Splines#Vignola divided µ into 12 parts for the #Tuscan and #Doric orders, and into 18 parts for the #lonic, #Corinthian and #Composite orders. To make it easier to move between the orders, it is helpful to choose a value for µ that is a multiple of both 12 and 18. We could start with module µ = 36, which is the least common multiple of 12 and 18.
However, in classical design, it's a lot easier if minor grid lines are multiples of 8 (as in music 'octaves' and poetry 'octets'). So, I use a grid with minor grid marks at 8 and major grid marks at 32, which leads us to choose µ = 144 as that is the least common multiple of 12, 18, 8, and 32.
This value of µ = 144 is still abstract because we haven't chosen a physical unit like mm or inch, but it does not matter at this point. Once the model is constructed with the chosen value of µ, it can be scaled up or down in software to get the desired column width or order height in physical units.
The total height of columns for the same µ are in the ratio 7:8:9:10 for the #Tuscan, #Doric, #Ionic, and #Corinthian, with #Composite the same as Corinthian. When µ = 144, the diameter is 288. So the height of the Ionic column is 288*9 = 2592 units.
The #pedestal, #column, and #entablature are always in 4:12:3 ratio across all orders. So the height of the Ionic pedestal is 864 and height of the entablature is 648, and the total height of the order is 4104 units.
Remember that in the Ionic order, µ is divided into 18 "parts". So each part is 144/18 = 8 units. With this we can convert between µ and parts in either direction.
Total order height of 4104 units is then 4104/8 = 513 parts. If you are trained in computer science, you will grow despondent that it is 513, not 512. Let go, and everything will be all right.
The pedestal is the easiest component, and we start with that.
Splines (@[email protected])
This is a sketch of the complete #IonicOrder, excluding #intercolumniation and #arches, which came later. Different people have different abilities and different levels of mathematical knowledge. I make few assumptions about the minimum knowledge one must possess to follow my posts. At a minimum, one must understand ratio, proportion, similar, congruent triangles, Pythagoras, and basic properties of circles, including radius, diameter, circumference, tangents, secants, and chords. No trigonometry or calculus is assumed, but people who have a knowledge of differentiable continuity, maxima, minima, and inflection points will have increased appreciation of the nuances of some designs featuring smooth curves and surfaces. I start with first principles, even if it might be a little boring for people with advanced skills. The most basic requirement is that one must be able to mark points on a 3D grid, draw a straight line between two points, and draw a circle or arc from the center. The CAD tools should help with the rest, for example, to find a point of tangency, draw a circle through three arbitrary points, or tangential to three curves (if possible). There are three components in the #Ionic order. Starting at the bottom is the #pedestal (which is optional), the #column, and the #entablature. Each of these three components has three subcomponents: — Pedestal has #basement, #dado, and #cap. — Column has #base, #shaft, and #capital. — Entablature has #architrave, #frieze, and #cornice. The pedestal, column, and entablature are always in 4:12:3 ratio. If all components are present, the total order height is divisible by 19. If there's no pedestal, the total height is divisible by 15. The entire order is parameterized by a SINGLE parameter — the radius of the column at its base. #Vitruvius called the radius a "module" (µ) — an abstract unit of measure independent of physical units. Components of Ionic column and entablature also have classic and modern variations.
Alankar