#ModernIonicCapital sketch

The modern #IonicCapital with curved faces and radial symmetry is a drop-in replacement for the classic Ionic capital with flat faces.

Unlike the classic variant, which has a rectangular footprint, the modern variant has a footprint that fits in a square. In the classic variant, the volutes and scrolls project out so that they are visible from the top. In the modern variant, there are no scrolls, the volutes have a curved face, and they are completely nestled under the top.

The sketch omits the #fillet at the bottom because we added that to the column #shaft in https://pixelfed.social/p/Splines/791794072490907090.

So, we start at the bottom with an #astragal which is exactly the same size as in the classic variant.

Next up from the bottom is the #ovolo which is shorter than in the classic variant. It still has a #tectonicSurface on which #decorativeElements rest, and a #virtualSurface that envelops the decorative elements. In this case, I chose a minimalist design with no #eggsAndDarts. Instead, I use another plain ovolo as a substitute that is offset from the tectonic surface by 1 part (or 8 units, when µ = 144).

Above the ovolo is the #channel, which in this case is a round slab whose surface matches the neck of the column with a radius equal to 5/6 of µ (120 units).

Above the channel is the #abacus which has a curved face that is repeated on all four sides. There is an abacus with flat sides in the classic variant as well, but it is not visible from the front because it is hidden behind the #volute slab. In fact, the vertical #braidsAssembly in the classic variant is attached to the abacus.

Above the abacus is a #reed, and above that, another small Ovolo that tops the modern capital.

The curved volutes follow the blue circular arcs at the bottom of the sketch. The volutes are shaped like a wedge, as can be seen more clearly in the corner facing the front. The portion of the wedge between the outer rims has a concave surface.

UPDATE: Alignment of various elements in the classic #IonicCapital

There is an error in the measurements for arc AD in https://pixelfed.social/p/Splines/792124787573855518 where it is shown concentric to the arc BC, with AD having a bigger radius than BC.

The two arcs are not concentric. Arc AD is shifted down and to the right by 1 part or 8 units and has the same radius as arc BC.

When revolved around the column axis, arc AD yields the #virtual surface that encloses #decorativeElements resting on the #tectonicSurface of the #ovolo. Revolving arc BC around the column axis gives the tectonic surface of the Ovolo.

The #eye of the #volute is centered exactly at µ = 144 away from the column axis and 1/2 µ, or 72 units directly below the #cymaReversa as shown by the orange crosshairs.

The top of the Ovolo's tectonic surface (shown in magenta) is tangential to the top of the tectonic surface of the curved #braids assembly. That latter surface is also shown in magenta.

The outer surface of the vertical braids assembly is 4 units inset from the cyma reversa and is also tangential to the outer surface of the curved braids assembly near the bottom of the Ovolo's tectonic surface.

The vertical braids assembly is 33 units tall, as described in https://pixelfed.social/p/Splines/799340150182400358. The bottom portion of it is shown buried 1 unit under the Ovolo surface.

#3StrandBraids #FlowOnSurface

In the top-left, the highlighted magenta portion shows the interface between the #braids assembly and the #IonicScroll from https://pixelfed.social/p/Splines/795276076797088402.

Extract the #profileCurve shown as ABC in the top-right where the interface meets the scroll.

In https://pixelfed.social/p/Splines/794199123072358090, we rebuilt curves from 2nd-degree arcs to 3rd-degree NURBS for smoothness and swept the scroll surface one set of arcs at a time.

Now we have to flow braids on a single surface in one operation. So we need to combine the separate segments into a single NURBS curve. To do that, #explode the profile curve into individual segments, discard the straight portion, and join the curved portions.

Curves and surfaces have a #direction that you can change in the #CAD tool. Check that the direction of the joined curve is A to C, not C to A, and flip it if necessary. Then divide the curve at 120 units starting at A. This is marked by point B. Split the curve AC at B so that AB is 120 units long.

At this point AB is still made up of 5 segments, and exploding it would again decompose the curve into separate segments. So #rebuild AB as a single NURBS curve with 32 sections.

#Extrude AB to get a 48 units wide surface shown in magenta in the top-right. Point D is at the midpoint of AE and lies on the XZ plane.

Slice the channel assembly so that it is 8 units tall, 6 of which will be above the #tectonicSurface for the braid and 2 below. The tectonic surface is shown in the bottom-left as the flat magenta surface on the channel and the curved magenta surface for the scroll neck.

Flow the entire braid and channel assembly along the curved surface lining up points A, D, and E. For the vertical part on the side of the capital, just use the 33 unit tall block from https://pixelfed.social/p/Splines/799340150182400358 and bury 1 unit inside the #ovolo.

This concludes 3-strand braids. Only the non-essential #column #flutes remain.

#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.

#Braids #3StrandBraids

After we #sweepOneRail with the blue #sweepingCurve on the orange #railCurve for the #braid strand, we cap #planarHoles to get a single #airtight strand.

As described in https://pixelfed.social/p/Splines/798015349727305297, the radius of the blue circle is 8 units, and the total height of a strand swept by that circle is 24 units. Half of that is above the #tectonic surface, which is still twice of what #Vignola documented in #RegolaArchitettura.

We will apply #nonuniformScaling in the Z direction to reduce the height in half while keeping the length (X) and width (Y) the same. But before we do that we split and discard some of the bottom portion of the strand that is below the tectonic surface and is not needed.

The top half of the figure shows the front view of a single strand. The bottom shows the perspective view of the same strand.

Note that the orange braid rail starts at the center of the blue sweeping circle which looks like an ellipse in the front view. The orange rail itself looks like a flat sinusoid in the front view, but its beautiful meandering shape is really apparent in the perspective view.

The orange rail curve is centered on the "ground" or XY plane, which is also where the #tectonicSurface for the braid is. The amplitude of the orange curve (maximum or minimum from axis of the curve) is 4 units. We need to preserve the geometry of the strand at least up to the bottom of the orange rail.

If we split the strand exactly at 4 units below the ground plane, we hit a limit that not all #CAD tools are able to handle. To get around, we extend it past that limit by an arbitrarily small fraction, like 0.2, and discard the portions below. We could have extended it by 0.1 or 0.3 but it wouldn't have mattered.

Next, we draw a profile curve for the braid channel which is 32 units wide and its groove is 12 units tall for now. The rims on both sides of the channel are each 8 units wide. Total height of rim and channel is 16.2 units