No you guys, most shapes do not have a center of mass with this property (but some do) 

but but but life would be so much easier if it WAS true!

I remember thinking to myself "I'm not sure if such a point exists, but if it did then we wouldn't have needed to do all this work, so it probably can't be proven to generally exist".

I love how confident all the commenters were that they figured it out

Shortcut on finding the areas at 2:27: The top triangle is similar to the whole triangle, with a ratio of 2/3. So the area ratio is (2/3)^2 = 4/9 (which is not 1/2).

Thanks for shouting STEMerch EU out, Zach

I was one of those people who commented something like this. For some reason it seems intuitive to me that a "magical point" would always exist, but my intuition was wrong. Super interesting!

Huh... So I've done a lot of simulation stuff in the past using boring filled solids as approximations. My immediate reaction to the premise was surprise, but the centroid doesn't just measure the volume, but the volume AND distance. A large lump of volume near the centroid can be balanced out by a small amount of volume far from it. Derp. Good to be humbled and remember the basics every once in a while. >_

People who commented on other video: "this seems easy just do this, why make a video. " Those same people this episode: "I totally agree, good thing I never said that last episode" *deletes comment*

Was wondering why there were no comments until I realised this was just posted. Darn, I can't read other people's insights until after I've watched it...

It's really good that you addressed this! Very often I read a neat proof but if I didn't try to solve it myself I'd have no clue why they chose to use that idea, and why other easier ones don't work.

For those who are still confused, the triangle balances across the horizontal line because the center of mass of the top triangle is farther away from the line than the center of mass of the bottom trapezoid. So long story short leverage is weird.

I think the simplest way of seeing how this doesn’t work is an L shape, when the centre of gravity is outside the shape. Or any shape with the centre of mass outside. It doesn’t cut the area into half because the centre of mass depends on the torque/moments instead of the distribution of mass.

Thanks for pointing out I was being dumb! You made me less dumb now. I didn't comment it, but I definitely thought it. My intuition was the same as everyone else's: "If I can balance a shape on a line, there's equal mass (and thus area) on each side". Cool, now try that with a spoon.

This is what happens when a core audience is comprised of engineers instead of physicists! Love your vids btw

This is just the odd number rule of symmetry. The odd number rule states that all polygons with odd number points/line segments will only have the same number as the number or points for symmetry. Irregular polygons are weird

A line intersecting the center of mass does not cut the shape into two parts of equal area. Instead, the invariant is going to be based on the area weighted by its position relative to the center of mass.

you should also explain WHY the center of mass does not have to split volume evenly. I believe it is about distance; a bunch of area (mass) close to the center can be balanced by a smaller area that is further away on the opposite side. Just like a heavy guy close to the fulcrum of a seesaw can be balanced by a lighter kid far away at the other end.

I've actually come to this conclusion for 3d space by asking a somewhat similar question regarding fluid containers: I was looking for some rule or algorithm by which I could quickly find the half-way point where a container is half-full. I started with the known case of a cylinder, since half-way up the cylinder is half, halfway deep is half, and halfway into the breadth is half, since all those planes intersect at right angles, I figured that for all possible orientations of a half-full cylinder, the fluid level would be tangential to a single point, being the center of volume for the cylinder. I continued that line of reasoning, thinking of what the easiest way to find the half-way point in a cup was. I started by thinking of cups as cones, and immediately came to the conclusion you stated upon discovery that cones have infinite orientations where a line plane through the center of mass would have been above the half-way point in the container. My question is if there is a name for the shape that's described as the region where all planes that bisect a polyhedral area in half are tangential to this shape. Literally realized this seconds after posting this comment, but the plane that bisects all three objects will simultaneously be tangential to some part of the surface of each of their respective 'half-oids'

No, quite simply, the masses don’t need to be equal the same on either side of a solid (or thin object). What needs to be same are the moments (integral *). You can have a ‘solid’ that is a 1kg, and a 10kg particle, exactly 11 m apart, and you will find that centre of mass is on the line joining them, 1m from the 10kg weight and 10m from the 1kg weight. For a simple case, consider the axis (of rotation) to be a line perpendicular to this system, of course passing through the centre of mass. One side obviously has ten times as much mass as the other, yet the moments are equal (1 kg*10 m = 10 kg*1 m).

The moduli space of oriented planes in R^3 is homeomorphic to S^2 × R in the obvious way. This maps into I^3, whose coordinates represent the portion of mass of each object on the positive side of the plane. This map extends to the compactification S^3, where one pole at negative infinity maps to (0, 0, 0) and the other pole at positive infinity maps to (1, 1, 1). It's now a trivial exercise in topology to conclude that there is a point that maps to (1/2, 1/2, 1/2). Hint: Excising this point leaves a target space with homotopy type S^2. Hint 2: Consider what happens when you reverse the orientation of a plane.

For that property to work you would need an object who's points mirror on it's centre of mass (ball, square, 6 sided dice, etc). Think of it as a see-saw, a father playing with his son and trying to balance things would try to move up the see-saw, his mass doesn't shrink, but it is balanced at some point.

This just seems so intuitive, but our brains are very good at... optimism i guess, when i was doing the questions for the math olympiad where i live i always had to double check a pattern by actively searching bad scenarios because otherwise i'd just filter out the best case scenarios.

The equilateral triangle's case is a particularly easy one to disprove using simple geometry. You can prove that there are five congruent equilateral triangles on the bottom and four on the top. No trig or irrational numbers necessary. It's quite beautiful how you can prove that 4 to 5 ratio with two different methods

For those confused how this is possible, the long story short is leverage.

It's not the area that is cut in half, it's the integral xdA over the area of the object, where dA is a small area segment and x is the distance coordinate of the small area segment perpendicular to the line that cuts through the centroid. This integral is actually the torque on the object generated by a uniform gravitational field, and it is always zero around the centroid (or center of mass in the case where we have variable density), which is why a gravity will never cause a free falling object to start rotating by itself.

An interesting observation I derived from this - while it is true not all shapes have a dot where any line splits the shape in half, you could derive a sort of "loop", where for any given point the tangent splits the given shape in half. Because making this without brute-forcing a bunch of points is beyond me, I'm not quite sure how this loop will look, but im sure that's a brilliant idea for another video!

Bro this channel is amazing. Im so glad I got recommended this channel.

It holds for shapes with 180 deg rotational symmetry around some point. when you're rotating the plane you ad the area that the plane passed at the one side of the point and subtracting at the other, only if they're the same the areas won't change after rotation

Great explanatory follow up! Thanks!

I think the property does apply to the example shapes used in your previous video. The shapes had 180 degree rotational symmetry in three perpendicular axes, basically stretched, puffed out cubes. My first thought was the center of mass as well. I think it would have helped your illustration to pick at least one 3D shape that would obviously not work that way.

The more interesting question is: How good is it to approximate the plane that cuts the objects in half by the plane that goes through the center of masses

Thank you for pointing it out, I did think it was the case and am glad I now know it is not. I can intuitively say that I know the problem was true, but I could not have come up with a true explanation that does not have a circular dependency

There's another useful "center" that helped me prove a fixed point theorem. Every nonempty set in a compact metric space has a minimum-radius "circumsphere".

I was kinda bugged about those peeps saying like it's just that intuitive cus their line of reasoning never really made much sense to me as well. Thanks for clarifying, Mr. Zach!

So, if we take say, a 2d perfect homogeneous fluid, and put it inside a triangle, find a line that divides its area in half, we won't be able to rotate that line with respect to the CoM axis, if the total amount of fluid is to be conserved inside the triangle, assuming the line permits no leaks, of course... So counterintuitive! Is there any other axis we can use to accomplish continuous rotations?

Hey mate! I subscribed to your channel I stumbled upon your really interesting video about how expected resuts didn't matched with reality like making parents pay for showing up late to take their kids from kindgergarden actually increased number of such cases, because fine was too low, parents felt that they pay for showing up late and it was ok in their eyes. Could you please make more videos like that? Something about probability (I find that subject really interesting) or touching upon psychology like in that video. I am not asking you to remake your whole channel, I just want to see more videos this from time to time. :)

Another usefull solid to think about to get an intuition on this is the following: Imagine a barbell with three weights on one side and one on the other. Assume the bar is of negligible volume and mass. |||----| Finding the center of mass is easy. It's like balancing a seesaw. There's three times more mass on one end so the other end must be three times further from the fulcrum. So the center of mass is 3/4 of the way along the bar towards the side with three weights. |||-*---| But obviously with four weights, half of the volume is when you have two weights on each side ||*|----| So center of mass =/= center of volume.

yes yes!! that was awesome. great video dude!!

Awesome! This made me realize I had a wrong intuition about CoM as well. I also wondered what is actually equal on both sides if a plane through the CoM. This quantity includes not just the mass but also its distance from the plane.

I think there's a big difference between a line which cuts the shape in half (resulting two shapes that have the same area) and a line that has the property of the following: if the shape is supported on such a line, the shape would be in equilibrium. This statement holds for any line through the center of mass right?

It would be really cool to see an animation of the set of lines that do cut an equilateral triangle in half, as well as a discussion of which 2D and maybe 3D shapes possess the point property

First video of yours i saw. I corrected one of my ideas this morning thanks to you.

Thanks so much for posting this video. I was one of the people who posted on the last video and I was wrong.

I really liked the proof for the triangle that you did with, but actually the first thing I thought of has more of a calculus flavor to it. I was thinking, if you took one of these lines that splits the triangle in two and passes through the "center of mass", then you rotated it by a super small amount around the center, you would be taking off more from one side of the line than the other because the slopes of the edges of the triangle--relative to the line--are different on each side of it. If I'm missing something, though, the reader may feel free to correct me

My physics textbook for 9th grade had an explanation of an experiment for finding the center of mass of any comparably flat object(eg a key which was used as test subject there) and it involved kind of pinning the object on some point, waiting until it stopped swinging and drawing a line through the pinned point straight down on the object itself and repeating this process one more time with a different pinned point and it stated that no matter what point you choose the perpendicular containing this point will also contain the center of mass

Bro you are so much underrated

but but but life would be so much easier if it WAS true! 2.2K Zach Star Knasterbart Knasterbart 1 month ago I love how confident all the commenters were that they figured it out 845 Flammable Maths Flammable Maths 1 month ago Thanks for shouting STEMerch EU out, Zach _< 112 Jade Bishop 1 month ago (edited) Was wondering why there were no comments until I realised this was just posted. Darn, I can't read other people's insights until after I've watched it... 243 LZH1703 3 weeks ago (edited) I think the simplest way of seeing how this doesn’t work is an L shape, when the centre of gravity is outside the shape. Or any shape with the centre of mass outside. It doesn’t cut the area into half because the centre of mass depends on the torque/moments instead of the distribution of mass. 32 mha hart. mah sole 4 weeks ago However, a cool fact is that all convex shapes have a point in their interior whereby any lime drawn through it will cut the shape into two pieces, each with area at least 4/9ths of the shapes original area. 28 Christopher Gibbons 1 month ago For those who are still confused, the triangle balances across the horizontal line because the center of mass of the top triangle is farther away from the line than the center of mass of the bottom trapezoid. So long story short leverage is weird. 95 John Chessant 1 month ago It's really good that you addressed this! Very often I read a neat proof but if I didn't try to solve it myself I'd have no clue why they chose to use that idea, and why other easier ones don't work. 45 Heph 3 weeks ago People who commented on other video: "this seems easy just do this, why make a video. " Those same people this episode: "I totally agree, good thing I never said that last episode" deletes comment 15 Vereinigte Religionen der Ente 1 month ago This is just the odd number rule of symmetry. The odd number rule states that all polygons with odd number points/line segments will only have the same number as the number or points for symmetry. Irregular polygons are weird 82 jucom 4 weeks ago (edited) This just seems so intuitive, but our brains are very good at... optimism i guess, when i was doing the questions for the math olympiad where i live i always had to double check a pattern by actively searching bad scenarios because otherwise i'd just filter out the best case scenarios. 9 Wafi Marzouq Mohammad 1 month ago For those confused how this is possible, the long story short is leverage. 44 Sean Sdahl 1 month ago The more interesting question is: How good is it to approximate the plane that cuts the objects in half by the plane that goes through the center of masses 34 cmilkau 4 weeks ago There's another useful "center" that helped me prove a fixed point theorem. Every nonempty set in a compact metric space has a minimum-radius "circumsphere". 1 Rafael Freitas 3 weeks ago (edited) So, if we take say, a 2d perfect homogeneous fluid, and put it inside a triangle, find a line that divides its area in half, we won't be able to rotate that line with respect to the CoM axis, if the total amount of fluid is to be conserved inside the triangle, assuming the line permits no leaks, of course... So counterintuitive! Is there any other axis we can use to accomplish continuous rotations? 2 Johnnie Chan 4 weeks ago A line intersecting the center of mass does not cut the shape into two parts of equal area. Instead, the invariant is going to be based on the area weighted by its position relative to the center of mass. 8 15:44 NOW PLAYING What if the universe had a higher dimensional twist in it? Zach Star 104K views 1 year ago 14:54 NOW PLAYING The intuition and implications of the complex derivative Zach Star 106K views 9 months ago 13:52 NOW PLAYING Gödel's Incompleteness Theorem - Numberphile Numberphile 1.7M views 4 years ago 13:49 NOW PLAYING Approximations. The engineering way. Zach Star 137K views 4 months ago 20:05 NOW PLAYING Hamming codes and error correction 3Blue1Brown 912K views 11 months ago 11:14 NOW PLAYING What is Computer Science? Zach Star 928K views 4 years ago 23:16 NOW PLAYING The things you'll find in higher dimensions Zach Star 4.1M views 2 years ago 22:09 NOW PLAYING The Simplest Math Problem No One Can Solve Veritasium 9M views 1 week ago 10:40 NOW PLAYING

I didn't see the first video. I'm onboard though. For any shape and any angle plane you can move the bisection until both sides have equal volume. And you can join the 3 shapes with a plane. But figuring out a formula depends very much on the shapes involved. I think what had people confused is that you use highly symmetrical shapes in your example (with some visible imperfections, sure). But using the center of mass would work for many perfect cylinder / sphere / cube style shapes which are mirrored in all axis

Man. Solved it with simple trig. Love it

I almost mistook this video as saying there is NO line that bisects the perimeter of an eqtri. I was gonna have a cow. I almost mentioned euclidea's theta 1 lvl for easy reference - then I realized I was tripping. Good video. Thanks.

Suppose you looked at each possible orientation of a line, and found the offset from the center of mass needed to make a line with that orientation bisect the shape, and plotted that point (rephrasing: the point where, if you start at the center of mass, and move perpendicular to the line orientation until the line passing through the point with the given orientation for the line, bisects the shape), and take the resulting curve. What would this curve look like? It should be a continuous curve, and I want to say also smooth, but maybe only smooth almost everywhere, idk. It should also have all the symmetries the original shape has. The smallest the area bounded by it can be is of course 0, because for the circle (and the square?) the path is a constant point. But what is the maximum area it can be for a shape with a given bound? (E.g. a shape that fits within the unit disk, or, as an alternative question, a convex shape with area at most 1)

All 2D shapes DO have a center of mass & it’s easy to find for any arbitrary shape. Say, a flat map of the US. Tape it to a piece of cardboard, cut out the shape. Get a pin, a string & a weight attached. Wrap the string around the pin. Push the pin (with string & weight thru the cardboard at ANY point near one edge of the shape & into a wall. Gravity will cause the cardboard shape to rotate until the centroid is directly below the pivot point (the pin). The string also hangs straight down. Make 2 marks on the cardboard directly under the string near the top & bottom og the string. Pul the pin, rotate the cardboard, & repeat the procedure. Join each set of 2 marks with a straight line. Where the two lines cross is the center of mass. You can prove this by pushing the pin thru the CoM & into the wall. Even with a very “low friction” pin to cardboard interface, the cardboard shape will be stable (not inclined to rotate) in ANY angular orientation.

Your globe looks so cool

For those who are still confused: imagine a big circle and a small circle that are far apart, but connected with a very thin ”line” (like a line that has thickness) so it’s all connected. The center of mass shall be somewhere on that line, but if you simply cut it in half through that point, you will end up with a heavy piece and a light piece. Why would it balance if you lift the whole thing from a point that divides the mass inequally? Because the small mass is further away and thus the torque is the same on both sides.

thank you for your video. I am glad that your channel exists. I am considering studying electrical engineering. Later I would like to deal with the topic of control engineering. Which course of study do you think is more suitable, electrical engineering or mechatronics engineering ? Many Thanks :)

For those who obviously don't know the definition of "center of mass" or "center of gravity," its position is X = (m_1 x_1 + m_2 x_2 +...+ m_N x_N)/(m_1 + m_2 +...+ m_N) and the same for Y and Z. This has NOTHING to do with volume or area. A support anchored there results in net zero torque and no angular acceleration. The sum of the gravitational TORQUES on the left side of the center of mass exactly cancels the torques on the right side of the center of mass. (And the same for before vs. behind and above vs. below.)

Oh wow, I have a degree in physics and didn't realize this. I always implicitly though the center mass was extra super duper special and every plane through it cut the object in half. What about when the object is hanging from one point? Isn't the center of mass always supposed to be underneath the hanging point? Wait, that's not just mass but also moment of inertia. Nvm. (Cue people telling me my degree was a waste of time)

Cool graph idea. Find the formula for a line that cuts the object in half at a given angle. Then find the point on that line that is closest to the center of that object. Now graph the path that point takes as you change the angle from 0 to pi. What would that look like for a triangle? What about different polyhedrons?

It seems as if the center point will move around if you went through many different lines where the 2 halves actually do remain equal, it would be cool to visualize all of those point positions, and see how they move as the dividing line moves.

centerpoint theorem guarantees that there is a centerpoint of an object, and every hyperplane going through that object cuts space in such a way that each half-space has at least 1/(d+1) of the volume. So it holds true for 1 dimensional objects.

The centre of mass makes it such the area further away has more “weight”. When cutting area in half, the distance between the cut and other areas does not matter. An example where this would be obvious is a T shape made from 2 of the same rectangles.

learned something new tonight....thank you

The centre of mass of a tetrahedron is at 3/4 of its height, so the volume of the "upper" tetrahedron (considering an horizontal plane through the centre of mass) would be (3/4)^3=27/64, NOT 1/2 of the volume of the whole tetrahedron (since the two are obviously similar. So yeah it doesn't work for a tetrahedron either.

So there must be a formula which describes the height of the center as a function of the angle it makes with the base of the triangle when cutting the triangle exactly in half and it must be root(3)/3 at 30, 90 and 150 degrees. Some sine function probably

Oh I see the problem you saw on the argument. I bet mine were in the middle of the bunch BUT look the point of this argument is that the point doesn't need to be at the same spot, if it moves continuously as the plane continuously turns at any moment you'll have 3 points to form a plane and this makes it feel like it obviously works. Like the 2 points are always inside the 2 first objects and the plane that cross the shape in half even if when turnining you have to ajust them slightly up or down you can see you can move them all across the 3rd shape and will eventually meet with the point that contains the "half plane" inside the 3rd object, still even if it moves slightly, as long as it does so continuously.

"they all look very... similar" made me chuckle

if the object (2d) has 180 degree rotational symmetry then it should have that property. An easy way to see this is that with any given cut, if you rotate the line say a miniscule amount, you must be gaining as much area on one part of the line as you lose on the other. hence the "centroid" must always be the midpoint of this line, implying 180egree symmetry. A simple way to disprove the 3d case is just with a prism of any 2d shape that fails, as any line can be projected to a plane that (supposedly) should cut the prism in half perfectly

1:15 Where did you get that floating globe?

No civil engineers commenting I see! The CoM balances the moments of all opposing sides, not the areas/volumes. I mean, that’s literally how the CoM is defined; as the balancing point and it should be immediately obvious that the balance point does not necessarily cut an object in half. Take a broom, or rake, slide your fingers together until they balance. They will be much closer to the side of the broom or rake with the head.

A lot of people may have jumped to this because there is a similar property the the center of mass does have. It cuts the First Moment of the mass of an object equally in every direction (thats pretty much the definition of the center of mass), but not the mass itself.

Do someone know which applications Zach use for geometric representation such as the one used to represent plane for Linear Algebra (3Blue1Brown may use the same) ?

I love this and all I want is to see the faces of the people who thought they had one sentence proofs.

intermediate axis. is possible. and its interesting to watch the center try to collapse into itself. a spinning triangle will flip .. that point moves on the object as it spins.. but it can still have one if spinning in sace

I think the best way to illustrate this is with a donut. No point on a donut will allow all lines through it to cut the shape in half, because a donut is a circle without a center, so it lacks that one point on a circle that would have such a property

I wonder what curve you would get if you plotted the area of the two parts of the triangle while rotating the line in a circle? There would be three places the line would cross over itself (since there are 3 lines where the two parts would be equal) but in-between that, one side would be less or more than the other's area. Do you get sine waves? Something else?

The thumbnail is so ambiguous. 1. Can the lines be curved? 2. Do the lines have to pass all the way through the shape? 3.??? 4.Profit. Anywayyys Good video, v. educational

Well my first explanation is they didn’t necessarily mean the centre of mass itself rather a central curve if you draw a shape in the middle of the triangle that represents the points at which you can cut the triangle and it would be considered evenly in half but I don’t know for sure I’d have to test it

my go-to proof of this statement is to imagine a large sphere with a negligibly thin long cylinder protruding several of the sphere's radii off to the side, and then a small sphere attached to the other end of the cylinder. A line cutting the shape in half, if the two asses are far enough apart, will not cut through either sphere, and therefore the masses on either side must necessarily be different.

It would be interesting to know what is the set of all shapes for which the property holds. Clearly a subset of it is all shapes with 180 degree rotational symmetry, but I wonder if there are any others, and how to prove it.

This video is solid golld!!!!

There is a point in any object called the centroid or centre of mass but they don't necessarily mean it's inside the object. Take a C for example. But for every object there is a single point inside the volume. No matter how small the object there lies at least one point in space. Any 3 objects will have this and you only need 3 points to make a plane. Easy done

Say you were to bisect an equilateral triangle evenly by area using a line segment, and then you gradually rotated that line segment keeping the area the same. What shape would be inscribed within the triangle by the axis of rotation?

I was a bit disappointed that the video just gave an example without diving into why so many people might have gotten this wrong. It leads to a discussion of moments of inertia, which is a useful concept. Here are my thoughts on this: I think the thought process of many commenters goes something like this (restricting the discussion to convex 2D objects in a uniform gravitational field for simplicity): 1. Any (convex, 2D) object can be balanced on any line through its center of mass 2. So, the net force on the object is zero when balanced on such a line 3. Therefore, each piece of the object experiences the same force (otherwise it'd tip to one side) 4. So, each piece must have the same area. The problem is it isn't the forces that are equal, it's the torque on each piece about the chosen line! So the quantity that is equal for both pieces is the moment of inertia about the line, not the mass. This made me wonder if there's some easy-to-state restriction on the shapes that requires the areas to be the same as well as the moments of inertia, for lines through the center of mass. My first thought is that any such partition must divide the shape into two pieces that are just translated, rotated copies of each other. While it's clear that that's sufficient, it's not obvious to me that it's necessary, or if there's a more elegant way to state it.

Hello Zack! I'm a big fan of your content. I'm curious about what software/tools you use for the animations. Thanks.

I believe that in general, a line would cut shape in half if: Center of mass of the object is located the same place as the center of mass of a circle (or an ellipse) drawn intersecting the edge of the object. Well i am an idiot so i might be wrong. Sorry for not using math jargon as well so it might be confusing

It’s different than the center of mass, but it is possible to find a point of this type in at least 3 distinct planes, if not all of them. Maybe there is still a proof there if you align your choice of planes, but its more complicated than trying the center of mass.

Simple proof : take two unequal circles and consider your object to be the sum of both. As you take the smaller circle farther and farther away, and keep the bigger circle at the origin suppose, then the COM eventually leaves the bigger circle and is in the middles somewhere. Clearly, a vertical line through this point does not split our "object" in half, as that would imply that both the circles are actually of equal size.

I shared that baseless intuition about the center of mass until I thought about it and then the difficulty became clear. But the video didn't really show me examples of shapes where it'd feel hard that you could figure out that you can cut them all in half. Like for instance three strings curled up into a mess. It's slightly too hard to keep track off but somewhere between that and the simple shapes there's something that'd show how amazing the proof in the previous video is.

and how you make this sort of super animation ,any software recommendation?

Whoah - you’re looking really healthy my dude; good job 👍

what happen when you cut thought the center of sierpinsik triangle with a horizontal line? how much of the mass are on the upper side?

A more obvious way to check if it holds would be to see if the left of the line and the right side of the line has the same length. Rotate by an infinitesimal angle, and you increment each side's area by that length^2 times 1/2 times the angle.

Thanks

Does there exist a shape with the property of the video( there exist a point where all line passing throught it divides the area in half) that doesn't have a central symmetry?

Easiest way to know immediately that the proposition is false is to take the earth-moon system (say connected by some very thin wire to actually make it a continuous shape); the center of mass would be somewhere in space, and clearly the mass of the moon is different from the earth, so a line between the two doesnt cut the system in half.

Is it true that if you try to balance a two dimensional shape of uniform thickness and density about a line through its centroid then it will balance, otherwise not. ??

alright, this makes a lot of sense. so much actually that I'm not sure why it wasn't immediately obvious. however, there is always a plane that can cut the shape in half. as you showed, the triangle can be cut on half if you move the line around. supposedly (i don't see how this isn't the case) this is true no mayter the angle the line is at, even if said line does not have any point in common with other lines of the same property. this should hold true to the third dimension, no? that any shape CAN be cut in half by a plane, no matter what the orientation of said plane? just move the plane along the perpendicular, and it will eventually be true.

But.... wrt the first video, there does exist within the solid SOME point where the plane would cut it in half. And since three such points define the plane, we know the plane exists. To find it, might require iteration or such, since the point in two of the solids would provide a line that the plane 'pivots' around until it bisects the third, and once pivoted to this position, might no longer bisect the first two and that would shift the plane again... and so on and so on ad infinitum. But we can still show that such a plane exists, and that it exists between the plane that touches the bottom of them all and the plane that touches the tops.

If you have a central symmetry (shape rotated 180° is no op) then every line going through that center will cut the shape int two surfaces of same area. Simply because those two surfaces can be superposed. It's possible that these people confused that center with the center of mass as both overlaps when they exists.

Can we characterize the shapes such that every line through the center of mass, cuts the area in half? Or, more generally, characterize the mass distributions such that any line through COM cuts mass in half?