Plan for the #ModernIonicCapital

If the design in https://pixelfed.social/p/Splines/807569519962747338 looks daunting, let me assure you it is far simpler than the work that went into the reconstruction of just the #scroll for the #classicIonicCapital. Be sure to check out #MileStone4 at https://pixelfed.social/p/Splines/795361973789834465.

With the modern #IonicCapital, the designers went back to the basics of using just straight lines and circular arcs to define the geometry of the essential elements of the capital. No #braids, #keystones, or #modillions, and no #helix curves or #sinusoids.

We start the floorplan for the modern ionic capital with a circle of radius 5/6 of µ (120 when µ = 144) which marks the neck of the #columnShaft.

Tangent to this circle is a large circle of radius 296 units centered on the X axis exactly 416 units from the column axis. This is the circle that marks the curve of the #abacus, which is always tangential to the column shaft at the neck. This circle also marks the curved faces of the interior portion of the #volute wedge. Without the raised volute spirals, the interior wedge appears flush with the abacus as they follow the same circular arc.

Concentric to this large circle is another circle with a radius of 280 units to mark the extent of the raised volute spirals which are 16 units thick. Another concentric circle of radius 266 units marks the outer edge of the top of the capital.

The gap between the outermost large circle and the innermost concentric circle is 30 units, and that is reflected in another pair of circles centered on the column axis with radius of 250 units and 220 units to define the four corners.

The capital footprint fits in a square 396 units wide — or 24.75 parts horizontally from axis, per #Scarlata in https://babel.hathitrust.org/cgi/pt?id=mdp.39015031201190&view=1up&seq=45.

Use this with the sketch in https://babel.hathitrust.org/cgi/pt?id=mdp.39015031201190&view=1up&seq=142

#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.

#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.

Made this for a banner thing I was working on last night that I didn't end up using. A rare 4:3 for ur Saturday. #hexagons #sinusoids