#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.

#Milestone9 - #ArcadeIntercolumniation #Keystone and #Modillion

#Arch without Pedestal https://pixelfed.social/p/Splines/803615973439041638

#Arch with #Pedestal and Keystone https://pixelfed.social/p/Splines/804537414363507454

#Keystone and #Dentil Details https://pixelfed.social/p/Splines/804548474524642209

#Modillion Details https://pixelfed.social/p/Splines/805587292338863257

#ModernEntablature with Detail https://pixelfed.social/p/Splines/805607059171193759

#Milestone8 - #SimpleIntercolumniation https://pixelfed.social/p/Splines/803106316515798367

#Milestone7 - Complete #IonicOrder https://pixelfed.social/p/Splines/800050647761776920

#Milestone6 — #Braids #3StrandBraids https://pixelfed.social/p/Splines/799602946527813102

#Milestone5 — #EggsAndDarts https://pixelfed.social/p/Splines/797069447808333887

#Milestone4 — #IonicScroll https://pixelfed.social/p/Splines/795361973789834465

#Milestone3 — #IonicColumn https://pixelfed.social/p/Splines/792803978865652429

#Milestone2 — Classic #IonicEntablature https://pixelfed.social/p/Splines/791021871062069787

#Milestone1 — #IonicPedestal https://pixelfed.social/p/Splines/790752092700055739

#Arch with #Ionic #Entablature and #Keystone Detail

The #dentils arrangement we saw in https://pixelfed.social/p/Splines/791013152244518907 goes well with the classic entablature #profile we saw in https://pixelfed.social/p/Splines/790888454384861893, and they both go well with #simpleIntercolumniation, also known as #architravato.

However, with arches, the entablature profile has to be adjusted a bit so that the dentils arrangement is as shown here. The shape, size, and gap between individual dentils remains the same, but a crucial difference is that the dentils at the #outer corners touch each other.

As I mentioned in https://pixelfed.social/p/Splines/803615973439041638, in #arcadeIntercolumniation, the entablature is repeated on the wall behind the half-column. It doesn't end at the columns and has two "outside" corners and one "inside" corner. While the dentils at the outer corners touch each other, there is a single dentil in the inside corner that is shared by both walls.

A bedrock principle of dentils (like that with #flutes and with eggs in the #EggsAndDarts motif) is that when viewed directly from the front or the sides, a dentil must be centered on the column axis. It is this principle that forces us to adjust the profile of the entablature in arcade intercolumniation giving us the arrangement shown here.

The image also shows the detail of the decoration in front of the #keystone. The most easily recognizable component of that is the large #volute, which is the exact same size as the ones on the #capital. The smaller volute is exactly half the size of the larger one. It is mirrored, rotated and put within a bounding rectangle whose height is exactly 2µ (288 units). The channels of both volutes are bridged with #sinusoids derived from half turn of #helix curves that have been flattened.

This motif in the keystone, where volutes of different sizes are combined with sinusoids is very common. It will be seen in the #modillions of the #modernEntablature.

#Milestone8 - #SimpleIntercolumniation

Floor Plan of archetypal Greek Temple https://pixelfed.social/p/Splines/803021258359555093

Front and back #intercolumniation https://pixelfed.social/p/Splines/803076419096100108

Side intercolumniation https://pixelfed.social/p/Splines/803089629244302486

#Milestone7 - Complete #IonicOrder https://pixelfed.social/p/Splines/800050647761776920

#Milestone6 — #Braids #3StrandBraids https://pixelfed.social/p/Splines/799602946527813102

#Milestone5 — #EggsAndDarts https://pixelfed.social/p/Splines/797069447808333887

#Milestone4 — #IonicScroll https://pixelfed.social/p/Splines/795361973789834465

#Milestone3 — #IonicColumn https://pixelfed.social/p/Splines/792803978865652429

#Milestone2 — Classic #IonicEntablature https://pixelfed.social/p/Splines/791021871062069787

#Milestone1 — #IonicPedestal https://pixelfed.social/p/Splines/790752092700055739

#Milestone7 - Complete #IonicOrder with fluted and unadorned #IonicColumn with details of #decorativeElements

#Milestone6 — #Braids #3StrandBraids https://pixelfed.social/p/Splines/799602946527813102

#Milestone5 — #EggsAndDarts https://pixelfed.social/p/Splines/797069447808333887

#Milestone4 — #IonicScroll https://pixelfed.social/p/Splines/795361973789834465

#Milestone3 — #IonicColumn https://pixelfed.social/p/Splines/792803978865652429

#Milestone2 — Classic #IonicEntablature https://pixelfed.social/p/Splines/791021871062069787

#Milestone1 — #IonicPedestal https://pixelfed.social/p/Splines/790752092700055739

#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.

#IonicColumn

#Flutes have a different configuration in the #IonicOrder than they do in the #DoricOrder. In #Doric, the flutes run right next to each other, dividing the circumference of the column into 24 equal sectors, or 15° each.

In #Ionic, there is a small gap between the flutes. This gap used to vary, but over time, Ionic designers seemed to have settled and standardized the measurements by splitting 15° in 4:1 ratio, giving 12° to a flute and 3° to the gap between flutes.

Because of this standardization, there would seem to be little room for variants, but there is. In his #RegolaArchitettura [see https://archive.org/details/gri_33125008229458/page/n37/mode/2up], #Vignola documented flutes with hemispherical tops but flat bottoms, as shown in the image here.

However, it is acceptable to have hemispheres at both top and bottom as long as they are consistently used within a #colonnade or #arcade.

Flute geometry is interesting. Just like the #IonicColumn #shaft, a flute also gradually tapers as it rises from bottom to top. Additionally, it bends along the shaft surface due to #entasis [see https://pixelfed.social/p/Splines/791794072490907090]. In other words, flutes hug the column shaft.

Unlike other decorative elements like #eggsAndDarts and #3StrandBraids, flutes are #subtractive, not #additive to the rest of the design. In other words, we have to carve the flutes out instead of adding them to the design.

#Milestone6 — #Braids #3StrandBraids

#Sinusoid from #Helix https://pixelfed.social/p/Splines/797893262102038801

Braid Geometry https://pixelfed.social/p/Splines/797916882329430160

Braid Strand https://pixelfed.social/p/Splines/798252244743520392

Braid Assembly https://pixelfed.social/p/Splines/799340150182400358

Braid #FlowOnSurface https://pixelfed.social/p/Splines/799514176049543252

#Milestone5 — #EggsAndDarts https://pixelfed.social/p/Splines/797069447808333887

#Milestone4 — #IonicScroll https://pixelfed.social/p/Splines/795361973789834465

#Milestone3 — #IonicColumn https://pixelfed.social/p/Splines/792803978865652429

#Milestone2 — Classic #IonicEntablature https://pixelfed.social/p/Splines/791021871062069787

#Milestone1 — #IonicPedestal https://pixelfed.social/p/Splines/790752092700055739

#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.

#3StrandBraids

#Braids are the last of the #decorative elements on the #IonicScroll, but like #EggsAndDarts, they are not specific to the #IonicOrder.

Braids are a popular design motif that find wide currency in modern #hairstyle, #fashion, and fashion accessories like #belts and #bracelets.

Braids come in infinite varieties with varying number of strands, thickness of strands, roundness or flatness of strands, and how tightly or loosely they are wound together. Here, I focus on the 3-strand variant mentioned in #Vignola's book and previewed in https://pixelfed.social/p/Splines/792015485979791089. The image here is brightly colored to draw attention to the 3 strands.

The geometry of braid strands is not at all obvious despite how familiar they look. Also, a braid strand is the only feature in the entire iconic order whose geometry cannot be captured with straight lines and circular arcs. Instead, a strand geometry must be defined in a series of steps starting with a basic #sinusoidal curve.

A sinusoidal curve or #sinusoid is a wave form whose function belongs to a family of functions known as #transcendentalFunctions that also include #logarithmic and #exponential functions. I mentioned #logarithmicSpirals in https://pixelfed.social/p/Splines/792499765146596723, and in a future post I will show how to construct one and compare it with the #spiral used in our implementation of #IonicVolute.

They are called transcendental functions because they transcend the math of finite algebraic polynomials and go beyond geometry into trigonometry. Fortunately, we don't have to go there.

Few #CAD tools have a direct primitive for a sinusoid, but almost all have a primitive for a 3-dimensional round coil shape called a #helix which we can use to create the sinusoids we need for a braid strand. To create a sinusoid, all we need to do is #project a helix on a flat surface to convert it into a 2D waveform.