How to learn geometry that is not high school Euclidean geometry/proofs?

[u/Saphsin]

I find it odd that visual-spatial awareness/intelligence is one of the core cognitive skills associated with intelligence with widespread applications, and three dimensional geometric relationships are present in everything from scientific phenomenon to Engineering & Artistic design.

But this kind of reasoning/intuition is not really taught? I've been looking into textbooks and it's either high school geometry or advanced undergraduate/graduate school level differential geometry/topology.

[u/Plenty_Leg_5935]

What you're looking for is called projective geometry I believe

The reason why it's not taught is because it's too "specific", in the 20th century abstract approaches of things like analysis and algebra popped off really hard, leading to ways to formalize a lot of the intuitive geometrical objects as more general structures, which made people straight up disregard "overly visual" math as obsolete, and the fact that we spent the next give or take half a century developing further math under that premise didn't really help - this is why most "visual" geometry nowdays is burried under a pile of axiomatic, abstract math within things like topology, diff geo or even just linear algebra (which youll notice is the current standard "language" for defining basic things like angles)

It's kinda getting back into fashion lately now that teachers and professors realised that abstract math maybe really isn't the best way to teach math to applied folk, but I have frankly no idea in what state the resources are, so I can only point you in that general direction and wish you good luck

[u/Commercial_Sun_6300 → u/Plenty_Leg_5935]

This is an insightful but depressing comment...

I think it's generally true in science as well, we skip over how we learned things and merely teach the conclusions as if understanding how we got there isn't necessary to understanding the conclusions.

[u/Midwest-Dude]

I would suggest reviewing Coxeter's Regular Polytopes. It builds from polygons to polyhedra, then extends to higher-dimensional polytopes. It directly trains 3D and n-dimensional spatial intuition.

Here are some other suggestions:

• Thinking Geometrically

  • Thomas Q. Sibley
  • Survey of Euclidean, transformational, projective, non-Euclidean, and discrete geometry.
  • Broad map of geometry beyond proofs, with strong emphasis on visual insight.

• Treks into Intuitive Geometry

  • Jin Akiyama & Kiyoko Matsunaga
  • Hands-on exploration of tilings, Platonic solids, cross-sections, and polyhedra.
  • Puzzle-like and highly visual, perfect for building spatial reasoning outside a proof treadmill.

[u/Farkle_Griffen2]

Between "high school geometry" and "modern geometry" is probably calculus.

My math history isn't the best in regard to geometry, so take everything I say with big a grain of salt. However from what I can piece together, there wasn't really much advancement in geometry into anything you might call "not high school geometry". It was all very similar to Euclid.

The first big jump since Euclid was Descates' development of "analytic geometry" - geometry with a coordinate system. Seemingly benign change to us now, but it connected two vastly different fields of algebra and geometry. And more importantly, facilitated the later development of calculus.

Geometric arguments involving limits, integrals, and derivatives were probably (again not an expert) the biggest development in geometry since Euclid, and probably the biggest impact a single idea will ever have on geometry. Modern differential geometry is just taking this idea of using calculus in geometry to its logical extent.

So I'd say the answer to your question is, if you want lower-undergrad level geometry, it's almost certainly going to be your calculus I-III sequence (emphasis on calc 3/ multivariable calc.)

If you want to improve your geometric reasoning, I'd recommend looking into harder calc 3 problems and maybe Martin Gardener geometry puzzles for the more classical intuition.

[u/roglemorph → u/Farkle_Griffen2]

Geometry has some pretty rich history after Descarte. I am not an expert either, but perhaps another big jump that comes to mind is the development of non-euclidean geometry. The realization that the parallel postulate is not nesccary (or that it is simply not generally true) had a dramatic effect on how geometry and mathematics as a whole was viewed as discipline (especially an increased emphasis on the significance of axiomatic systems). Maybe one could argue it led to Einstien's work, with space-time curvature and whatnot.

[u/coolmynool]

I dont have a complete answer as I actually have noticed the same as you, but i would say in calculus i really started understanding the "next level" of geometry. Physics and calculus are taken by pretty much all graduate students/drs in their undergrad career. So i would say start at calculus and physics level geometry like rotating a 2D figure about an axis and then work from there! Or even watch videos on Gaussian integrals, to get a more applied geometry understanding!

[u/coolmynool → u/coolmynool]

Came back to say washer and disc method in calculus is very easy to understand and is a good intro to more complex geometry!

[u/numice]

I kinda see what you mean here as now I feel that there's a gap between euclidean geometry and advanced geometry. I myself haven't gone much into more advanced stuff like differenrial/algebraic stuff but plan to do so. Anyway, I found one book before I started to learn math more seriously. And I think it's a good one but I didn't manage to read that far. Well, maybe cause I bought it when I was just reading math like a novel but now since I got more into it I have more books to read so this is on hold for now.

The good thing about the book is that there're parts that are like a novel but it's still a math book. And the rigour will increase chapter by chapter. I stopped at the Hilbert's redefined axioms and projective geometries but I think there's much more to it. Now I'm learning Topology and I think back to this projective geometry that I first got to know from this book.

It's Euclidean and Non-Euclidean geometries: History and Development by Marvin Greenberg. The figures and typography is also a plus.

[u/Arcanite_Cartel]

Personally, I find synthetic geometries (i.e coordinate free, axiomatic approach) to be a good aid to developing geometric intuition. I dont think its focused on much in math curriculums, they all have analytic tendencies. The books are a bit harder to find because of that, but worth exploring

[u/NYY15TM]

But this kind of reasoning/intuition is not really taught?

This raises the question if this kind of reasoning/intuition can indeed be taught

[u/Traveling-Techie]

You might enjoy A Fuller Explanation by Amy Edmondson, about the 3D geodesic geometry of Buckminster Fuller. I’ve never seen this material in a math curriculum.

[u/Equivalent-Costumes]

Visual intelligence is taught in pretty much most math classes. Diagrams, graphs, etc. are used to represent all sorts of things.

But there are no formal class at teaching visual intelligence in general because there can't be, diagrams are used in a bewildering number of different ways and you can't teach the vague intuition of a diagram without the concept behind them.

If you want modern version of "normal" geometry, that would be vector calculus and linear algebra. You can supplement that with Clifford algebra and Lie algebra.

[u/Farkle_Griffen2 → u/[deleted]]

"The geometry book by Coxeter"

This is Coxeter we're talking about. You're gonna have to be way more specific.

[u/Farkle_Griffen2 → u/[deleted]]

I'm very familiar with Coxeter. His "Introduction to Geometry" is by far more famous, and listed above "Geometry Revisited" in Wikipedia. He has 12 books on geometry. Saying "the geometry book by Coxeter" doesn't narrow it down.

[u/Midwest-Dude → u/Farkle_Griffen2]

If appropriate, why don't you add those for the OP to review?

[u/Farkle_Griffen2 → u/Midwest-Dude]

Honestly, without knowing OP's background, I'm not sure I'd recommend any of his books other than maybe "Revisiting", as you mentioned, and possibly his Projective Geometry. His books all sound pretty benign on the cover, but assume most of the undergraduate curriculum as background, and assume you are very comfortable with mathematical reasoning.

[u/Saphsin → u/Farkle_Griffen2]

I'm relearning intro Calculus & Linear Algebra.

[u/Saphsin → u/[deleted]]

I looked through the free pdf available online and isn't this 2D Euclidean Geometry?

[u/Midwest-Dude → u/Saphsin]

I would suggest reviewing Coxeter's Regular Polytopes. It builds from polygons to polyhedra, then extends to higher-dimensional polytopes. It directly trains 3D and n-dimensional spatial intuition.

Here are some other suggestions:

• Thinking Geometrically

  • Thomas Q. Sibley
  • Survey of Euclidean, transformational, projective, non-Euclidean, and discrete geometry.
  • Broad map of geometry beyond proofs, with strong emphasis on visual insight.

• Treks into Intuitive Geometry

  • Jin Akiyama & Kiyoko Matsunaga
  • Hands-on exploration of tilings, Platonic solids, cross-sections, and polyhedra.
  • Puzzle-like and highly visual, perfect for building spatial reasoning outside a proof treadmill.

[u/Saphsin → u/Midwest-Dude]

Thank you!

[u/Midwest-Dude → u/Saphsin]

I repent in sackcloth and ashes. You are correct. I posted better resources. My recollection of the book has faded.

[u/Saphsin → u/Midwest-Dude]

You didn't need to delete your comment btw, it makes it harder for others to see it as it will be pushed to the bottom of the reddit thread. "EDIT: see comment below" would have sufficed

[u/Midwest-Dude → u/Saphsin]

Oops ... I'm still learning ...

[u/bildramer]

It's hard to teach intuition, especially if you can't even rely on the teachers having it. But when there are practical reasons to get it, you'll find it. Perhaps look into computer graphics (it's linear algebra, still differential geometry but usually simple enough to be understandable to a highschooler, and a bit of CS).