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.

#IonicColumn #Flutes

In https://pixelfed.social/p/Splines/799864068250003272 I mentioned rounding off the radius of the bottom circle, but you don't have to. #CAD tools are perfectly happy working with 15.0728 or even higher precision as they are with 15.

After placing the two circles as described in that post, use the full #primaryProfileCurve of the shaft from https://pixelfed.social/p/Splines/791794072490907090 as a #sweepingRail and the two circles for the flutes as the #sweepingCurves, and #sweepOneRail for the body of a single flute. Close #planarHoles on both ends to get an #airtight solid.

Then draw a sphere at the center of the top circle using the same radius as the circle, and perform a #booleanUnion between the sphere and the flute body.

If you want a round bottom for the flute, repeat the sphere at the center of the larger circle using the same radius (15.0 or 15.0728) and perform another boolean union to get one flute.

Switch to the top view and make 24 copies of the flute (including the original) centered at the column axis and #group the 24 flutes.

Finally, perform a #booleanDifference with the flutes group on a copy of the solid #unadornedShaft to get a fluted variant.

The result is a column shaft with flutes carved out. Save the flutes separately for future reuse.

This concludes the entire #IonicOrder, including all #decorativeElements.

Now we pause and reflect: The whole exercise seemed like one of #art and #sculpture. Where is the #architecture in all of this?

Without a ceiling or a roof, there is no building. Without additional columns or walls, there is no ceiling. So, while we have completed the Ionic Order itself, we only have the first #buildingBlock — a single column.

Next step is to repeat the columns to create a #colonnade, which together with supporting walls or additional colonnades can support a ceiling.

Just like with everything else in design, there are rules of proportion for #intercolumniation, or space between columns.

#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

#Braids #3StrandBraids

With the #rail curve for a #braid strand in https://pixelfed.social/p/Splines/797916882329430160, we can start the construction of a braid that matches the sketch in #Vignola's #RegolaArchitettura shown in https://pixelfed.social/p/Splines/793215298082967733.

The strands for this braid have a radius of 1 part or 8 units. So their diameter is 16 units, and the braid itself is 32 units wide.

To create one strand, we start with a circle of radius of 8 units perpendicular to the tip of the orange curve. We use the orange curve as a #railCurve and the blue circle as the #sweepingCurve in the #sweepOneRail operation.

Note that the circle appears distorted like an oval in all views — front view is on top-left, top view on top-right, right view in bottom-left, and perspective view in bottom-right. That is because the circle is perpendicular to the rail curve, not to any of the "world" coordinate planes. If the sweeping curve does not appear like an angled line in the top view, something is wrong.

After the sweep, close #planarHoles to get a solid strand and, as always, check for #nakedEdges and #nonManifoldEdges to ensure an #airtight object.

Refer again to the middle portion of the top diagram in https://pixelfed.social/p/Splines/793215298082967733 between the two bell shapes of the scrolls. The total width of the #tectonic surface on which the braid will be laid is 4 parts or 32 units wide. The braid has a rim 8 units wide on both sides that rises 6 units above the tectonic surface. Half of the braid should be above the tectonic surface, meaning that the total height of the braid should be 12 units.

If you check the bounding box of this strand, you find that it is indeed 32 units wide as needed, but the height is 24 units. So we have to apply #nonuniformScaling to keep X and Y scale intact but halve the scale in the Z direction. This will reduce the roundness of the strand and cut its height in half to 12 as needed.

#Braids #3StrandBraids

We are finally ready to convert the two #sinusoids from https://pixelfed.social/p/Splines/797893262102038801 into a single 3D curve that captures the essential geometry of a #braid strand.

First extrude the blue sinusoid into a surface that extends past the magenta sinusoid on both sides. Then draw a bounding box around the blue extrusion and trim the magenta sinusoid that falls outside the bounding box.

Discard the bounding box, and extrude the trimmed magenta sinusoid into a surface that extends past the blue extrusion on both sides.

Then split either surface with the other. It doesn't matter which surface is split and which is used as a cutting surface. The braid strand lies literally at the intersection of both surfaces.

I trimmed the magenta surface with the blue one and deleted the top portion to reveal the curve at the intersection — shown here in orange. In perspective view this curve continuously swerves from left to right and simultaneously from top to bottom as it progresses along the X axis.

This single curve has the characteristics of both sinusoids as seen in front and top views. In the side view, this looks like the #infinity symbol. So we have progressed from zero (with #helix), to plus (with #sinusoid), to infinity (with intersection of two #sinusoidal surfaces).

Once we have this curve, we can sweep a circle around it to make a round strand. We can change the radius of the circle to make thinner or thicker strands. We can slant the circles to give a "calligraphic" look to the strands. We can use ovals, rectangles, squares, stars, or any closed shape to give different surface properties to the strands — the possibilities are endless.

Once you have a closed #airtight strand with capped #planarHoles, make 2 more copies of the same strand. Shift the first copy by 1/3 the wavelength of the magenta sinusoid (48/3 = 16 units) and shift the second copy by 2/3 (48*2/3 = 32 units) while leaving the original one in its place.

See https://pixelfed.social/p/Splines/796798349526747214 and https://pixelfed.social/p/Splines/796786779066451143 for detail.

Just like #ArcZero in the #spiral for the #IonicVolute, the plan for #EggsAndDarts starts out larger than what is eventually used.

The outermost frame is 7.5 parts or 60 units tall (from A to D), and 5 parts or 40 units wide when µ = 144. The innermost frame is 6 parts or 48 units tall (from B to C), and 4 parts or 32 units wide.

The height and width for both inner and outer are in 3:2 ratio. The difference in height is also split in the ratio 3:2. So, the gap between A and B as 12*2/5 = 4.8, and the gap between C and D as 12*3/5 = 7.2. We then divide both of these gaps in 5 equal segments shown by the dots between them.

Create an ellipse to fill the outer frame, and another to fill in the inner frame. Then interpolate them to create 4 more through the dots, only to delete the 3rd ellipse. This gives us the 5 #rails for sweeping the rim of the egg.

Create circles perpendicular to and touching adjacent pairs of rails, and then #sweepTwoRails using the circles to get the rim shown in the top-right. The red cutting plane marks tentative slicing for eggs.

The bottom left image shows a convex egg created with an #ellipsoid that is 48 units tall, 32 units wide, and 32 units deep. This is equivalent to #revolving the innermost ellipse around its major axis. Only half of this egg is above the oval slab.

The bottom right image shows the concave variant which is created by performing a #booleanDifference between the oval slab and the round egg created above. The top portion of the egg is removed in the concave variant, but the convex variant must still account for the bulge of a fully round egg. As such, the slab heights of the convex and concave variant are different. I will show the measurements later.

To create the slabs for the convex and concave variants, #extrude the outermost ellipse according to their measurements and #cap #planarHoles.

#IonicColumn #VignolaBase and #AtticBase #CAD Plans

Both #Vignola base and #Attic base have the same square footprint of 400 units x 400 units. The #plinth for both is 48 units (6 parts, or µ/3) tall, and the total height for both is 144 units (18 parts, or exactly µ). As such, they are easily interchangeable.

In the Vignola variant, we start at the plinth with a #fillet 2 units tall and a classic #scotia 18 units tall gouging out part of the fillet.

Then there is another fillet 2 units tall, followed by two #reeds, each 8 units tall, followed by another classic scotia as described above.

This is followed by yet another fillet 2 units tall and topped off with a #torus 40 units tall. A Torus is the same as a reed, except larger. When we reach the neck of the shaft, we will see another molding called #Astragal which has the same profile as reed and torus, but sits in the middle in size. Think of reed, astragal, and torus as small, medium, and large of the same profile.

The modern Attic variant is more elegant with fewer moldings. It also gives the impression of more heft for more stately columns. It starts at the plinth with a torus 36 units tall, followed by a fillet 4 units tall, followed by a modern scotia 24 units tall, followed by another fillet 4 units tall, and topped off with another torus 28 units tall.

As in the construction of #IonicEntablature [https://pixelfed.social/p/Splines/791013152244518907], split the construction of the #columnBase into two steps.

Just as we extruded #dentils separately, we extrude the plinth separately. First draw a square 400x400 in the top view. Then extrude the square 48 units in the front view.

For the rest of the base, we need a new 3D operation — #revolve around an axis. Instead of extruding the #primaryProfileCurve, we revolve it around the #columnAxis, and cap the #planarHoles on both ends before performing a #booleanUnion with the plinth. Finally check edges of the solid for #nakedEdges and #nonManifoldEdges.

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.

From https://pixelfed.social/p/Splines/790645054230337543, we now have an open surface for the #IonicPedestal. To finish this, close the #planarHoles at the top and bottom with flat caps and join everything. And just like that, we have finished 1/3 of the complete #IonicOrder with very little work.

To ensure that your finished object is amenable to #3DPrinting or #CNCMilling, always check the edges of your object after all surfaces have been joined. Do this EVERY time you join surfaces to create a closed object.

Most CAD programs will offer edge analysis tools that let you detect #nakedEdges or #nonManifoldEdges. If you have either of those, your object is not #airtight, and you will not be able to physically realize it.

This version of the pedestal uses the classic variation of #CymaRecta and #CymaReversa. If you want to remain faithful to the original, then you are done.

However, designs are rarely static and they continue to evolve. There is an opportunity for a slight refinement at the top and bottom of the pedestal without compromising the integrity of the order, but it requires the introduction of a new kind of curve — a #helix, which is a coil-shaped 3D curve.

I will discuss the #helixVariation later. For now, look closely at the #basement and notice how pronounced the turns of the cyma recta are. Instead of using elliptical arcs in the #primaryProfileCurves of the cyma recta and cyma reversa, it is possible to substitute a half-turn of a helix that has been flattened to a 2D shape. The result is a softer, more gradual profile curve that produces a very refined shape.