Tunnelling through a Mountain - Numberphile
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- Опубліковано 29 вер 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
OMG numberphile with Hannah Fry. Christmas came very early!
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.
@@ГеоргиГеоргиев-с3г 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.
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.
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.
maybe they took a giant square cloth and pinned it up on one side of the mountain, built a contraption with a viewing frame several meters from the viewer at the top of the mountain which, when properly aligned, allowed the cloth square to fit perfectly into the frame and then did a similar process on the other side of the mountain to ensure the distances were equal from the top of the mountain.
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.
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.
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.
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.
The 'they have to start at the same height' problem can be rather easily solved by digging a small canal around the mountain, and using that as a water level. Just start somewhere, dig a few meters around the mountain, make the water level in it, dig some more, till you got where you want the exit to be. As long as its water level all around, they are of the same heights. Then you can erase the water-level-canal (or let it erode), since it does not have to be that big. 20 cm wide would do the trick.
If I wanted to get the same altitude on both sides of a mountain, I would start at the top with two long sticks, but only like 5 m long. Then I would extend one of the sticks out horizontally from the top keeping it perfectly level (using a spirit level) and use the othe stick to measure the distance from the ground to the end of the horizontal stick.
Then knowing I was, say 3m down from the top, I would do the same thing again, then again, and again and again, stepping all the way down to my entry hole.
Then I would add up all the vertical distances I had recorded, and do the same on the other side until I had reached the same distance down, giving me the altitude of the exit hole.
Hannah is a great educator and has beautiful personality, looks and voice. A pleasant person!
EASY MAN
To keep same altitude at both holes:
Send a rope with one man up to the top of the hill, (starting from the first hole location). Make that length your distance. Also use an angle measurement from level ground at the first hole up to the hilltop to discover the first angle to the top from said hole. Don't get the angle by way of rope angle because it will have droop. Use a sextant style angle man to hilltop man.
If terrain hits the rope, a tower at the top of the hill is required and you start over. In fact doing this more than once to confirm numbers is part of this too.. Keep rope the same tension at each destination so as to allow dropping to be similar in both cases, thereby it could be discarded such that distance is unchanged.
Walk around the hill, using the length of rope to keep you at the same distance from the top. (The man at top remains there as a kind of rope-pivot). Once at your destination, check to see if you have the same angle to the top as originally. If you do, then you are at the same altitude as the original hole.
If not, use mathematics and geometry to adjust for the new rope length using the new angle:
If the angle to the top of the hill at the second point is greater than the original angle, then you are currently lower than the first hole. By using the new angle and the original rope distance or length, you can calculate how much rope length needs to be removed..
Step inside and diagram all the information once you have it. Using this diagram, you should see the angles and the first rope distance, now on paper.
The removed rope length will be based upon right triangles.
Al la there you are. Well, not really, but detailing the details here is going to be long and messy. I hope I've laid down enough, though, such that you can see where I'm going.
Height doesn't seem like an issue to me. Just need a fixed value L shaped tool that is say, half a meter high, and count from sea level how many you have to place in height to get to the entrance. You don't need to worry about x,y position, just the height and apply that to both sides of the mountain. So if you know your enterance is say 50 meter high, and you know x,y positions from triangles, then you know exacly the tunnel position. -- Another solution would be a level tool angle with a fixed lenght rope. You hold the level tool and you friend just go higher on the mountain holding the rope with you. Say the rope is 10 meter long, you can calculate the angle it makes with the level tool and the lenght of the rope to get the height of the formed triangle. Repeat until you get to the entrance and add all values.
-- Another solution is dig a small trench all around the mountain at the height you want your entrances and fill with water the same way they kept height in the tunnel. Probably less efficient than rope calculation.
for the altitude match the easiest way should be digging a moat around the mountain. and if you make it rectangular on edges, no need to walk around too. free straight line.
Never thought I lived in a mountain as when I am on my balcony I have an entry to my mountain and an exit out of my mountain.
I dug a kilometer-long tunnel in Minecraft once. I was tired of riding my horse all the way around the mountain every time I had to get to the villager village.
It doesn't matter if the rope is a little slack because you just pull it with the same force on the other side and you know both sides are the same.
If each of those small steps is made using a stick of some length (say 10m) and that stick also functions as a spirit level by having a vessel of water (perhaps like a long half pipe) then it would be easy for them to maintain the same altitude as they progress around the mountain counting out the North and West steps.
When making the angle steps around the mountain, the height difference could be measured as well (of each segment). When the other side has been reached the height difference would be known. So you could see this as not only measuring the changes of X and Y but also the changes of Z.
All you need is two reasonable sized poles, one long pole and two right angle trangles.
The technique that was allegedly used, could also be deployed from just one side (digging all the way in one direction). Maybe they would not have come out in exactly the right spot at the other end, but since they were just centimeters off at the meeting point, I guess the deviation would be acceptable?
"You're unlikely to find each other in the middle of a mountain" I guess that's mostly right, especially when no tunnels are involved :-)
the height you can measure from sealevel (make sure you measure at the same time though, so you have the same tide)
If they could walk around the mountain level, they could have built a flume around the mountain and filled with water to similarly use to find a level entrance.
Why would you dig from both sides and not just tunnel from one? Makes it far easier.
Hannah. Cool. I'm loving the maths and the explanation. If you have to dig a tunnel from both ends that meets in the middle then this makes sense.
As an engineer, I have to ask, why can't you just dig a tunnel from one end?
It seems unnecessary complicated to pick both ends of the tunnel at the start when you could just start tunnelling at one end and keep digging until you come out at the other end.
Imagine the tree the kilometer long stick came off, wow. Lol
Amazing, I'm left wondering about this 1km stick! Handy
Tom Davies' Geowizard UA-cam channel has five "straight line" missions where, with the aid of GPS, he attempts to walk in a completely straight line through various countries, mostly Wales. Everybody talking about how you can just walk X number of steps due north, then Y number of steps due west (etc.) should maybe check out how actually HARD it is to stay on a perfectly straight line if you're going through any sort of terrain. Granted Samos isn't Wales, but my guess is that going up and around a mountain or even a modest hill you're going to have extreme difficulty staying within even tens of meters of where you might think you are. Let alone 60 cm!
I think the idea is that the surveyor/pacer team make their way around the mountain the easiest way possible and just measure what that is. That approach is the opposite of deciding that you *will* follow a path determined by some criteria (like a straight line) and then showing how difficult it is to do so.
I loved this video
The high problem could fairly easy be done by using 4 poles.
2 poles as legs in each end of a long leveled pol (5-10 yard). And the last pole as a height-stick. How high the height-stick is, is totally irrelevant.
You do now position the height-stick in the start end of the long pole and makes sure you know how much space there is between the height-stick's top and the pole, let's say 5 pole widths. You do now go to the other end and measure the height there, is maybe 3 pole widths.
You do now move your long pole to where the height-stick are now, move the height-stick again and measure the différance again.
All these measurements do you add and subtract, just in the same way you did to triangulate the hole in the leveled plane, to get exact high of the hole on the other site of the mountain. :-)
If the mountain is just a hill like in the picture, you can just build a canal around it out of wood first and then fill with water to make sure it's level
And as a nifty byproduct, you also have your aquaduct. No tunnel needed!
If they had compasses, why couldn't they have just dug following a compass heading from one end and the 180 degree opposite from the other? But I think the answer is that compasses didn't make it to Europe until around the year 1190. So the only alignment tools would have been the stars and sun, both of which they would have quickly lost tract of underground. I guess the geometry of triangles really does win the day here.
What if you had a 3 foot rope and two poles that are the same length and a line level? Level the line between the poles then move a pole 3ft forward.
If the line level isn’t level the the tunnel is going up or down. Just keep doing it.
My brain explodes
If they had a compass why couldn't they dig along a bearing while staying level?
Theoretically (not feasible), they could have also used a verrrrry long hose, filled with water, and look at both ends and every tunnel entrance. The water would level out, marking the entrance points at both ends of the mountain. I am not at all a physicist, but this seems like it would have to work, at least from a theoretical standpoint.
Polycrates - man of many boxes.
Love your videos Hannah
I got the pronunciation of "Polycrates" correct, when I remembered how to say "Socrates".
How about the Siloam tunnel and the zidah they found in the middle?
Maybe they could've dug a well and used the water table to get the correct elevation on each side.
Maybe they do the same things for altitude
Just measure with triangle. But count forward and down from the top instead
thanks mommy for these videos :)
If you had 2 towers, a known distance apart at the top of the hill, you could do the triangle thing again.
This explanation relies on maintaining elevation throughout the survey process, how did they do that?
I'm no tunnel doctor, but this technique of walking and measuring a zigzag line halfway around the mountain seems like it would be wildly inaccurate. Why wouldn't they have just had a surveyor stand at the top of the mountain to sight a single straight line between the two reference points and the two tunnel entrances?
A sensitive enough barometer could solve your altitude problem, bút did they have barometers?
Who would want to tunnel through a hat?
i do not think i got what the kilometer stick and rope theory was for, but could a reflection of light be used similarly?
They must have had some more sophisticated surveying equipment than we give them credit for because sort pacing it out would never be accurate enough.
Easy. For the altitude, they just built a moat around the mountain. The water would remain level in a ring around the mountain. Your welcome.
great question, another interesting question is: is Hannah Fry related to Stephen Fry?
Where’s the pile of mashed potatoes to simulate the mountain?
I see Hannah i click it. :D hiii!!
now, how many dwarfs do you need to dig such tunnel, assuming standard granite.
How about using a compass? It will also tell you if you're digging in the right direction or tilted a bit. Your partner must try to be on the same trajectory and you'll find each other as well. Or was the compass not found back then?
Iron in the mountain could throw a compass off as you dig into the hill.
Mirrors, for the straight line?
Well you set down a pole at one entrance point then you walk around the mountain to the other point, and the triangle are at a given angle and are at the point you started and stopped your measurement, so you don’t need a huge stick. Thats the whole point os it not?
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
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".
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 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.
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'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."
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
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.)
The sexual tension in this video is through the roof. Felt like a 3rd wheel
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 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
Hannah Fry is so lovely, I really like her ❤️
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.
She's a great teacher
Ikr
And more.
@@Bill_Woo Bonk
nudge nudge wink wink. Great teacher I know what you mean
@@An.Individual 🤤
5:50 They didn't have magnetic compasses in Ancient Greece. The lodestone compass was invented during the Chinese Han Dynasty (202 BC - 220 AD)
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?
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.
"You'd like to dig a tunnel through a mountain."
GET OUT OF MY BRAIN
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...
All of a sudden I'm attracted to tunnels
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
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.
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 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
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_ ♫♩
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.
Simple leveling survey, from the sea level at both sides, using simple tools available in the ancient times (libels, sticks and tubes), would be much easier...
It is always so pleasurable listening to Hannah Fry
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.
The pronunciation of "Poly Crates" reminded me so much of "So Crates" in Bill & Teds Excellent Adventure.
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.
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?"
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.
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.
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.
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