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.
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.