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I just finished a year of numerical analysis and fell in love with the subject. Here is the book we used; I personally found it to be a wonderful text. I'm sure you can find it online somewhere if you looked hard enough. Maybe others can recommend a better one.

This is a physical way of demonstrating one (of many) definition of scalar curvature. Wikipedia has a good formula and explanation. https://en.wikipedia.org/wiki/Scalar_curvature#Direct_geometric_interpretation

Hopefully a more intuitive explanation is the following. Let A(r) denote the area of a circle (the set of all points in the surface at distance r to p) around a point p on the surface. Express A(r) as a Taylor expansion around r = 0. At orders 0 through 3, the series matches the formula one would get in flat space. At fourth order, it stops matching (in flat space the series is just \pi r\^2), and the coefficient of r\^4 is scalar curvature.

What this means is that for surfaces of positive curvature, there is less area than there should be in the plane, so when you flatten it into a plane there's a gap. For surfaces of negative curvature, there is more area than there should be in the plane, so you get a ripple refusing to flatten down. I like to think of a kale leaf.

When I learned this, my prof brought a kale leaf to class. I have a favorite tool for demonstrating this to students: https://www.amazon.com/ETA-hand2mind-85104-Polydron-Triangles/dp/B06XJH14M8

Thanks. The HJE is usually included in a course on advanced classical mechanics. Landau and Lifshitz do a great job with it, but I actually prefer a more direct derivation.

Interesting, I have not heard it called that, and don't seem to be able to find other references. I do know that Conway calls the structure Hexasticks (or hexastakes if the pencils are sharpened, which changes the symmetry group). It is discussed for example in The Symmetry of Things. My understanding is that the design comes from George Hart, though I do not think he claims to have invented it.

sejongmall, you can buy it from them on amazon!