Mastodawn

#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

Splines (@Splines@pixelfed.social)

#Arcade #Intercolumniation without #Pedestal In https://pixelfed.social/p/Splines/803089629244302486, we saw #simpleIntercolumniation, also known as #Architravato. Roman architects combined columns with walls thick enough to bury half of the column width inside the walls and added arches to them for better load distribution. An arcade (multiple arches) can be run in series along a single wall, or in parallel to form a walkway. They can also be combined in both series and parallel configurations, perhaps the most famous of which is the #Colosseum in Rome. In the Colosseum, the outer walls follow an elliptical curve (even though it looks circular from the outside), and it has multiple tiers of arches in series. The interior has arches in concentric passageways in the lower tiers giving it a lattice-like design. Because arches distribute the load from above, they allow for wider intercolumniation. The rules for #ArcadeIntercolumniation differ depending on whether the columns have pedestals or not. Besides the arch itself, which is part of the wall, the figure shows some new architectural elements. The narrow part of the wall immediately behind a column is known as a #pier. The visible face of a pier between a column and the opening under the arch is known as #alette. The base of the pier has a molding, the flat part of which has the same height as the column base (µ) while the rest follows the #fillet and #cavetto or #conge of the #shaft. As we move up the pier, there is a horizontal molding known as #impost just below where the arc of the arch starts. The impost wraps around on the sides of the pier. Around the arc is a circular molding known as #archivolt, the bottom portion of which has a #fascia that is aligned with the face of the wall. The wall itself extends all the way to the top of the #entablature. It is worth noting that the entablature is repeated on the wall. It doesn't end at the columns and has two "outside" corners and one "inside" corner.

Pixelfed

Splines Mar 6

Splines (@Splines@pixelfed.social)

Floor Plan of https://pixelfed.social/p/Splines/802974815166948953 showing #intercolumniation. Greek architects classified temples and public buildings based on number of columns in front, number of columns in both front and rear, as well as interior columns. The simplest buildings are those with walls on three sides, and partial walls called #antae (singular #antis) in front, flanked by just two columns. Buildings with 2 columns in front are #distyle, 4 columns are #tetrastyle, 6 are #sexastyle, 8 #octastyle, 10 #decastyle, and those with 12 columns would be #dodecastyle. The classifications are refined further. Those with 2 columns flanked by antae are called #inAntis. These never have any columns in the back or sides. Tetrastyle buildings with 4 columns only in the front are #prostyle, and those with 4 columns in both front and back are #amphiProstyle. Sexastyle buildings like those in the previous post are called #peripteral. Octastyle buildings with densely arranged internal rows are called #dipteral, and when some internal columns are removed, the sparse structure is called #pseudoDipteral. Decastyle buildings are also called #hypaethral. With 10 columns in front and rear, these are noteworthy for their width. In fact, they are so wide that they don't have a roof in the middle, which is open to the sky. Only the four sides have roofs supported by columns. In addition to the number columns, there is a further classification based on #intercolumniation. #Vitruvius described five classes of temples, designated as follows: "#pycnostyle, with the columns close together; #systyle, with the intercolumniations a little wider; #diastyle, more open still; #araeostyle, farther apart than they ought to be; #eustyle, with the intervals apportioned just right." The building in my previous post (shown without walls) has 6 columns in front and back — so, it is sexastyle. It is also known as a "Peripteral Eustyle," with column "intervals apportioned just right."

Pixelfed

Splines Feb 25

#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

Splines (@Splines@pixelfed.social)

#Milestone 6 — #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 #Milestone 5 — #EggsAndDarts https://pixelfed.social/p/Splines/797069447808333887 #Milestone 4 — #IonicScroll https://pixelfed.social/p/Splines/795361973789834465 #Milestone 3 — #IonicColumn https://pixelfed.social/p/Splines/792803978865652429 Milestone 2 — Classic #IonicEntablature https://pixelfed.social/p/Splines/791021871062069787 Milestone 1 — #IonicPedestal https://pixelfed.social/p/Splines/790752092700055739

Pixelfed

Splines Feb 25

#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 25

#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.

Splines (@Splines@pixelfed.social)

#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.

Pixelfed

Splines Feb 24

#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.

Regola delli cinque ordini d' architettura : Vignola, 1507-1573 : Free Download, Borrow, and Streaming : Internet Archive

48 leaves : 44 cm (fol.)

Internet Archive

Splines Feb 24

Splines (@Splines@pixelfed.social)

#Braids #3StrandBraids After creating the two #helix curves as described in https://pixelfed.social/p/Splines/797732962403957263, switch to the front view and #project the smaller blue helix on the vertical "wall" of the XZ plane. Hide the original helix. Then switch to the top view and project the larger magenta helix on the "ground" or XY plane and hide the original helix. Now compare the figure in this post with that in the previous post. Both curves have now been #flattened from 3D helix to 2D #sinusoid. When viewed from the front (top-left portion of the diagram), the blue curve is still visible as a #sinusoidal waveform but the magenta appears as a straight line flattened on the ground. When viewed from the top, the magenta curve is still visible as a sinusoid but the blue appears as a straight line clinging to the vertical wall. In the view from side (bottom-left portion of diagram), neither waveform is apparent, and both curves appear as perpendicular straight lines. Only in the perspective view you can see both waveforms, but even here it is clear that they are both flat 2D curves oriented perpendicular to each other in 3D space. Our goal is to convert these two flat sinusoids back into a single composite 3D curve that shows the smaller waveform in the front view and larger one in the top view. In acoustics, a sinusoid represents a pure tone with a single frequency. The tone varies with frequency and its perceptibility varies with amplitude. Musicians and people familiar with acoustic physics will immediately recognize that the blue curve has twice the frequency (or pitch) of the magenta curve, while the magenta curve has twice the amplitude (loudness) of the blue curve. We can divide the period or wavelength into phases. For the blue one, we divide the wavelength into 4 phases of 6 units each and shift the magenta curve left by that amount. Later, we will divide the magenta one into 3 phases — one for each strand, and shift each rightward by that.

Pixelfed

Splines Feb 17

Splines (@Splines@pixelfed.social)

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.

Pixelfed

Splines Feb 17

#EggsAndDarts continuation from https://pixelfed.social/p/Splines/796961505955555432

The slab height depends on the roundness of the egg and whether we have a concave design or not. If we are using a concave base, then top half of the egg is eliminated. For a fully round egg, that means the concave variant must scoop out up to 16 units deep. The dart slab will match the egg slab in depth.

To create the 3-dimensional shape of the dart, first #rotate the fin profile 90° in 3D space along the straight line at the bottom of the fin so that the rotated profile is perpendicular to the two #sweepRails for the dart.

Using the two sweep rails and the perpendicular fin profile, #sweepTwoRails to develop the surface of the dart. Remember to close the planar hole at the end of the fin to get a solid #airtight object. As always, check for #nakedEdges and #nonmanifoldEdges to stave off problems later.

#Extrude the bottom of the dart until it is flush with the bottom of the oval slab.

Two details worth noting in the dart design are:

1. The most exaggerated portions of the dart fin are sliced off when the eggs are sliced. After slicing, the size of the fin is roughly in proportion to the rims of the eggs on both sides.

2. There is a gap between the dart arrow and the oval slab. See the gap between points R and T in https://pixelfed.social/p/Splines/796961505955555432. This gap is necessary and will automatically close when we transfer the egg and dart to the #doublyCurved surface of the #ovolo on the #capital of the #IonicColumn. That is because the Ovolo is shaped like a bowl whose top has a bigger radius than the bottom. As a result, the motif will be warped, and its bottom will be condensed to fit the smaller radius at the bottom, closing the gap in the process.

If you plan to use the eggs and darts motif on a linear surface where there is no warping, experiment with the arrow and tip for a pleasing result.

Splines (@Splines@pixelfed.social)

#EggsAndDarts continuation of https://pixelfed.social/p/Splines/796958366767133979 Successive egg slabs are 1/2 part or 4 units away from each other. So the thinnest part of the dart is 4 units. The tip of the dart is at point P, which is 22 units from the major axis for the egg slab. The inside egg is 4 parts or 32 units wide, and we will start with the dart fin also at 32 units wide, even though a portion of the fin will be sliced off when the egg is sliced by the red cutting plane. To find the start and end of the fin, draw a vertical line from P of length µ/2 or 72 units. Offset that line on both sides by 16 units. Mark the intersections of these lines with the outline of the largest ellipse with points A and B. Draw an oval centered on AB with major radius 16 units and minor radius 2/5 of that, or 6.4 units. Divide each arc of the oval into 5 equal parts and mark AD and EF at 2/5 of that. Copy the oval from its top point C to D as well as E. Mark the intersection of the copies at F. Trim the 3 ovals to get 4 arcs AD, DF, FE, and EB. Join them and close the curve with a straight line from A to B. Draw a square 12 x 12 whose bottom edge is centered on P. From each of the top 2 corners, draw a circle with diameter 4 units, shown by RS. #Reflect point P using the major egg axis as a #mirror to get point Q. Draw a #circleThrough3Points P, Q, and R. The origin of the circle, point O should be 3.5 units directly below the base of the large oval. Draw a vertical line up from S to where it is tangent to the side of the large oval. Trim the straight line and arcs to get the left profile of the dart starting with A, passing through S and T, and ending at P. Join all 4 segments and reflect them using the line PC as the mirror. These mirrored copies are the #rails along which we will sweep the fin of the dart with #sweepTwoRails operation to create the dart, but we must orient the fin to be perpendicular to the rails first

Pixelfed

Splines Feb 12

Splines (@Splines@pixelfed.social)

#IonicVolutes are the sinews of #IonicScrolls. Without #volutes, there would be scrolls, but not #Ionic Scrolls. Each scroll starts with a volute in front and is modulated by as many as six volutes of different shapes and sizes as it reaches the back, with the scroll surface tightly hugging the volutes at each contact point in ALL 3 dimensions. This is a key point to remember before we start #reverseEngineering the first #primaryProfileCurves from old image scans. This diagram shows the #scaffolding we will construct using straight lines and rectangles, first in 2 dimensions, then place them front-to-back in 3 dimensions using precise markers, and finally scale and superimpose the volutes on this scaffolding. All of this will be done before we derive the primary profile curves from the image scans. How did I know about this scaffolding? I didn't. It is not documented anywhere that I'm aware of. I developed this after years of striving to derive the correct shape, and I hope that there are people who can still "see" things I might have missed and help improve the design. So, the actual process went like this: I drew outlines from 2D image scans in the top view, getting close to #Vignola's detailed sketches. Then, I did the same thing with image scans in the side view, and I found that the designs didn't line up. After several iterations, I got the designs to line up in both views, and it was obvious that the bell shape of the scroll would follow the large volute in the front. So, I used the large volute as a "rail" and tried to sweep the primary profile curves on one rail. Big mistake! The undulating shapes of the primary profile curves wobbled wildly on the single rail — The middle, 3/4, and back of the scroll were twisted out of shape! Instead of trying to #sweepOneRail, I decided to clamp down wobbling with another operation called #sweepTwoRails, using volutes at both front and back ends as rails with less wobbling. You will need a #CAD tool to practice.

Pixelfed