Stone archways, as a means of architecture, represent a great many things beyond the confines of masonry. The idea of a keystone is something that more people would recognise in a different context entirely. Rather, most people with whom the words has come up in conversation strangely don't actually know what a keystone

*is*. However, our use of stones to hold open doorways in an otherwise unbroken wall dates back to, at least in bridges, the Sui Dynasty in China in the form of the Anji Bridge constructed over the period of a decade starting in 595 A.D.

My concern with the archway extends a little farther back into the history of builders to the Roman Empire who implemented their learnings of the assimilated Etruscan Empire to build architectural archways that did not rely on piled earth to compensate for the lateral thrust resultant of the radial nature of the stone's arrangement.

While there are literally millennia that span between the first arches and today, I found it difficult enough to locate any mathematical documentation for their design and implementation that I disregarded all of that and started from scratch. Or mostly, anyway.

First off, there are substantially more types of arches than I knew about. Of the ones which I had no aesthetic desire to explore, there are round arches, trefoil arches, three centred arches, elliptical, inflexed, ogee,

*reverse*ogee, tudor, parabolic, etc. etc. Each of these has their own design characteristics which likely corresponds to some far distant cultural revolution and style. Good for them, not so much for me.

Before diving into this, I will say that while drawing these all on onionskin paper was great for transferring lines, it was extraordinarily difficult to photograph for a number of reasons. For that, I will include only a few of the iterations to illustrate what these are all about.

The first and most quickly discarded of the lot were Lancet Arches. These, like the rest of the class that I am dealing with, is designed from two points like foci on an ellipse. Only instead of making a continuous curve, the arcs intersect somewhere and that somewhere is the top of the arch. For the Lancet Arch, the two foci are

*outside*the bounds of the stone, residing somewhere in what will later be the abutment (stones that reside one layer outside the structural arch). To form the intrados and extrados (inner and outer dimensions of the springers) I based these all of of the arbitrary metric of making the arches 5 'units' tall to for the rise and 6 units to the crown.

For the first iteration of the Lancet Arch, I placed the foci at 4 units from the centre, which had a intrados arc radius of 6,4 and the extrados arc radius of 7,2 units. Because the two foci are relatively close together for the length of the arch radius, the sides seemed to come past the vertical tangent and curve back inwards. No good.

To compensate for this, I moved the foci an extra unit farther apart to 5 from centre. This made a more pleasing and more pointy arch, but for a standard doorway dimension of 32" this makes the thing disproportionately tall (an extra 87" on top of an already 80" door). I'll spare you the subsequent iterations of this style of arch but spoiler, they get narrower and pointier.

*on*the bounds of the intrads and extrados, it becomes and equilateral arch. Here, however, they are inset and provide a significantly more favourable ratio of rise to span. Because this will be used for a doorway that is more likely than not going to be at least width and a half if not double wide, the lower the rise for a given span the better. A combination of aesthetics and math secretly drove me to optimise this quality throughout the arch exploration that I unwittingly undertook.

Again using the magic constraints of 5 units for the rise and 6 for the crown, the rest is fairly simple. For the Gothic arch, the closer the foci move towards the centre (then becoming a round or semicircular arch), the closer the rise to span ratio moves towards 1. In the above drawing, the top arch has a focal distance of 1 2/3 while the bottom has a focal distance of 1 1/3. You can clearly see how the arch becomes less pointy and closer to a single curved half circle as the distance from the centre decreases.

Ultimately, I liked the look of the arch which had an approximate focal distance of 1,25 units from the centre. but not the constraint of the rise to span ratio it presented. The solution? A Two Centred Pointed Segmental Arch of course! The term Segmental means that the arc length is less than 180 degrees, or is less than a full semicircle. Pointed is a bit obvious, but means that the two Segments meet at some angle that is not parallel tangent. This style combines a few things from various other arch types and, at the expense of introducing considerably more variables than I really wanted to explore, produces wider arches that do not come out round at the crown. While there is nothing wrong with round arches, and with a proud keystone you would probably never notice the difference anyway, the simplicity of that style defeats the purpose of trying to learn more about them.

The Two Centred Pointed Segmental Arch is similar to the Two Centred Drop Arch in that the foci lie within the intrados. However, this is the first one that has them

*offset*from the spring line (where the arch stops and the imposts or skew backs start). Moving the foci above the spring line would give us the same problem that the Lancet Arches have where the shape comes back in and makes it bulbous. Rather than that, the foci drop below the spring line.

Insofar as I am aware, the two most significant variables here are the distance the foci are offset from the centre and how far below the spring line they are. To determine the distance from the centre, I did not just set the spacing at 1 or 1,25 or however many units because there are a few more alignment problems that come from that. So instead, I set a distance of 1 or however many units offset, then drew a series of arcs at a fixed length of 2 units towards the inside of the arch. This formed my sort of anchor point. From the centre of this point I drew another arc, this time at a fixed length of 2,5 units, and intersected it with a circle drawn from the centreline at that original radius of 1 or however many units long. In the top drawing, that radius is 1,25 and in the bottom it is 1 unit. Through a bit of math and a lot of trial and error, I found that doing it this way helps control how much the rise to span ratio changes. Because I did not want a large number for that, the way the foci move by constraining two separate but seemingly unrelated variables keeps it from changing too dramatically while allowing for a fine adjustment of the keystone profile.

Since it is likely none of that made any sense, I'll move on. For the arch I am actually designing (and yes, none of those were actually arches yet), I chose the top one in the previous drawing. It has a rise:span of 1.11:1 and also has a cleaner division of units being that the span is exactly 9 for a crown of 6.

From here, I redrew the intrados and extrados, leaving everything but the foci and spring line out. Because I do not have an accurate and repeatable way of measuring arc lengths, I used the good old guess and check method to divide the arch into a number of equally sized stones. The only difficult part of this is getting the spacing right when the two halves meet at the keystone. I tried a number of divisions, and I found I liked the size best when there are 15 total, 7 on each side plus the keystone. This gives almost square sprigners, but because this is a secondary constraint to the dimensions of the arch itself, it can be changed at any time prior to the stones actually being cut.

From the foci, I took the lines from centre to intersect each of the divisions. Although this is nothing more than an educated guess, like almost everything else so far, the lines must be drawn this way for the stones to actually sit correctly once placed. While there are styles of arch (or pseudo-arches) that do not rely on the transformation of radial force into lateral force via wedges, I can only assume that incorrect geometry here would ultimately lead to failure.

Once all the springers and keystone are drawn, I moved on to the imposts. These are the things the arches sit atop of and support the weight of the stone in the arch. Columns or stacked stone or any number of things can serve as the sides of an archway, but for the sake of completeness I decided to continue with the design experiment.

In order to determine the height of the subsequent imposts and stone beneath, I tried a few things. First, as shown above, using the height of the springer at the intrados. This gives the smallest relative dimension, whereas using the height of a springer at the extrados would give the largest. Using the height of a springer midway between the two is what I ultimately used because it made sense.

And there it is, a finished arch. In later iterations, I will almost certainly change the shape of the keystone as depicted in the bottom half of the previous drawing, but that is a problem for another time. The primary design challenge now is how to integrate the arch into a wall and the adjoining of dissimilar materials. To fill a stone arch with a wooden door, for example, requires fastening one to the other in such a way that it will not be mutually destructive or cause eventual instability.

As I am not an architect, the more I learn about seemingly the simplest of processes, the more I come to appreciate the complexities that secretly reside in untold centuries of our great builders' minds.

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