Acutely Triangulated, Stacked, and Very Ununfoldable Polyhedra
CCCCG 2020, Saskatoon, Canada, August 5–7, 2020
Acutely Triangulated, Stacked, and Very Ununfoldable Polyhedra
Erik D. Demaine ∗ Martin L. Demaine ∗ David Eppstein † Abstract
We present new examples of topologically convex edge-ununfoldable polyhedra, i.e., polyhedra that are combi-natorially equivalent to convex polyhedra, yet cannot becut along their edges and unfolded into one planar piecewithout overlap. One family of examples is acutelytriangulated , i.e., every face is an acute triangle. An-other family of examples is stacked , i.e., the result offace-to-face gluings of tetrahedra. Both families achieveanother natural property, which we call very unun-foldable : for every k , there is an example such thatevery nonoverlapping multipiece edge unfolding has atleast k pieces. Can every convex polyhedron be cut along its edgesand unfolded into a single planar piece without overlap?Such edge unfoldings or nets are useful for construct-ing 3D models of a polyhedron (from paper or other ma-terial such as sheet metal): cut out the net, fold alongthe polyhedron’s uncut edges, and re-attach the poly-hedron’s cut edges [25]. Unfoldings have also proveduseful in computational geometry algorithms for find-ing shortest paths on the surface of polyhedra [3, 5, 9].Edge unfoldings were first described in the early 16thcentury by Albrecht D¨urer [16], implicitly raising thestill-open question of whether every convex polyhedronhas one (sometimes called D¨urer’s conjecture). Thequestion was first formally stated in 1975 by G. C. Shep-hard, although without reference to D¨urer [17, 23]. Ithas been heavily studied since then, with progress oftwo types [15, 22]:1. finding restricted classes of polyhedra, or general-ized types of unfoldings, for which the existence ofan unfolding can be guaranteed; and2. finding generalized classes of polyhedra, or re-stricted types of unfoldings, for which counterex-amples — ununfoldable polyhedra — can beshown to exist. ∗ Computer Science and Artificial Intelligence Laboratory,Massachusetts Institute of Technology, { edemaine,mdemaine } @mit.edu † Computer Science Department, University of California,Irvine, [email protected]. This work was supported in part bythe US National Science Foundation under grant CCF-1616248.
Results guaranteeing the existence of an unfolding in-clude: • Every pyramid, prism, prismoid, and dome has anedge unfolding [15]. • Every sufficiently flat acutely triangulated convexterrain has an edge unfolding [21]. Consequentially,every acutely triangulated convex polyhedron canbe unfolded into a number of planar pieces that isbounded in terms of the “acuteness gap” of thepolyhedron, the minimum distance of its anglesfrom a right angle. • Every convex polyhedron has an affine transforma-tion that admits an edge unfolding [18]. • Every convex polyhedron can be unfolded to a sin-gle planar piece by cuts interior to its faces [3, 14]. • Every polyhedron with axis-parallel sides can beunfolded after a linear number of axis-parallel cutsthrough its faces [10]. • Every triangulated surface (regardless of genus) hasa “vertex unfolding”, a planar layout of trianglesconnected through their vertices that can be foldedinto the given surface [13]. • For ideal polyhedra in hyperbolic space, unlike Eu-clidean convex polyhedra or non-ideal hyperbolicpolyhedra, every spanning tree forms the systemof cuts of a convex unfolding into the hyperbolicplane.Previous constructions of ununfoldable polyhedra in-clude the following results. A polyhedron is topolog-ically convex if it is combinatorially equivalent to aconvex polyhedron, meaning that its surface is a topo-logical sphere and its graph is a 3-vertex-connected pla-nar graph. • Some orthogonal polyhedra and topologically con-vex orthogonal polyhedra have no edge unfolding,and it is NP-complete to determine whether anedge unfolding exists in this case [1, 8]. • There exists a convex-face star-shaped topolog-ically convex polyhedron with no edge unfold-ing [20, 24]. a r X i v : . [ c s . C G ] J u l nd Canadian Conference on Computational Geometry, 2020 • There exists a triangular-face topologically convexpolyhedron with no edge unfolding [7]. • There exist edge-ununfoldable topologically convexpolyhedra with as few as 7 vertices and 6 faces, or6 vertices and 7 faces [4]. • There exists a topologically convex polyhedron thatdoes not even have a vertex unfolding [2]. • There exist domes that have no
Hamiltonian un-folding , in which the cuts form a Hamiltonian paththrough the graph of the polyhedron [12]. Simi-larly, there exist polycubes that have no Hamilto-nian unfolding [11]. • There exists a convex polyhedron, equipped with3-vertex-connected planar graph of geodesics par-titioning the surface into regions metrically equiv-alent to convex polygons, that cannot be cut andunfolded along graph edges [6].In this paper, we consider two questions left openby the previous work on edge-ununfoldable polyhe-dra with triangular faces [7], and strongly motivatedby O’Rourke’s recent results on unfoldings of acutely-triangulated polyhedra [21]. First, the previous coun-terexample of this type involved triangles with highlyobtuse angles. Is this a necessary feature of the con-struction, or does there exist an ununfoldable polyhe-dron with triangular faces that are all acute? Second,how far from being unfoldable can these examples be?Is it possible to cut the surfaces of these polyhedra intoa bounded number of planar pieces (instead of a singlepiece) that can be folded and glued to form the polyhe-dral surface? (Both questions are motivated by previ-ously posed analogous questions for convex polyhedra,as easier versions of D¨urer’s conjecture [15, Open Prob-lems 22.12 and 22.17].)We answer both of these questions negatively, by find-ing families of topologically convex edge-ununfoldablepolyhedra with all faces acute triangles, in which anycutting of the surface into regions that can be unfoldedto planar pieces must use an arbitrarily large numberof pieces. Additionally, we use a similar constructionto prove that there exist edge-ununfoldable stackedpolyhedra [19], formed by gluing tetrahedra face-to-face with the gluing pattern of a tree, that also re-quire an arbitrarily large number of pieces to unfold.We leave open the question of whether there existsan edge-ununfoldable stacked polyhedron with acute-triangle faces.
Our construction follows that of Bern et al. [7] in beingbased on certain triangulated topological disks, which Figure 1: Combinatorial structure of a hatthey called hats . The combinatorial structure of a hat(in top view, but with different face angles than the hatwe use in our proof) is shown in Figure 1: It consistsof nine triangles, three of which (the brim , blue in thefigure) have one edge on the outer boundary of the disk.The next three triangles, yellow in the figure, have avertex but not an edge on the disk boundary; we callthese the band of the hat. The central three triangles,pink in the figure, are disjoint from the boundary andmeet at a central vertex; we call these the crown of thehat.In both the construction of Bern et al. [7] and in ourconstruction, the three vertices of the hat that are inte-rior to the disk but not at the center all have negativecurvature, meaning that the sum of the angles of thefaces meeting at these vertices is greater than 2 π . Thecenter vertex, on the other hand, has positive curvature,a sum of angles less than 2 π . When this happens, wecan apply the following lemmas: Lemma 2.1
At any negatively-curved vertex of a poly-hedron, any unfolding of the polyhedron that cuts only itsedges and separates its surface into one or more simplepolygons must cut at least two edges at each negatively-curved vertex.
Proof.
If only one edge were cut then the faces sur-rounding that vertex could not unfold into the planewithout overlap. (cid:3)
Lemma 2.2
Let D be a subset of the faces of a polyhe-dron, such that the polyhedron is topologically a sphereand D is topologically equivalent to a disk (such as ahat). Then in any unfolding of the polyhedron (possiblecutting it into multiple pieces), either D is separated CCG 2020, Saskatoon, Canada, August 5–7, 2020
Figure 2: Two paths from a boundary vertex of a hat,through all three negatively curved vertices, to the cen-ter vertex into multiple pieces by a path of cut edges from oneboundary vertex of D to another or by a cycle of cutedges within D , or the set of cut edges within D formsa forest with at most one boundary vertex for each treein the forest. Proof.
If the cut edges within D do not form a forest,they contain a cycle and the Jordan Curve Theoremimplies that this cycle separates an interior part of theboundary from the exterior. If they form a forest inwhich some tree contains two boundary vertices, thenthey contain a boundary-to-boundary path within D ,again separating D by the Jordan Curve Theorem. Theonly remaining possibility is a forest with at most oneboundary vertex per tree. (cid:3) Lemma 2.3
For a hat combinatorially equivalent tothe one in Figure 1, with positive curvature at the centervertex and negative curvature at the other three interiorvertices, any unfolding that does not cut the hat intomultiple pieces must cut a set of edges along a singlepath from a boundary vertex to the center vertex.
Proof.
By Lemma 2.2, each component of cut edgesmust form a tree with at most one boundary vertexwithin the hat. But every tree with one or more edgeshas at least two leaves, and every tree that is not a pathhas at least three leaves. By Lemma 2.1, the only non-boundary leaf can be the center vertex, so each com-ponent must be a path from the boundary to this ver-tex. (cid:3)
Up to symmetries of the hat, there are only twodistinct shapes that the path of Lemma 2.3 from theboundary to the center of a hat can have (Figure 2).These two cuttings differ in how the crown triangles areattached to the band and to each other, but they bothcut the brim and band triangles in the same way, intoa strip of triangles connected edge-to-edge around theboundary of the hat.Our key new construction is depicted in unfolded (butself-overlapping) form in Figure 3. It is a hat in whichall triangles are acute and isosceles: • The three brim triangles have apex angle 85 ◦ andbase angle 47 . ◦ . • The three band triangles have base angle 85 ◦ andapex angle 10 ◦ . • The three crown triangles are congruent to theband triangles, with base angle 85 ◦ and apex angle10 ◦ .As in the construction of Bern et al. [7], this leaves neg-ative curvature (total angle 425 ◦ from five 85 ◦ angles)at the three non-central interior angles of the hat, andpositive curvature (total angle 30 ◦ ) at the center vertex,allowing the lemmas above to apply. The cut edges ofthe figure form a tree with a degree-three vertex at oneof the negatively curved vertices of the hat, and a leafat another negatively curved vertex, the one at whichthe self-overlap of the figure occurs, So the cutting inthe figure does not match in detail either of the twopath cuttings of Figure 2. Nevertheless, the brim andband triangles are unfolded as they would be for eitherof these two path cuttings. It is evident from the fig-ure that this unfolding of the brim and band trianglescannot be extended to a one-piece unfolding of the en-tire hat: if a crown triangle is attached to the middleof the three unfolded band triangles (as it is in the fig-ure) then there is no room on either side of it to attachthe other two crown triangles, and a crown triangle at-tached to either of the other two band triangles wouldoverlap the opposite band triangle. We prove this visualobservation more formally below. Lemma 2.4
The hat with acute triangles describedabove has no single-piece unfolding.
Proof.
As we have already seen in Lemma 2.3, any un-folding (if it exists) must be along one of the two cutpaths depicted in Figure 2. As a result, the unfolding ofthe brim and band triangles (but not the crown trian-gles) must be as depicted in Figure 3. In this unfolding,the three base sides of the unfolded band triangles forma polygonal chain whose interior angles (surroundingthe central region of the figure where the pink crowntriangles are attached) can be calculated as 105 ◦ .A regular pentagon has interior angles of 108 ◦ , andhas the property that each vertex lies on the perpendic-ular bisector of the opposite edge. Because the interiorangles of the chain of base sides of band triangles are105 ◦ , less than this 108 ◦ angle, it follows that the bandtriangle at one end of the chain extends across the per-pendicular bisector of the base edge at the other endof the chain. Further, it does so at a point closer thanthe vertex of a regular pentagon sharing this same baseedge (Figure 4).If a crown triangle were attached to one of the twobase edges at the ends of the chain of three base edges,2 nd Canadian Conference on Computational Geometry, 2020
Figure 3: A hat made with acute isosceles triangles. Un-like Figure 2, the cuts made to form the self-overlappingunfolding shown do not form a path.Figure 4: Each vertex of a regular pentagon lies on theperpendicular bisector of the opposite side; in a path ofthree equal edges with the tighter angle 105 ◦ , the lastedge overlaps the perpendicular bisector of the first.its altitude would lie along the perpendicular bisector ofthe base edge. And because the crown triangle has anapex angle of 10 ◦ , sharper than the angle of an isoscelestriangle inscribed within a regular pentagon, its altitudeextends across the perpendicular bisector farther thanthe regular pentagon vertex, causing it to overlap withthe band triangle at the other end of the chain of threebase edges.Therefore, attaching a crown triangle to either thefirst or last of the band triangle base edges in the Figure 5: Tetrahedron with faces replaced by hatschain of these three edges necessarily leads to a self-overlapping unfolding. However, these two ways of at-taching a crown triangle are the only ones permitted bythe two cases depicted in Figure 2. Attaching a crowntriangle to the middle of the three base edges, as in Fig-ure 3, can only be done by cutting along a tree that isnot a path. Therefore, no unfolding exists. (cid:3) The following construction is straightforward, andwill allow us to construct polyhedra with multiple hatswhile keeping the hats disjoint from each other.
Lemma 2.5
The hat with acute triangles describedabove can be realized in three-dimensional space, lyingwithin a right equilateral-triangle prism whose base isthe boundary of the hat.
We now use these hats to construct a topologically con-vex ununfoldable polyhedron.
Theorem 3.1
There exists a topologically convex un-unfoldable polyhedron whose faces are all acute isoscelestriangles.
Proof.
Replace the four faces of a regular tetrahedronby acute-triangle hats, all pointing outward, as shownin Figure 5. Because each lies within a prism havingthe tetrahedron face as a base, they do not overlap eachother in space. By Lemma 2.4, no hat can be unfoldedinto a single piece, so any possible unfolding (even oneinto multiple pieces) must cut each hat along some pathbetween two of its three boundary vertices (at least;there may be more cuts besides these). The four paths
CCG 2020, Saskatoon, Canada, August 5–7, 2020
Figure 6: Hat for stacked polyhedra (top view, left, andexploded view as a stacked polyhedron, right)formed in this way are disjoint except at their ends, andconnect the four vertices of the tetrahedron, necessarilyforming at least one cycle that separates the tetrahedroninto at least two pieces. (cid:3)
Like the examples of Tarasov, Gr¨unbaum, and Bernet al. [7, 20, 24], the resulting polyhedron is also star-shaped, with the center of the tetrahedron in its kernel. A stacked polyhedron is a polyhedron that can beformed by repeatedly gluing a tetrahedron onto a singletriangular face of a simpler stacked polyhedron, startingfrom a single tetrahedron [19]. To make ununfoldablestacked polyhedra, we use a similar strategy to our con-struction of ununfoldable polyhedra with acute-triangle-faces, in which we replace some faces of a convex poly-hedron by hats. However, the acute-triangle hat that weused earlier cannot be used as part of a stacked polyhe-dron: in a stacked polyhedron, every non-face triangle issubdivided into three smaller triangles, but that is nottrue of the outer triangle of Figure 1. Instead, we use thehat shown in Figure 6. As before, it has three brim tri-angles, three band triangles, and three crown triangles,but they are arranged differently and less symmetrically.We make the brim and band triangles nearly coplanar,with shapes approximating those shown in the figure,but projecting slightly out of the figure so that the resultcan be constructed as a stacked polyhedron. We choosethe crown triangles to be isosceles, and taller than theisosceles triangles inscribed in regular pentagons, as inour acute-triangle construction, so that (as viewed inFigure 6) they project out of the figure. Lemma 4.1
The hat described above has no single-piece unfolding.
Proof.
As with our other hat, the center vertex of thishat has positive curvature, and the other three interiorvertices have negative curvature, so by Lemma 2.3 anyunfolding of the hat that leaves it in one piece mustform a path consisting of a single edge cutting from the boundary to the crown, two edges cutting between theband and the crown, and one edge cutting to the centerof the crown.There are many more cases than there were in Fig-ure 2, but we can avoid case-based reasoning by arguingthat in each case, the brim and band triangles unfold insuch a way that the three edges between the band andcrown triangles form a polygonal chain with interior an-gles less than the 108 ◦ angles of the regular pentagon(in fact, close to 60 ◦ , because of the way we have con-structed this part of the hat to differ only by a smallamount from the top view shown in Figure 6. There-fore, just as in Figure 4, each edge at one end of thischain of three edges overlaps the perpendicular bisectorof the edge at the other end of the chain.Cutting along a path from a boundary edge of the hatto its center vertex forces the three crown triangles tobe attached to the unfolded brim and band triangles onone of the two edges at the end of this path. However,our construction makes the three crown triangles tallenough to ensure that, no matter which of these twoedges they are attached to, they will overlap the edgeat the other end of the path at the point where it crossesthe perpendicular bisector. (cid:3) Theorem 4.2
There exists an ununfoldable stackedpolyhedron.
Proof.
We replace the four faces of a regular tetrahe-dron with the hat described above. Each such replace-ment can be realized as a stacking of four tetrahedraonto the face, so the result is a stacked polyhedron. Asin Theorem 3.1, each hat lies within a prism having thetetrahedron face as a base, so they do not overlap eachother in space; and the set of edges cut in any unfoldingmust include at least four paths between the four tetra-hedron vertices, necessarily forming a cycle that cutsone part of the polyhedron surface from the rest. (cid:3)
A stacked hat with the same combinatorial structureas the one used in this construction, with the center ver-tex positively curved and the surrounding three verticesnegatively curved, cannot be formed from acute trian-gles, because that would leave the degree-four vertexwith positive curvature. We leave as an open questionwhether it is possible for an ununfoldable stacked poly-hedron to have all faces acute.
Both families of examples above can be made into veryununfoldable families. In both cases, the approach is thesame: instead of starting from a tetrahedron, we startfrom a polyhedron with many triangular faces, and showthat attaching hats to more and more triangles requiresmore and more unfolded pieces.2 nd Canadian Conference on Computational Geometry, 2020
Theorem 5.1
There exist topologically convex polyhe-dra with acute isosceles triangle faces such that any un-folding formed by cutting along edges into multiple non-self-overlapping pieces requires an unbounded number ofpieces.
Proof.
For any integer k ≥
1, refine the regular tetra-hedron by subdividing each edge into k equal-lengthedges and subdivide each face into a regular grid of (cid:80) ki =1 (2 i −
1) = k equilateral triangles of side length1 /k , for a total of 4 k faces and (by inclusion-exclusion) (cid:80) k +1 i =1 i − k + 1) + 4 = 2 k + 2 vertices. Replaceeach equilateral triangular face by an acute-triangle hatpointing outward. As in Theorem 3.1, each hat lieswithin a prism having the face of the tetrahedron as abase, so they do not overlap each other in space; andany unfolding into multiple pieces must, in each hat, ei-ther cut along a cycle within the hat or cut along somepath connecting two of its three boundary vertices. Let c be the number of cycles within hats cut in this way, sothat we have a system of at least 4 k + c disjoint pathsconnecting pairs of subdivided-tetrahedron vertices.Now consider cutting the polyhedron surface alongthese paths, one by one. Each cut either connectstwo subdivided-tetrahedron vertices that were not pre-viously connected along the system of cuts, or twosubdivided-tetrahedron vertices that were previouslyconnected. If cutting along a path connects two ver-tices that were not previously connected, it reduces thenumber of connected components among these vertices;this case can happen at most 2 k + 1 times. If cut-ting along a path connects two vertices that were previ-ously connected, then that path and the path throughwhich they were previously connected form a Jordancurve that separates off two parts of the surface fromeach other. Because there are 4 k − c paths connectingpairs of subdivided-tetrahedron vertices, only 2 k + 1 ofwhich can form new connections, this case must happenat least 2 k − − c times. Because the surface startedwith a single piece and undergoes at least 2 k − − c separations, it ends up with at least 2 k − c pieces, whichtogether with the c additional pieces formed by cycleswithin hats, form a total of at least 2 k pieces. (cid:3) Theorem 5.2
There exist topologically convex stackedpolyhedra such that any unfolding formed by cuttingalong edges into multiple non-self-overlapping pieces re-quires an unbounded number of pieces.
Proof.
For any integer k ≥
0, refine the regular tetra-hedron by choosing any face and attaching to the facea very shallow tetrahedron whose apex is near the in-center of the face, effectively splitting the face into threefaces, and repeating this process a total of k times. Be-cause each attachment increases the number of faces by2 and the number of vertices by 1, the result is a stacked polyhedron with 4 + 2 k triangular faces (not necessarilyequilateral) and 4 + k vertices. Replace each trianglewith a version of the hat from Section 4 pointed out-ward, using the availability flexibility to make the inter-face between the band and crown an equilateral trianglenear the in-center of the original triangle. As in Theo-rem 4.2, the result is a stacked polyhedron; each hat lieswithin a prism having the face of the tetrahedron as abase, so they do not overlap each other in space; and anyunfolding into multiple pieces must cut each hat alongsome path connecting two of its three boundary vertices.As in Theorem 5.1, at most 3 + k such paths can de-crease the number of connected components among the4 + k vertices, leaving at least 1 + k paths that separatethe surface into at least 2 + k pieces. (cid:3) Acknowledgments
This research was initiated during the Virtual Workshopon Computational Geometry organized by E. Demaineon March 20–27, 2020. We thank the other participantsof that workshop for helpful discussions and providingan inspiring atmosphere.
References [1] Zachary Abel and Erik D. Demaine.Edge-Unfolding Orthogonal Polyhedra is StronglyNP-Complete. In
Proceedings of the 23rd AnnualCanadian Conference on Computational Geometry(CCCG 2011)
Proceedings ofthe 23rd Canadian Conference on ComputationalGeometry (CCCG 2011), Toronto, August 10–12,2011 , 2011. URL: https://cccg.ca/proceedings/2011/papers/paper85.pdf.[3] Pankaj K. Agarwal, Boris Aronov, JosephO’Rourke, and Catherine A. Schevon. Starunfolding of a polytope with applications.
SIAMJournal on Computing , 26(6):1689–1713, 1997.doi:10.1137/S0097539793253371.[4] Hugo A. Akitaya, Erik D. Demaine, DavidEppstein, Tomohiro Tachi, and Ryuhei Uehara.Minimal ununfoldable polyhedron. In
Abstractsfrom the 22nd Japan Conference on Discrete andComputational Geometry, Graphs, and Games(JCDCGGG 2019) , pages 27–28, Tokyo, Japan,September 2019. URL: https://erikdemaine.org/papers/MinimalUnunfoldable JCDCGGG2019/.
CCG 2020, Saskatoon, Canada, August 5–7, 2020 [5] Boris Aronov and Joseph O’Rourke. Nonoverlapof the star unfolding.
Discrete & ComputationalGeometry , 8(3):219–250, 1992.doi:10.1007/BF02293047.[6] Nicholas Barvinok and Mohammad Ghomi.Pseudo-edge unfoldings of convex polyhedra.
Discrete & Computational Geometry , 2019.doi:10.1007/s00454-019-00082-1.[7] Marshall Bern, Erik D. Demaine, David Eppstein,Eric Kuo, Andrea Mantler, and Jack Snoeyink.Ununfoldable polyhedra with convex faces.
Computational Geometry: Theory & Applications ,24(2):51–62, 2003.doi:10.1016/S0925-7721(02)00091-3.[8] Therese C. Biedl, Erik D. Demaine, Martin L.Demaine, Anna Lubiw, Mark H. Overmars,Joseph O’Rourke, Steve Robbins, and SueWhitesides. Unfolding some classes of orthogonalpolyhedra. In
Proceedings of the 10th CanadianConference on Computational Geometry (CCCG1998) , 1998. URL: https://cgm.cs.mcgill.ca/cccg98/proceedings/cccg98-biedl-unfolding.ps.gz.[9] Jindong Chen and Yijie Han. Shortest paths on apolyhedron. In
Proceedings of the 6th AnnualSymposium on Computational Geometry (SoCG1990) . ACM Press, 1990.doi:10.1145/98524.98601.[10] Mirela Damian, Erik D. Demaine, Robin Flatland,and Joseph O’Rourke. Unfolding genus-2orthogonal polyhedra with linear refinement.
Graphs and Combinatorics , 33(5):1357–1379,2017. doi:10.1007/s00373-017-1849-5.[11] Erik D. Demaine, Martin L. Demaine, DavidEppstein, and Joseph O’Rourke. Some polycubeshave no edge-unzipping. Electronic preprintarxiv:1907.08433, 2019.[12] Erik D. Demaine, Martin L. Demaine, and RyuheiUehara. Zipper unfoldability of domes andprismoids. In
Proceedings of the 25th CanadianConference on Computational Geometry (CCCG2013), Waterloo, Ontario, Canada August8th–10th, 2013 , 2013. URL: https://cccg.ca/proceedings/2013/papers/paper 10.pdf.[13] Erik D. Demaine, David Eppstein, Jeff Erickson,George W. Hart, and Joseph O’Rourke.Vertex-unfoldings of simplicial manifolds. In
Discrete Geometry: In honor of W. Kuperberg’s60th birthday , volume 253 of
Pure and AppliedMathematics , pages 215–228. Marcel Dekker,2003. [14] Erik D. Demaine and Anna Lubiw. Ageneralization of the source unfolding of convexpolyhedra. In Alberto M´arquez, Pedro Ramos,and Jorge Urrutia, editors,
Revised Papers fromthe 14th Spanish Meeting on ComputationalGeometry , volume 7579 of
Lecture Notes inComputer Science , pages 185–199, Alcal´a deHenares, Spain, June 2011.doi:10.1007/978-3-642-34191-5 18.[15] Erik D. Demaine and Joseph O’Rourke.
Geometric Folding Algorithms: Linkages,Origami, Polyhedra . Cambridge University Press,July 2007.[16] Albrecht D¨urer.
The Painter’s Manual: AManual of Measurement of Lines, Areas, andSolids by Means of Compass and Ruler Assembledby Albrecht D¨urer for the Use of All Lovers of Artwith Appropriate Illustrations Arranged to bePrinted in the Year MDXXV . Abaris Books, Inc.,New York, 1977. English translation of
Unterweysung der Messung mit dem Zirkel unRichtscheyt in Linien Ebnen und GantzenCorporen , 1525.[17] Michael Friedman.
A History of Folding inMathematics: Mathematizing the Margins .Birkh¨auser, 2018. See in particular page 47.doi:10.1007/978-3-319-72487-4.[18] Mohammad Ghomi. Affine unfoldings of convexpolyhedra.
Geometry & Topology , 18:3055–3090,2014. doi:10.2140/gt.2014.18.3055.[19] Branko Gr¨unbaum. A convex polyhedron which isnot equifacettable.
Geombinatorics ,10(4):165–171, 2001. URL:https://sites.math.washington.edu/ ∼ grunbaum/Nonequifacettablesphere.pdf.[20] Branko Gr¨unbaum. No-net polyhedra. Geombinatorics ∼ grunbaum/Nonetpolyhedra.pdf.[21] Joseph O’Rourke. Edge-unfolding nearly flatconvex caps. In Bettina Speckmann and Csaba D.T´oth, editors, Proceedings of the 34thInternational Symposium on ComputationalGeometry (SoCG 2018) , volume 99 of
LIPIcs ,pages 64:1–64:14, 2018.doi:10.4230/LIPIcs.SoCG.2018.64.[22] Joseph O’Rourke. Unfolding polyhedra.Electronic preprint arxiv:1908.07152, 2019.2 nd Canadian Conference on Computational Geometry, 2020 [23] G. C. Shephard. Convex polytopes with convexnets.
Mathematical Proceedings of the CambridgePhilosophical Society , 78(3):389–403, 1975.doi:10.1017/s0305004100051860.[24] A. S. Tarasov. Polyhedra that do not admitnatural unfoldings.
Uspekhi Matematicheskikh Nauk , 54(3):185–186, 1999.doi:10.1070/rm1999v054n03ABEH000171.[25] Magnus J. Wenninger.