The Brick Factory Problem - Numberphile
HTML-код
- Опубликовано: 27 сен 2024
- Featuring Dr James Grime... See brilliant.org/... for Brilliant and 20% off their premium service & 30-day trial (episode sponsor)
More links & stuff in full description below ↓↓↓
James Grime: www.singingban...
His RUclips channel: / singingbanana
More James on Numberphile: bit.ly/grimevideos
Numberphile is supported by the Simons Laufer Mathematical Sciences Institute (formerly MSRI): bit.ly/MSRINumb...
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. www.simonsfoun...
And support from The Akamai Foundation - dedicated to encouraging the next generation of technology innovators and equitable access to STEM education - www.akamai.com...
NUMBERPHILE
Website: www.numberphile...
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberph...
Videos by Brady Haran
Animation by Pete McPartlan
Thanks Michael Colognori from the Numberphile Society for error spotting!
Patreon: / numberphile
Numberphile T-Shirts and Merch: teespring.com/...
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanb...
Sign up for (occasional) emails: eepurl.com/YdjL9
To me, the obvious solution is to fix the track crossings rather than minimise the amount you use them. But maybe that's why I'm an engineer rather than a mathematician...
Yes that was my intuitive approach also.
Or use both techniques to minimise cost/effort/power/etc
How hard you try to avoid crossings surely depends on how much more expensive the improved crossings are compared to the extra track needed to avoid the crossing.
Or make each stopping point a through-station instead of a terminal.
I think fancy crossings would be expensive in build time and capital, longer routes expensive in efficiency and (therefore) cash flow. Case by case decision I guess.
The story of working mathematically in such adverse conditions is inspiring.
Gets you through it.
@@Sonny_McMacsson Exactly, anything to distract your mind from the horrors all around you will only help to endure them. Still inspiring and admirable, of course.
Mathematically minded individuals will spend their time musing over math problems even if they are in a garden of eden.
He was probably mentally grumbling about the stupidity of the idiots who built the place...
he was a collaborator to the system that oppressed him, by doing actions that benefit a system you are helping to support that system. the bricks get used to make bunkers, the more bricks the more bunkers, the more bunkers the more attackers it will take to over come them, by improving efficiency he likely caused greater death of those that would free him. He deprived those that were knocking over the cart and slowing down the system, the chance to fight those that in enslaved them. is it all about you and what gets you threw it, about the entertainment of your own mind, what helps you pay the bills, get a head over the next slave. NEET, lying flat, atlas shrugged, quiet quitting, the cheap slave with to many whip scares, the lazy. these are the actions of those that sacrifice their own benefit to hold to their own morals and beliefs.
Another obvious solution is to utilize the fact we're working in three dimensions, and have a bridge over the track.
i think it is outrageous that they didnt use x(n) for the minimum number of crosses. Absolute tragedy!
viewer who crosses their z's: *sweats nervously*
But all these mathematicians draw their 'x's like ")(", so there's no cross.
@@topherthe11th23 I guess people call it a cross 'cause one of the lines crosses over the other, and/or 'cause it's a cross rotated a little ways.
Yeah, this is, like, the one time when x would be the best.
@@wbfaulk rolles theorem: what doesn't cross?
For those who don't know: these graphs that are split in two parts with mutual edges between their vertices but not within same part are called "bipartite graphs".
Complete bipartite graph reminds me of videos on planar graph
Also for those who don't know: the graphs in my phone gallery are called photographs, and they are used to depict naked images of your mother.
Pretty cool! Graph Theory is fascinating.
@@aceman0000099 lol
@@hafizajiaziz8773 yea it's all graph theory.
Three kilns and three storage units is literally the three utilities problem!
That was my thought as well!
I couldn't help reciting the dedication of a graph theory book I had years ago: "To Kazimierz Kuratowski who gave K5 and K3,3 ..."
@@pierreabbat6157 Yeah, without him we still would have no graphs anything comparable to those. What an achievement indeed!
Yes, but it is asking for minimizing the number of crossings and not only proving that the number is not zero.
“Just put it on a bagel!”
To anyone who thinks this kind of maths is a bit abstract - I used to use a lot of these graph algorithms in the EDA (electronic design automation) industry - when you get down to really low level problems, this sort of stuff is invaluable and using it is the only way to make many things realistic and/or feasible.
What firm are you working in? (Working in EDA too)
@@anujthakur614 I worked for a startup called Arithmatica - later acquired by one of the big boys after I'd left. We were developing synthesis tools that specialised in producing low area/high speed gate level descriptions.
Its just graph theory
The premise for this one reminds me of Futurama, when Hermes was in a prison camp and optimazed the labor so that it could all be done by one Australian man.
"Quiet mate! Hauling the empty carts is the closest thing we get to sleep."
The obvious solution is to place each brick onto a blockchain and 3D print it on location.
And also throw AI at it
The obvious solution is to place the kilns on the trolleys and rolling them to where they are needed, no need for storage units
Am I the only one who feels that the story of the proofs will even be MORE interesting? Why 6? Why 12? What was the mistake that no one noticed for more than 10 years?
agreed, I want to see the story of the proof now!!
The reason that we know that the conjecture for complete graphs is correct up to 12 is because it has been checked computationally. The reason we don't know any more is a because it would take too long to calculate.
Finding the actual minimum is pretty interesting, because it takes so long to check every possibility it is instead rewritten as an integer linear optimisation problem solved using the associated optimisation algorithms.
+
Ooh, I love that application of minimizing the number of layers on computer chips! It's really awesome how these problems that seem like idle curiosities eventually find unexpected real-world relevance.
A bit late, but it's also a distinct problem for electricity production and use, and from that, communicating. Ring bus, token passing, TCP/IP. All related.
I don't know why, but as soon as I saw thumbnail and title I felt this video features Dr Grime
He looked so happy when he got to name the variables after (k)ilns and (s)torage units
In practical terms there would be a no crossing, just need to converge all tracks to a rotary platform.
Imagine he comes back to his boss like: i solved it but it only works in 27 dimensions
mathematician: "Need to minimize crossing." engineer: "Need to fix crossing."
I'd be interested to hear exactly where some of these proofs that were later proven wrong were discovered to have issues
+
From the sounds of it, a point of singularity are considered as many points has have already passed through it instead of "one".
So computers chips are full of kilns, bricks, storage units and railway tracks? I've always suspected as much.
This is just like that famous puzzle of connecting three houses to the three utilities. In fact, for the 3K3S example, it's quite literally the same!
And it can be solved the same way: just add a third dimension, by making the railroads elevate above or tunnel under each other, you can make it have no crossings.
Alternatively, stepping outside the realm of abstract maths, you can connect a kiln to just one storage, and then connect the storages so they can exchange stock when needed.
Or fix the damn crossings.
Missed opportunity to bring the complete graph segment back into the kilns and storage segment. The kilns and storage part is trying to make a complete bipartite graph with minimum crossings.
A bipartite graph is a graph where there are 2 sets of vertices (e.g. kilns and storage units) where members of either set only connect to members of the other set, for those wondering.
Another fantastic Numberphile video. I am so glad I found this channel 6 years ago.
The engineer in me can't help but think "If intersections are so bad, why don't you just design the rail system with switched junctions instead?" You could have all the kiln routs converge to a single line that then diverges to the storage units, the key difference is that the forks in the line would be controlled by switches that would make it so the carts are only ever contacting one set of rails a time. Perhaps this would reduce or eliminate the instability caused by the intersections. Another possible solution: use rubber tired carts not rail carts. Well anyway, the point is to make a mathematical puzzle, and even if that problem isn't actually practical to the brick carts, it can apply to other situations like the circuitry problem.
I noticed that the 3 kiln to 3 storage unit case of the problem is identical to the 3 houses and 3 utilities unsolvable puzzle. I remember as a kid my uncle in Russia posed to me that puzzle, and very quickly I conjectured it unsolvable (the goal being to have 0 intersections), at least on the Euclidean plane, but I really wanted to mathematically PROVE it was impossible and felt frustrated that I didn't have the mathematical tools to do so.
My first thought is to connect to a turntable, next would be switches.
My first thought is "don't ask a mathematician to do engineering"
Maybe that was the plan, and lots of bricks are needed to buy that upgrade.
As i said to the other commenter with the same idea, having tracks converge is actually more difficult to do compared to a somewhat orthogonal track crossing. So if the cart tracks at the work site were so rudimentary, that they could not get the crossings right, they for sure could not get tracks to converge. Remember, this is not a rail way for freight trains, this is a work site, possibly even just temporary, and since it's forced labor, they won't really mind some worker there having to pick up a cart and pick up bricks, it's certainly cheaper than having to install proper tracks.
@@yt.personal.identificationSecond thought: don't ask an engineer to do mathematics.
I know the main point is the graph theory problem but the engineer in me is screaming the whole time about the 2-crossing solution for arbitrarily many kilns/storages by merging and then unmerging tracks
10:28 this crossing, if we draw the full graph on a donut, is the crossing passing through the hole of the donut we need to make a map that needs at least 5 colours to colour every adjacent country with a different colour.
Well I timed that refresh perfectly...
Hi Squonk 😊
same!
first comment?!
Wow.
what refresh?
If a track merger was a viable intersection for this problem, I think you could bring it down a lot by having them all merge down to one and then split back up.
Pretty sure those were rudimentary tracks, hence the original problem, a track merger should be even more difficult to get right compared to a (somewhat) orthogonal crossing.
That would be a modified Steiner tree problem
I would put kilns on one side and storage units on the outside of a loop, with each kiln and storage unit having an entry/exit ramp.
The number of junctions is just kilns+storage units.
Only true mathematicians are crazy enough to think of maths problems while working in life threatening situation .
@@Holofractalius Totally agree .Thanks for suggesting.
If you want the most efficient way to do a task, ask a lazy person to do it.
@@Holofractalius Yep, if the lazu person is too lazu to use their brain, nothing will happen. You need to force them to do it!
@@Holofractalius My 6 year old grandson is very bright. So bright he'd rather use your brain to solve problems, not his own. I'm wondering if he has a future in AI now...
Hungary sided with nazis in ww2. He was manager of some sorts, I suppose
It's nice that we humans can think of things to distract ourselves from horrible things going on externally. There are many ways to fix the brick foundry
Just build a bridge at each crossing; or have a 1-way loop so there are no collisions, just onramps and offramps. But that isn't a math problem. It's a civil engineering problem.
I knew K_3,3 and K_5 were nonplanar, but it had never occurred to me to think how many crossings were required to draw them on a plane. Has this problem been studied on any other surfaces? I know K_3,3 and K_5 can be embedded on a torus (and hence any surface of higher genus), and as for nonorientable surfaces, the Möbius band (and the Klein bottle) and the projective plane.
Just a thought: Despite not having the crossing number cr(G) for a given graph G, an upperbound N will tell you that G can be embedded in a surface of genus N. Since such a surface will have N "handles" which can serve as bridges when you embed the graph G, which allows you to avoid crossings. So to me it seems the problem of crossing number is equivalent to finding "the best surface" it can be embedded in.
Robertson and Seymour actually have one of the best theorems of graph theory (I think): For any surface T, there is a finite collection of graphs H such that a graph G is T-embeddable if and only if it has no h-minor for any h in H. For example, on the plane (S^2), H=K_3,3 and K_5.
Following this theorem, if you look at minimal surfaces a graph is embeddable on (characterized by the number of holes in the minimal surface), this problems and the one in the video are equivalent.
@@beardedboulderer2609 Thanks for the heads up! I will absolutely check out that theorem.
There is something called the "toroidal crossing number," which is what it sounds like. If you look at papers on that subject, you'll also find some work that has been done for other surfaces.
@@beardedboulderer2609 came here to mention the Robertson-Seymour excluded minor theorem! Nice work.
It's incredible how I could listen to James "Weasley" Grime all day long
The optimum solution to the 3+3 case is to build your brick factory on the surface of a coffee cup.
What I would do is make it so instead of multiple tracks crossing, I would have the spot where the crossings would happen cut out and replaced with a single track on a turntable so that you just orientate the track on the turntable to align with the track you need to use.
James has the most clickbaity problems. He seems like the most interesting mathematician. Can't believe i've watched him for over 10 years.
taking notes for my next Factorio run
The crux of the story is that mathematicians should do more manual labor - this will generate new interesting problems and sometimes solutions for them.
Absolutely!
Love it when Dr Grime talks about classic problems and graphs! Bonus point if it's not proven yet.
I like to watch the post-credit (post-sponsor?) scenes showing outtakes or bloopers. Too bad this one has none, haha...
After Pal Turan was finished, all of the forced labour in that camp was done by a single Australian man (Futurama reference)
3K,3S is no problem if you have a large enough coffee mug to build on.
You’re kiln-ing me with these extremely fascinating videos!
You have only so much space for a factory. Paving everything with rails is a bad idea.
1. More wheels under the trolleys will make them more stable. Redesigning the track may also help.
2. Dividing the factory into several parts, with their own furnaces and storage (2x2). Making them back to back in an alternating pattern (checkerboard) will add more flexibility, but won't create any crossings. No chaotic tracks. Easy to expand in the future if needed.
3. Discarding the rails altogether.
4. Making storage units bigger to limit their number.
edit:
A checkerboard pattern makes one kiln for 4 surrounding storage units, and vice versa. IMO more than enough flexibility to cover all the needs.
Anything: (exist)
Mathematics: "And that's a problem"
Bro's been watching Numberphile for at least the past 5 years.
The obvious solution is to have storage units linked by a single track so you deliver to the first unit and if required the cargo is transferred to another unit. Same deal with kilns.
There's only a couple of tracks that don't need to cross
Then the mathematicians would have nothing to eat!
There is a mistake in the problem description or a better solution is possible. At 2:39 we see that multiple crossings over the same point are only counted once. At 3:17 we see that curved tracks are possible. Given these two precedents, the partial "optimal" solution at 5:45 can be beaten by (for example) getting rid of intersection #7 by running the 4-5-6 track through intersection #2 instead of creating a new intersection.
I've just looked at the original problem. Pál Turán would have counted eight crossings at 2:39. This is a happy mistake because it gives rise to other rich problems. For example - for a bipartite graph with straight edges what are the minimal number of "crossing-vertices" created? At 2:39 we count seven of these "crossing-vertices."
This is kinda trivial but you can always bound the number from above by creating a roundabout, but only if you allow that type of intersection.
At 2:37, the count of crossings is reported as 7, but typically I think of such problems as disallowing three edges crossing at a single point - is that actually part of the problem, or just a simplification?
That must be disallowed, otherwise all graphs could be solved with only one or two crossings. Just send everything into the rail vortex.
Just connect all points in a circle and have the train drive around in circles and make it stof wherever it needs to load/unload
If you have more than six kilns and more than six storage units, you’re probably firing the kilns continuously enough to designate half the storage units (rounded up) for unfired bricks and the other half (rounded down) for fired ones.
Haha, but the problem is if the 6 kilns are all specialised to one type of brick, and all the storage units needed an equal mix of all 6 brick types. I don't know who would ever do this insanity though.
Dude was forced into slave labor and still managed to turn it into a mathematical exercise, absolute legend
Using your brain like that helps keep you sane. There are PLENTY of stories from POWs about such distractions.
If you connect all of the Kilns and Storage Units to a large Round About (does not have to be circular or symmetrical), then there are no track crossings. However, this introduces other issues, such as the travel distance will be greater between many locations.
I love how they just casually have their channel's stats on the wall behind them
Crazy that in 3D space it’s always possible for lines to never cross. Only 2D plane has this crossing problem.
The realistic solution is to just reuse the tracks. You can just make a roundabout or go through another point.
Yep. Use a combination of wyes and through stations. Loading and unloading is done on a siding, not the main line. Even including a freight station for raw material input and finished product output, you can solve this with no diamond crossings at all.
It's not really the spirit of the mathematical question though.
When you want to learn so bad you’re willing to help the enemy directly
Ouch! But who says he told The Management about his thoughts? Perhaps he only spoke about this after the war. (I got that impression from the intro, but when this problem was first published wasn't stated in the video.)
@@Cellottia the implementation of it kinda implies what he did with the info
Oh snap, Singing Banana is still at it, all these years later! I'm so happy about that :)
Excellent explanation thank you so much.
3 kilns and 3 storage units is a tradition around here. Grant Sanderson enters the the video with his coffee mug!
We should use the z axis, use a little tunnel or bridge over the crossing points!
I appreciate the mathematic angle. Though, the first thing I thought of to solve an increasing number of kilns/storage units was either using the locations as stations which could be passed through, or to add a main line where most travel would take place. At that point, however, it'd be an issue of throughput instead of an increased chance of mishaps(Accounting for the possibility of operator error, as well).
Lovely video, regardless.
the rubiks cube has been unsolved ever since James started on numberphile...
Would love to see a sequel to this that includes more dimensions, or plots the scenario on the surface on a sphere or manifold.
add sidings to each node, add switches where needed, you'll need 0 crossings.
Crossings themselves can be nodes of their own for larger cases
You know, it's kinda nice to see floor division outside of programming.
Not everyone's paths crossing matters, just a few. It's a matter of who's paths are crossing where because you can't eliminate paths crossing everywhere, but not all paths need to cross either. ✌️
Why couldn’t you arrange them in a large circle, make a smaller circle of track inside that circle, and then have every kiln/storage unit attach to the circle? Then the number of crossings is exactly equal to the number of kilns and storage units, and you can get to every point on the graph.
I guess we can say, going "through" another kiln-storage is not allowed, you can only have point intersections with other paths.
I think you can bring the number always down to 1 (counted naively) by doing that. A column of kiln and a column of storage with all of their connections routed through a point in the middle.
It's implied that only two tracks can ever intersect at a given point. Or, equivalently, you're counting the number of times tracks cross, not the number of places where there are 1 or more crossings.
@@QuantumHistorian I’m not understanding why this applies. If you have a circle with spokes coming off to each destination, you still only have a series of single points of intersection. They’re not all intersecting at the same point - they’re all converging to the same circle. Like a roundabout.
@@titleloanman Think about it this way, do the paths between different kiln/storage pairings have to share a portion of track/circle? If so, that is not allowed.
@@titleloanman I think to see QuantumHistorian's point, draw the various travelling paths from kiln to storage in different colors and see that the colors intersect many times. By his implied rule those would add to the intersection number.
Though these implied rules do feel a little adhoc from the viewpoint of the original problem and also because these rules are only telling you after you've finished your drawing. In order check pairwise intersection completely you kind of have to look "globally".
I love it when mathematicians make theoretical conjectures out of real life problems, rather than addressing the practical issue and fixing the rail switch mechanisms
This is basically all we ever do. Even with math. We'll see a math problem, think about a pattern it has and formulate a new problem and get lost trying to answer that one instead.
I would desing the factory such that one kiln would be connected to one storage unit, and then the storage uniits would have separate tracks to connect between other storage units. I dont think you need so many tracks in brick factory :D
I love visual math videos better than the numerical ones
mathematician's solution: graph theory
real world solution: adding a lid over the carriages
You're the first person to say this! You must have a unique but very sensible brain!
or just add a third node, as a distribution point. so every kiln goes to that one point, and from that one point to every storage. you could unload and reload carts at the distribution point, or have some sort of rotating mechanism that can seamlessly transfer a cart from one rail to another. like a carousel. maybe a crane. who cares, there's plenty of ways. and this way there would be literally zero crossings.
alternatively, make the rails and carts more robust so that they dont topple at the crossings in the first place.
the perks of being an engineer over being a mathematician
I wounder why they did not use a relay point. Basically a storage in the middle where all kilns connect to. A set of workers who were tasked with unloading stuff that arrives from the kilns and distributing it into the storage units. or...just have a bigger storage unit.
This is not an indictment on the channel, presenters nor the intent of this video - merely a personal reflection. For me, hearing about WWII forced labor and kilns, followed by someone gleefully attacking a maths problem made it extremely difficult to focus on what was being presented. Still, I very much appreciate the initial attribution.
The description wasn't very clear about whether you're trying to minimise the number of _track crossings_ (i.e., the number of places where tracks cross each other) or the number of times _carts_ have to pass over a crossing.
But seriously.... if you introduce the length of the vertices (tracks) as being important (minimize the time it takes to move the bricks) combined with the time penalty of navigating the cart over an intersection (node).... The problem becomes more practical perhaps. There may be scenarios where the length time value exceeds the node crossing time value, so crossing is the better option. And vis-a-vis where the length time value is more efficient, so perhaps going a longer distance is less time expensive than another crossing.
Would love to see that model! 🙂
the obvious solution to me is to have the kilns connect to a ring that disperses to every storage unit so you would have x number of crossings as number of units at least for the bricks to storage units the number of carts in the ring doesn't matter as long as you loop around the storage units everything can run in a single direction and there will be no collisions.
Stack them in a spiral staircase like design. With a center elevator
Solution: Put tracks in 3D pipes and you can connect as many as your need.
For tracks a turntable would have been handy.
Wow, I was just reading about this problem yesterday!
@9:00 The computer chip seems a more reasonable application of these maths. The diagram @2:33 counted the 3 way crossing is counted as 1 intersection and the original problem was a derailing solution attempt. In a computer chip, that crossing would require 2 intersections, but 3 layers worth if confined to the same place. It could be lowered back to 2 layers if one circuit moved back down and to the side.
For the original problem though, wouldn't the invention of the track turntable have been more practical? You could just have all the tracks meet in the middle, and build one turntable. Turntables don't have the problem of derailment when well constructed and well operated.
Note: I know this is a maths channel and I love the channel. My point wasn't to question the maths, but the initial problem.
Solving this is easy, describing it (as I can't drag and drop a picture) will be more difficult. Draw two circles one inside the other, make the inner circle much smaller than the outer circle. Place as many kilns around the inner circle as you wish and place as many storage buildings on the outer circle as you wish. Connect the inner circle to the outer circle with as many tracks as you wish but 5 or 6 would probably be sufficient. The connecting tracks should be arched in such a fashion that they leave the inner circle at a tangent and connect to the outer circle at a tangent, they merge with both circles but never cross. Think of the connecting tracks as being half of the dividing line on a yin yang symbol. The exact angle that the connectors meet each circle can be tweaked in such a manner as to always provide 3 wheel stability minimizing the possibility of the brick carts becoming unstable. (Similar to driving your car of a curb, you do it at an angle allowing one wheel to drop at a time rather than both if you approached at 90 degrees) This hub and spoke design will allow any number of kilns to connect to any number of storage buildings with minimal connecting tracks and ZERO crossings. Efficiency is increased as multiple facilities can be accessed by a single connector by simply continuing around either circle allowing you to chose the most expedient routes. The connecting lines can be accessed for travel in either direction by simply pushing past a junction point then reversing.
This is a lot of work because the tracklayer did a shoddy job. the solution here isn't to break your brain with crossings, but to punish the tracklayer and make him do it again, but better.
LOL I love this answer
Defeats the nuance of the problem. But imma just build my factory along a straight line, storage and all
How many studios Numberphile have? They're always in random different places. There's no formula to calculate this chaotic pattern.
Imagine a prisoner coming up to a guard and showing him a mathematical model to make the work camp more efficient.
Mathematicians: People who solve lots of useless number problems but sometimes they accidentally end up being very useful
2:39, it's 9 intersections, not 7, no? the cross in the middle is three tracks crossing at one "point" if you allow that you can always do it in 1 crossing.
Better crossings? No.
Bridges? No.
Cranes which can reliably transfer carts between tracks? No.
Track which can rearrange itself to accommodate cart destinations? No.
_Catapults to launch bricks to storage instead of using carts and tracks?_ *Yes.*
For computer chips you just install little silicon hands to flick the electrons between circuits.
As many have already said make a better crossing, problem solved.
But aside from that why not use the 3rd dimension and make bridges? Or Instead of individually connecting each pair with its own track, maybe run a single primary track that can be branched off of.
Trivial solutions: If all points are allowed to be degenerate, zero crossings. Otherwise, put them all in a line. Also zero.
Tracks can also split into Ys, which makes this issue trivial.
2:00 "maybe i can put them in a circle" as he makes a square 😂
While not a mathematical solution… the practical solution is a central turntable.
Forced labor camp in WWII? How do you MAXIMIZE the number of crossings to slow everything to a crawl?
Also, maximise the length of the tracks, and make sure you use the storage spaces as inefficiently as possible without obviously taking up more room, such as having to move fired bricks out of the way to load up the unfired ones.
They (the nazis) probably did want the bricks though. They (the nazis) just didn't care about the working conditions (of the workers).
Edited for clarity.
@@phizc
The people who were forced to work there would have had other priorities than their captors.
@@ragnkja I edited my post for clarity. It definitely wasn't obvious who "they" were. 😄
The hexagon calculation at about 11:00 does not make sense to me. The slope of the line joining 2 points depends on which end you start at so how do you decide ?
You always choose the leftmost point as the starting point. If there is no leftmost point in a pair, you start at the lowest point and that is defined as infinite positive slope.
because the Romans were backwards and decided reading left to right was correct so now all maths happens that way
@@joleneonyoutube
It’s not so backwards if you remember that the script was developed for ink rather than stone tablets, and that most people are right-handed.
Hi idk if you’ll see this but thank you for your videos and keep up the great work
The brickyard is a poor selection to illustrate the "connections and crossing" problem.
The layout with hundreds (not 5 or 6) of brick molding houses, raw material storage houses, kiln houses and finished brick storage houses is to make a large circular track around the circumference of the buildings or through the middle of each building which are also arranged in a circle. The rail cars can then be loaded and unloaded inside or just outside each of the buildings. The track will be close enough to each of the building regardless to their function to allow for service, picking up materials, dropping off materials, baking the bricks in the kiln-house then offloading them again to the storage house. This loop of a track would never cross itself, while allowing for continuous brickmaking. It will also merge off and onto the main railway so rail cars loaded and staged on the circular track can be pushed onto the mainline track and picked up by the scheduled engine. This model seems to be the most efficient, produce the least number of crossing for any n building count.
What if you could merge lines rather than cross them?
Fantastic graph theory video analysing non planar complete and bipartite graphs. It's curious that at the solution for the crossings of the complete graph problem, it tends to a somewhat arbitrary denominator of z(n) -> n^4/2^6 as n -> inf..