Talk:Chazelle polyhedron

vague

Chazelle polyhedron is a non-convex polyhedron constructed by removing pieces of wedges from both top and bottom of a cube's sides, leaving the notches. Its saddle surface can be considered as the set of line segments that lie forming the hyperbolic paraboloid with an equation $z=xy$.

This is baffling. It needs to be made more explicit. A surface consisting of the set of line segments that lie in a hyperbolic paraboloid is a hyperbolic paraboloid.

The image (which spins too quickly) does not match that in the first reference, which shows narrow notches that leave most of the cube's surface between them; and in that diagram the upper notches are perpendicular to the lower. —Tamfang (talk) 05:14, 24 March 2025 (UTC)

@Tamfang, ask @David Eppstein for the image's spinning the polyhedron. Speaking of the definition, it is already mentioned in the source. Dedhert.Jr (talk) 09:35, 24 March 2025 (UTC)

[1]: "The upper wedges have their edges on lines of the form $y=xz+\varepsilon$, $z$ = const and the lower wedges have their edges on lines of the form $y=xz-\varepsilon$, $x$ = const, so the hyperboloid $y=xz$ is approximated on one side by ridges parallel to the xy plane, and on the other side by ridges parallel to the yz plane.

The povray code that generated it is https://ics.uci.edu/~eppstein/junkyard/untetra/chazelle.pov —David Eppstein (talk) 16:48, 24 March 2025 (UTC)

The first source mentions about the upper wedges with the form $y=xz+\varepsilon$. I could not find the lower one $xy-z$. Dedhert.Jr (talk) 01:22, 30 March 2025 (UTC)