We prove that every configuration of four through eight points in three-dimensional space, with no four points coplanar, has a connected full geometric flip graph under $2 \leftrightarrow 3$ flips. Every point remains fixed and present throughout the sequence. The proof brings each tetrahedralization into placing form at a convex hull vertex: the tetrahedra not incident to that vertex fill the convex hull of the remaining points. This reduction allows us to establish connectivity by induction, using the connectivity of regular tetrahedralizations. The main geometric tool is radial projection, which turns the tetrahedra incident to a hull vertex into a planar triangulation. When this planar triangulation is regular, varying its lifting heights produces legal spatial flips that progressively shrink the region occupied by the incident tetrahedra and reach placing form. Planar lifting criteria and compatibility constraints between the projections at different hull vertices resolve the remaining small cases. For eight points, at most 35 flips are needed to reach placing form.
We report a novel polyhedral surface of genus~3 embedded in $\mathbb{R}^3$ with eight planar, simple, non-convex nonagonal faces, 24 vertices and 36 edges, in which every two faces share at least one edge: 20 pairs of faces share one edge and 8 pairs share two collinear edges. Its face planes are $3x-4y-2z=5$, $-2x+5y-...
Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum i...
We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph $H$ with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph...
B. Bhattacharya, Sandip Das, Sk Samim Islam et al.· 0 citations
An edge card of a graph is obtained by deleting one edge, and a class edge-reconstruction number asks for the fewest carefully selected cards that identify the graph when its class is known. We determine the sharp universal bound for maximal planar graphs. Two selected cards always suffice, and the octahedral graph sho...
Each of the 12 known conjecturally optimal kissing configurations in dimensions 5, 6, and 7 carries a contact-graph polytope: a polytope whose vertices are the configuration points and whose edges are exactly its contacts. Eight of these polytopes are non-convex and distinct from their convex hulls, which include longe...
We determine the exact probability that the closed hexagonal polygon obtained by cyclically joining six independent points uniformly distributed on the unit sphere is knotted. The only possible nontrivial knot is a trefoil. Almost surely, the convex hull of the six points has one of two simplicial combinatorial types:...
Joseph Geisz, C. Peterson, Clayton Shonkwiler· 0 citations
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