What novel results are there in Euclidean geometry in the last 50-100 years?

Question

Most summaries of the history of geometry conclude with an overview of developments in non-Euclidean geometry over the 18th-20th century. Some mention reformulations of Euclidean geometry based on different postulates with similar implications, more sophisticated treatments might mention algebraic approaches. But I'm wondering what new theorems (ideally proven via Euclidean techniques) there may be in the last century or so, or if there are current questions practitioners may be pursuing answers to.

Answer 1

I'd suggest that the theory of convex polytopes has been one of the principal advances. They do not strictly need to be constructed in Euclidean spaces, but most are. The topic might seem an unpromising one, the regular 3D examples having been constructed by Euclid himself (though named after Plato), but it has found modern application in an extraordinary variety of mathematical disciplines from linear programming to topology to fundamental physics (checkout for example the amplituhedron) and I don't know what all else.

Grünbaum's 1967(-ish) book on Convex Polytopes has become a classic introductory text and is currently in its second edition.

Answer 2

I'd start by looking in

• the works of Donald Coxeter;
• textbooks on computational geometry.

Max Dehn's solution of Hilbert's third problem appeared more than 100 years ago but since you said "or so" in your phrase "in the last century or so", I will list it here.

There are unsolved problems in computational geometry that are entirely within Euclidean plane and solid geometry.

Answer 3

Here are a couple examples involving the properties of polygons:

1. We can divide a square region into smaller square regions that are all of different size. The first published division dates to the 1930s.

2. It is readily proven, in both Euclidean and non-Euclidean geometries, that the area of a convex polygon inscribed in a circle depends only on the lengths of its sides and not on how the sides are ordered. Quantifying the area is another matter; for five or more sides the high-degree polynomial equations that govern the area began to be uncovered only by Robbins in the 1990s. Wolfram Mathworld, in a brief article on the pentagonal case, cites this work [1,2].

References

1. Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Discr. Comput. Geom. 12, 223-236, 1994.

2. Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523-530, 1995.