@yoink6830
Holy Shit! He found the password to the channel!
@Bigdog5400
Oh yea, Zach is an engineer, as well as a comedian
@minecat1839
I realze the mall is just a convenient way to teach the math, but it us fun to hear about and good to know that people still know what stores actually are. Imagine the problem wittern as "How to design an online shopping eebsite to minimizw the furthest dsytance between 7 buttons within a circle"
@StevenSiew2
sounds like the distribution of cell phones towers for covering a small town.
@jimi02468
It's funny how intuitively you would think seven circles would have the most difficult solution (since people commonly think seven is the 'most random' number). But it turns out it has the simplest solution of all. Or at least the simplest non-trivial one.
@scibanana3542
Divide the mall into sevenths, separated from one another, then flood them all with water going in and out for 7 mega fountains that everyone has to wade through to get around.
@kayleighlehrman9566
It would be a more practical scenario if the circular area represented a park, since most buildings aren't circular but outdoor park areas can be
@Qermaq
The cases for 5 and 6 are actually kinda fun except for the end bit, I wish you at least glossed over them more. For 5, we follow the same strategy. We want 5 equally-spaced intersection points along the circumference, so they subtend 72 degrees. We want to find a point which is (1) along a radius that passes exactly between two of these intersection points, so a line connecting the two will be perpendicular to this radius, and (2) for now at least, equidistant from the center and an intersection point. We find that this forms an isosceles triangle 36-36-108 degrees with long side R and the similar sides (root5R -R1)/2, or 0.618...R. This is close to optimal. The idea now is to shrink the circles, maintaining the intersection at the center, until the circles are less overlapping and are tangent, bringing the distance to more like 0.6094. That math is wicked hard, but visually it's easy to see what is done.
@Madtrack
Man this is so simple yet still so hard. Surely a predicted population density map as well as a layout of walls would make it more realistic and applicable. And of course as a classic engineer I will solve it via computationally brute force.
@m.h.6470
What is very interesting in the final solution: It is not only the outer perimeter, that is covered exactly by the outer 6 fountains, it is also the perimeter of the inner fountain, that is exactly covered by those 6 fountains. If you connect the intersections of the circles, you get perfect hexagons of side length 0.5km and diagonals of 1km. And hexagons (or 6 equilateral triangles) are the best shape to fill a flat plane.
@B3Band
I asked GPT-4, and it knew to put the 7th fountain in the center, but told me to put the other 6 on the perimeter of the circle. It said the minimum distance was 1.15 divided by 2. When I asked if that is true, it apologized and then told me that the fountains go slightly inside the perimeter and the minimized distance is 0.5
@coulie27
Intuitively figured a hexagonal setup with a fountain in the middle was optimal. Turned out to be right, neat. 👍 ✌️
@DrZygote214
At the start of this video, my instinct said a hexagon with a center point too. That is 7 points and i just know of it from a little reading about circle packing a long time ago. But i would have no idea how to prove optimality.
@Woodside235
Thinking about how you could make an approximation using a simulation where the fountains can move around and are repelled by one another.
@hydrashade1851
i found this guy with his skits, so him teaching complex math problems hit me like an educational freight train. nice.
@jackrain0461
I wonder how this can relate to the traveling salesman problem
@belgaer4943
It was explained a bit at the end, but the motivation for why a circle should be put in the middle was a bit lacking, especially as the minimal example provided to help solve the problem didn’t need a point in the middle. Cool problem, and generally well explained
@MrDannyDetail
As other people have also already said, my intuition was to have one central one, and the others arranged in a hexagon, which turned out to be right.
@airtoumfake
I could definitely see this problem becoming way harder with like 46 water fountains
@katakana1