Joseph O'Rourke
Mathematician and Computer Scientist, Smith College, USA.
cs.smith.edu/~jorourke/
Polyhedron displayed in banner has max volume of all foldings from a square.
- Among every set of six points in 3-space (in general position) are two linked triangles: The Conway-Gordon-Sachs theorem. General position excludes three points collinear and four points coplanar. #MathSky #Geometry
- Saturn's North pole hexagon. Still not thoroughly understood. Multiple Earths could fit inside. en.wikipedia.org/wiki/Saturn%... #MathSky #Geometry #Astronomy #Planets
- Happy Easter from the Stanford Bunny! (en.wikipedia.org/wiki/Stanfor...) Developed by Stanford researchers in 1994 as a test bed model for computer graphics algorithms. This version: 2,503 vertices. #MathSky #Geometry #Graphics
- A cube can be reoriented so that it can pass through a hole carved in a congreunt cube: Prince Rupert's cube (1693!) "It is unknown whether this is true for all convex polyhedra"! (en.wikipedia.org/wiki/Prince_...) #MathSky #Geometry
- Correcting an earlier post (thanks Sophie Huiberts) to show off this attractive image. It shows the shortest paths from the white dot to each of 260 points on the genus-6 surface: 56 vertices, and regularly spaced points on edges. With student Biliana Kaneva. #MathSky #Geometry
- This open question was resolved ~20 yrs ago: Can one tie a knot with one foot of one-inch diameter rope? The answer is No: Diao, Yuanan. "The lower bounds of the lengths of thick knots." *Journal of Knot Theory and Its Ramifications* 12, no. 01 (2003): 1-16. #MathSky #Geometry #Topology
- There are three simple closed geodesics on a cube. Two are planar: an equatorial band and the hexagon slice. The 3rd is a bit more unusual. It angles off the front bottom edge at arctan(2) = ~63 deg, forming a nonplanar hexagon of length 2 sqrt{5}. #MathSky #Geometry
- A Collatz-like function f(n) that bifurcates on the primes, I posed a decade ago. It remains unknown if it always falls into a 2/4 cycle. For n=229, f(n) shoots off to 10^{376} before returning to that cycle after 6309 iterations. mathoverflow.net/questions/20... #MathSky
- Unfoldings of the hypercube. There are 261 unfoldings of the 4D hypercube into 3D. It is known since 2021 that none self-overlap (doi.org/10.37236/9796) and each tiles 3-space (Moritz Firsching). The most famous is the Dali cross, immortalized in his painting *Corpus Hypercubus*. #MathSky #Geometry
- The attractively intricate optimal tiling of an equilateral triangle by the trapezoid polyiamond (pentiamond), found in 2019 by an anonymous user ('theonetruepath') on MathOverflow. #MathSky #Geometry
- There exists a minimum area convex k-gon Q circumscribing a convex n-gon P (n>k) that has at least (k-1) edges flush with P and includes the midpoint on the k-th edge. A theorem of Adlai DePano. In the example, k=4 and 3 edges are flush, with the 4th balanced on a midpoint. #geometry #mathsky
- On a 𝘡𝘰𝘭𝘭 𝘴𝘶𝘳𝘧𝘢𝘤𝘦, every geodesic is simple (non-self-intersecting) and closed. Aside from the sphere, remarkable that such surfaces exist! Zoll was a 1903 student of David Hilbert. Image by Konrad Polthier and Markus Schmies. #Geometry #MathSky
- The traditional art gallery problem asks for the fewest guards to guard a polygonal art gallery. Through the work of Abrahamsen, Adamaszek, Miltzow, it is now known that if the polygon coordinates are rational, the guards may need irrational coordinates. [Wikipedia image] #geometry #mathsky
- "Dudeney’s Dissection is Optimal." It took 120 yrs. I posed the question 13 yrs ago here: mathoverflow.net/q/80101/6094 #mathsky #geometry
- A tetrahedron for 2025 Joe Malkevitch suggested exploring tetrahedra with edge lengths 3,3,3,3,5,5, in light of 3^4 5^2 = 2025. #geometry #mathsky #math
- Curved crease origami from Vesica Piscis Image by Klara Mundilova, from "On mathematical folding of curved crease origami..." *Computer-Aided Design*, Vol. 115, Oct. 2019, pp. 34-41. #geometry #origami
- "All Polyhedral Manifolds are Connected by a 2-Step Refolding": Any polyhedral manifold A can be unfolded to B, which can be refolded differently to manifold C. So the space is connected via unfold/refold operations, answering a 2007 question *Geom. Folding Algs*. arxiv.org/abs/2412.02174 #geometry
- Finally settled: "Optimality of Gerver's Sofa": arxiv.org/abs/2411.19826
