Mastodawn

#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.

Splines (@Splines@pixelfed.social)

#IonicColumn #Flutes This diagram shows the 2D geometry of an #Ionic #flute. The larger blue circle shows the flute outline near the #base of the #column. The smaller blue circle shows the flute outline near the #neck of the #shaft. Both subtend a 12° angle at the center of the column. Like an egg in the #EggsAndDarts motif, a flute must be centered on the column axis when viewed directly from the front, back, or the sides. This is why the 12° are split into 6° on either side of the X axis. The center of the larger circle is µ = 144 units from the origin on the X axis. The center of the smaller circle is 5/6 of µ, or 120 units from the origin. In https://pixelfed.social/p/Splines/799340150182400358, I mentioned working at sub-micron precision, and you might wonder where that came from when we have been using abstract units like µ without specifying any physical units. My apologies for not making it clear that I had assumed 1 unit was equal to 1 mm. If that assumption holds, then µ = 144 mm gives a total order height of 4104 mm, that is 13.46 ft. At smaller scales, the precision is even higher than 1/10 of a micron. With that said, here the radius of the larger circle is 15.0728 units and that of the smaller circle is 12.5606 units, with sub-micron precision if 1 unit = 1 mm. Refer to https://pixelfed.social/p/Splines/791399680747885646 and place the center of the smaller circle exactly at point J on the neck line. Later we will draw a sphere at the same location with the same radius. If you want a flat bottom for flutes, place the center of the larger circle at exactly 28 units (12 for the #fillet and 16 for the #cavetto or #conge) above point A in that figure. If you want a round bottom, then further move the larger circle up by the size of its radius. Nobody would quibble if you used a radius of 15 units instead of 15.0728 units, but it would make it easier to switch from flat to round bottom or vice-versa by simply moving the circle up or down 15 units.

Pixelfed

Splines Feb 23

#Braids #3StrandBraids #MulticoloredBraids

To preserve the ability to print different strands in different colors when #3DPrinting, we must keep them separate. When #CNCMilling a block of wood or other material, we don't need to keep the strands separate.

To accommodate both kinds of output, I suggest that you keep the strands separate until the very end, and perform a #booleanUnion at the last possible stage after making a copy of the separate strands.

The topmost part of the diagram shows what the strands look like after a boolean union. Much of the internal structure is absorbed in the channel block, and overlapping parts of individual strands are eliminated.

The magenta curve from https://pixelfed.social/p/Splines/798252244743520392 is also shown here. Note that the location of the red cutting planes has changed slightly — Instead of 40 units from the origin, the first cutting plane is located at 39 because I ran into another limit that we must avoid.

Also, we need two blocks 120 units and 32 units long (not 24 units as was erroneously mentioned earlier). Turns out that cutting the strands at 32 units from the first cut puts us at 71 units from origin, and we run into another limit that destroys the #airtight properties of the cut solids. To get around that, we place the second cutting plane at 72 from origin to get a block 33 units long. The last cutting plane is at 159 units from origin, and when used with the first cutting plane it gives us a block 120 units long.

The lower portion of the diagram shows individual strands cut using the cutting planes as described above.

Depending on precision, you might or might not see a #nonmanifoldEdge on the second strand when cutting a length of 33 units. With precision set to 1/10 micron, which is ~100 times finer than current high-end #3DPrinters, I got a non-manifold edge.

Sometimes the fix is easy — Just #explode the solid, and rejoin the tiny surface fragments. Experiment with different precision settings.

Splines (@Splines@pixelfed.social)

#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

Pixelfed

Splines Feb 5

We completed the #primaryProfileCurves for the classical flat #IonicVolute in https://pixelfed.social/p/Splines/792616677005177924.

To create a 3-dimensional slab with a recessed #channelGroove for the volute, you will need an outline of the volute without the inner #spiral arms.

To create the outline, make a copy of the spiral curves and work on the copy so that you don't destroy the originals. Drop a straight vertical line from the start point of outer Arc 1 of the spiral to the maxima or horizontal tangent of outer Arc 5. Trim away all other interior spiral lines and close the curve as shown in the left figure.

#Extrude the closed outline curve front to back by 1 part or 8 units in the side view. Extrude the #closedCurve of the inner and outer spirals by 2 parts or 16 units in the side view (but without the 6 unit extention on the top, which is only used when integrating the volute face with the #capital). Perform a #booleanUnion of both solids, and remember to check for #nakedEdges and #nonManifoldEdges.

The #volute design can be used outside of the #IonicColumn, such as in a #medallion. For a medallion, you have two options regarding the size of the enclosing circle.

You can either use the circle that Arc Zero lies on, or you can use the circle that Arc 1 lies on. Obviously, the latter is more compact. Just remember that the center for the larger circle is #groundZero or point 4 and the center for the smaller circle is point 1. In either case, inset the chosen circle with a concentric circle whose radius is 1 part or 8 units less.

The figure on the right shows the outlines of the enclosing circles based on the size of Arc 1 with center at Point 1.

Splines (@Splines@pixelfed.social)

We saw how to create the #outerSpiral for the #IonicVolute in https://pixelfed.social/p/Splines/792511464365923534 and the #innerSpiral in https://pixelfed.social/p/Splines/792561721929860260. Create a 270° circular arc of radius 1 part (24 units at 3x scale), spanning quadrants 2, 3, and 4 as shown in orange for the #eye of the volute. The arc for the eye intersects arc 12 of both inner spiral (shown in green) and outer spiral (shown in magenta). Outer arc 12 makes a kink where it meets the orange arc as seen in the left diagram. Discard the magenta arc 12. We can do better. Trim both the inner arc 12 (green) and the orange arc for the eye where they meet and discard the right portions of both. Finally, perform an #arcBlend between points A and B as shown in the right diagram. Arc blend is a new operation we are seeing for the first time. Previously we used #tangencyBlend to blend various sections of the #primaryProfileCurves for the #shaft of an #IonicColumn [https://pixelfed.social/p/Splines/791723063470910081]. Arc blend also maintains tangency, but instead of generating freeform #NURBS curves for blending, it exclusively uses one or more sections of circular arcs to blend the ends. Join all segments of the inner spiral, outer spiral, eye, blended arcs, and straight lines near the top-left of the volute to create a single #closedCurve. Mark the center of the eye as the origin or base point for #moving, #scaling, and other #transformations, and don't forget to scale the entire design to 1/3 using a scaling factor that has a high degree of precision, e.g., 0.33333333. I mentioned that Dürer's approximation of a #logarithmicSpiral is close, but doesn't fit perfectly. So far there's nothing that doesn't fit. The fit issue only comes up during #scroll construction. This concludes the task of volute construction. Next, we will look at creating the 3-dimensional volute slab using this closed volute curve and adapt it for the recessed #channelGroove in the slab.

Pixelfed

Splines Feb 1

#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.

Splines (@Splines@pixelfed.social)

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.

Pixelfed

Splines Jan 31

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.

Splines (@Splines@pixelfed.social)

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.

Pixelfed