Revolutionary #Volutes

The two large blue circles at the bottom marked with A and B are from the floor plan for #ModernIonicCapital shown in https://pixelfed.social/p/Splines/807782440025967685. The larger one has a radius of 296 units, while the smaller one is 16 less, at 280. Both are centered at x = 416. Both are then rotated ±45° about the column axis to give us the four red circles. The small blue circle with radius 120 units is for the column neck.

The two red circles in the back are shown extruded vertically as cylinders. The other two haven't yet been extruded because we want to see the rest of the structure from the front.

The two tube or #torus-like structures are obtained by revolving the trimmed #spiral #volute outlines from https://pixelfed.social/p/Splines/808043616946914228 about the vertical axis located at x = 416. Only the bottom tube maintains the interior shape of the spiral.

With this setup, vertically extrude the two remaining red curves in the front so they are at least as high as the top of the upper tube.

Next, perform a #booleanDifference between the top tube and the two extrusions marked A. Keep the wedge shape of the top tube on the left side and discard the remaining portion of the tube from the right.

Then, perform a boolean difference between the bottom tube and the two extrusions marked B. With the wedge shape of the bottom tube on the left side, perform a further #booleanIntersection with the two extrusions marked A. This will produce two curved spirals that are 16 units thick. Discard the remaining portions of the bottom tube as well as all the red extrusions.

After these operations, we are left with one wedge shape with curved faces and two spirals that are 16 units thick, also with curved faces.

The next step is to convert the outer surface of the wedge shape from convex to concave.

#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.

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.

#EggsAndDarts is a common classical design motif with endless variations, two of which are shown here — the top-left variant has convex eggs and the bottom-left variant has concave eggs. The sketch on the right shows the bottom view of the concave variant.

This motif is neither specific to the #IonicOrder, nor limited to the #ovolo of the capital. It is common to find it laid on linear #moldings like #cymaRecta or #cymaReversa of a #cornice.

The egg shape, the dart shape, the degree of convexity or concavity, and so on, are infinitely variable from subtle to pronounced. Designers are not limited to convex or concave, and it is possible to combine both in a single design. Also, it is not necessary to use the eggs and dart motif at all. There are infinite possibilities. However, when the eggs and darts motif is used, it is almost invariably sliced off at the top, as the bottom view of concave variant on the right reveals.

The concave version here is quite subtle, but a more pronounced version can be really eye-popping. I will show how to construct one using just straight lines and circular/elliptical arcs exclusively as I originally promised in https://pixelfed.social/p/Splines/789956327130679640.

As usual, we start with a flat 2-dimensional plan with lines and ovals to use as #sweepingRails. Then, we add circles and arcs as #sweepingCurves to define the cross-sections. After sweeping the cross section curves on the rails, we create the eggs.

Simply #revolve an ellipse on its major axis to get the convex version of an egg. To get the concave version of an egg, simply create a flat slab and perform a #booleanDifference on that slab using a convex egg.

Once we have all of this preparatory work done, we have to transfer the 3-dimensional design from the flat surface it was originally created on to the #doublyCurved surface of the Ovolo. This requires some elementary calculations using circle geometry.

Previous— https://pixelfed.social/p/Splines/795361973789834465