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Sunday, January 28, 2018


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.

The Two Centred Drop Arch, or as it is affectionately known the Gothic Pointed Saracenic Arch, is more or less the opposite of the Lancet. Here, the foci are inside the bounds of the arch. In case you were curious, when the foci are 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.

Friday, December 22, 2017

Heaven's Reach

or, The Ha'ikū Stairs

It seems like an incredibly long time ago now that I stood at the top of the fabled 'Stairway to Heaven' nearly three months past. As with all things, the experience comes and goes and the pictures superpose memory.

In the early hours of dawn, we set out to catch sunrise at the summit. Arguably, we took an admirable go about it but ultimately fell short by an hour or two. Not knowing the way or even where to begin, the first moments echoed everything we saw on the internet regarding the ascent. From the north, the stairs themselves are very illegal due to the state of decay that grew upon the since their use in World War II as a pathway to a secret radio network. From the south, the trail itself is not necessarily illegal, depending on how you go about it. At nearly three in the morning, we thought it would be a safe venture. To our education, there is no end to the tenacity of disgruntled neighbours. Within less than five minutes of coming to the trailhead, we were joined by another pair of hikers and, even shorter thereafter, the police.

Sympathetic to our good nature and general disposition that must have been contrary to the other sorts of blokes who frequent the area for early morning adventures, they seemed relieved that they would not have to detain anyone. Even better, a minute after they arrived, they jumped back into their cars and took off to the background noise of a radio squelching someone's stolen vehicle. As soon as that, they were gone (and so were we).

The trial itself snaked its way across the valley between ridgelines for a handful of miles before taking a great leap upward and through the bristling slopes of O'ahu. That it was still dark served as a testament of how laborious the process was; stars themselves over the edge of the horizon seemed to sweat in the fall of early dawn. That mist which lingers in all cool places which know a striking heat to the day blanketed the cloudforest and, as we scrambled our way higher, grew subsequently thinner.

After a time, the first light of dawn broke over the eastern mountains and cascaded the world in a brilliantly green light. The light, it turns out, of billions of living things drinking it in at the very first opportunity. Seeing the scale of the canyons and the lushness of the slopes in spite of their offensive steepness was breathtaking. Or maybe it was the general ascent. In either case, the shedding of night's veil was a welcome relief even though we knew we would not reach the tower in time for true sunrise.

Eventually, the old radio tower appeared on a distant peak. Another good mile or so of twisting, muddy trail lay between us and it yet the end was near. All along the upper pass, a series of precarious ropes dangled on the slick mud faces for hikers to climb. More often than not, the top of the rope was hidden around a bend or so far overhead that it disappeared in the sheerness of the slope. More often than not, it looked like a better idea to not use it. Until trying. Thankfully, they all held to their purpose and everyone returned safely.

It was around this point in the hike that we realised the severity of how much the islanders did not want anyone climbing the stairs themselves. Out of the fog, we heard the sound of a helicopter on the north side of the trail which holds the entire length of the stairs. As it turns out, they were surveying the stairs for anyone who managed to sneak past the security stationed everywhere withing creeping distance of the entrance.

In the end we were, as some old nefarious vandal noted, 'On Top'. The tower itself was a neat combination of derelict history and waystation to rest. A sizeable congregation of hikers and campers (who slept inside it) waited there for the morning and so we joined their ranks and looked down over the stairs. A quarter mile down or so held the iconic scenery where the stairs descend at a frighteningly steep angle and the rising mist obscures the world below.

From above, it looks almost like a miniature model but in reality plummets a few thousands of feet down to the lowlands and the ocean.

Down the back side lay a collection of ruins which once housed other communications stations and outbuildings for the radio tower. Along the stairs in the brush, a pair of thick cables runs parallel to the ridgeline where the nation's most secret of secrets once travelled. In a sense it is remarkable how simple and how resilient the way has become. While a good portion of it is in severe disrepair and wholly not recommended for casual travel, the nature of it is supremely interesting. When coupled with the views, I am considerably glad and fortunate to have walked a part of the stairs before they are either removed or some more serious effort is taken to prevent expeditioners from pursuing them.

And, of course, I was compelled to find the US Geodetic Survey marker at the summit, an old habit from my long days in the mountains in the east.

Happy trails!

Thursday, November 16, 2017

Across the Seas

16,000 endless leagues of open ocean pass beneath the keel, grey as slate and blue as glass, each one preserved in the same rhythmic imbalance of a crew without landfall in as many days as pass between two distant shores. Were it not for the changing of the moon each night might pass undistinguished from the last and as easily forgotten. Hours stretch into weeks, and in the blink of an eye it seems a month has come and gone before the sun meets noon. And yet, to the eye of western civilization, it is all dreamlike and distorted through a lens unappreciated through modern perspective.

Pago Pago Harbour, American Samoa
Nations take to the seas, their population more heavily laden on the decks of trawlers than the wharf from where they hail. To live for years at a time aboard a vessel no longer than a hundred yards, never mooring and never returning until the voyage is at an end, it would be to another mariner unfathomable. And yet that is a way of life. From the early hours of manhood until the seas harbor safety no more, there is a people who become strangers to land.

Port Villa, Vanuatu

Entrance to Port Villa, Vanuatu

Port Villa Harbour
For a day or three spanned by months upon months on either side, the Pacific's fishermen know only what touches the ports. Shirts on the backs of the deck hands once belonging to defeated sports champions and collapsed cultural icons, modern phones out of context from use, food for the bait of the catch and little else that cannot fit inside a footlocker is all that there is to sustain a life at sea. Perched on the tops of some, there stand greenhouses to grow fresh vegetables and herbs where otherwise there would be none. A pack of anxious dogs lie in a litter while their mother barks at foreign scent. Freezers to store the catch with fans and frost so blisteringly cold that a person would be frozen through in a quarter hour. Hands and feet calloused by working lines and scarred from the barbs and hooks that pass between them. There is a sovereignty in the life of the high seas no longer concerned or consumed by modern society.

Magnetic Island, Australia

Townsville, Australia
When the sun passes behind the horizon, an unbroken line of blue stretches across the sky 800 miles from the nearest rock. On those rare and hypnotizing nights where the moon has taken instead to the daylight, all manner of stars blaze across that midnight abyss. More lights appear as the eye drinks in the night than the busiest cities, each shimmering with their own hidden mysteries. Constellations that guided ancient mariners and storytellers return to the wanderer, cleft by the haze of the Milky Way. There it hangs in the sky and spans from one end of the horizon overhead and back down again to meet the earth. Even in the darkest places of the land it has never been so clear. Tendrils of its enormity fan out from the core, slight tints of colour painting the otherwise two tone heavens.

O'Ahu, Hawai'i
And again when daylight comes, lingering cloud cover broken in spectacular hues and crepuscular rays, only to reveal those rolling waves of blue and grey. When those rare days come between and land falls on the distant shore, it is with the awe of a world unchanged. So many thousands of islands and cultures and peoples lie on the edge of the modern world's chart that each one bears that same ancient wonder of exploration and mystery. And when the mountains and jungles and reefs drift into view, suddenly all that compels man to venture across the seas becomes understood.