[OB] Dimensions of Urithiru

[ccstat he/him]

EDIT: Updated with some suggested numbers from the thread

It is a little hard to grasp the scale of the tower city. Even Shallan can't capture it on the page! (To talk about that issue and whether it is driven by scale, psychology, or magic, see this thread.)

Here are all of the physical descriptions we have of Urithiru. (If I've missed something you consider important, include a quote and I'll add it to this post. My chapter 87 quote is incomplete at the moment.)

Spoiler

WoR Interlude I-10 (Szeth)

Spoiler

 

The highest tower in the world, hidden in the tops of the mountains, was perfect for his contemplation. […]

He did not look up. He would not meet the gaze of the God of Gods. But it was good to be in the sunlight. There were no clouds here to bring the darkness. This place was above them all. Urithiru ruled even the clouds.

The massive tower was also empty; that was another reason he liked it. A hundred levels, built in ring shapes, each one beneath larger than the one above it to provide a sunlit balcony. The eastern side, however, was a sheer, flat edge that made the tower look from a distance as if that side had been sliced off by an enormous Shardblade. What a strange shape.

He sat on that edge, right at the top, feet swinging over a drop of a hundred massive stories and a plummet down the mountainside below. Glass sparkled on the smooth surface of the flat side there.

WoR Ch 86

Spoiler

The plateau everyone was crowding onto was relatively small, by the scale of the Shattered Plains—but it was still several hundred yards across. [...]

Beyond them spread the peaks of an unfamiliar mountain range. It was the same plateau, and here was in a ring with nine others. To Shallan’s left, an enormous ribbed tower—shaped like cups of increasingly smaller sizes stacked atop one another—broke the peaks. Urithiru.

WoR Ch 87 (Shallan)

Spoiler

Shallan stood at the front gates of Urithiru, looking up, trying to comprehend.

Inside, voices echoed in the grand hall and lights bobbed as people explored. [...]

The large stone field in front of the city bore very few rockbuds, and the ones that did grow were tiny, smaller than a fist. They would provide little wood for fires.

The field was ringed by ten columnar plateaus, with steps winding around their bases. The Oathgates. Beyond that extended the mountain range.

Crem did cover some of the steps here, and dripped over the sides of the open field. There wasn’t nearly as much as there had been on the Shattered Plains. Less rain must fall up here.

Shallan stepped up to one edge of the stone field. A sheer drop.

WoR Ch 89 (Dalinar) 

Spoiler

 

For such an enormous tower, the roof was actually relatively small, and not that encrusted with crem. This high, less rain likely dropped during highstorms—and everyone knew crem was thicken in the east than in the west.

Storms, this place was high. His ears had popped several times while riding to the top, using the fabrial lift that Navani had discovered. She spoke of counterweights and conjoined gemstones, sounding awed by the technology of the ancients. All he knew was that her discovery had let him avoid climbing up some hundred flights of steps.

He stepped up to the edge and looked down. Below, each ring of the tower expanded out a little farther than the one above it. Shallan is right, he thought. They’re gardens. Each outer ring is dedicated to planting food. He did not know why the eastern face of the tower was straight and sheer, facing the Origin. No balconies along that side.

He leaned out. Distant, so far down it made him queasy, he picked out the ten pillars that held the Oathgates.

 

 WoB (source) 

Spoiler

 

FLETCHERSHAIR

Why is Urithiru safe from the Everstorm?

BRANDON SANDERSON (PARAPHRASED)

It's on the east side of a mountain, and it might be high enough to be above the storm. So it's not SAFE, it's safER.

 

Oathbringer Ch 2 (Dalinar)

Spoiler

 

Above him rose the enormous tower city of Urithiru, a strikingly high structure built up against the mountains. Created from a sequence of ten ringlike tiers—each containing eighteen levels—the tower city was adorned with aqueducts, windows, and balconies like this one.

The bottom floor also had wide sections jutting out at the perimeter: large stone surfaces, each a plateau in its own right. They had stone railings at their edges, where the rock fell away into the depths of the chasms between mountain peaks. At first, these wide flat sections of stone had baffled them. But the furrows in the stone, and planter boxes on the inner edges, had revealed their purpose. Somehow, these were fields. Like the large spaces for gardens atop each tier of the tower, this area had been farmed, despite the cold. One of these fields extended below this balcony, two levels down.

Dalinar strode up to the edge of the balcony and rested his hands on the smooth stone retaining wall.

 

 Oathbringer Ch 2 (Adolin)

Spoiler

 

They were deep within the tower, for all the fact that this cavern was as large as a small town.[…] Oil lamps on the ground did little to push back the darkness of the enormous room, which had a ceiling that went up four stories.[…]

“Section two,” Adolin said, rapping the knuckles of his left hand against the wagon. “Taverns are being set up along the central corridor with the lifts, six crossroads inward. My aunt expressly told your highlords this.”

 

Oathbringer Ch 4 (Dalinar)

Spoiler

 

He passed Navani—who sat at the writing desk working on her memoirs again—and stepped onto his balcony, which hung straight out over the cliffs beneath Urithiru. […]

The Everstorm appeared in the distance, its black clouds lit from within by crackling red lightning. It was low enough in the sky that—fortunately—its top wouldn’t reach Urithiru. It surged like a cavalry, trampling the calm, ordinary clouds below.

Dalinar forced himself to watch that wave of darkness flow around Urithiru’s plateau. Soon it seemed as if their lonely tower were a lighthouse looking over a dark, deadly sea. [...]

Oathbringer ch 4 (Dalinar)

Spoiler

Dalinar was surprisingly nervous as, several hours later, he rode one of Urithiru’s strange fabrial lifts toward the roof of the tower. The lift resembled a balcony, one of many that lined a vast open shaft in the middle of Urithiru—a columnar space as wide as a ballroom, which stretched up from the first floor to the last one.

The tiers of the city, despite looking circular from the front, were actually more half-circles, with the flat sides facing east. The edges of the lower levels melded into the mountains to either side, but the very center was open to the east. The rooms up against that flat side had windows there, providing a view toward the Origin.

And here, in this central shaft, those windows made up one wall. A pure, single unbroken pane of glass hundreds of feet tall. In the day, that lit the shaft with brilliant sunlight. Now, it was dark with the gloom of night.

The balcony crawled steadily along a vertical trench in the wall

The balcony ground its inexorable way toward the top of the tower. Only a handful of the dozens of lifts worked; back when Urithiru flourished, they all would have been going at once. They passed level after level of unexplored space, which bothered Dalinar. Making this his fortress was like camping in an unknown land.

Although each tier was smaller than the one below it, this roof was still over a hundred yards wide. 

 

Oathbringer Ch 8 (Shallan)

Spoiler

 

A simple landscape; she should be able to draw a simple, calming landscape. She sat on the edge of one of the ten Oathgate platforms, which rose ten feet higher than the main plateau. […]

The city dominated her view. It stretched impossibly high, and she struggled to contain the enormous tower on the page. […]

If she locked it down into a sketch, would she finally be able to grasp its incredible size? She couldn’t get an angle from which to view the entire tower, so she kept fixating on the little things. The balconies, the shapes of the fields, the cavernous openings—maws to engulf, consume, overwhelm.

 

Oathbringer Ch 15

Quote

Radiant stood in what felt like a very stiff pose, Blade held before herself in two hands. She’d only scraped Pattern on the ceiling two or three times; fortunately, most of the rooms in Urithiru had high ceilings.

Oathbringer Ch 18 (Shallan)

Spoiler

Most rooms on the outside had these small balconies, but hers on the second level was particularly advantageous. It had steps down to the field below. Covered in furrows for water and ridges for planting rockbuds, the field also had boxes at the edges for growing tubers or ornamental plants. Each tier of the city had a similar one, with eighteen levels inside separating them.

 

Summary:

• There are 10 tiers of 18 levels each, seemingly carved into the mountain.
• Each level is larger than the one above it, at least by the width of a balcony.
• The whole thing looks like 
• The roof of the top level is ~100 yards across.
• The lowest level has a large plateau in front with the 10 oathgate platforms
• The oathgate platforms are raised 10 feet over surroundings, and are large enough to hold three armies worth of Everstorm survivors
• The lowest level also has large gardens/farms surrounding it.

Assumptions: We have to make a lot of these. I'll update them as we get more information, or as others make convincing arguments, but for now let's go with:

• A Rosharan yard is close to 1 meter.
• Each balcony is 1.5 meters wide.
• Each new tier extends 20m 50m past the bottom of the one above, to accommodate gardens.
• Each floor is 4.5m 5.5m high, including the thickness of stone that supports the floor above. 
• 20,000 people crowded onto the oathgate plateau at the end of WoR. (Does someone have a better number for this?) They pack closely but don't stab each other, about ~4 people/sq meter. This gives us an 80 meter diameter platform. The plateau is described as several hundred yards across, so we'll assume 300m for now. 

Conclusions: Using these numbers, we get a tower that is 1.5km wide at the base, and 1km high. Here's how that looks when compared to tall buildings on Earth (modified from this image on Wikipedia). A circular arrangement of ten 300m Oathgate platforms has an inner diameter of ^700m, and an outer diameter >1300m.

To my mind, these are probably underestimates. As has been mentioned, Burj Khalifa is the tallest building on earth at 830 meters, and it has 160 floors to Urithiru's 180. At 4.5m per level+ceiling, I don't believe that our assumed thickness of stone has the structural integrity to hold up that massive a structure when it is filled with tunnels and cavernous spaces. Yes, surges could augment that strength, but I'm skeptical of a fabrial that would function to keep the city standing for millenia. I think we need to add more thickness to these load-bearing floors, and probably more width to the tiers. But before I modify the above assumptions, I'd like to see what the rest of you think.

EDIT: Here is a perspective view of the tower and the oathgate complex. There is some disagreement in the thread about the arrangement of oathgates relative to the city, so I've included three versions here. Please comment to make your case for the arrangement you think is correct (one of these, or something different).

Previous images:

Spoiler

 

First estimate:

Overhead view:

 

 

 

 

Edited October 3, 2017 by ccstat

[The Invested Beard]

So it's really really really big then.

[Runyan Firetree]

> the eastern face of the tower was straight and sheer, facing the Origin. No balconies along that side.

[asterion137 he/him]

Rosharans are taller than people on earth, so 4.5 meters is likely a bit low for a room+ceiling. And I agree that the stone ought to be thicker to account for the weight of the tower. The gardens ought to be bigger too, since they presumably can feed the population of the city.

[ParadoxicalZen he/him]

Also, just to point out; the Szeth passage says a 100 levels, possible typo?

[+Extesian he/him]

This is brilliant @ccstat. A silly point regarding your diagram - Urithiru isn't symmetrical coz it's against a mountain 

Quote

The easter side, however, was a sheer, flat edge that made the tower look from a distance as if that side had been sliced off by an enormous Shardblade. What a strange shape.

There's another unreliable WoB

Quote

ALTERODENT

What are the dimensions of Urithirurelative to something like the Sears Tower?

BRANDON SANDERSON

That's a question for Peter and Isaac. They work out the math. It's bigger! More area, but maybe not taller. I'm not sure

Source

 

Edited September 13, 2017 by Extesian

[WhiteLeeopard she/her]

Great post, I'd upvote but I'm out of them.

I personally think they can fit all or a good chunk of Roshar's population into Urithiru. They get in 8 warcamps worth of alethi in (maybe half of Alethkar?) , and they barely make a dent in the space of Urithiru. They also probably have enough farming lands that when added to soulcasters humanity could take refuge there indefinetely. Each floor will probably be very tall, 5m minimun, after all it may be a refuge but its also the center of the KR power, stately, graceful, awe-inspiring, etc. So, refuge/city/fortress/palace all into one.

[Ciridae]

@Extesian Is is described as right up against the mountain? I thought it was freestanding but sheltered by the mountains around it. The sheer side faces the origin, so do the tiers merge with the mountains? I hope Kaladin does end up flying Shallan to another peak so we can get a canon image that isn't surrealistic.

And great work @ccstat! +1 when I have some to give again

[Michael Portz he/him]

34 minutes ago, ParadoxicalZen said:

Also, just to point out; the Szeth passage says a 100 levels, possible typo?

I think not: its 10 tiers with 18 levels each, ie 180 levels; I think Szeth just didnt bother counting ....

[ParadoxicalZen he/him]

23 minutes ago, Michael Portz said:

I think not: its 10 tiers with 18 levels each, ie 180 levels; I think Szeth just didnt bother counting ....

That's why I found it odd

[Starla]

@ccstat Thank you for this post! I am fascinated by the Urithiru architecture, and have been thinking about it's design and layout. I have collected all the references to it with the intent to attempt a drawing when I get time. Fortunately, I think we'll have a drawing in Oathbringer. I'm pretty sure I saw it confirmed somewhere recently, perhaps via WOB or WOP on reddit.

I am guessing the tower itself is a natural mountain with a sheer cliff face on the eastern side, sitting within a ring of taller mountains. The mountain has been carved into a semi-circle stepped pyramid shape. I would guess that each floor + ceiling is taller than you suggest, and the base and terraces for each level are wider than your drawing. I imagine it like a giant wedding cake, where each 18-level tier has a wide terrace with gardens. I believe it is much bigger than any man-made structure we have on Earth. If you think of a standard size mountain sitting within a mountain range high enough to be hundreds of feet above the cloud line.... it's pretty darn big.

I also love the descriptions of the stripes and swirls of multi-colored strata on the interior. And the open central shaft with the balcony lifts and gigantic single-paned window. Hopefully we'll also get some drawings of the inside.

I look forward to expanding on your post as we get more chapters released!

[Michael Portz he/him]

6 hours ago, ParadoxicalZen said:

That's why I found it odd

Always keep the POV in mind! The 100 comes from Szeths estimates, and mathematical precision is not something important to him. 

[ccstat he/him]

The consensus seems to be that each level+ceiling should be more than 4.5 meters. I've upped it to 5.5, which rounds us out nicely at 1km high.

I agree that the terraces should be wider to accommodate more garden area. For some reason I second-guessed myself on there being gardens on every tier, but that is how it is described. Does 50m per tier sound reasonable? More?

Several of you commented on the shape. As far as symmetry, yes, Urithiru is sheer on the eastern side. If viewed from the north or south, you would only see half of the presented silhouette. The shape I drew should match what you would see if you were looking from the west, which I believe is where the Oathgates are located. I'm not entirely sure how the mountain fits into things, though. Initially I was imagining the city up against the mountain, with more uncarved cliffs abutting it. However, after compiling these quotes it sounds more like the city is alone on top of the peak.

I also think I've severely underestimated the size of the Oathgate platforms. Looking at the Battle of Narak map from WoR and reading the descriptions of the number of soldiers, I'm inclined to revise it up to at least 120m across, probably more. I'm tempted to go as high as 300m, mostly based on the map, but if you have 10 of those platforms arranged in a circle around a field the whole area is between 4 and 6 times larger than that each individual platform, depending on how closely they are spaced. I highly doubt the Oathgate entrance field is 2 km across. I think I will wait for more descriptions/art before I to try to pin that part down.

 

[redbishop]

This is really impressive, though part of me wishes you had included a banana for scale.  Hard to visualize, harder still to contextualize.

Two questions:

1. Do your calculations for required thicknesses account for 0.7g surface gravity?  Rock would not need to be as thick on Roshar to provide similar relative strength.
2. In a similar line of thought, is 5.5m the average, or a set number?  It seems like the upper levels, supporting as they are significantly less weight, would require less material and could be stacked closer together.

Overall, very impressive work!

[WhiteLeeopard she/her]

Could just be 4m in most floors, with some floors having 100m height. Wouldn't be hard to imagine Windrunners and Skybreakers would have a floor to train/play.

[Steeldancer he/him]

I'm betting when we get the book we will get pictures

[+Wax he/him]

Fix reply later

Edited September 14, 2017 by axcellence

[Calderis he/him]

44 minutes ago, axcellence said:

ccstat - you forget that there is also a mechanism to transport the whole thing  very long way away.

What? 

[+Wax he/him]

Just now, Calderis said:

What? 

Sorry... accidentally hit post of an incomplete thought.  Will fix it later...

[ccstat he/him]

3 hours ago, redbishop said:

1. Do your calculations for required thicknesses account for 0.7g surface gravity?  
2. In a similar line of thought, is 5.5m the average, or a set number?  

Good questions! The height value is intended as an average, but I think it is probably pretty uniform for the purposes of this discussion. Adolin describes his enormous chamber as being four stories high, which makes me think that within each tier there is some communication between levels (vaulted ceilings, etc.) but the levels themselves are more-or-less consistent.

So, here's the thing about material strength. I'm not an engineer. I spent way too long this afternoon trying to research tunnel strength in stone mines, pillar stress, and engineering constraints of stone. I can share my calculations if you care about the details, but the upshot is that allowing for ideal design (think perfect arches and flying buttresses) and using a very strong stone (like granite), in Roshar's 0.7g gravity the building could be self-supporting for at least 2 km of vertical height, possibly more than double that. So we aren't even close to maxing out the foundation.

The thickness of the floors becomes important for two things. The less  demanding task is supporting the contents of all the rooms and tunnels. That's no big deal with the right design, and the original 4.5m level+ceiling guess would easily cut it. The bigger issue is transferring loads and forces between pillars and walls that aren't precisely aligned between levels. For that, thickness matters, and I would like to go with at least 5m for that. However, this is the part where I don't have a great grasp of the forces involved, so I'm ready for an engineer or stonemason to jump in and correct my guesses with their expert knowledge.

Edited September 14, 2017 by ccstat

[Hero-of-Pages he/him]

Now we just need to estimate the volume.

[+Wax he/him]

Trapezium volume reducing for each level... excel gurus go!

[ccstat he/him]

16 hours ago, heroofpages said:

Now we just need to estimate the volume.

Using the numbers so far, the total volume of the structure is very close to 1/3 of a cubic kilometer. More meaningfully, we can guess at the usable area, comparable to the square footage of a house. If the entirety of Urithiru (on all levels) was open space, we'd have 62 square km (24 square miles). That's almost exactly the land area of Manhattan. Clearly there are walls taking up some portion of that space, so depending on how much of the tower you think is rock vs. usable space, you could guess that our heroes have access to anywhere from 1/5th to 2/3rds of that area.

[The Invested Beard]

17 hours ago, heroofpages said:

Now we just need to estimate the volume.

 

[Michael Portz he/him]

On 15.9.2017 at 6:39 PM, The Invested Beard said:

 

Took me some time to get this ... then i lolled REALLY LOUD