Understanding classical column entasis

- In this movie we're going to look at the shaft for our medium-detail column. Now this shaft family is perhaps a little bit more complex than it looks at first. There's actually a couple of things going on here. First of all, I've got two windows tiled on screen. I'm looking at a Front Elevation Simple, and I've adjusted the crop region so that we can see the top, bottom, and middle of the shaft a little bit more clearly. And the first thing that you notice when looking at that elevation is the tapering or the entasis of the column. And so that's one of the things that we're going to discuss here in this movie. The other thing is of course I'm currently in the massing environment as you can see from the 3D view on the right. So, I'm using the massing environment because these lines that I have drawn here that represent the fluting on the column, they actually required a feature that is available in the massing environment, but we'll talk about that in a future movie. What I want to talk about right now is actually the shaft itself. Now when I highlight it and select it, you'll notice that the shaft itself is actually a nested family. Now I've done that for a couple of reasons. One of the reasons is I can reuse this shaft family in other families, so if I want to use it in a Doric order, if I want to use it in an Ionic order, I have it available as a separate family that I can use and just flex to match the proportions of those other orders. So that makes it a little bit more convenient to reuse the pieces. The other reason that I did that is because actually it didn't really need the massing environment, so I created that in the traditional Family Editor. So I can click Edit Family here and open that family up directly. Now you remember you can take a traditional family and nest it into a massing family, but not the reverse. You cannot take a massing family and nest it into a traditional. So the reason that I built this as a traditional family first is, again, for that flexibility. I can use it in other traditional families, or I can use it in massing families. So that's why I did that. Now, the shaft itself here is actually built from two pieces of geometry. There's a sweep, and then inside the middle of that sweep you can see that there's a cylinder that kind of fills in that hole. Now the reason there's a hole there is because the way this sweep looks, is there's a round path here, and if I zoom in a little you can see that a little bit better. And then this profile, which sweeps around that circle. And that leaves that void in the middle. So then you just come back and fill that in with a solid extrusion. Now when you create the profile for your sweep, you have two options. You can either draw that profile directly using the By Sketch option, then you'd have to actually draw the sketch directly in that sweep. Or, you can load in another family that's a profile family for that purpose. Now I've opted to do that and again, there's a couple reasons why I've chosen to do that. One reason is the same as why I've built the shaft family separately, and that is it makes it more reusable. I've got this profile family and I can load it into other families that need a similar shape, and it's just more convenient to build it that way. The other reason that I've done that is actually this profile family is fairly complex, so by putting all that complexity in a separate, nested family, it allows me to compartmentalize and I can work on the profile, get it all correct, and then when I'm done with it just simply load it in and build the sweep. So it's just more of a management tool that makes a little bit of sense, and so that's why I rely pretty heavily on profile families when I'm building sweeps. Now, I always name my profile families with some sort of prefix, like this PRF so that when I'm looking at them in a list in Windows Explorer I can easily spot where those profile families are. So here on Browser, on the Families branch, I'm going to select the shaft profile here and right-click it and choose Edit. And that takes me into the third family now. So I'm now three families deep, and that's the only part that you have to really pay close attention to so that you don't get confused. Now, you can already tell that this family is a little bit complex. There's a lot going on here, and we're zoomed out pretty far away. And that's because there's actually some fairly large distances here, so this dimension right here, if I zoom in, you can see that that's a pretty long distance, and that's accounting for some of the reason why this family is so far away. There's also several small pieces, and several distances, and you can see by all the different parameters that are in this family that are flexing those different distances, that helps to make it a little bit complex. But what I really want to talk about in the remainder of this movie is the entasis. So, again, that's that small tapering curve that you can kind of see here on screen with this curve right here. Now, the way that the entasis works is it applies to the top two-thirds. So why don't we leave the family for a moment and just talk about entasis, so make sure that we understand exactly how that works. So first of all, what is entasis? Well, entasis is quite simply the process by which a column will taper as it moves up to the top of its shaft length. And so, classical columns are not straight up and down cylinders, they always have this sort of tapering that goes on, up to the top. And it's sort of an aesthetic consideration or an optical illusion. There's some theories that the ancient Greeks did this in order to visually increase the length of the columns or just because it created a more visually pleasing composition, and so on. So it's just a visual trick that we see in the columns to make them appear more aesthetically. But let's actually dig in to how we create the entasis, okay? So what I've got here is two diagrams. I've got a diagram of the column shaft on the left side, and then a blow-up of it on the right, but I've taken that blow-up on the right and I've actually squished it down in the vertical dimension so that we can understand the subdivisions a little bit more clearly. So let's start with what we know. What do we know about this column shaft, and how are we going to put this together? Well, the first thing that we know is the entasis height, and I've kind of indicated where that height occurs. It's really that top two-thirds of the shaft. So the method that I'm using, which I've borrowed from Chitham here, to calculate the entasis says take that height and divide it into an equal number of segments. Now it doesn't say how many segments. You can use four, or five, six, whatever. I've chosen six here. So we've taken our entasis height and we've divided it into six segments. The next thing we do is we take our base diameter. Now you remember our base diameter is measured at the base of the column, but because the first third of the shaft is cylindrical, it actually occurs at that first third as well. So there's the base diameter right there. So that's the next piece that we know. Now the next thing we can do is we can divide that base diameter in half. So if we just take the distance between those two portions, that's the radius of the shaft, or half of the base diameter. So we know that as well, that's easy for us to calculate. The other thing that we have to determine is how much do we want the column to taper at the top? So, what do we want the total tapering to be when it terminates at the top? Now, traditionally that's somewhere in the neighborhood of about 85% of the overall base diameter, so here in the diagram I'm using 85. With the sone column, it'll be a little bit different than that, but it'll be close, okay, so it's in that neighborhood. Alright, so those are the three things that we know. Now the way that the construction technique works is that you create a semi-circle at the base diameter and you can see that in the diagram. And then where that semi-circle intersects the lines drawn from the radius and the top radius, those two reference planes that I just highlighted a moment ago, that gives us a small arc there that I've highlighted here, and we divide that arc into the same six segments that we used to divide the entasis height, and these are six equal wedges that we divide that in. That creates points along the arc, and if you project those points up, as straight lines, you get the points that your entasis curve needs to pass through. And it turns out that when I was analyzing this, that that curve is really just an ellipse. So what we actually have here, this curve for the entasis is really just a long, thin elliptical arc. So that being the case, we now just need to figure out mathematically how to construct the required ellipse. Now, to construct an ellipse, there's a few things we need to know. We want to know what the semi-minor axis is. Well, that's easy enough because it basically matches our radius there. And we need to know our semi-major axis. Well, that's actually the point that we don't know. So, why don't we consider the standard formula for an ellipse. Now you can look this up online, and there's a standard ellipse formula, and it looks like that. And the way the standard formula for an ellipse works is, there's actually four variables that you need to know. So to kind of understand what it is we're trying to do, let's take a diagram in the middle here, which represents the shaft that we're trying to create. The diagram on the left is more simplified, but in the middle is actually what we're trying to create, this sort of long, thin ellipse. The standard formula says, you can find any point on an ellipse as long as you know these four variables. Well, we know the points, right? We've got the point already because we know several of the key variables. What we're trying to do is make sure that our ellipse actually passes through that point. Now, I had the help of an engineer friend of mine to figure out how to reduce this formula and translate it into Revit format, so let me just kind of walk you through it. So if we review again what we know, we know the base radius, that's half of the base diameter. That's letter B in the formula, so we've got that. We know the entasis total. That's letter Y in the formula, that's the top two-thirds of the shaft, so we know what that is. We also know the top radius, because that's that 85% or so of the base radius, and that's letter X in the formula. So we've got three of the four pieces of information. Well, mathematically, if you've got three of the four required, you can solve for the fourth one in the formula. So, it turns out that letter A, the missing variable that we need, is going to give us a point way up here. Now, that's the point that we need to calculate in order to create the ellipse properly. And that's why when opened up the family in Revit it was zoomed out so far away, because that point is actually way off in space. So, if we just take our mathematical formula and we solve for A, it generates this kind of scary looking square-root formula here. Well, if we translate that into Revit format it looks like this. Now, rather than make you feverishly scribble and write that down, I have provided this formula in a text file that you can copy and paste to use in the Family Types dialog in Revit, and we will look at doing that and using the formula and recreating the ellipse in Revit in the next movie.