1mo ago

Everyone talks about the 4th dimension, BUT why is 3D geometry so hard for the average person compared to 2D ?

I think 3D geometry has a lot of quirks and has so many results that un_intuitively don't hold up. In the link I share a discussion with ChatGPT where I asked the following:

assume a plane defined by a point A=(x_0,y_0,z_0), and normal vector n=(a,b,c) which doesn't matter here, suppose a point P=(x,y,z) also sitting on the space R^3. Question is:
If H is a point on the plane such that (AH) is perpendicular to (PH), does it follow immediately that H is the projection of P on the plane ?

I suspected the answer is no before asking, but GPT gives the wrong answer "yes", then corrects it afterwards.

So Don't we need more education about the 3D space in highschools really? It shouldn't be that hard to recall such simple properties on the fly, even for the best knowledge retrieving tool at the moment.

63 comments

Back in 2001, I wrote my own 3D graphics engine, down to the individual pixel rendering, shading, camera tracking, Z buffer, hell even error diffusion dithering for 256 color palettes.
And I still don't know half the terms you just used.
I do know points, polygons, vectors, normals, roll, pitch, yaw, Lambert's Law shading, error diffusion feedback..
And my Calculus 2 teacher admired my works and told me I had the understanding of a Calculus 4 student.
- impressive, I'd like to ask abou stuff like how long it took you and stuff. But in this discussion I'd like to mention that I didn't use any complicated terms, only orthogonal projection (middle school) and perpendicularity (elementary school).
  
  I started from the ground up in December 1998 with a bare wireframe engine, largely inspired from a demo wireframe engine from another developer. I was 17 years old then so it was basically my after school project, not a school assignment, but my teachers were impressed.
  I didn't quite just copy/paste his code though, I carefully read over his code and comments to the point that I understood how it all worked, and rewrote a much cleaner wireframe engine of my own that supported colored lines and even loading from files, which the original demo didn't support.
  Later on I came across another demo, from the same developer I think, that demonstrated rendering solid triangle shaded 3D models. Again, I read over everything and rewrote everything from the ground up, largely looking to optimize the rendering technique for the highest number of polygons per second, and of course to be able to load different models from file.
  Then I just started having a bit of fun with the polygon rendering, starting with an optimized integer based greyscale gouraud shading algorithm, which ran way faster than any similar demos I could find at the time. Note that this was all CPU driven, no fancy GPU at the time, the 3Dfx Voodoo was still a pretty new thing I couldn't afford..
  Then I got the idea of trying to bring color to the project via error diffusion, since I was basically limited to 320x200x256 color display mode, unless I wanted to run a high end video mode at a snail's pace LOL! Error diffusion is slow though, so how did I speed that up?
  Well, I did away with the gouraud shading and went back to treating each polygon as a single solid RGB color, shaded using the Lambert's Law technique. To speed up the error diffusion process, I'd only process 8 pixels into the diffusion algorithm, then as the polygon rendered, it would just pick randomly from that 8 pixel buffer.
  Since I was programming in QuickBasic, arrays were limited to 64KB each, meaning that memory was very tight, and I actually had to allocate two arrays for the Z Buffer, one for the top half of the screen and another for the bottom half.
  The inspiration for the camera tracking came from a rather unexpected source, a simple mouse string toy demo of all things LOL! I realized that if I used just one segment of that string algorithm, I could link the viewing angle to follow a point in the model, or with some creative adjustments, basically follow any arbitrary point.
  I also made a side project crude CAD scripting thing of sorts, mainly meant to render a torus or sections of a torus with whatever dimensions I wanted. With the right inputs, that also allowed me to easily generate spheres, cylinders, cones and tubes.
  I think I finished the original wireframe engine within just a couple or few days, but the other versions that had filled in polygons probably took me a week to start with, and the more advanced techniques probably took me around 2 months each, all in my spare time of course.
  I didn't really have any final product in mind, I was just experimenting and learning ya know. When 3D GPUs started becoming a big and common thing, I didn't see much future for my little project, but I sure did learn a lot!

so you asked an LLM a question and then asked if we should adjust our schooling based on that?
you're the one who might need schooling again, bruh
- yeah, I'm starting all over again with university, so hopefully this will be eventually fixed. About the rest of the population though ...
  
  Good luck.
  Keep in mind there's no way to rush hours. Whenever you're learning something, you can ask yourself roughly how many hours you've spent on it. That should reflect your skill and there's no way around it.

We percieve the 3 dimensions we exist in, through a 2d mapping, i.e. our retinas. So I think we are limited in how much of 3d we can really grasp at a time.
- Came here to say this. Even "3D movies" are actually just stereoscopic 2D (meaning two ever-so-slightly-different 2D images, one for each eye). True 3D vision would be, for lack of better term, x-ray vision.

You should read Flatland it's an awesome book and blew my mind as a young textile geek.
- You just reminded me of the planiverse which was inspired by flatland.
  Is a simple concept, thoroughly thought through, with coool drawings in it.
- I also recommend checking out this series by 10th dim: https://www.youtube.com/watch?v=gg85IH3vghA
- 3 years ago, a university teacher proposed it to me on facebook and added it to "the list", but still didn't go back to

I can't quite tell what the question is by your image. I have done a lot of descriptive geometry prior to CAD tools coming on the scene, and now work a lot with 3D geometry/topology problems, but what you describe is not going to be taught in schools because 95% of people will never need to know it. Honestly half, to three-quarters, of the people that run 3D CAD don't really understand the geometry; its just a result they get
- Spoiler alert: the image has no relation to the question, it's just something OP picked to elicit that "3D geometry" feeling. There is a point A', a point B, point C, and point D, but no point P or H or n.
  
  my lazyass had it hard to put correct labels. But judging by how many people ignored the proble an are just scolding me for using AI, fair is fair.
- @TauZero@mander.xyz It is a Geogebra drawing I did to reason with the problem, I took a screenshot of the drawing to attached it.
  In the drawing, the labels are different from the problem, but I just made a sphere whose diameter is [AP] (here point P has label A, while A has label A'),
  then constructed the plane using A and two other points of the sphere (C and D in the picture),
  I thought like "if that property from 2D geometry holds in 3D then any point in the intersection of plane and the sphere will satisfy the perpendicularity, and thus two of them will do for a counterexample".
  And It is exactly what happened: Using Geogebra's tool of measuring angles it shows that the two points, C and D, that I picked up both satisfy the orthogonality condition (in the picture angle(A,C,A')=90°=angle(A,D,A'), but they can't be both the projection of P, right ? Counterexample! (the hypothesis was that a point on a plane that satisfy that condition is immediately THE projection of the point that isn't on the plane)Yeah It is not the best thing but I wanted to attach something, and the drawing that I used was the best thing at hand.
  
  Ah, I can see OP's line of thought now:
  you have a point A' on a plane and a random point A
  you find a midpoint B and draw a sphere around it. A and A' are now a diameter of the sphere
  pick two random points D and C at the intersection of the plane and the sphere
  by the "triangle inscribed in a circle/sphere where one side is a diameter" rule, such a triangle must be a right triangle
  therefore both angles ACA' and ADA' are right angles
  thus C and D both satisfy the conditions of the initial question (with all points renamed: A=P, (C or D)=H, A'=A)
  OP never defined what a projection is, it being "4th grade math", but one of the requirements is being unique
  C and D cannot both be the projection, therefore the initial question must be answered "false": just because AH is perpendicular to PH doesn't make H a projection.
  I like treating posts as puzzles, figuring out thread by thread WTF they are talking about. But dear OP, let me let you know, your picture and explanation of it are completely incomprehensible to everyone else xD. The picture is not an illustration to the question but a sketch of your search for a counterexample, with all points renamed of course, but also a sphere appearing out of nowhere (for you to invoke the inscribed-triangle-rule, also mentioned nowhere). Your headline question is a non-sequitur, jumping from talking about 4D (never to be mentioned again) into a ChatGPT experiment, into demanding more education in schools. You complain about geometry being hard but also simple. The math problem itself was not even your question, yet it distracted everyone else from whatever it is you were trying to ask. If you ever want to get useful answers from people other than crazed puzzleseekers like me, you'll need to use better communication!

So Don't we need more education about the 3D space in highschools really?
How often do you think most people need to know the advanced mathmatical properties of 3d space?
- I think it is a shame that I'm a math student in university and needed to verify about such a thing. And if we're talking about people doing physics it might be even worst if they suck like me at 3d geometry.
  
  Math students in university need to verify basically everything, that's a lot of what the career is about. I remember being humbled when asked to prove something as familiar to everybody as -1 * -1 = 1
- I mean, we live in it. It comes up in practice fairly often.
  
  Let me guess, you're the kind of person who thinks we need to understand gravity to make use of it.
  I really wish people like you could just have their mouths taped shut and their fingers glued together.
  
  We use plenty of simple geometry everyday, sure, but you don't need to be able to even understand what OP's example says to engage with the world. Like you don't need to provide a mathematical proof to put a shelf up properly.

I think adding a bit of curvature to the six surfaces of a regular cube can throw off many. Then there's scale. Astronomical scales and milli or micro meter scales adds its own complexity by the simple fact that we lack regular language tools to capture the ideas and express them completely.
Where do we see curved surfaces? Everywhere from flight routes to space flight to deep sea diving.
Though I am not all tbat clear where we apply 3d geometry at micro scale or smaller, just a hunch that we may need them.
Language plays catch up. Is.

No for an orthogonal projection, because literally every point in the plane centered at H and normal to (AH) (so dihedrally perpendicular to the plane given in the problem) could potentially be P. In other words, it could project to H, or a point off of P perpendicularly to (AH)
You don't really need math for that one, it's just spacial reasoning, which you can't really directly teach. I suppose just the concept of solid angle vs. dihedral angle vs. face angle would be good for everyone to know. To formally prove it, it seems like you'd need linear algebra, which they don't usually teach in high school anyway.
Now, if you can use oblique projections as well, it's pretty trivial to find one that's "tilted" such that any point not already in the plane maps to a given H - the projection can proceed along any set of parallel lines through the space, and there's always a line between any point X and H. Mathematically, you use the fact that X-H must be in the kernel space of the projection, and the standard formula for constructing a projection operator from a basis complementary to the kernel space and one in the plane it projects to.
- I couldn't make sense of the first paragraph, are you sure it is right ?
  
  Pretty sure, yes. I'm probably just explaining badly.
  There's a full 360 degrees of rays perpendicular to (AH) starting at H. That would be true of line to a point in 3D. In 2D there would be exactly 2 possibilities (left and right), while in 4D they would correspond to an ordinary sphere, and hyperspheres in higher dimensions yet.
  Together, they take up a plane. Only points on a certain (infinite) line going through this new plane and H will actually orthogonally map to H, and it's the same one that's normal to to original plane. Let's call the line L.
  If point P wasn't in this plane, (PH) couldn't be perpendicular to (AH). It is in the new plane, but we still don't know for sure it's on line L, so it's not true that that implies it projects to H.

Wait is that not true? Why wouldn't H form a right angle with P and A?
AH would be perpendicular to n, and PH would be parallel to n, making them perpendicular to each other? Or am I misunderstanding the definition of a plane projection?
- AH and PH do form a right angle, that's postulated in the problem. But P is only the projection of H onto the plane if PH is indeed parallel to n. Which is not necessary.
  Imagine a nail patrols hammered into a piece of wood at an angle. The wood surface is the plane, the entry point is H and the head of the nail is P. A is anywhere on the line perpendicular to the nail on the board.
  If you shine a light from above, you can see P', the projection of P as the end of the shadow cast by thaw nail. Unless the nail is straight, P' != H.
- PH would be parallel to n
  The question doesn't posit that.
  
  OH! I see now. Perpendicular-ness is not commutative in 3d. Gotcha, thank you!
- if (PH) is perpendicular to (AH) and n is perpendicular to (AH) ==> it doesn't really follow that (PH) is parallel to n, unlike in 2D geometry. ChatGPT also got the wrong implication at first.
  Props to you for being one the few comments who actually understood the problem from my horrible statement/language though.

While 3D geometry is more difficult for me than 2D, I could almost immediately tell that the answer is no, there are infinitely many points H that satisfy this. The reason it's unintuitive is that our intuition about what "perpendicular" means comes from 2D and poorly translates to 3D.
The most intuitive explanation I can muster is this: imagine all possible planes that pass through both A and P. It should be obvious that there are infinitely many of them (I visualize it as a plane "rotating" around the AP axis). Each of these planes intersects the given plane since it passes through A. Think of the intersection line. It never passes through P (unless P is on the plane), so it is always possible to draw a perpendicular line from P to that intersection line. With one exception (when the perpendicular line falls on the A point), the point where the perpendicular falls satisfies the conditions for H. (I think all such points actually form a circle with AP' as the diameter, where P' is the parallel projection of P to the given plane, but I'm not 100% sure)
- DUH! If this was math.stackexchange I'd choose this as answer

An argument could be said that 3D space is infinitely more complex than 2D space.
As with most things, skill comes with practice. It's not really worth forcing people to learn something that has a low likelihood of being relevant to their lives, but simple exposure to something like Blender or Godot would be great for making more people familiar with 3D software.

Well see, here you have good proof that chatGPT isn't actually "the best knowledge retrieving tool at the moment". ChatGPT (and every other LLM) suuuucks at complicated math, because these text extruders don't reason. Seriously, try out some more complicated math problems. I think you'll find chatGPT gets most of them wrong, and in infuriating ways that make very little sense.
I don't disagree that we need better math instruction for students. I've been saying this since I was a student. But using chatGPT being horrible at math as evidence of this is, well, ridiculous, frankly. ChatGPT's performance isn't based on how well your average high schooler understands something, and I don't know why you're trying to tie those two very different things together.
- ChatGPT is trained based on forum discussions and pretty likely pirated books. If it found the idea in a previously established text it would have answered correctly. That's why I DO think it is representative of what the average good student was taught (not how smart, or good at problem solving they be). What's funny is that after reasoning it found the right answer, which is counter intuitive, since ChatGPT is supposed to be good at retrieving information, not at reasoning!
  
  I'm trained on pirated books and I do alright.
  Believing forum shitposts would definitely poison the well though.

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