcade #Intercolumniation without #Pedestal. Here we see it with pedestals. The previous post was missing #dentils but they are included here.

Arches are made up of wedge-shaped blocks known as #voussoir. The middle one at the top of the arch has a special name — #keystone — and it is the stone that supports the most vulnerable part of the beam above and distributes its load to the adjacent blocks, which, in turn, do the same to the next lower block until the load is transferred to the #impost above a #pier.

When the arch does not include a pedestal, the arch opening is closer to the entablature, and there is no special decoration on the keystone. When there are pedestals, they add 6µ (864 units) to the total height, but the height of the opening goes up only by 5µ. So there is greater separation between the entablature and the top of the arch. In that case, there is a decoration on top of the arch to close the gap.

With pedestals, everything added for the arch has different measurement. Because of the increased height, the whole wall behind the columns is thicker — 2µ instead of 1.5µ when there is no pedestal. The pier is also wider, but its base molding is shorter. The impost profile is same, but it is wider because the pier is wider. The #archivolt is also wider, almost reaching the column shaft.

#Scarlata's book https://babel.hathitrust.org/cgi/pt?id=mdp.39015031201190&view=1up&seq=47 has all the measurements. So, I won't repeat them here.

In this post as well as in the previous one, I showed only a single arch on a flat wall with two columns half buried in the wall. When there are multiple arches running along a curve, the wall, entablature, archivolt, impost, pier, and its base are flexible and bend along the curve, but the column and pedestal are rigid. Most #CAD tools offer the option of flowing rigid bodies along a curve or curved surface — Just flow the columns and pedestals separately.

#3DPrinting

Early #3DPrinted prototype of #Classic #IonicCapital from a few years back using a #Prusa #MK4S #3DPrinter.

The length of the #volute from end-to-end is more than 10". I rotated the #voluteSlab 45° on the printer bed, and by the grace of #Pythagoras, I was able to print it inside the 8" build envelope of the MK4S.

This prototype has many imperfections, and some elements are just plain "wrong" -- like the curved inside of the volute channel (instead of flat).

The #ovolo is lacking #eggsAndDarts because I had not yet made the distinction between a #tectonicSurface on which the #decorativeElements rest, and a #virtualSurface that encloses the decorative elements. My calculations at the time always seemed to differ from those in #Scarlata's book. But as the physical #3DPrint shows, it is perfectly OK to go for a minimalistic look — even if it means a ribbon shorn of #braids. It's the distinctive design of the core tectonic elements that has the greatest impact.

I used brown silk filament for the braids, because, why not? For this prototype, I printed them separately and glued them to the #scroll.

Also, the scrolls are misshapen because I had not yet figured out the correct geometry. It would be another 4 years of tinkering with my #CAD design to finally get my #Eureka moment — and that's when I decided to start this series.

#Ovolo #TectonicSurfaces for #EggsAndDarts

Continuation of https://pixelfed.social/p/Splines/796857354690493749

Reconcile this figure with the figure in https://pixelfed.social/p/Splines/792124787573855518. The points A, B, C, and D are the same in both. The arc from A to D is the profile for the invisible virtual surface that encloses the decorative elements. Arc BC is start of the #tectonic surface where decorative elements rest. Points K, L, M, P, and Q are new here.

Point K is the center of the Ovolo and lies on the #columnAxis. KD is the upper radius of the Ovolo including the decorative elements, and its length is 22 parts of 176.

The horizontal distance between C and D is 1 part (8 units), and the vertical distance between C and P is also the same. PQ is where the eggs and darts are sliced off exposing the wall CP around the entire Ovolo and making it visible.

Normally, the decorative elements rest on the surface swept by arc BCP. In the case of https://pixelfed.social/p/Splines/796786779066451143, we have a fully round egg shape half of which is buried behind the arc AQD. In the concave variant, this means part of the tectonic surface is also carved out.

In other words, instead of limiting ourself to the range CD (or PQ), we are taking liberty to gouge out portions as far back as arc LM, and that's OK. Neither #Vignola nor #Scarlata mention tectonic surfaces or how far back they should or would be. The actual depth will depend on the choice of our decorative elements — in this case, concave eggs that are fully round like natural eggs.

The concave version of the egg is shown on the bottom right and it lines up with the orange wireframe with the round hole on top. The wireframe of the convex egg is superimposed on the concave portion to show where it will be placed if both are used together.

The depth of the dart slab is the same as that of the concave egg. The slab of the convex egg is thinner because we have to leave room for the bulge of the egg.

#ReverseEngineer #ImageScans

We now dig into the archives and resurface old sketches for #restoration. This one is from #Vignola's #RegolaArchitettura at https://archive.org/details/gri_33125008229458/page/n39/mode/2up. This lavishly illustrated book with copious notes that also flaunt his #calligraphy was written (in Italian) when America was still a British colony. The book went out of copyright a long time ago.

Straighten the image as much as you can in an image editor and crop it before bringing it into a #CAD tool.

Then, stare at the image for a while and squint occasionally until you "see" crucial features and patterns emerge, while ignoring the "noise."

Finally, try #curveFitting with the simplest of curves — straight lines, circular arcs, ellipse, and so on to get as close an approximation as possible. Remember that with hand-drawn sketches, the fit will rarely be perfect. So use some structure as a guide or #scaffolding as I laid out in https://pixelfed.social/p/Splines/792966507797633558.

In the top left of the diagram, I show the measurements that I was satisfied with after a lengthy process of trial and error because the numbers comport with my understanding of the proportions the original designers intended — many, but not all of which are documented in #Scarlata's #PracticalArchitecture with #VignolaProportions in tabular form.

For measurements that are missing, use plausible heuristics to fill in the blanks and try to justify your choices using simple rules. In this case, the bedrock rules are:

1. The entire #volute is exactly µ = 144 units wide, including #ArcZero, which extends 32 units beyond the portion of the volute that is actually used in the design.

2. The portion of the volute that is actually used in the design is 112 units wide, same as the height of the unadorned #capital.

3. Width of the #scroll bell shape as seen from the bottom is 112 units in front, 56 units in the middle and 28 units in the rear — all in #geometricSequence.

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.

The bottom 1/3 of the #columnShaft for an #IonicColumn is a perfect cylinder. So the line below point B is a straight line.

In https://pixelfed.social/p/Splines/791723063470910081, we blended the bottom end of the 60° arc and the top end of the long interpolated curve between points J and K. Now blend the bottom end of the interpolated curve and the top end of the straight line between points B and C to obtain the 3rd and final #NURBS segment for the #primaryProfileCurve of the shaft.

Just like there's a #cavetto and #fillet near the #neck of the shaft, there is a fillet and cavetto near the foot of the shaft. However, there is a subtle difference between the two. The cavetto near the neck is tangential to the blended #NURBS curve that is not a straight line. The profile curve for the cavetto near the foot is tangential to a straight line.

There is a special name for a cavetto that is tangential to a straight line or flat surface, like the two cavetto moldings in the #dado of the #pedestal. It's called a #conge. Another alternate name for the cavetto molding is #cove, which is evocative of "cave" because of its concave profile curve.

Above the neck is a fillet 8 units tall and an #astragal 16 units tall that #Scarlata puts in braces in the column shaft section within his tables of #VignolaProportions, with a note saying they are not counted as part of the shaft but are accounted for as part of the #capital.

I decided to include the top fillet as part of the shaft and keep the astragal with the capital. It does not change the design or alter the proportions in any way, but the inclusion of the fillet makes it more practical for #3DPrinting and #CNCMilling of the neck. This concludes the profile curve for the shaft with a height of 291 parts or 2328 units + 8 for fillet.

The column shaft is tapered in the upper 2/3 due to #entasis whose purpose is to make optical corrections to the shape of the column which, without correction, appeared concave near the top.

Arcs and lines toil for #splines

2500 years ago, when they had neither computers nor #CAD tools, designers and architects relied on knowledge of algebra, geometry, and trigonometry for their daily work. It was a mere 350 years ago that Leibniz and Newton brought calculus as a new mathematical tool for design and engineering.

Before computers arrived, artists, designers, and architects toiled with manual drafting tools to engineer breathtaking masterpieces. "Toil" is not an exaggeration to describe that endeavor, even though I suspect some of them really enjoyed what they were doing.

#Scarlata compiled an entire book on #VignolaProportions with painstaking accuracy and high precision before there were calculators and spreadsheets, making it "easy" to convert from µ to physical units in both English and Metric systems, but the world has moved on, his work is forgotten, and nobody is thankful for his contributions.

If you have a CAD tool, you need not toil. Simply draw an arc of radius µ = 144 that is centered on the #columnAxis and passes through point B. Then draw a vertical line parallel to the column axis at x = µ * 5/6, or 120 units. Use this line to split the arc and trim away the left portion of the arc. Next, divide the length of the remaining portion of the arc into 8 equal portions using your CAD tool to mark points 1 through 8 as shown. If your CAD tool is able to divide the leftover arc this way, you can just ignore the angular lines radiating from the center. Otherwise, I will show you how to use them as a fallback.

Now look at point C, which seems like it is vertically above point B, but it is not. It is actually vertically above point 1.

Draw 7 more vertical lines starting with point 1, then point 2, and so on. Mark point C at 192 units vertically above on line 1, D at 192*2 on line 2, E at 192*3 on line 3, and so on until you reach point J.

Select these 8 points and use your CAD program to interpolate a free-form NURBS curve to fit these points.

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.