Zipper Unfoldings of Polyhedral Complexes
Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit
Thursday, August 12, 2010 1 Unfolding Polyhedra—Durer 1400’s
Durer, 1498 snub cube
Thursday, August 12, 2010 2 Unfolding Polyhedra—Octahedron all unfoldings
Thursday, August 12, 2010 3 Thursday, August 12, 2010 4 Zipper Unfoldings of Polyhedra—Octahedron
Thursday, August 12, 2010 5 Zippers
separating zipper
multiple toggles (cosmetic)
Thursday, August 12, 2010 6 Zippers
• 1891 patent by Whitcomb Judson • novel, but not practical (“If skirt is to be washed, remove fastener.”) • named “zipper” by B.F. Goodrich company in 1920’s • ubiquitous but super!uous
Thursday, August 12, 2010 7 Edge Cuts versus Face Cuts
edge cuts face cuts
Zipper edge cuts = Hamiltonian unfolding [Shepherd ’75]
Thursday, August 12, 2010 8 Zipper Edge Cuts (Hamiltonian Unfolding)
What is a zipper unfolding of a polyhedron?
on the polyhedron the cut is a simple path on the polygon
this is forbidden
Nick Chase
Thursday, August 12, 2010 9 Outline of Talk
Convex Polyhedra
Platonic Solids Archimedean Solids
Polyhedral Manifolds
Polyhedral Complexes
Thursday, August 12, 2010 10 Platonic Solids
Thursday, August 12, 2010 12 Platonic Solids
"ese are doubly Hamiltonian—the cut is a path and faces are joined in a path.
Thursday, August 12, 2010 13 great rhombicosi- Archimedean Solids dodecahedron
truncated truncated tetrahedron dodeca- hedron
truncated truncated icosa- cube hedron
great truncated rhombicub- octahedron octahedron
small cubocta- rhombicosi- hedron dodecahedron
small snub cube rhombicub- octahedron
icosidodeca- hedron snub dodeca- hedron Thursday, August 12, 2010 14 great rhombicosi- Archimedean Solids dodecahedron
truncated truncated tetrahedron dodeca- hedron
truncated truncated icosa- cube hedron
great truncated rhombicub- octahedron octahedron
small cubocta- rhombicosi- hedron dodecahedron
small snub cube rhombicub- octahedron
snub icosidodeca- dodeca- hedron hedron
Thursday, August 12, 2010 15 great rhombicosi- Archimedean Solids dodecahedron
Thursday, August 12, 2010 16 great rhombicosi- Archimedean Solids dodecahedron
truncated truncated tetrahedron dodeca- hedron
truncated truncated icosa- cube hedron
great truncated rhombicub- octahedron octahedron
small cubocta- rhombicosi- hedron dodecahedron
small snub cube rhombicub- octahedron
snub icosidodeca- dodeca- hedron hedron
Thursday, August 12, 2010 18 What next? ⎧ these have zipper unfoldings⎨ ⎩ ⎧ ??⎨
⎩ Peda Software Polyhedron Poster
Thursday, August 12, 2010 19 Not all convex polyhedra have Hamiltonian unfoldings
rhombic dodecahedron net
its graph has no Hamiltonian path
OPEN: #nd a convex polyhedron with no Hamiltonian unfolding but whose graph has a Hamiltonian path.
Thursday, August 12, 2010 20 Unfolding Convex Polyhedra
unfolding zipper unfolding
YES OPEN face Every convex polyhedron Does every convex polyhedron cuts has an unfolding— have a zipper unfolding? star, source unfolding.
OPEN NO edge Does every convex polyhedron Not every convex polyhedron cuts have an edge unfolding? has a Hamiltonian unfolding.
Thursday, August 12, 2010 21 Polyhedral Manifolds
polyhedral manifold—a #nite union of planar polygons in 3D s.t. every point has a neighbourhood homeomorphic to a disk
may be non-convex
may have genus ≠ 0
Magnus Wenninger
Figure 2: An nonorthogonal polyhedron composed of rectangular faces.
(and symmetry) of a regular octahedron, with each octahedron vertex replaced by a cluster of five vertices, and each octahedron edge replaced by a triangular prism. Let us fix the squares at each octahedron vertex to be unit squares. The Thursday, August 12, 2010length L of the prisms is not significant; in the figure, L = 3, and any length 22 large enough to keep the interior open would suffice as well. The other side lengths of the prism are, however, crucially important. The open triangle hole at the end of each prism has side lengths 1 (to mesh with the unit square), √3/2 and √3/2; see Fig. 3. This places the fifth vertex in the cluster displaced by 1/2 perpendicularly from the center of each square, forming a pyramid, as shown in Fig. 4.
3/2 L
1 1/2 3/2
1/2 2/2 1 1
Figure 3: Right triangular prism, Figure 4: The cluster of five vertices re- used twelve times in Fig. 2. placing each octahedron vertex form a pyramid.
The central isosceles right triangle (shaded in Fig. 4) guarantees that two oppositely oriented incident triangular prisms meet at right angles, just as do the corresponding edges of an octahedron. The result is a closed polyhedron of V =6 5 = 30 vertices, E = 84 edges, and F = 42 faces. The 42 faces include 6 unit· squares, 12 L 1 rectangles, and 24 rectangles of size L √3/2. × ×
3 22.4. Unfoldable Polyhedra 317 using the labeling shown in Figure 22.18, to reach the conclusion that there is a path of T in the hat connecting two hat corners A,B,C . { } The hat peak x must connect via T to outside the hat, and so there must be a path from x through a hat corner, say B. If this path passes through all three spike base vertices a, b, c , then any additional edge of T (and there must be at least one to avoid Figure 22.16) { } will connect two corners. So suppose that the path from x to B passes through just one (b: Figure 22.18(a)) or two (a and b: Figure 22.18(b)) base vertices. In the former case, a and c need to be spanned by T , but neither can be a leaf (because both have negative curvature). So there must be a path from A to C as shown. In the latter case, c needs to be spanned by T , but again it cannot be a leaf. The path through c could go either to C or to A (or to both), and in either case, we have a path between two corners. Imaging maths ! Unfolding polyhedra Now the hat corners are the four corners of the tetrahedron. So we have these four corners connected by four corner-to-corner paths inside the hats. Viewing each of these four paths version (643K) as an arc connecting four corner nodes, we can see that it is impossible to avoid a cycle, for a tree on n nodes can have only n 1 arcs. This violates Lemma 22.1.2, andUnfoldings establishes exist for many more polyhedral surfaces. For example, the torus shown below has a planar − Polyhedral Manifolds—Unfoldingunfolding to a single component such that no two faces overlap. (For experts, this polyhedral torus is the that the spiked tetrahedron has no edge unfolding to a net. Clifford torus obtained from a non!standard parameterization using Hopf fibers in the 3!sphere.) One might wonder if the spiked tetrahedron can be unfolded without restricting the cuts to edges. Figure 22.19 illustrates a strategy (for a differently parametrized version of the
polyhedron) which throws the four spikes safely out to the boundary of the tetrahedron some higher genus polyhedral6
unfolding. 6. Fig.
a oyernin polyhedron nal manifolds with edge unfoldings nonorthogo the to folds that polygon orthogonal An 7: Figure In fact, no example is known that cannot be unfolded without overlap: a “nice” non-convex The Clifford torus is generated by a family of circles. The torus unfolds nicely polyhedron with no edge despite the negative curvature of the inner region ! view the animated version unfolding (805K) It is a challenging task in combinatorial geometry to find polyhedral models of a given topological shape which use the minimal number of faces or vertices. The Clifford torus shown above uses about 200 vertices, which is surely more than necessary. A torus with the least number of vertices, 7, was found by Császár [6]. F.H. Lutz [7] produced the model of the unfolding below.
Donoso & O’Rourke 2. Fig. to cubes Adding 6: Figure Figure 6: Adding cubes to Fig. 2.
Figure 22.17. Two views of an ununfoldable triangulatedBern, Demaine, polyhedron. Eppstein, The Császár torus is an embedded polyhedral torus (i.e. one Kuo, Mantler, Snoeyink, 2003. Császárwithout self torus!intersection) (net bywith Lutz, the least picture number ofby vertices Polthier) ! view the animated version (364K)
Now we are ready to understand an even more complicated unfolding. The Boy surface mentioned in the OPEN: Does every polyhedralintroduction is a modelmanifold of the projective have plane a . [general]One model of the unfolding? projective plane can be obtained by taking x x the upper hemisphere of a ball and identifying opposite points of the equator. Alternatively, you can take a
planar disk and pairwiseFigure identify 7: An orthogonal opposite polygon points that folds on to thethe nonorthogo boundarynal polyhedron circle. in Since these are topological constructions you canFig. take 6. any piece of the plane which is bounded by a single closed curve ! such as the C C unfolding of the Boy surface. Thursday, August 12, 2010 23 c b c b 6 a a A A (a)B (b) B
Figure 22.18. Possible restrictions of a cut tree to one hat. The discrete Boy surface unfolds to a simply connected disk. Opposite vertices of the boundary are identified during refolding ! see the animated version (565K)
Geometric Origami 4 Polyhedral Manifolds—Zipper Unfoldings
use separating zipper
Thursday, August 12, 2010 24 Polyhedral Manifolds—Zipper Unfoldings
"eorem. If P is a polyhedral manifold that has a zipper unfolding to a planar polygonal region F, then either P is a polyhedron and F is a polygon, or —in the case of a separating zipper—P is a torus polyhedron and F is an annulus (with outer perimeter = inner perimeter).
not possible
Nick Chase
Thursday, August 12, 2010 25 Polyhedral Manifolds—Zipper Unfoldings
"ese have no zipper edge unfoldings.
Lemma. A zipper edge unfolding of a torus polyhedron is an annulus with faces forming a cycle with trees attached.
Thursday, August 12, 2010 26 Polyhedral Complexes
polyhedral complex—a #nite union of planar polygons in 3D s.t. intersections are edge-to-edge joins
only one face at an edge
3 faces meeting at an edge
Weaire–Phelan structure
Thursday, August 12, 2010 27 906 W. Liu, K. Tai / Computer-Aided Design 39 (2007) 898–913 318 Chapter 22. Edge Unfolding of Polyhedra
906 W. Liu, K. Tai / Computer-Aided Design 39 (2007) 898–913
906 W. Liu, K. Tai / Computer-Aided Design 39 (2007) 898–913
(a) Hyper-common edge set with FAG. (b) Invalid spanning tree. (a) Hyper-common edge set with FAG. (b) Invalid spanning tree.
(a) Hyper-common edge set with FAG. (b) Invalid spanning tree.
Polyhedral Complexes—Unfolding (c) Valid spanning tree. (d) Detecting topological validity. Figure 22.19. A general unfolding of a spiked tetrahedron. Fig. 12. Topological validity of spanning tree.
be made up of one single face. There are in fact three such v pairs in this example (faces 1 and 2, faces 3 and 4, and faces (c) Valid spanning tree. (d) Detecting5 and 6). For reasons of structural strength/integrity or ease cannot be topologicalof folding, validity. it may sometimes be desirable to have any such unfolded Figure 22.20. An open surface that haspair no generalof faces formed unfolding by one to single a net. face instead (as indicated (c) Valid spanning tree.Fig. 12. Topological validity of spanning (d) tree. Detectingin Fig. 13(b) for face pair 3 and 4, and face pair 5 and 6). topologicalHowever, validity. it is not always desirable to have any such pairs be made up of one singleof faces face. defined There as are a single in fact face three before such applying the basic Bern, Demaine, Eppstein, pairs in this example (facesunfolding 1 and procedure 2, faces because 3 and this 4, may and reduce faces the number of Fig. 12. TopologicalKuo, Mantler, validity Snoeyink, of spanning 2003. tree.5 and 6). For reasonsresults of structural generated, strength/integrity thereby missing out or on ease some potential flat of folding, it may sometimespattern designs be desirable where the two to have faces are any not such situated adjacent to each other but are in fact separated (as in Fig. 13(c) where face pair of faces formed bypair one 3 and single 4 and face face pair instead 5 and 6(as are indicated separated). be made upin ofFig. one 13(b) single for face pair face.Therefore, 3 and There the 4, and strategy face are adopted pair in 5 in and this fact work 6). threeis to always such Open Problem 22.14: General NonoverlappingHowever, Unfolding it is not of always Polyhedra.define desirable each anda every toDoes have pair every anyof such such faces pairs as two individual pairs in this example (facesfaces 1 in and the 3D structure, 2, faces but with 3 an and optional 4, switch and for the faces closed polyhedron have a general unfolding toof a faces nonoverlapping defined as a single polygon? face before applying the basic unfolding procedure becausedesigner this to choose may whether reduce to the merge number any such of pairs of faces 5 and 6). For reasons ofin structural the results. If the strength/integrity choice is to merge, then for every pair or of ease results generated, therebysuch missingfaces that happen out on to some be adjacent potential within flat the flat pattern, the a Bern et al. (2003). of folding, itpattern may designs sometimes where thetwo two faces faces be will automatically are desirable not situated be merged adjacent to ashave one to face. any For any such each other but are in factpair separated of such faces (as which in Fig. are 13 not(c) adjacent where within face the flat pattern, pair of facespair formed 3 and 4 and by face one pairthey 5 will and single of 6 course are separated). have face to be instead kept as separate. (as In this indicated way, no in Fig. 13(b) forTherefore, face the pair strategypotential 3 adopted and flat pattern in 4, this results and work will face be is left to out.always pair 5 and 6). define each and every pair of such faces as two individual Liu & Tai,However, in Computer-Aided it is easy to Design constructFig. 13., Merging2007 of open some pairs polyhedral of coplanar adjacent surfaces faces. that5. have Optimality no net. criteria Figure for flat 22.20patterns However, itfaces is not in the always 3D structure, desirable but with an optional to have switch for any the such pairs is a simple example.than Its one one hyper-common interior vertex edge, thenv has bothdesigner conditionsnegative to choosemust curvature be whetherDepending and to merge so on cannot the any number such be of pairs a faces leaf as of well faces as the connectivity of a cut tree. So thesatisfied leaves for of eachof the and faces cut every tree such defined must edgein before lie the aon results. spanning as the a If boundary. tree the single choiceamong is Because face them, to merge, a 3D before a foldedthen tree for structure has every applying at can pair have of thousands the or basic is certified as valid. Thursday, August 12, 2010 such faces that28 happen tomillions be adjacent of valid within non-overlapping the flat pattern, flat patterns. the However, the least two leaves, the surface mustunfolding be disconnected procedure by the because cuts. designer this would may usually be reduce trying to select the just number one that is of 4.4. Merging of coplanar adjacent facestwo faces will automaticallyoptimal according be merged to some as one criteria. face. Very For often, any as the flat pattern results generated,pair of such thereby faces whichis missing are a single-piece not adjacent layout, out within there on theis a some desire flat pattern, for it potential to be as compact flat Fig. 13(a) shows a 3D folded structurethey will and ofFig. course 13(b) haveas to possible be kept because as separate. compactness In this makes way, for no easier handling shows one possiblepattern way of unfolding designs itpotential to form where its flat flat pattern. pattern the results twoand folding will faces be and left also are out. has not advantages situated in terms of adjacent storage and to 22.5 Special ClassesThis example of Edge-Unfoldable illustrates an instance where Polyhedra a pair of coplanar transportation. However, there is no single universal measure adjacent faceseach within the other 3D folded but structure are can alternatively in fact separatedof compactness, so (as three in differentFig. measures 13(c) of compactness where are face Fig. 13. Merging of some pairs of coplanar adjacent faces. 5. Optimality criteria for flat patterns One might think that, althoughpair no 3 one and has 4 been and able face to pair prove 5 that and every 6 are convex separated). polyhe- than one hyper-common edge, then both conditions must be Depending on the number of faces as well as the connectivity satisfieddron for each is edge-unfoldable and every such edge to before a net, a spanning surelyTherefore, many tree specialamong the them, infinite strategy a 3D classes folded adopted of structure polyhedra incan have this thousands been work or is to always is certifiedestablished as valid. as edge-unfoldable.define Such is each not themillions and case. every of Results valid non-overlapping pair here of are such sparse. flat patterns. faces An under- However, as two the individual graduate thesis (DiBiase 1990) established thatdesigner all polyhedra would usually of 4, be 5, trying or 6 verticesto select just may one be that is 4.4. Merging of coplanar adjacent faces edge-unfolded to a net, but, asfaces mentioned in the earlier, 3Doptimal the structure, according proof does to some not but criteria. seem with to Very give an often, insight optional as the flat for pattern switch for the is a single-piece layout, there is a desire for it to be as compact Fig. 13larger(a) showsn. For a 3Dsome folded classes structure of polyhedra,designer and Fig. 13 a(b) nonoverlapping to chooseas possible whether because unfolding compactness is to obvious merge makes and forany therefore easier such handling pairs of faces shows oneuninteresting. possible way of For unfolding example, it to form allin itsprisms the flat pattern. results.—formedand If by folding the two andcon- choice also gruent, has is advantages parallelto merge, in convex terms then of poly- storage for and every pair of This examplegons illustrates (the base anand instancetop where) and a rectanglepair of coplanar side facestransportation. (see Figure However, 22.21)—can there is no be single unfolded universal by measure adjacent faces within the 3D folded structuresuch can alternatively faces thatof compactness, happen soto three be different adjacent measures within of compactness the flat are pattern, the two faces will automatically be merged as one face. For any pair of such faces which are not adjacent within the flat pattern, they will of course have to be kept as separate. In this way, no potential flat pattern results will be left out.
Fig. 13. Merging of some pairs of coplanar adjacent faces. 5. Optimality criteria for flat patterns than one hyper-common edge, then both conditions must be Depending on the number of faces as well as the connectivity satisfied for each and every such edge before a spanning tree among them, a 3D folded structure can have thousands or is certified as valid. millions of valid non-overlapping flat patterns. However, the designer would usually be trying to select just one that is 4.4. Merging of coplanar adjacent faces optimal according to some criteria. Very often, as the flat pattern is a single-piece layout, there is a desire for it to be as compact Fig. 13(a) shows a 3D folded structure and Fig. 13(b) as possible because compactness makes for easier handling shows one possible way of unfolding it to form its flat pattern. and folding and also has advantages in terms of storage and This example illustrates an instance where a pair of coplanar transportation. However, there is no single universal measure adjacent faces within the 3D folded structure can alternatively of compactness, so three different measures of compactness are 906 W. Liu, K. Tai / Computer-Aided Design 39 (2007) 898–913
Polyhedral Complexes—Unfolding
906 What does unfolding mean(a) Hyper-commonwhen multiple edgeW. set faces with Liu, FAG. meet K. Tai at / Computer-Aided an edge? Design (b) Invalid 39 (2007) spanning 898–913 tree.
valid
906 W. Liu, K. Tai / Computer-Aided Design 39 (2007) 898–913 (c) Valid spanning tree. (d) Detecting topological validity. (a) Hyper-common edge set with FAG. (b) Invalid spanning tree. invalid Fig. 12. Topological validity6 of spanning tree. 1 be made up of one single face. There are in fact three such 3 2 pairs in this example (faces 1 and 2, faces 3 and 4, and faces 5 4 5 and 6). For reasons of structural strength/integrity or ease of folding, it may sometimes be desirable to have any such (a) Hyper-common edge set with FAG. (b) Invalid spanning tree. pair of faces formed by one single face instead (as indicated Liu & Tai, in Computer-Aided Design, 2007 in Fig. 13(b) for face pair 3 and 4, and face pair 5 and 6). However, it is not always desirable to have any such pairs
Thursday, August 12, 2010 of faces29 defined as a single face before applying the basic unfolding procedure because this may reduce the number of results generated, thereby missing out on some potential flat pattern designs where the two faces are not situated adjacent to (c) Valid spanning tree.each other but are in (d) fact Detecting separated (as in Fig. 13(c) where face pair 3 and 4 and facetopological pair 5 and 6 validity. are separated). Therefore, the strategy adopted in this work is to always (c) Valid spanning tree. (d) Detecting topological validity. define each and every pair of such faces as two individual Fig. 12. Topological validity offaces spanning in the tree. 3D structure, but with an optional switch for the Fig. 12. Topological validity of spanning tree. designer to choose whether to merge any such pairs of faces be madein the results.up of If one the choice single is to face. merge, There then for are every in pair fact of three such be made up of one single face. There are in fact three suchpairssuch in this faces thatexample happen to (faces be adjacent 1 and within 2, the faces flat pattern, 3 and the 4, and faces pairs in this example (faces 1 and 2, faces 3 and 4, and faces two faces will automatically be merged as one face. For any 5 and 6). For reasons of structural strength/integrity or ease5 andpair 6). of such For faces reasons which are of not structural adjacent within strength/integrity the flat pattern, or ease of folding, it may sometimes be desirable to have any suchof folding,they will of it course may have sometimes to be kept as be separate. desirable In this to way, have no any such pair of faces formed by one single face instead (as indicated potential flat pattern results will be left out. in Fig. 13(b) for face pair 3 and 4, and face pair 5 and 6).pair of faces formed by one single face instead (as indicated However, it is not always desirable to have any such pairs Fig. 13. Merging of some pairs of coplanar adjacent faces. in Fig.5. Optimality 13(b) for criteria face for pair flat patterns 3 and 4, and face pair 5 and 6). of faces defined as a single face before applying the basicHowever, it is not always desirable to have any such pairs thanunfolding one hyper-common procedure because edge, this then may both reduce conditions the number must be of Depending on the number of faces as well as the connectivity satisfiedresults generated, for each and thereby every missing such edge out before on some a spanning potential tree flatof facesamong defined them, a 3D as folded a single structure face can before have thousands applying or the basic ispattern certified designs as valid. where the two faces are not situated adjacentunfolding to millions procedure of valid non-overlapping because this flat patterns. may reduce However, the the number of each other but are in fact separated (as in Fig. 13(c) where face resultsdesigner generated, would usually thereby be trying missing to select out just on one some that ispotential flat 4.4.pair Merging 3 and 4 and of coplanar face pair adjacent 5 and 6 are faces separated). optimal according to some criteria. Very often, as the flat pattern Therefore, the strategy adopted in this work is to alwayspatternis a designs single-piece where layout, the there two is a desire faces for are it to not be as situated compact adjacent to defineFig. each 13(a) and shows every a 3Dpair folded of such structure faces as and twoFig. individual 13(b)eachas other possible but because are in compactness fact separated makes (as for in easierFig. handling 13(c) where face faces in the 3D structure, but with an optional switch for the shows one possible way of unfolding it to form its flat pattern. and folding and also has advantages in terms of storage and designer to choose whether to merge any such pairs of facespair 3 and 4 and face pair 5 and 6 are separated). This example illustrates an instance where a pair of coplanar transportation. However, there is no single universal measure in the results. If the choice is to merge, then for every pair of adjacent faces within the 3D folded structure can alternatively Therefore,of compactness, the so strategy three different adopted measures in of compactness this work are is to always such faces that happen to be adjacent within the flat pattern, the two faces will automatically be merged as one face. For anydefine each and every pair of such faces as two individual pair of such faces which are not adjacent within the flat pattern,faces in the 3D structure, but with an optional switch for the they will of course have to be kept as separate. In this way, nodesigner to choose whether to merge any such pairs of faces potential flat pattern results will be left out. in the results. If the choice is to merge, then for every pair of Fig. 13. Merging of some pairs of coplanar adjacent faces. 5. Optimality criteria for flat patterns such faces that happen to be adjacent within the flat pattern, the than one hyper-common edge, then both conditions must be Depending on the number of faces as well as the connectivitytwo faces will automatically be merged as one face. For any satisfied for each and every such edge before a spanning tree among them, a 3D folded structure can have thousandspair or of such faces which are not adjacent within the flat pattern, is certified as valid. millions of valid non-overlapping flat patterns. However, thethey will of course have to be kept as separate. In this way, no designer would usually be trying to select just one that is 4.4. Merging of coplanar adjacent faces optimal according to some criteria. Very often, as the flat patternpotential flat pattern results will be left out. is a single-piece layout, there is a desire for it to be as compact Fig. 13(a) shows a 3D folded structureFig. and 13.Fig. Merging 13(b) ofas some possible pairs because of coplanar compactness adjacent makes faces. for easier handling5. Optimality criteria for flat patterns shows one possible way of unfolding it to form its flat pattern. and folding and also has advantages in terms of storage and This example illustrates an instance where a pair of coplanar transportation. However, there is no single universal measure adjacent faces within the 3D foldedthan structure one can hyper-common alternatively of compactness, edge, then so three both different conditions measures must of compactness be are Depending on the number of faces as well as the connectivity satisfied for each and every such edge before a spanning tree among them, a 3D folded structure can have thousands or is certified as valid. millions of valid non-overlapping flat patterns. However, the designer would usually be trying to select just one that is 4.4. Merging of coplanar adjacent faces optimal according to some criteria. Very often, as the flat pattern is a single-piece layout, there is a desire for it to be as compact Fig. 13(a) shows a 3D folded structure and Fig. 13(b) as possible because compactness makes for easier handling shows one possible way of unfolding it to form its flat pattern. and folding and also has advantages in terms of storage and This example illustrates an instance where a pair of coplanar transportation. However, there is no single universal measure adjacent faces within the 3D folded structure can alternatively of compactness, so three different measures of compactness are Polyhedral Complexes—Zipper Unfolding
not allowed
4 1
2 3
on the polygon
Thursday, August 12, 2010 30 Polyhedral Complexes—Zipper Unfolding
Thursday, August 12, 2010 31 Polyhedral Complexes—Zipper Unfolding
Polyhedra Sharing Faces
2 tetrahedra
what will this zip into?
Thursday, August 12, 2010 32 ¶ ), or by allowing ¶ polyhedral complexes Jonah L. Shallit icosahedron dodecahedron ), or by allowing § polyhedral complexes Jonah L. Shallit icosahedron polyhedral manifolds dodecahedron Arlo Shallit § CCCG 2010, Winnipeg MB, August 9–11, 2010 octahedron tetrahedron ‡ polyhedral manifolds The physical model of a zipper provides a natural Section 2 contains our results on Hamiltonian unfold- Arlo Shallit Figure 2: DoublyPlatonic Hamiltonian solids. unfoldings of the other extension of zipperwhich unfolding generalize polyhedra, to eithergenus by (these allowing are positive an edge to being incident to a more boundary than twoWe edge faces, elaborate that or on is allow- thetion incident model 1.1. to of Figure only zipper 3 onetetrahedra shows unfolding joined face. a in at zipper a Sec- edge face. unfolding of two Figure 3: Zipper edgeing unfolding of a two tetrahedra face.shaded shar- paths lie The on zipper the back starts of the at surface. the circle. Lightly ings of polyhedra:solids we have show Hamiltonian thatprove unfoldings. all that the polyhedral Infoldings Archimedean manifolds have Section genus that 0 3 have orspecialized. we zipper 1, In and un- Section that 4per the we latter edge give are unfoldings some quite of examples polyhedral of complexes. zip- CCCG 2010, Winnipeg MB, August 9–11, 2010 octahedron Anna Lubiw tetrahedron ‡ edge The physical model of a zipper provides a natural Section 2 contains our results on Hamiltonian unfold- † zipper Figure 2: DoublyPlatonic Hamiltonian solids. unfoldings of the other extension of zipperwhich unfolding generalize polyhedra, to eithergenus by (these allowing are positive an edge to being incident to a more boundary than twoWe edge faces, elaborate that or on is allow- thetion incident model 1.1. to of Figure only zipper 3 onetetrahedra shows unfolding joined face. a in at zipper a Sec- edge face. unfolding of two Figure 3: Zipper edgeing unfolding of a two tetrahedra face.shaded shar- paths lie The on zipper the back starts of the at surface. the circle. Lightly ings of polyhedra:solids we have show Hamiltonian thatprove unfoldings. all that the polyhedral Infoldings Archimedean manifolds have Section genus that 0 3 have orspecialized. we zipper 1, In and un- Section that 4per the we latter edge give are unfoldings some quite of examples polyhedral of complexes. zip- Anna Lubiw edge [email protected] [email protected] † zipper Hamitonian unfoldings Martin L. Demaine
. A zipper unfolding of a Polyhedral Complexes—Zipper Unfolding Zipper Unfoldings of Polyhedral Complexes as introduced by Shephard in ∗ [email protected] [email protected] [email protected] Polyhedra Sharing Faces Hamitonian unfoldings it to a planar shape. When cuts are Martin L. Demaine . A zipper unfolding of a
Zipper Unfoldings of Polyhedral Complexes 2 cubes—seems impossible as introduced by Shephard in zipper unfolding ∗ unfold [email protected] . It is an open problem whether every convex . When cuts are restricted to the polyhedron Erik D. Demaine of the polyhedron. Nets of Platonic solids and it to a planar shape. When cuts are Massachusetts Institute of Technology, Massachusetts Institute of Technology, University of Waterloo, [email protected] [email protected] Figure 1: All Hamiltonian unfoldings of the cube. † ‡ § ∗ ¶ We give examples of polyhedral complexes that are, In this paper we replace glue by a zipper, i.e., we limit net 2 squashed cubes Abstract We explore which polyhedracan and be polyhedral formed complexes byand folding fastening up it aprocess with planar polygonal one a region zipper.polyhedron is We a callto path the a cut reverse planar that polygon;Hamiltonian unfolds in unfoldings the the case polyhedron 1975. of We edge show cuts, that all thesehave Platonic are and Hamiltonian Archimedean unfoldings. solids and are not, zipperamples [edge] include unfoldable. a polyhedral Thejoined positive at torus, an ex- and edge two or tetrahedra at a face. 1 Introduction A common waydron to is to build fold and aa glue paper a planar model polygonal shape, of called a polyhe- unfoldings edges, the cut path must formgraph a of Hamiltonian path the in polyhedron. the were studied Such byHamiltonian Shephard unfoldings [4]. of the Platonic Figures solids. 1 and 2 show other polyhedra with regularby in faces D¨urer 1525, were and first haveever been published since—see a the source book ofThe by fascination reverse Demaine process and involves slicing O’Rourke the [2]. hedron surface of to a poly- limited to theunfoldings edges of thepolyhedron polyhedron has an these edge are unfolding [4, 2]. the slicing ofpath the that polyhedron’s does surface not to self-intersect. a We single call cut these zipper unfolding unfold . It is an open problem whether every convex . When cuts are restricted to the polyhedron Erik D. Demaine of the polyhedron. Nets of Platonic solids and Massachusetts Institute of Technology, Massachusetts Institute of Technology, University of Waterloo, [email protected] [email protected] Figure 1: All Hamiltonian unfoldings of the cube. † ‡ § ∗ ¶ We give examples of polyhedral complexes that are, In this paper we replace glue by a zipper, i.e., we limit net Abstract We explore which polyhedracan and be polyhedral formed complexes byand folding fastening up it aprocess with planar polygonal one a region zipper.polyhedron is We a callto path the a cut reverse planar that polygon;Hamiltonian unfolds in unfoldings the the case polyhedron 1975. of We edge show that cuts, all thesehave Platonic are and Hamiltonian Archimedean unfoldings. solids and are not, zipperamples [edge] include unfoldable. a polyhedral Thejoined positive at torus, an ex- and edge two or tetrahedra at a face. 1 Introduction A common waydron to is to build fold and aa glue paper a planar model polygonal shape, of called a polyhe- unfoldings edges, the cut path must formgraph a of Hamiltonian path the in polyhedron. the were studied Such byHamiltonian Shephard unfoldings [4]. of the Platonic Figures solids. 1 and 2 show other polyhedra with regularby in faces D¨urer 1525, were and first haveever been published since—see a the source book ofThe by fascination reverse Demaine process and involves slicing O’Rourke the [2]. hedron surface of to a poly- limited to theunfoldings edges of thepolyhedron polyhedron has an these edge are unfolding [4, 2]. the slicing ofpath the that polyhedron’s does surface not to self-intersect. a We single call cut these
Thursday, August 12, 2010 33 Polyhedral Complexes—Zipper Unfolding
Polyhedra Sharing Edges chain of tetrahedra sharing adjacent edges
can extend to n tetrahedra
Thursday, August 12, 2010 34 Polyhedral Complexes—Zipper Unfolding
Thursday, August 12, 2010 35 Open Problems
• Does every convex polyhedron have an edge unfolding?
• Does every polyhedron have a [general] unfolding? Every polyhedral manifold? New:
• Does every convex polyhedron have a zipper unfolding?
• Show it’s NP-hard to recognize torus polyhedra with zipper edge unfoldings.
• Which polyhedral complexes have [zipper/edge] unfoldings?
Thursday, August 12, 2010 36 Zipper Unfoldings of Polyhedral Complexes
Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit
Thursday, August 12, 2010 37