#ModernIonicCapital #Volute Construction

As mentioned in https://pixelfed.social/p/Splines/807933255910367093, we mentally rotate the floor plan of the modern #IonicCapital 45° so that the volute #spiral curves lie flat on the XZ plane.

The top-left diagram shows the original volute spirals from https://pixelfed.social/p/Splines/800383518517869430. If you haven't already rebuilt them from disjointed arcs to seamless #NURBS curves as described in https://pixelfed.social/p/Splines/794199123072358090, do so now. Explode the whole spiral into constituent arcs, select all arcs for the outer spiral and join them separately, and repeat this step for the inner spiral. Then, rebuild both spirals with 256 segments each.

Volute spirals for the classic and modern variants are the exact same size, but the ones in the modern version appear smaller. That's an optical illusion because the spirals are rotated 90° in the modern version as shown in the bottom left diagram.

The bottom right diagram shows the placement of the volute spirals relative to the #profileCurves of the lower portion of the modern capital. The #voluteEye is slightly above and far to the right of the #astragal profile in the modern version compared to the classic version.

At this point, we make two copies of the volute spirals and trim them to the top of the #ovolo. In one copy we only trim away the outermost arms of the spirals while maintaining the inner spirals, as shown in the bottom right figure. In the other copy, we trim away the inner spirals as shown in the top right image.

The bottom right figure in https://babel.hathitrust.org/cgi/pt?id=mdp.39015031201190&view=1up&seq=142 shows the eye 6.5 parts (52 units) from column shaft. That's not an error, but poor documentation. See floor plan in https://pixelfed.social/p/Splines/807782440025967685 where x = 198 on the square is homologous to x = 250 on the circle. To reconcile, scale 52 by 250/198 = 65.656, or 66, and offset by 120 to get 186 units.

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.

#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.

We have now looked at every nook and cranny of the complete #IonicOrder in microscopic detail.

Here is one more look at the underside of the #ovolo to revel in the splendor before we move on to the macro level, the first step of which is arranging columns in a row to create a #colonnade according to spacing rules known as #intercolumniation.

Colonnades need not be straight and can follow arcs or other (preferably loose) sweeping curves.

As mentioned in https://pixelfed.social/p/Splines/790357912719769731, a #pedestal is always optional.

Greek designers designed the Ionic Order with full columns. The Romans introduced half columns on surfaces of walls, with or without arches. Multiple columns with arches, whether in series or in parallel, are called #arcades.

The rules for #arcade intercolumniation are different from those for simple intercolumniation, varying even by whether pedestals are present or not.

More details on alignment of various elements in the classic #IonicCapital in https://pixelfed.social/p/Splines/800161382832200305

Here, we zoom in on the two #braids assemblies — the straight vertical one on the side of the #capital and the curved one around the neck of the #scroll.

In this diagram, the magenta curve for the curved braids is extended on both sides to show the full original #modulatingSpiral for the rear end of the scroll.

The diagram also shows the lines tangential to the full modulating spiral. As previously mentioned, the top tangent is coincident with the magenta line for the top of the #ovolo. Additionally, the bottom tangent is tangential to the bottom of the #eye of the #volute.

The right tangent of the modulating spiral bisects the curved braid assembly with 4 units on either side of the tangent, and the magenta #tectonic surface further bisects the gap between that tangent and the "underside" of the braids assembly on the right.

The top gap between magenta and blue arcs is split into 6 units and 2 units — same as the proportion of the braids channel above and below the tectonic surface.

Moving on to the bottom of the vertical braids assembly, follow the lines that divide the depth of the assembly (8 units) into 4 portions. The leftmost 2 units are, of course, sub-surface, buried inside the vertical wall of the capital.

The middle line is tangential to the left side of the eye of the volute. The next line, moving right, is tangential to the left side of the modulating spiral. The rightmost line is tangential to the outer surfaces of both braids assemblies, as already mentioned in the previous post.

Note the symmetry of 2 units and 4 units near the left side of the eye.

These meticulous details are what I call pure #poetryInGeometry.

In https://pixelfed.social/p/Splines/792499765146596723, I wrote that Dürer's approximation of #logarithmicSpirals comes close, but still doesn't fit the measurements of the #IonicCapital. This is why.

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.

#EggsAndDarts

To transfer the egg and dart in https://pixelfed.social/p/Splines/797038670230603707 to the #doublyCurved surface of an #ovolo is a multistep process.

There are 24 eggs and darts around the entire Ovolo. So each egg and dart nominally occupies 360°/24 = 15°.

#Revolve arc AD in https://pixelfed.social/p/Splines/792124787573855518 about the #columnAxis to get the virtual surface. Points A and D are same in both figures. Then use cutting planes as described in https://pixelfed.social/p/Splines/790645054230337543 to get a wedge-shaped segment whose angle is 15° and one side of which is marked with A and D.

This is a doubly curved surface. Unlike a cylinder, where one side is straight while the other is curved, the surface of an Ovolo is curved in both horizontal and vertical directions.

To transfer the egg and dart to a doubly curved surface, we need a new operation called #UnrollSurface, which unrolls the wedge shape into a flat surface whose top-left corner is marked Q. Note the top is wider than the bottom.

Place the flattened wedge between the rim and the flat slab and align the top of the flat portion of the slab with the center of the top edge of the flattened wedge. Temporarily move the dart to align its center too, but only move it in the horizontal direction

If the flattened wedge were flexible and if the eggs and darts were flexible, we could flip all of these over, flex and squeeze the solid shapes, and line up Q with D so that the rim and dart appear "outside" the original wedge while the slab remains "inside" (or toward the center of the Ovolo).

Fortunately, in a #CAD tools, solids don't always have to be treated as rigid. Now we will use another new operation called #FlowOnSurface to flow the egg and dart on the wedge.

We then slice off the top of the egg and dart at the location PQ in https://pixelfed.social/p/Splines/796958366767133979.

Finally, we separate the dart, which we had aligned with the egg to minimize distortion

#EggsAndDarts continuation from https://pixelfed.social/p/Splines/796961505955555432

The slab height depends on the roundness of the egg and whether we have a concave design or not. If we are using a concave base, then top half of the egg is eliminated. For a fully round egg, that means the concave variant must scoop out up to 16 units deep. The dart slab will match the egg slab in depth.

To create the 3-dimensional shape of the dart, first #rotate the fin profile 90° in 3D space along the straight line at the bottom of the fin so that the rotated profile is perpendicular to the two #sweepRails for the dart.

Using the two sweep rails and the perpendicular fin profile, #sweepTwoRails to develop the surface of the dart. Remember to close the planar hole at the end of the fin to get a solid #airtight object. As always, check for #nakedEdges and #nonmanifoldEdges to stave off problems later.

#Extrude the bottom of the dart until it is flush with the bottom of the oval slab.

Two details worth noting in the dart design are:

1. The most exaggerated portions of the dart fin are sliced off when the eggs are sliced. After slicing, the size of the fin is roughly in proportion to the rims of the eggs on both sides.

2. There is a gap between the dart arrow and the oval slab. See the gap between points R and T in https://pixelfed.social/p/Splines/796961505955555432. This gap is necessary and will automatically close when we transfer the egg and dart to the #doublyCurved surface of the #ovolo on the #capital of the #IonicColumn. That is because the Ovolo is shaped like a bowl whose top has a bigger radius than the bottom. As a result, the motif will be warped, and its bottom will be condensed to fit the smaller radius at the bottom, closing the gap in the process.

If you plan to use the eggs and darts motif on a linear surface where there is no warping, experiment with the arrow and tip for a pleasing result.