Ghomi proved that every convex polyhedron could be stretched via an affine transformation so that it has an edge-unfolding to a net [Gho14]. A net is a simple planar polygon; in particular, it does not self-overlap. One can view his result as establishing that every combinatorial polyhedron has a metric realization that allows unfolding to a net. Joseph Malkevitch asked if the reverse holds (in some sense of “reverse"): Is there a combinatorial polyhedron such that, for every metric realization P in R^3, and for every spanning cut-tree T, P cut by T unfolds to a net? In this note we prove the answer is NO: every combinatorial polyhedron has a realization and a cut-tree that unfolds the polyhedron with overlap.

A quasigeodesic is a curve on the surface of a convex polyhedron that has ≤π surface to each side at every point. In contrast, a geodesic has exactly π to each side and so can never pass through a vertex, whereas quasigeodesics can. Although it is known that every convex polyhedron has at least three simple closed quasigeodesics, little else is known. Only tetrahedra have been thoroughly studied. In this paper we explore the quasigeodesics on a cube, which have not been previously enumerated. We prove that the cube has exactly 15 simple closed quasigeodesics (beyond the three known simple closed geodesics). For the lower bound we detail 15 simple closed quasigeodesics. Our main contribution is establishing a matching upper bound. For general convex polyhedra, there is no known upper bound.

A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P, if there is a convex polyhedron that has P as one face and all the other faces are convex polyiamonds, then we say that P can be domed. Our main result is a complete characterization of which equiangular n-gons can be domed: only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the integer edge lengths.

On a convex polyhedron P, the cut locus C(x) with respect to a point x is a tree of geodesic segments (shortest paths) on P that includes every vertex. We say that P has a skeletal cut locus if there is some x in P such that C(x) subset Sk(P), where Sk(P) is the 1-skeleton of P. At a first glance, there seems to be very little relation between the cut locus and the 1-skeleton, as the first one is an intrinsic geometry notion, and the second one specifies the combinatorics of P. In this paper we study skeletal cut loci, obtaining four main results. First, given any combinatorial tree T, there exists a convex polyhedron P and a point x in P with a cut locus that lies in Sk(P), and whose combinatorics match T. Second, any (non-degenerate) polyhedron P has at most a finite number of points x for which C(x) subset Sk(P). Third, we show that almost all polyhedra have no skeletal cut locus. Fourth, we provide a combinatorial restriction to the existence of skeletal cut loci. Because the source unfolding of P with respect to x is always a non-overlapping net for P, and because the boundary of the source unfolding is the (unfolded) cut locus, source unfoldings of polyhedra with skeletal cut loci are edge-unfoldings, and moreover "blooming," avoiding self-intersection during an unfolding process. We also explore partially skeletal cut loci, leading to partial edge-unfoldings; i.e., unfoldings obtained by cutting along some polyhedron edges and cutting some non-edges.

This note proves that every polar zonohedron has an edge-unfolding to a non-overlapping net.

We prove that every positively weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T edge weights matching C(x) edge lengths. If T has n leaves, P has (in general) n+1 vertices. We show there is in fact a continuum of polyhedra P each realizing T for some x∈P. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem, and a new cut-locus partition lemma. The construction of P from T is surprisingly simple.

We explore an Art Gallery variant where each point of a polygon must be seen by k guards, and guards cannot see through other guards. Surprisingly, even covering convex polygons under this variant is not straightforward. For example, covering every point in a triangle k=4 times (a 4-cover) requires 5 guards, and achieving a 10-cover requires 12 guards. Our main result is tight bounds on k-covering a convex polygon of n vertices, for all k and n. The proofs of both upper and lower bounds are nontrivial. We also obtain bounds for simple polygons, leaving tight bounds an open problem.

This note establishes that every polyhedron that has a Hamiltonian quasigeodesic can be edge-unfolded to a net, and shows that the class of such polyhedra is infinite.

We introduce a notion we call quasi-twisting that cuts a convex polyhedron P into two halves and re-glues the halves to form a different convex polyhedron. The cut is along a simple closed quasigeodesic. We initiate the study of the range of polyhedra produced by quasi-twisting P , and in particular, whether P can “quasi-twist flat,” i.e., produce a flat, doubly-covered polygon. We establish a sufficient condition and some necessary conditions, which allow us to show that of the five Platonic solids, the tetrahedron, cube, and octahedron can quasi-twist flat. We conjecture that the dodecahedron and icosahedron cannot quasi-twist flat, and prove that they cannot under certain restrictions. Many open problems remain.

Pogorelov proved in 1949 that every convex polyhedron has at least three simple closed quasigeodesics. Whereas a geodesic has exactly a π surface angle to either side at each point, a quasigeodesic has at most a π surface angle to either side at each point. Pogorelov’s existence proof did not suggest a way to identify the three quasigeodesics, and it is only recently that a finite algorithm has been proposed. Here we identify three simple closed quasigeodesics on any tetrahedron: at least one through one vertex, at least one through two vertices, and at least one through three vertices. The only exception is that isosceles tetrahedra have simple closed geodesics but do not have a 1-vertex quasigeodesic. We also identify an infinite class of tetrahedra that each have at least 34 simple closed quasigeodesics.

Anyone browsing at the stationery store will see an incredible array of pop-up cards available for any occasion. The workings of pop-up cards and pop-up books can be remarkably intricate. Behind such designs lies beautiful geometry involving the intersection of circles, cones, and spheres, the movements of linkages, and other constructions. The geometry can be modelled by algebraic equations, whose solutions explain the dynamics. For example, several pop-up motions rely on the intersection of three spheres, a computation made every second for GPS location. Connecting the motions of the card structures with the algebra and geometry reveals abstract mathematics performing tangible calculations. Beginning with the nephroid in the 19th-century, the mathematics of pop-up design is now at the frontiers of rigid origami and algorithmic computational complexity. All topics are accessible to those familiar with high-school mathematics; no calculus required. Explanations are supplemented by 140+ figures and 20 animations.

22 We study the possible moves and reachable space by rolling a 3D convex polyhedron on a 2D periodic 23 tessellation in the xy -plane, where at every step a face of the polyhedron must coincide exactly with 24 a tile of the tessellation it rests upon. We topple the polyhedron around one of the edges of the 25 grounded face toward a neighboring face until it hits the xy -plane on a neighboring tile, only if the 26 new face and the new tile also coincide. We observe the space that can be reached by succession of 27 such rolling moves. If the whole plane can be reached, we call the polyhedron a plane roller for 28 the given tessellation. We further classify polyhedra that only reach a limited strip or a bounded 29 area on given tessellations as band rollers and bounded rollers respectively. We present a 30 polynomial-time algorithm to determine the set of tiles reachable from a given starting position, 31 which in particular determines the roller type of the given polyhedron and periodic tessellation. 32 Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on any 33 regular-tiled ( ≤ 4)-uniform tessellation. Finally, we suggest how to employ these findings in puzzle 34 games. 35

We present nonoverlapping general unfoldings of two infinite families of nonconvex polyhedra, or more specifically, zero-volume polyhedra formed by double-covering an n -pointed star polygon whose triangular points have base angle α . Specifically, we construct general unfoldings when n ∈ { 3 , 4 , 5 , 6 , 8 , 9 , 10 , 12 } (no matter the value of α ), and we construct general unfoldings when α < 60 ∘ ( 1 + 1 / n ) (i.e., when the points are shorter than equilateral, no matter the value of n , or slightly larger than equilateral, especially when n is small). Whether all doubly covered star polygons, or more broadly arbitrary nonconvex polyhedra, have general unfoldings remains open.

Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geodesic segments that share endpoints, and can then zip closed. In the first part of this monograph, we primarily study properties of the tailoring operation on convex polyhedra. We show that P can be reshaped to any polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings. This investigation uncovered previously unexplored topics, including a notion of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto P, and to continuously folding P onto Q. In the second part of this monograph, we study vertex-merging processes on convex polyhedra (each vertex-merge being in a sense the reverse of a digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to produce non-overlapping polyhedral and planar unfoldings, which led us to develop an apparently new theory of convex sets, and of minimal length enclosing polygons, on convex polyhedra. All our theorem proofs are constructive, implying polynomial-time algorithms.

We prove that every tetrahedron T has a simple, closed quasigeodesic that passes through three vertices of T. Equivalently, every T has a face whose exterior angles are at most pi.

It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we construct polycubes that have no *edge zipper unfolding* where the cut edges are further restricted to form a path.

Given any convex polyhedron P of sufficiently many vertices n , and with no vertex’s curvature greater than π , it is possible to cut out a vertex, and paste the excised portion elsewhere along a vertex-to-vertex geodesic, cre-ating a new convex polyhedron P 0 of n + 2 vertices. polyhedron

Given any two convex polyhedra P and Q, we prove as one of our main results that the surface of P can be reshaped to a homothet of Q by a finite sequence of "tailoring" steps. Each tailoring excises a digon surrounding a single vertex and sutures the digon closed. One phrasing of this result is that, if Q can be "sculpted" from P by a series of slices with planes, then Q can be tailored from P. And there is a sense in which tailoring is finer than sculpting in that P may be tailored to polyhedra that are not achievable by sculpting P. It is an easy corollary that, if S is the surface of any convex body, then any convex polyhedron P may be tailored to approximate a homothet of S as closely as desired. So P can be "whittled" to e.g., a sphere S. Another main result achieves the same reshaping, but by excising more complicated shapes we call "crests," still each enclosing one vertex. Reversing either digon-tailoring or crest-tailoring leads to proofs that any Q inside P can be enlarged to P by cutting Q and inserting and sealing surface patches. One surprising corollary of these results is that, for Q a subset of P, we can cut-up Q into pieces and paste them non-overlapping onto an isometric subset of P. This can be viewed as a form of "unfolding" Q onto P. All our proofs are constructive, and lead to polynomial-time algorithms.

The construction of an unbounded polyhedron from a jagged'' convex cap is described, and several of its properties discussed, including its relation to Alexandrov's limit angle.