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.

  • scratsearcher 🔍🔮📊🎲@sopuli.xyzEnglish
    1·
    4 days ago

    I believe this has multiple reasons in education. Most 3D stuff is assumed to be self-taught later by the interested learner in private somehow.

    • the cross product for 3D is introduced within 1 day. No tasks or assignment where given to me that improve my intuition and proofs on its properties. a × b. Tbh the Cartesian product and determinants of 3x3 matrices are way more important than the time dedicated to them in average education systems.
    • rotation in 2D space is very intuitive “clockwise” and “counterclockwise” everyone learns that at age 11 minimum. While in 3D you suddenly need gimbals and f**** quarternions, sphere coordinates, which are also never properly explained to the average person. I needed to teach myself why sphere coordinates work the way they do, my teachers think it was obvious I think. Coding something up in a 3D game engine helps with this immensely imo.
    • right hand rule and its connection to the cross product. Its connection to physics and magnetic fields, forces that work on every charged particle in our universe. 🌌

    Personally I am interested in a concept I call prime spaces. This to me means an intersection of geometry and number theory. Every entry in Matrix or Vector has to be a prime number. Geometry is connected to every other field of math.