Sometimes you get the most unexpected design ideas from the most unexpected source.

Here, I took the top portion of the #braid strand sliced horizontally in https://pixelfed.social/p/Splines/798252244743520392 and extracted the surface borders of each "bean" shaped segment of the strand.

The extracted curves shown in the top portion of the diagram look like 4 pairs of tiny bathroom slippers, each with a single strap near the front where the toes would be.

Fun activity for the entire family: Use #uniformScaling or #nonuniformScaling to fit every feet and be creative with the straps.

#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.

Classic #IonicScroll #Scaffolding

Take the time to understand the measurements of this scaffolding plan in the broader context of the #IonicCapital measurements, as this is the bedrock on which the #scroll reconstruction rests.

First, note that there are 5 rectangles labeled M, N, P, Q, and R, where M and N are coplanar with the large #volute at the front of the scroll. P and Q are congruent, with P exactly midway between the front and the back of the scroll, and Q exactly 3/4 of the way from front, or 1/4 from back. R is the smallest of them and lies at the back of the bell-shaped part of the scroll, but ahead of the ribbon bearing the 3-strand #braid.

M completely encloses the volute, including #ArcZero, but much of Arc zero is discarded later. So, the part of the volute that really matters is enclosed by N, whose width is 112 units, height the same as M at 128 units, and the width of M itself is exactly µ or 144 units. So these measurements are in the ratio 7:8:9.

The width of P and Q is 56 units, which is exactly half the width of N. Recall from the post on #IonicCapital #Tectonic Surfaces [https://pixelfed.social/p/Splines/792124787573855518] that the unadorned capital is also exactly 112 units, divided into two halves of 56 units each.

R is concentric with P and Q and its width is exactly half of the width of P and Q, which are exactly half the width of N.

Top of R is 32 units from top of N and 16 units from top of P and Q. Bottom of P and Q are 32 units from bottom of N and 16 units from bottom of R.

Finally note the diagonal line from the origin to the #eye with a point in the middle. That middle point is the center of rectangle labeled N. It shows that the volute #eye, the center of N and the centers of concentric rectangles P, Q, and R would be collinear if these rectangles were coplanar.

All of these constraints point to warrantable consistency and coherence of the scaffolding measurements, justifying their use in scroll reconstruction.

In https://pixelfed.social/p/Splines/792511464365923534 we created the outer spiral for the #IonicVolute. We will now create the inner spiral. Remember that we're temporarily using a 3x scale.

Recall from https://pixelfed.social/p/Splines/792124787573855518 that the vertical gap between the maxima for the 2nd turn of the spiral and that of the first turn shown by points G and H, respectively, is exactly 4 parts or 32 units when µ = 144. With a 3x scale, it is still 4 parts, but each part is 24 units, so gap is 96 units now. Of these 4 parts, 1 part (now 24 units) is the #fillet that gets progressively narrower as it follows the spiral, and 3 parts (now 72 units) form the #channelGroove.

The volute extends 3/4 parts or 18 units now (6 units in the original) to the left of the point of horizontal tangency of the outer spiral. This is the amount needed to accommodate the ribbon and 3-strand #braid on the flat vertical wall on the sides of the #capital as shown earlier in the same post by points F and H.

Now focus on the square where we have the 12 numbered vertices. At present scale, the square is 1 part or 24 units wide. Now inset each concentric square by 1 unit on each side, effectively dividing the gap between the original squares in a 1:3 ratio — same as the fillet height to groove size ratio.

Move to new point 1 in the square and sweep arc 1 of the inner spiral using a radius of 181 units so that the vertical gap between the top of original arc 1 and new arc one is exactly 1 part (24 units at present scale), and the vertical gap between the start of new arc 1 and the old arc 5 is exactly 3 parts or 72 units.

Continue with inner arc 2 from where inner arc 1 ended, using new point 2 as the center. Finish drawing the entire inner spiral exactly as described earlier for the outer spiral. This time you don't need arc zero or the outer circle unless you are contemplating a #medallion.

All that's left now is to complete the #eye of the volute, which I show next.

Classic #IonicCapital #Tectonic Surfaces Plan

We already made the 8 unit tall #fillet at the bottom of the #capital a part of the #shaft in https://pixelfed.social/p/Splines/791794072490907090. So, excluding that, the remainder of the capital is 14 parts or 112 units tall, for the bottom half of which we use the #revolve operation (like the #columnBase and #columnShaft), and for the top half we use the #extrude operation (like the #pedestal, #entablature, and #plinth).

Starting at the bottom, we have an #astragal that is 2 parts or 16 units tall and has the same profile as a #reed and #torus, falling in between the two in terms of size. The arc AD is shown in gray because it is an invisible #virtualSurface that envelops the decorations like #eggsAndDarts on the #ovolo. This is the measurement that is given in #Scarlata's #PracticalArchitecture, but it makes no mention of the #decorative and #tectonic surfaces. Arc BC with a radius of 4 parts or 32 units is the tectonic surface on which the Ovolo decorations rest. Such decorations have a variable or uneven surface which may not exceed 1 part or 8 units.

Points E and F mark the horizontal tangent or maxima of the second spiral and the first (outermost) spiral, respectively. The gap between them is exactly 4 parts or 32 units. GH is the profile for the vertical side surface on which part of the #ribbon and #braid lie flat, protruding exactly 6 units to coincide with the invisible virtual flat surface through EF.

The #cymaReversa is 2 parts or 16 units tall and 1.5 parts or 12 units wide. It starts 4 units to the right of F and stops 4 units short of the top fillet, which is one part or 8 units tall and 20 parts or 160 units from the #columnAxis.

Of the 4 parts or 32 units between G and H, the lower 3 parts or 24 units are part of the #voluteChannel groove and the top 1 part or 8 units is a fillet that follows the curve of the #volute and progressively gets narrower until it converges with the #eye of the volute.