Tunnelling through a Mountain - Numberphile
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- čas přidán 27. 08. 2024
- Featuring Professor Hannah Fry - more details on her work below.
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Another interesting fact is that the meeting point in the middle isn't perfectly straight, Eupalinos was afraid they wouldn't intersect if they had done the math wrong, so to make sure he instructed to bend both tunnels to the same side so that a crossing point was guaranteed, even if both tunnels were originally parallel to each other.
that is also genius
I initially thought that would be the solution when I heard triangles
That is some fantastic geometry-ing.
ahh very tricky!
Yes, the same technique was used in the Swiss alps for train tunnels on the 1850s but vertically instead of horizontally :
As both entrances were not at the same altitude, they bored with a 2% slope upwards. So that they could adapt the crossing point when they would meet each other
I found the best technique is using F3 to get the coords of both ends.
and remember to always put torches on the right side so you know which way you're going
@@jasondeng7677 F3+F4 also works I think
@@jasondeng7677 Naw. Left side. That way they lead you "right out of the mine"! :p
@@satisfiction I just say "Red Right Return", and ignore the "Red" part.
When I was watching the video my first thought was "oh i can totally do this in minecraft" but then I realized, yeah, F3.
As a Greek person, I can tell you that"Polycrates'" name means, "a whole lotta crates."
It's similar to that other name that's like "SO crates!"
It is, indeed, so crates, mai dood.
I approve of the pun but for anyone who doesn't get that it's a pun, as another greek person, I'll have to add the technical correction that it more precisely means "one who holds many (things)"
As an American person, I can tell you that "politics" means "a lot of blood-sucking insects."
That's a _crate_ pun
I believe the main reason to excavate from both ends is to gain time. When opening a tunnel, you are limited by the number of people that can work at the same time. To increase the number of workers you need a bigger tunnel, so you gain nothing. But working both ends you half the time for a given tunnel size 😊
considering how long it would take to dig a tunnel, wasting a few days trying to figure out how to dig from both sides is worth the effort.
@@danilooliveira6580 For sure, to chisel 1 km of rock by hand it's estimated it took about 8 years. I visited it in Samos and it's actually an aqueduct, very impressive engineering.
When talking about underwater yes, but on dry land you have chutes along the tunnel path and several crews at different points excavating multiple segments.
@@eleSDSU What do you mean by that, vertical holes from the top?
Can take 1/3 the time if you start a crew in the middle! :D
As an old tunnel surveyor I know how difficult this is, even with modern equipment! Impressive. Great video as always.
And no, GPS won’t work in a tunnel. We use modern total stations that measures angles and distances very accurate. For very long tunnels we also use gyro theodolites to help find the correct direction. :)
When they were building the tunnels from the UK to France they discovered that there was a significant error due to just having the instruments on one side of the tunnel. The differences in temperature close to the tunnel lining caused a slight bend in the light, as in mirage, so the tunnels started to deviate off the intended line. They switched to having instruments mounted on opposite walls to compensate.
Nowadays we could just use lasers to find a straight line within a tunnel, can't we? But with ancient technique could they use a lamp and two holes several feet apart?
@@kwzieleniewski Laser beams do not go in straight lines if there is a temperature variation. They are used but you can not just assume that they are straight.
So was the method in the video just a demo of a mathematical principle, or a serious theory of how it practically could have been done? Because to me the accumulated errors of a bunch of zig zags mapped out around the mountain, seems overwhelmingly likely to add up to too much. It seems infinitely easier, for example, to have a marker at the top and drop two plumb lines (pair near each entrance) to define a no-parallax eyeline toward it, and then dig on that same line. And probably a dozen other solutions that a regular joe like can’t think of off the cuff.
I'm guessing that we know, from archaeological research, that these guys had access to certain methods of construction planning. Then we extrapolate the construction method from there.
My grandfather was a civil engineer on the team that built the tunnel through Zion National Park (in Utah). The story I heard was that using only slide rulers to calculate, they dug from both ends and were only an inch or two apart when they met. The "Mt Carmel Tunnel" is 1.1 miles long (1.6 km) and is not a straight path through the mountain, which I always thought was a pretty impressive feat.
i've been through zion, way impressive, i say the grand canyon is too big to appreciate really, but zion, huge rocks overrhanging the roads are scary impressive.
I've driven through that tunnel many times!
Love that tunnel!
Have a 2 km local tunnel here in NZ, built in the 60s which was reputed have less than 4 cm error at the end
That's so cool! Props to your grandpa!
when doing those right angle steps around, because the steps are relatively small, we could possibly build planks between the marking poles and use the water level test.
Yeah, considering the effort involved in the tunnel and the measurements before hand, digging a trench around the mountain on the surface to ensure both sides are level seems to be a great way to start....
That said, if there is a way AROUND the mountain, then just build your road there. I would only consider a tunnel if the mountain were long and going around weren't an option to start with. Kind of invalidates the premise a bit.
@@jimbrookhyser Apparently, the tunnel was used as an aqueduct and was run through a tunnel in order to better guard it against attackers trying to cut off their water supply. So that's why it doesn't go around the mountain.
@@HermanVonPetri that makes a lot more sense! Thanks!
Edit: plus you’d probably get less water loss as a bonus
Why not fill a hose with water. The water on both ends will have the same level.
@@paulf5351 hanna mentioned the “bucket test” so bucket on long plank is closer to the context and doable. not sure about the hose, because a hose sacks. unless it is rigid, but then we would call it a pipe
Looking forward to a future video where Matt Parker and Hannah Fry build a kilometre long stick to dangle from a mountain 😅
I'd like to see them try to take that up The Shard!
I could literally SEE Matt Parker doing that, with highly-skeptical Hannah somehow persuaded to follow along! 😂
Now THAT’S ENTERTAINMENT!
Except it would be confiscated at the bottom of the mountain, and Matt would be like "well, here's a 30cm ruler I brought along just in case. Hannah, you go stand at the bottom of the mountain and we'll estimate..." 😂
*2273 cubits
@@GamerSloth2275 Thus was born the discipline of "Parker surveying".
It's because of that 60 cm mismatch that we even know that they used this ingenious method. If they had gotten it perfectly then the "likely" explanation would've been that they dug it from one side all the way through
So this is the 500 BCE way of showing their work. They could have nailed it precisely, or at least tiled over the mistake - but instead they were like, "Nah lets leave this here to wow future engineers" :)
That 60 cm mismatch is probably forged to make them look clever. :)
I’m wondering why they left the 60cm step at all, you could cover it up by expanding the tunnel’s dimensions very slightly in both directions and tapering back to the normal dimensions so that it would end up being a slightly wide spot.
Another commentator suggested they added a slight angle to each side to ensure they’d meet instead passing through each other side a side.
@@lordook5413 having come up with the idea "oh, we could do this, get it done in half the time."
"Nah. Let's just busk it. But we could include a small step in the middle to convince people we were crazy accurate."
Solution to the level problem: temporarily build a trough around the mountain, filling it with water as you build it. The water level in the trough at the tunnel entrance will always be at the same altitude as the level at the tunnel exit. The trough could be as simple as a clay-lined ditch in the ground.
Nice idea, but in practise a mountain can stretch pretty far in some direction. Usually it's exactly those mountains that warrant a tunnel. If you can dig around in an adequate amount of time you probably can just walk around it too.
Read that again but slowly
@@elephantchessboard9060 Tunneling through solid rock takes a very long time, especially without the aid of explosives or machinery. Digging a shallow ditch around the perimeter of the mountain would be like 1% of the work of actually making the tunnel.
Alternatively after getting you triangles you could use a trough of water say 20 meters long, go over the mountain in the above the proposed tunnel measuring the difference in height between each end of the trough. Sort of the same principle as the triangles but using up and down instead of north and sound.
A tube on a string and your mate with a long pole with measurements marked off would be the simplest way. They had functional theodolite type devices 500 BCE so it's not a huge stretch to think that prototypes of this kind of device, the forerunner to the Groma or Dioptra would have been available around 600 BCE.
She is fantastic! She speaks so clearly! I'm a non-native English speaker but she's so easy to follow... Love her!
It's just nice to hear Brady's voice in a video once in awhile.
Subscribe to Objectivity then? he talks more there
@@Dankey_King Or any of his other 10 channels. If nothing else, the guy is prolific in his video making. ;)
I used this technique 5 months ago in Minecraft for a 300 block tunnel in the nether
absolute madlad
BASED
this must be a joke, because in minecraft you have console telling you what elevention you are at :)
@@dospy1 I used it in the x z plane. I calculated the triangle to be roughly 280x and 70z. Simplifying the numbers the ratio was 4x : 1z. Then from each starting point I dug 4 blocks straight and one to the side. Worked like a charm
How did you make sure it was level though, you can't use water in the Nether.
It is always so pleasurable listening to Hannah Fry
With a plum line and a right angle attachment, you could just sight out points of same elevation around the mountain, until you get to the other side.
Or you could make the tower a giant sundial to create straight line shadows on the mountain at certain sunsets
Or, since this is an island, do the trigonometry with a sextant (or whatever the ancient equivalent was) and landmarks of known height from sea-level.
OP has it correct. We know the ancient Romans had survey tools just like that, a crossed stick with a plumb line. Works just like modern survey tools, sighting angles and doing trigonometry. It's how the Romans kept their roads straight and level, so it's hardly a stretch to imagine that the Greeks had the same tools.
I envisioned a series of posts leading all the way to the summit (or directly over the midpoint (or whatever spot is convenient) like a series of rifle iron sights of exactly the same height. Then again down the other side.
I think I know a decent solution to make sure both entrances are the same altitude. Dig a smallish trench outside from Point A to Point B, and use the water trick you described to keep the trench level. Once you have your reference trench, you can use that to get roughly equal starting points.
That might work.
No because you might aswell not have a tunnel if you have to build a level trench that is more than 1.5 times the length of the tunnel pi d = circumstance so half the circ might as well just build the tunnel
@@djscottdog1 Trench might be overstating the size of what I envision here. It could be something along the lines of placing three bricks side by side and removing the middle in terms of scale and work just fine for leveling purposes.
That pronunciation of "Polycrates" reminds me of the way Bill & Ted said "Socrates".
(Watching Bill & Ted's Excellent Adventure as a child spoiled my mental pronunciation of Greek names ending "-es" for life.)
She's a great teacher
Ikr
And more.
@@Bill_Woo Bonk
nudge nudge wink wink. Great teacher I know what you mean
@@An.Individual 🤤
A kilometer long "stick" would have just as much sag as a kilometer long rope, no matter how taut both are kept, owing to the similar cross-sectional area and the fact that the sag is essentially transverse to that cross section. The coefficient of restitution of the stick would have to be massively greater, for the mass, than the rope for it to matter across such a distance. And, it's just not.
Doesn't matter because that is not how they did it. The same technique for elevation as is used for the other 2 axes. Surprising they don't know this.
@@aculasabacca I don't buy it yet. Elevation is arguably more difficult to measure by walking around a mountain. Huge steps you might be able to notice, but tiny slopes you won't. Even a ball or water wouldn't help if the underground is mostly dirt or grass. So how would they do it?
@@lonestarr1490 All three axes require the exact same amount of accuracy. There is literally no difference between going up, down or side to side. No difference what so ever.
@@aculasabacca There is a clear difference in measuring accuracy between going up and down or side to side, because you cannot go "up" by a fixed amount (say "one step") whenever you want, for you would come back down immediately due to gravity. So how do you measure it?
Well bridges sag too. So they're simply supported every 'X' meters.
Who said that the stick had to be self supporting or free standing?
I love these history related maths problems like this and Josephus problem! Please do more of these, Brady!
Check out TedEd maths vids! They do lots of history related
Oh this is one of my favourite facts about Ancient Greek engineering! Thank you so much for covering this, I think Ancient Greek uses of maths are wild
The stick thing sounds like a misunderstanding. A stick going from a point on top of a mountain to the base of the mountain, while being impractical in general, also won't necessarily generate the same height if the different sides have different steepnesses. More plausable is having a stick going straight up and noticing when this is level with a structure on top of the hill.
When I have seen this subject visited elsewhere there was not mention of that "stick" theory. All three axes are done the same way. There is zero problem with including elevation as you survey, it would be silly not to.
Right maybe they made vertical structures at the entrance and checked that they were level at the top then simply measured down the same height.
If they measured the same angle of the stick relative to the vertical tower on both sides of the mountain then the end of the stick would be at the same altitude.
I think the prof read a poorly worded description and miscommunicated the "stick" idea. I'm pretty sure it was meant to be a description of ancient surveyor's tools, which are not unlike today's tools. You use a level and a plumb bob to get a stick perfectly vertical, and spot the angle from your stick to the identical stick that your assistant is holding. Measure the distance accurately across the level stretch and you have a nice triangle to tell you all the measurements. If you put your assistant at the top of the hill, you just have to spot the same angle from the bottom of the hill on both sides and you're at the level. Surveying a straight line across the top of the hill will also give you a reference to line up your sighting points at the bottom so that they're nice and straight.
When you have two points at different height in the distance (e.g. a long pole on top of the mountain), it is trivial to calculate your vertical distance to them by doing the "find the height of something by measuring the angle" twice. You don't even need any absolute measurements (like distance or the length of the stick), you calculate with placeholders at one entrance and then find a point near where the second entrance should be that gives the same value.
2 right-angled triangles where the vertical sides are in line, the top point of one triangle is a fixed length below the top point of the other and the far points are identical. Once you have measured two angles at the far point (one entrance), you can calculate any pair of angles for a second pair of triangles (other entrances) with the base in the same plane.
OMG numberphile with Hannah Fry. Christmas came very early!
Please do more videos with Hannah Fry :)
The tunnel was actually used to carry water with pipelines from the springs to the city of Samos
If they can make pipes, they can use some of those, round the mountain, to level the entrance and exit
but then it would have to/should be slightly sloped towards one end, no? or did they have pressurized pipes?
Water you put in at one end will still flow to the other end if the tunnel is flat, the pressure is provided by the spring itself rather than by gravity. Although what you've got in this case is actually a channel along one edge of the tunnel carved into the floor like a rain gutter, not any kind of pipe or aqueduct. They weren't terribly concerned with the sanitation of the water because it was already straight out of a hole in the ground anyway, and germ theory is still a couple of millennia off.
@@rbettsx not efficiently though maybe they were more perceptive than you
@@mimimi3440 I'm sure you're right. A primitive theodolite is probably much easier to make and use than plumbing / hose / channel. Although there are many examples of ancient contour-following channel around the world... just fantasizing, really...
Hannah Fry is such a clear educator…
@@aboudifortechit9459 im sorry brother but im afraid you might have some issues.
my fellow brother here is just praising our teacher and there are no explicit perverted phrases whatsoever in said comment.
please refrain yourself from bashing each other in the comments as it contributes nothing positive to the conversation and pointless.
are you trying to feel superior over a stranger on the internet just by critiquing ones 'ambiguous' comment? if so, please dont do so on educational videos, as we are here to study and make progress. or, better yet, stop.
please, my dear brother. i do not want to live in a world where admiring and complimenting your teachers is a crime and must be punished.
@@aboudifortechit9459 The deflection tactic smell is so strong that I'm smelling it from my screen
@@aboudifortechit9459 a sense of humor only picks up on funny things...
@@aboudifortechit9459 Maybe that smell you sensed was coming from your own mouth, since you use disgusting words like "simp". Is it really asking too much to think twice before picking up "incel" lingo?
People really love controversy
I don't care what the video is about, I see Hannah, I click.
I see numberphile, I click
Can you imagine how many views and subscribers Numberphile would have if Hannah was in every video?
Indeed, Hannah Fry is the new Carol Vordeman
Me too, but instead of clicking I cry for the rest of the entire day over the fact that I will never have a Hannah Fry GF
You can use the same zig zag pattern to find the elevation. Your "compass" is then a water level and a plum bob line, those form the 90 degree angles, referencing the center of gravity.
A water level is a far more plausible explanation than a km long stick.
Yup, This is a simpler and more effective solution.
what a great timing , I was going through Tobler’s hiking function few days ago and now this is so cool!
The pronunciation of "Poly Crates" reminded me so much of "So Crates" in Bill & Teds Excellent Adventure.
I've used this method to teach about vectors. I designate two points on the opposite corners of a large building on campus and challenge students to determine the straight line distance between the points. They use measuring tapes to record the perpendicular distances for each step and ultimately construct their large triangle, calculating the distance as the hypotenuse. They can then compare their answers to the correct answer via GPS.
To get the height, you could follow a similar principle as for the triangles. Take a long trough capable of holding water over a distance of a few metres. Put one end on a rock at the top of the hill, then measure how far off the ground you need to tilt the other end to get the water to line up to a certain marker. Repeat as many times necessary, marking out particular altitudes with marker rocks as you go.
Polycrates , so named because of the many empty beer crates scattered about his palace.
You can still have an inclined tunnel using a water level by using triangles in the vertical plane
That's right, but you would have to know the difference in elevation of the two entry points. And while huge steps are fairly easy to notice while walking around the mountain, a tiny slope isn't. Even a ball wouldn't help, if the underground is mostly dirt or grass.
This tunnel was built to deliver water to the city, so an inclination was necessary.
What they did was, after the completion of the flat tunnel, they dug a narow channel on the side of the main tunnel that had a gentle slope
If they knew the speed of sound(or just know it's constant), they could have used some noise to count the seconds to/from the top, to the two points and eyeballing the angle with waterlevel and with some meter sticks. You can scream/whistle a kilometer away. There just needs not to have trees in the vicinity blocking sound and vision.
@@user-zn4pw5nk2v Seconds weren't invented yet.
And how would you count seconds without synced clocks anyway?
@@lonestarr1490 by dripping water the Egyptian way. And if you want accuracy use more lengths two echoes twice the accuracy.
Actually in ancient rome they sometimes stood on the top of the mountain and drilled down at regular intervals. Since you can see the entry and exit from the top you can precisely place the holes on a straight line
A similar method was used by the ancient Jews
@@WagesOfDestruction what do you mean by ancient jews
Brady "as long as you can see the opening you know you are digging in a straight line". Hannah "they had to dig around hard rock at the entrance".
Recently I saw a YT about the Roman Roads in Britain that raised the point that I had never thought of, "how do you create a straight road to a town you can't see?"
The problem with the same altitudes at both ends of the tunnel, they could have solved with triangles as well, right?
Here's how they could've done it; Erect a giant pole on the highest point of the mountain. The top of the pole is a vertex of a triangle. Then they erect two smaller poles at the opposite sides of the mountain, but in the same vertical plane. With the help of trigometry, they calculate the altitude of each smaller pole. Then they move one pole up- or down the hill until both smaller poles have the same altitude after calculations.
As they count steps, they count elevation also, no problem.
In this technique small measuring errors would take huge effect. So it's not likely.
@@lonestarr1490 It can be done with relatively few measurements by sighting long distances. Also there is no reason to assume all errors would go in the same direction. It's really not too hard to do.
@@aculasabacca Since we're talking about digging a tunnel with ancient equipment - thus an undertaking that could easily occupy the workforce of a settlement for several years - I wouldn't be content with "maybe errors don't accumulate". I would want to be really really sure about that.
@@lonestarr1490 Well they did it. It's not that hard. If you are an expert in measurements, like I am, it's not hard at all. It's really not.
i had to laugh so much at the theory with the long stick.
they determined the height of the entrances in exactly the same way back then as we still do today.
step 1: you decide on an entrance. then you take a stick and drive it into the earth. this stick may have had a notch as a marker.
step 2: then you take 2 cups and a long hose, about 1 m long and make a "water scale" out of it.
you hold one cup at the mark and at the other cup you hammer the next stick into the earth and level the two. this is how you transfer the height measurement over any distance. for ages.
You could have a water scale where the hose sits on a swinging wood joint, which would set the length of each level measurement and could easily be squared. This device could take measurements for the angle at the same time as it's keeping you level with the entrance.
what would they have used for a hose back in the day?
@@lousypirate Animal intestine is what I would do.
What you say sounds interesting, but I'm not quite sure how it works. Googling it seems to find something else.
Would you mind directing me to an example of the method of which you speak?
Could you expand on this? Your explanation requires previous knowledge of the explanation
Hannah Fry is so lovely, I really like her ❤️
For accurate altitude we can do it by water level:-
●Start and build water level from one side of the mountain to another side.(just like how like they measured the sides of the triangle)
Easy right( it might take some time but it will be done)
Professor Fry is a favorite of mine. Always a fascinating look at numbers and their applications
"You'd like to dig a tunnel through a mountain."
GET OUT OF MY BRAIN
About knowing where to put the triangles and how to angle them, it doesn't even need to be based on north-south and west-east. It just has to use the same right angle you used to take your measurements. It's not about cardinal directions, it's about right angles.
Amazing!! I used this to introduce angle relationships and the importance of triangles in real life to my 8th and Algebra classes.
Elevation: 1- maybe dig down on each side until you reach water, then measure up from there? 2 - Maybe use a tall vertical stick and see where shadows fall at sunrise/sunset (works for East/West)?
If the sun were to pass directly overhead, you could use the shadow of a mast atop the hill in the morning vs the evening to set your direction at each end.
Unfortunately it's above the Tropic of Cancer so the sun never passes directly overhead
that assumes your tunnel is roughly east-west as well
The "really big stick" method doesn't need an actual stick. I'm sure they knew the height of the mountain, and could use the angle to each end point from vertical from the top of the mountain and some other location they also knew the distance to, by sighting along a much shorter stick.
yep, and the tower on the top that they both can see, is the easiest way to do it. Plus they can make it in a slope like the actual one.
Yay! Hannah is back!
To find a East-West line in Egypt they put a stick in the ground and marked where the shadow ended. They then waited a few hours and again marked the end of the shadow. By connecting these two points you get an exact East-West line, which they bisected to get the North=South line.
Couldn't they have preemptively constructed the triangles, then used the walking around the mountain trick to instead determine where the other end of the tunnel needs to be? That would also help explain why one end had to maneuver around tough rocks, as the location would have been determined after the more ideal entrance had been determined.
It was an aqueduct moving water to a city that had outgrown its local wells and looking at the elevation maps, my guess is the apertures are determined by the cities location and the elevation at the entrance.
did they have clinometers? if yes its a possible way to ensure same elevation. Put a tall stick on the top of the mountain and make sure that both of you have the same vertical distance from the top of the stick. you can easily make a clinometer by a protractor, string and a weight.
i love hannah's videos!
To get the entrances level, they could run a pipe from one side of the mountain, over the top of the mountain to the other side. Then angle the ends of the pipe up and fill the pipe with water (from the top). Where the water levels out each side is where both points are level.
Although, this would probably cause a vacuum to appear at the top and may not work.
They probably kept track of how much they went up or down as well, as they went around the mountain, and then adjust the entrance and exit accordingly. A rudimentary clinometer can be used to measure the angle of inclination.
And how would you track how much you went up or down?
I'm thinking they could've had the tower at the top of the mountain, and then measure the angle from each entrance to the top of the tower. If they can calculate the distance to the tower, they'll know the elevation of each entrance.
@@chengong388 They know the distance from one right angle to the next, and by measuring the inclination between those two points, they can calculate the difference in height between the two points.
instead of using a long rod or rope on the top of the mountain they could just use a mirror and reflected sunlight to point the spot.
That only works if the peak has an unobstructed line of sight to both tunnel entrances to be, right? And the rope would only work if the peak is in the exact middle, which is even less likely.
0:47 "So let's say you want to go in here, doo doo dooo ♫♩
and you want to come out here" _squeak_ _squeak_ _squeak_ ♫♩
Normally in a subway tunnel you want the middle to be deeper than the terminals. Especially if it is a railway tunnel. Because: The acceleration leaving a terminal will be gravity assisted and you get up to speed quicker and when you approach the other terminal, you go uphill, thus braking the train - which you are supposed to do. Jumping on and off a moving train is generally discouraged.
Perhaps the "massive long stick" was in fact the shadow of... well, a very long stick.
Doubt, diffraction will ruin the shadow too much for a measurement over that distance
Here's how I'd have done it in ancient Greece: Just dig from one side to whatever direction I think is correct, then wherever I reach the other side build a road there and call it a day.
The tunnel wasn't for traveling. It was to bring water from a spring to a town.
It would take twice as much time to do that, so they would call it a day sooner than you
@@the_Acaman Yes, but I'm more of a practical thinker.
That's actually how I think they did it. Just rearrange everything around that entrance and pretend that this is what was intended. And digging in a straight line isn't very hard: just look straight into the tunnel and you'll immediately see.
Construction set-out profiles (may be called fence) must have been used to achieve the mentioned tunnel, which remains standard procedure to this day : Once the entrance and the exit points are established, you form a "3/4 finished" RECTANGLE on the south side of the hill, out on the plain, the two start corners of which would be the exact entrance and exit points. As squaring up and levelling methods had long been perfected then, you obtain and mark two short sides and one long side of the rectangle. When you start excavating on the entrance side, you maintain the square angle between the established short side of the rectangle and the alignment of the tunnel in process. The same thing will be done on the other side of the hill : by maintaining the tunnel alignment square to the short sides of the rectangle, they would have created a tunnel alignment ( fourth line of the incomplete rectangle) parallel to the long side of the profile rectangle laid out initially. In other words two excavated lines meeting in the middle.
In an existing building with no available plans or drawings, full of irregular rooms and corridors and various barriers, the easiest way to reach an exact spot is still the profiling method I mentioned above. "Square up an outside fence". Cheers!
If you know the line you have to follow, then you could use a stick of some known length, and start moving along the line towards your goal as you lay the stick of known length on the hillside. This stick forms the hypotenuse of a right triangle. The rest of the triangle can be constructed by fixing a string or a rope to the end that is higher up, and keeping it tense as you raise a square (measuring tool) vertically up from the lower end of the stick. You raise the square up to the height where the elevated side of the square exactly touches the rope. This means you will have created a right triangle. The rope forms one of the cathetuse and the stick or square or whatever, forms the other. By recording the length of at least one of the cathetuse-s, one is able to figure out how much they have gained in elevation (and incidentally, also in distance, as well as the slope of that part of the hill). Once you reach the crest of the hill, you just need to make sure to go down exactly as much as you went up, to end up at the same elevation. Do this procedure like 10 times and take an average, and you would probably be pretty darn close even with ancient tools and precision.
If one were to put a pole with a target on the top of it (say a ball for ease of sight). You could measure your way down the mountain, and use the same Pythagorean theorem to develop an entrance and exit at the same level, you would also have the distance across between the entrance and exit, assuming the same heading was made between the two points. The same could be done today much easier using the same target and a rangefinder and impromptu sextant to get angle using the distance to target multiplied by the cosine of the angle you measured. Could then just use the compass vector, or use a laser level set to that vector and still use water to maintain level if desired, or just a yardstick (t square might be a nice way to get level and centered as you went along to that laser mark set to a known height.
Now that I've spent too much time pondering this, off to something completely different.
All of a sudden I'm attracted to tunnels
To get the two entrances at same height just establish the high side and dig straight through. If this system is accurate enough to meet in the middle it should be accurate enough to go all the way through. The height of the new end will become apparent when the crew breaks through.
With GPS: you put antennas on the Top of the mountain, drill a straight small hole from the top into where your tunnel is and hang a plumb bob from the antenna down into the tunnel. The surveying equipment can then use the plumb bob to have a precise 2D coordinate for this very point.
Great to see Hannah again; one of the best!
2:24 "Keep looking behind you" _moves camera behind him_
I liked that a lot haha
I always wondered how they intended to tunnel from Dover to Calais in 1880. They just needed to paddle across the Straits of Dover to map the route.
Thinking about it I might use the following technique:
Establish a reference point on top of the mountain (a tower or post) visible lower down. Or probably a straight line of posts if the top is set too far back.
Sight this from each side of the hill, (getting the posts in line if there's more than one up top), and using a plumb-bob to drop down that down to determine where the entrance or exit should be (again mark with posts). Leave a post at each lower sighting point for taking back-sights during tunnel digging.
Sighting from the top of the mountain can be used to ensure the sighting points lower down are co-linear with the top. For a few km semaphore style signalling can allow communication between three teams of surveyors to coordinate this.
I can see some long poles are probably part of the story for back-sighting from the tunnels to the lower down posts - once the tunnels got going you can only take back sights horizontally, so may need to erect tall poles to remain visible.
They could have dug a trench or built a trough around half the perimeter of the mountain and filled it with water to find two points at the same level for the entrance and exit.
I was thinking about making a rope of people that stretches around the mountain on exactly the same level (maybe 20-30 m per person so you can stay same level headed). Would require only about 400 people per 10 km
This is super cool from a math perspective!
although, the engineer in me says to just dig it from one side so that you only need to get it about right. I guess that would take nearly twice as long, though. :P
And you can't trust a single engineer working alone.
The point of going from both sides is so that you can make sure it comes out exactly where you want, instead of being off by whatever error margin you have.
Yea if the goal is to run from the spring to the city, the margin of error on hitting the general direction of the city is pretty lenient. So just start a tunnel from the spring-side and tunnel toward the city. Even if your off by like 100 feet side to side, You're still on track for a city aqueduct. The complicated water trick seems unnecessary to me too. Just make a long open pipe, fill with water, check the level - like a normal household level except say 10m long, you'd easily be able to eyeball 1-2mm inaccuracy, and over a 10m length that's a trivial incline. Plus this way you solve the hardest problem first - you know the tunnel entrance/exit is level because you built from one to the other level the whole way.
That said the mystery of how to measure equal elevation on opposite sides of a mountain is a fun brain teaser so maybe that's the real point.
Pretty shitty engineer if you neglect an easy solution to halve the construction time.
Remember, every meter you tunnel in, you add 2 meters of distance that workers have to traverse hauling the debris out. Coming from only one side, every bucket needs to be carried 2km as you approach completion. It would increase construction time by way more than double.
To determine consistent altitude, you could build a trench around the side of the mountain filled with water and make sure the water stays level all the way around.
Step 1: dig shallow channel around base of mountain
Step 2: add water, establishing a local "sea level"
Step 3: make 2 sticks of equal length from water level to desired tunnel opening
Step 4: hope
Step 5: meet in the middle?
depending on the altitude above sea level it might have been easier to measure up than down. Say it was only 30 meters above sea level you could much more easily make a "big long stick" or a keep a rope taut over 30m.
The sexual tension in this video is through the roof. Felt like a 3rd wheel
1st thought without seeing vid: dig a 6-foot pit at 1 side of mountain you are starting at. From there, start digging for the opposite side of mountain at a slight upward angle. Once started, use high pressure water hose to blast away the dirt ahead of you, which is washed downward and away from you as you move through the tunnel you are creating. A side benefit of this is that water pressure will keep the trailing hose straight and keep your tunnel straight. A beam-shoring team can work up from behind the "pumper", as he goes, if it's that sort of a project. In this way, NO time or effort is wasted getting rid of the excavated dirt. (now to watch vid).
Hannah Fry's voice gives me a warm feeling
Hannah is a great educator and has beautiful personality, looks and voice. A pleasant person!
EASY MAN
How would it be done if both extremes must be at different heigth?
Maybe also counting up/down steps and doing the triangle in 3D (polar coordinates)?
That is how it's done, simple tools, no problem. Not sure how they miss that but this is old news.
They didn't use triangles, they used many crates.
What they did there was more elementery, after the completion of the flat tunnel, they dug a narow channel on the side of the main tunnel that had a gentle slope.
@@myownsite On the video (4:00) is shown how triangles are used, by counting the base and the heigth of the triangle yo can make the angle to go.
Im guessing taking steps up and down to get the height you’ve travelled in vertical axis isn’t quite the same as taking steps south and west lol. They don’t have potential energy compasses. Im guessing they could just trace a path around the montain at the same level as they did here, but from both ends. You would get two parallel paths (in space) with a given distance apart from each other. Then u can get a right triangle with height (z), horizontal distance (xy) and the given distance as hypotenuse.
If their is a vertical pole at the top that is plumb and can be seen from both sides then each side can with a level and protractor determine all the sides and angles of a right triangle which can then be adjust as necessary until both sides are at matching elevations.
To find the elevation of the two entrances, a very well balance scale with a longish stick pointing downhill at the same angle from vertical on the scale. The viewer looks down the stick each direction and sees how high up the slope the entrance should be. Its hard to see that far away, but if another person had a mirror at the entrance location they could be told to move around a bit until they lined up perfectly with the stick.
I think they've used a sea level to calibrate the initial entry height to the tunnel from both sides
And how do you reference sea level if the sea is on the other side of a mountain?
@@dielaughing73 I do believe the sea would be on both sides of an island.
@@ronj9592 Cute. And have you ever heard of flat land? There is usually some of that on an island, so how did they reference the sea from miles away with millenia-old technology? What if (gasp!) there was _another hill_ in the way?
@@dielaughing73 ??? The high tide mark would be the same all along the coast which continues all around the island. There are no hills in the ocean.
@@ronj9592 on the *island*, genius. We're talking about people with none of our modern technology and you're acting like referencing sea level would be a trivial matter. The sea could be many miles from your tunnel, with any kind of terrain intervening. 'Just reference sea level lol' is beyond facile if they don't have the means to do so
Isn't measuring altitude vis-a-vis a fixed reference point rather easy? Hannah and Matt did it last year. As long as both ends of the tunnel can see the top of the mountain (or even the sea) then it wouldn't be hard. Walking exactly along cardinal directions seems like a MUCH bigger source of error to me.
Right angles only, required for all three axes, no compass necessary.
I know I'm missing something, but this feels far more complicated than it needs to be.
1) Straight Line Entrance Points
Starting at the top of the mountain where you can see what you're doing, pick your line of direction you want the tunnel to go. Place a series of stakes (or fence posts) down. As long as you can see (or touch with a taught rope) two posts, you can place a 3rd down in a straight line. Basically, just build a straight "fence" over the path of where you want the tunnel.
2) Elevation
Starting at one entrance point, use a simple theodolite (a board with "gun" sighting) and survey your way around the base. Plumb the theodolite to ensure it is level. And have a target post of equal height. Or more likely, use two equal theodolites looking at each other until both agree they are lined up. Now you can line of sight in way points around the base of mountain that are level. Terrain in between the way points is irrelevant.
Where your elevation line intersects your straight post line, you've got it. Dig in line with your fence posts. Ensuring it's straight and level as they described.
I don't think you'd even need to take measurements or do any trig. You could if you wanted to, but I don't think its necessary.
Some (large) assumptions of accommodating terrain and runaway measurement error do apply.
You can use a spirit level to find the relative elevations of two points close together. Do this over and over along any path and you will establish the relative elevations of each end of your path. You probably want to repeat this to correct any error that crept into your measurements.
Use the level to sight along a string held taught. Nowadays we use a scope on a tripod that can be set up so that it only points horizontally and you aim it a a tall pole with measured markings on it. You must also be careful to measure how heigh your level is obove the ground over which it is set.
I just got this notification after watching a scene in a video by pbs spacetime where they talk about quantum tunneling lol
They might just used mirrors ? knowing that light runs in a straight line, they could just align mirrors to establish a line between the foot and the top of the hill? just asking… Love your videos Hanna 👍🏽👍🏽
Exactly! I expected this video to retell the technique that I heard, years ago: place one or more mirrors, and angle them so that you can stand at one tunnel endpoint and see the reflection of a bright light displayed at the other endpoint. Then get out your sextant and measure all the angles involved as precisely as you can. Then use trigonometry to compute the exact direction of each endpoint from the other endpoint (relative to the mirror and other visible landmarks). Finally, shine a light in the appropriate direction and only dig the illuminated bit. Every night, cross-check by spying the lit end of one tunnel from the unlit end of the other.
Trying to survey the contours of the mountain surface is just asking for a ton of measurement error.
@@madrigal1213 I think you mean "metal polish". It isn't as if they needed a clear image, or anything more than a shiny flat surface.
@@AdrianColley Okay, all they would have had to do was invent time travel to get to 1759 when the sextant was invented. Pretty sure they stuck with the laborious manual survey method, which we know was how the Romans laid straight, level roads. A couple sticks, a plumb bob, and a protractor solve the problem.
@@johnladuke6475 Well, you're right. A sextant is a much more complicated device than the simple angle-sighting instrument I had in mind which doesn't seem to have a name I can find easily.
@@AdrianColley Protractor.
hannah fry could teach anything and i'd be willing to keep my attention levels high.
i know how they did the level. they built a set of stairs going up the mountain but not to climb. they measured the distances and used the verticals on both sides to get to their horizontal goal
Regarding altitude - would the triangle principle not hold there too? If they had a common high point to observe, and could establish a local horizontal plane, it’s just a case of knowing how far from the common high point you are, and the angle?
In this technique small measuring errors would take huge effect. Therefore I don't deem it likely.
Is it possible they measured the vertical angle to both the peak of mountain and some other known object from both sides? That would be a way to establish elevation. Or they spirit leveled from some common point.
Angle from the horizon at sunrise, maybe? (it is an island after all)
I literally had just gone, "Man, I miss Hannah's old videos on Numberphile. She should show up more often..."
...and then I looked at the release date.
Yay!!!
You can measure the elevation using a tube, water (or mercury or other liquid), and the air pressure. Equal elevation will have approximately equal air pressure.
It looks like there's sea water visible from both sides of the mountain. Could they not have calculated their altitude using the angle to the horizon?
The security guards will confiscate your home-made protractor.
Holly Fry’s videos are always massively entertaining, and this was no exception
To find the exact elevation they could use a series of water levels, which is just a bucket filled with water with a clear hose coming out of the bottom. You could use some kind of glass attachment so as to see the water line, given that clear plastic might be a bit hard for the ancient Greeks to have come by. You simply raise or lower the bucket to get the water at the desired level, and always measure from the same length of hose.
They probably used the water trick again, just dug a long narrow channel around the mountain. Should be easy compared to the actual tunnel.
No problem for a mountain that you can circumnavigate in 85 steps.
Confession time: I wish I was Hugh Grant in some other parallel universe where Dr Fry and I fall hilariously in love. There… I said it.
They could have used the same technique as you use to determine the height of a tall building or tree. By raising a flag on the mountain top, you could use a clinometer to measure the slope of the imaginary lines from both entraces to the flag. This slope together with the horizontal distance between the entrances and the flags would give you the difference in altitude of the flag and the entrances.
You don't need to know the height of the end, you just want it to be at the same level as the beginning. So when you are counting your steps, you plant two sticks in the ground. Then you attach a rope between the sticks. Then you use a third stick with a string and a heavy weight at the end. Once you have that, you just have to check that the angle between your string and the rope stays at 90°. Then you continue until you reach the end of the tunnel.
All they needed to do was to press F3 and make sure they're at the same XZY levels
People still play minecraft?