Universal Hinge Patterns for Folding Orthogonal Shapes

Transcription

1 Unversal Hnge Patterns for Foldng Orthogonal Shapes Nada M. Benbernou Erk D. Demane Martn L. Demane Avv Ovadya 1 Introducton An early result n computatonal orgam s that every polyhedral surface can be folded from a large enough square of paper [DDM00]. A recent algorthm for ths problem even attans practcal foldngs [DT10]. But each polyhedral surface nduces a completely dfferent crease pattern. Is there a sngle hnge pattern for whch dfferent subsets fold nto many dfferent shapes? Our motvaton s developng programmable matter out of a foldable sheet [HAB + 10]. The dea s to statcally manufacture a sheet wth specfc hnges that can be creased n ether drecton, and then dynamcally program how much to fold each crease n the sheet. Thus a sngle manufactured sheet can be programmed to fold nto anythng that the sngle hnge pattern can fold. We prove a unversalty result: an N N square tlng of a smple hnge pattern can fold nto all face-to-face glungs of O(N) unt cubes (polycubes). Thus, by settng the resoluton N suffcently large, we can fold any 3D sold up to a desred accuracy. The proof s algorthmc: we present the Cube Extruson Algorthm whch converts a gven polycube nto a crease pattern (a subset of the unversal hnge pattern) and a 3D folded state n the shape of that polycube, wth seamless faces. Fgure 1 shows a smple example. At the core of our algorthm s the noton of a cube gadget, whch folds a cube n the mddle of a sheet of paper. Such foldngs of a sngle cube MIT Computer Scence and Artfcal Intellgence Laboratory, 32 Vassar St., Cambrdge, MA 02139, USA, {nbenbern,edemane,mdemane,avvo}@mt.edu Partally supported by NSF CAREER award CCF , DOE grant DE-FG02-04ER25647, and AFOSR grant FA Correspondng Author 1

2 Fgure 1: Foldng a bend-shaped polycube wth a square base va the Cube Extruson Algorthm. For smplcty, the mountan-valley pattern n ths and other fgures does not exactly show the reflected creases when multple layers are folded. have been ndependently developed by orgamsts over the years; the frst documented desgn we are aware of was created by Davd A. Huffman n The novelty s the way we combne multple cube gadgets to form a desred polycube. We present three dfferent cube gadgets, one of whch s the gadget ndependently created by Huffman, each wth ts own advantages and dsadvantages when combned to fold a polycube. We also descrbe an mplementaton of the algorthm whch can be used to automate expermentaton and desgn of geometrc orgam usng a cuttng plotter or laser cutter to score the paper. 2 Defntons We start wth a few defntons about orgam, specfed somewhat nformally for brevty. For more formal defntons, see [DO07, ch. 11]. For our purposes, a pece of paper s a connected collecton of flat polygons n 3D joned along shared edges (a polyhedral complex; note that we can have multple polygons n the same place but wth dfferent connectons, and wth a specfed stackng order). A notable specal case s a sngle m n rectangle of paper for ntegers m and n (but n general, a pece of paper does not have to be flat; for example, a polyhedron s a pece of paper). We ndex the unt squares of such a rectangle n the style 1 Personal communcaton wth the Huffman famly. The fourth author ndependently developed ths gadget n mddle school crca

3 of matrces: s,j refers to the unt square n the th row and jth column, and s 1,1 s n the upper-left corner. A hnge s a lne segment drawn on a pece of paper whch s capable of beng creased n ether drecton. A hnge pattern s a collecton of hnges drawn on a pece of paper. The hnge patterns we consder n ths paper are all based on subdvsons of the unt-square grd, addng a fnte number of hnges wthn each unt square. The unt squares of the hnge pattern correspond to the unt squares of a rectangle of paper. An example of a hnge pattern s the box-pleated pattern (known n geometry as the tetraks tlng) whch s formed from the unt-square grd by subdvdng each square n half vertcally, horzontally, and by the two dagonals, formng eght rght sosceles trangles. The upper-left corner of Fgure 3 shows an example for four unt-squares. An angle pattern s a hnge pattern together wth an assgnment of a real number n [ 180, +180 ] to each hnge, specfyng a fold angle (negatve for valley, postve for mountan). We allow a hnge to be assgned an angle of 0, n whch case we call the hnge trval, though we do not draw trval hnges n most fgures. A hnge wth a nonzero angle s called a crease. The crease pattern s the subgraph of the hnge pattern consstng of only the creases. An angle pattern determnes a 3D geometry called the folded geometry, whch maps each face of the crease pattern to a 3D polygon va a Eucldean sometry (by the composton of rotatons at creases). More explctly, a folded geometry s a map from all ponts of the pece of paper to R 3 that satsfes constrants as specfed n [DO07, ch. 11] and there s an obvous mappng from angle patterns to folded geometres. A folded state conssts of such a folded geometry together wth an orderng λ, whch s a partal functon over the touchng ponts n the folded geometry, n our case descrbng the stackng relatonshp among polygons of the crease pattern that touch n the folded geometry. Defne the startng sheet of a folded state to be the orgnal pece of paper, that s, the doman of the folded geometry. 2 A foldng sequence s a sequence of folded states F 1, F 2,..., F k from the same startng sheet. The last folded state n a foldng sequence s called the fnal folded state. Defne the number of layers at a pont q to be the number of noncrease ponts n the pece of paper that get mapped to q by the folded geometry. The number of layers of a folded state s the maxmum number of layers over all ponts. 3 Next we defne a noton of coalescng whch lets us gnore certan 2 Note that sheet s not to suggest that the pece of paper needs to be flat; t can be any polyhedral complex. 3 Ths measure s a smple way to bound the effect of paper thckness, but n practcal orgam there are other quanttes that could be measured. 3

4 detals of a folded state. A coalesce folded state s a folded state augmented wth a coalesce set whch s a subset of the startng sheet. If we take the startng sheet and dentfy (glue together) all pars of ponts n the coalesce set that are collocated by the folded geometry, then we obtan a metrc space called the coalesce result. Ths coalesce result s also a pece of paper under our defnton and therefore can be the doman of a new coalesce folded state. A coalesce sequence s a sequence C 1, C 2,..., C k of coalesce folded states, where each C k s a foldng of the coalesce result of C k 1. One can generate a foldng sequence from a coalesce sequence by lettng F 1 = C 1 and F k = F k 1 C k, and then composng the geometry and orderng functons n the obvous way. Note that the startng sheet of each F k s the startng sheet of C 1, whle the startng sheets of the other C k s can be any shape folded from that startng sheet. The fnal folded state of a coalesce sequence s the fnal folded state of the generated foldng sequence. We say that a foldng sequence or coalesce sequence folds a pece of paper π nto a shape σ f the startng sheet of the frst folded state s the pece of paper π and the mage of the last folded geometry s the shape σ. In ths paper we allow all but the last folded state n a foldng sequence or coalesce sequence to have crossngs. By [DDMO04], all folded states are reachable from the startng sheet by the contnuous foldng moton, so the fnal folded state s stll reachable. We use foldng sequences as a tool to construct the fnal folded state, not as nstructons for foldng. Now that we can descrbe how to fold a shape, we defne our target shapes. A polycube P s a unon of unt cubes on the unt-cube lattce wth a connected dual graph; the dual graph has a vertex for each unt cube and an edge between two vertces whose correspondng cubes share a face. The faces of the polycube are the (square) faces of the ndvdual cubes that are not shared by any other cubes. A foldng of a polycube s a folded state that covers all faces of the polycube, and nothng outsde the polycube. In fact, some of our foldngs of polycubes wll also nclude the nternal squares, the faces shared by multple cubes, and some of our foldngs wll not put anythng else nteror to cubes, but n general we do not requre ether property. A face of a folded polycube s seamless f the outermost layer of paper coverng t s an uncreased unt square of paper. Our foldngs wll generally be seamless. 3 Cube Gadgets We now ntroduce the noton of a cube gadget; refer to Fgure 2. For postve ntegers r and c, an [r, c]-cube gadget s a method of extrudng a cube from a rectangular pece of paper at a specfed locaton. The nput to the cube gadget s an m n rectangle of paper, for ntegers m > 2r and 4

5 Before cube gadget applcaton s 1,1 After cube gadget applcaton 1 + 2r = 3 layers 1 + 2c = 5 layers c = 2 c = 2 s 1,1 s 1,1 n = 6 r = 1 s 4,5 n = n 2r = 4 s 3,3 s 3,3 r = 1 m = m 2c = 3 m = 7 Fgure 2: Abstract effect of applyng a [1, 2]-cube gadget at square s 4,5 of a 6 7 rectangle of paper. The two leftmost dagrams are top vews before and after foldng. The rght dagram s a stylzed perspectve vew of the folded state. n > 2c, as well as a unt square s,j on the paper, where r < < m r and c < j < n c. The output of the cube gadget s a foldng of the m n rectangle nto the shape of a cube sttng on a smaller, (m 2r) (n 2c) rectangle of paper. The cube sts on the square s r,j c n the smaller sheet of paper. All sx faces of the cube are seamless except for the bottom face. The top face of the cube s covered by square s,j from the orgnal pece of paper. The boundary of the orgnal m n rectangle paper s mapped onto the boundary of the smaller (m 2r) (n 2c) rectangle. The cube gadgets n ths paper acheve the foldng by makng horzontal pleats n the r rows above and below row, and makng vertcal pleats n the c columns left and rght of column j. The pleats are called half-square pleats because they are composed of unt squares folded n half. Each pleat adds two layers to the row or column t s under, so the number of layers of the folded state s at least max{r, c} (one for the row or column plus two for each pleat). Another property of our foldngs s that all foldng s wthn the 2r + 1 rows and 2c + 1 columns surroundng square s,j. Thus, the quadrant of paper consstng of rows < r and columns < j c s not folded and s ncdent to the top-left corner of the cube, and smlarly for the other four quadrants. In ths paper, we gve three dfferent cube gadgets based on three dfferent hnge patterns, as shown n Fgure 3. The three cube gadgets are based, respectvely, on the box-pleated pattern, the slt pattern, and the arctan 1 2 pattern. The arctan 1 2 gadget and slt gadget are [1, 1]-cube gadgets, whle the box-pleated gadget s a [1, 2]-cube gadget. The advantage of the box- 5

6 Box-Pleat Pattern Arctan(½) Pattern Slt Cube Pattern 1 unt square Box-Pleat Gadget Arctan(½) Gadget Slt Cube Gadget Mountan Valley Slt Hnge Unt Square Border (also Hnge) Fgure 3: The hnge patterns (top), mountan-valley patterns (mddle), and semtransparent folded states (bottom) for the three cube gadgets. The hghlghted regon of each mountan-valley pattern has dmensons 2r + 1 2c + 1 and s the regon used to actually fold the cube (half-square pleats extend out from those regons). pleated and slt gadgets s that the hnge pattern s smpler: box pleatng has all creases wth angles at nteger multples of 45. The slt gadget attans hgher effcency than seems possble wth regular box pleatng by addng a regular pattern of slts n the paper. The arctan 1 2 gadget attans hgher effcency usng more hnges some of whch are at angles of arctan 1 2. We use the arctan 1 2 gadget n all fgures for consstency. Next, we show how a cube gadget can be used to modfy an exstng foldng, whch wll be the key constructon n our foldng of general polycubes. Fgure 4 provdes some ntuton for how exstng cubes move as new cubes are folded and Fgure 5 provdes a formal example of the lemma below. Lemma 1 (Gadget Applcaton) Let C P be a coalesce sequence for a polycube P from an m n rectangle of paper. Let f be a face of the polycube 6

7 Before Foldng Durng Foldng After Foldng Fgure 4: Gven a pece of paper wth a cube already folded on t, ths dagram shows n an abstract manner how the paper moves when a new cube s folded dependng on whether the old cube s on the top face (above), or a sde face (below) of the new cube. P that s seamless n the fnal folded state of C P. Then there s a coalesce sequence C P for the polycube P, consstng of P plus a cube extruded from face f, from an (m + 2r) (n + 2c) rectangle of paper. The constructon s parameterzed by a cube gadget. Proof: Let C P = C1 P, C2 P,..., Cq P, and let Fq P be the fnal folded state (for P ). Let s,j be the square n the startng sheet of Fq P that s mapped to f va Fq P. We use σ to refer to ths square n an abstract sense when we add rows or columns to the start sheet σ wll move wth the nsertons so the square coordnates referred to by σ may change. We construct a new coalesce folded state C P 1 that we wll prepend to C P. Defne C P 1 to have the same startng sheet as that of Fq P, except that we nsert r rows above σ, r rows below σ, c columns left of σ, and c columns rght of σ. So the startng sheet s a rectangle of paper of sze (m + 2r) (n + 2c) and σ refers to the square s +r,j+c n ths enlarged sheet of paper. Defne ths entre enlarged rectangle to be the coalesce set of C P 1. Now we defne the folded state of C P 1 to be the gven cube gadget appled at σ. The result looks lke a cube sttng on the square s,j of an 7

8 П П О Δ П σ О Δ σ C P 1 C P 1 О Δ П О C P 2 Δ C P 2 Δ О C P q О Δ f C P q О Δ C P 3 Fgure 5: Gven a coalesce sequence C P, shows the applcaton of Lemma 1 to generate C P wth an addtonal cube. The purple regons show where addtonal rows and columns are nserted. We use the arctan 1 2 cube gadget n all fgures from here onwards for consstency. m n rectangle of paper. (The paper does have addtonal layers n some places from the pleats, but t folds flat except at the cube, and these layers are all coalesced because the entre sheet s n the coalesce set.) Now, for each coalesce folded state Ck P, we create a modfed coalesce folded state C P k+1 wth σ replaced by a cube of paper. Here we use the fact that σ never gets folded throughout the sequence (snce t s always the face of a cube), and thus corresponds to a seamless square of paper n the startng sheet of each coalesce folded state Ck P. Note that we add the fve new faces of the cube to the startng sheet, but we do not add these faces to the coalesce set of C P k+1 ; the latter wll reman a rectangle. We also add any polygons of paper nternal to the cube that appear n C P 1, n the same orentaton. Because the coalesce result of C P k s the startng sheet of C P k+1, we have thus generated a new coalesce sequence 8

9 Fgure 6: Ths sequence of folded states for a partcular coalesce sequence shows an example of self ntersecton n part of a coalesce foldng sequence. The self ntersecton occurs at the hghlghted cube and s resolved n the fnal step. The self ntersecton can be avoded n ths case by makng addtonal smple folds. C P = C P 1, C P 2,... C P q, Cq+1. P Note that these added cubes may create ntersectons n the coalesce folded states C P k (as mentoned n Secton 2). However, the fnal folded state F P q+1 of C P (as well as C P q+1 tself) s guaranteed not to have ntersectons. Ths follows because Fq P had no self ntersectons, the applcaton C P 1 of the cube gadget has no self ntersectons, and addng the cube of paper to make P nto P cannot create ntersectons. 4 Foldng Polycubes Our Cube Extruson Algorthm for foldng any polycube s parameterzed by an arbtrary cube gadget, and conssts of repeated applcaton of the gadget accordng to Lemma 1. We descrbe the recursve algorthm by way of an nductve proof: Theorem 2 (Cube Extruson Algorthm) Any polycube of N cubes can folded wth all faces seamless from a (2rN +1) (2cN +2) rectangle of paper by a sequence of N applcatons of an [r, c]-cube gadget plus one addtonal fold. Proof: We prove by nducton that any polycube P of N cubes can be folded seamlessly by a coalesce sequence from a (2rN + 1) (2cN + 2) rectangle of paper. Arbtrarly choose a bottom face f b of P, and let b be the unque (bottom) cube havng f b as a face. 9

10 The base case s N = 1, when P conssts of the sngle cube b. We can use the cube gadget drectly at square s r+1,c+1 to obtan a foldng of the sngle cube from a (2r + 1) (2c + 2) rectangle of paper. The folded state s a cube next to a (pleated) unt square of paper. Note that the bottom face of the cube corresponds to f b, and s adjacent to the square of paper. By defnton of cube gadgets, all faces of the cube except the bottom face are seamless. We fold the extra square of paper over to seamlessly cover the bottom face, thus makng a seamless one-cube polycube. The resultng folded state forms the frst and only step n a coalesce sequence. It remans to prove the nductve step. Let T be a spannng tree of the dual graph of P. Because every tree has at least two leaves, T has a leaf correspondng to a cube l b. Let u be the unque cube sharng a face wth l, and let f ul be the face shared by u and l. Now consder the polycube P = P \ {l}, wth N 1 cubes. Because l b, f b remans a face of P. By nducton, there s a coalesce sequence C P that folds a (2r(N 1) + 1) (2c(N 1) + 2) rectangle of paper nto P. By Lemma 1, we extrude from f ul to obtan a new coalesce sequence C P for P from a rectangle of sze ((2r(N 1) + 1) + 2r) ((2c(N 1) + 2) + 2c) = (2rN + 1) (2cN + 2). The fnal folded state of the nductvely obtaned coalesce sequence s the desred folded state from the rectangle of paper nto the polycube. Wthout the concern for a seamless bottom face, we can reduce the +2 n the rectangle bound down to +1. The algorthm runs n polynomal tme. The bottleneck s n convertng the coalesce sequence nto ts fnal folded state. Each of the N cube gadgets causes the creaton of at most O(N) creases, because the pece of paper at that pont has sze O(N) O(N) wth O(1) exstng creases per square. The sze bound of an O(N) O(N) rectangle of paper s tght up to constant factors for square paper. Specfcally, foldng a 1 1 N tower of cubes requres startng from a square of sde length N n order to have dameter N, as foldng can only decrease dameter. 4.1 Hnge Pattern Completeness Next we show that the Cube Extruson Algorthm does not create creases that stray from the gven hnge pattern. For a rectangular pece of paper, the tle t,j of a crease pattern s the set of creases wthn the unt square s,j. The tle set of a cube gadget s the set of all dstnct tles that can be generated by the cube gadget. The hnge pattern generated by a tle s the result of replcatng the tle n a unt-square grd. Proposton 3 Gven a cube gadget wth a fnte tle set and half-square 10

11 pleats as the only folded structure outsde of the cube, f we add to an empty tle a hnge for every crease of each tle n the tle set for every 2D orthogonal orentaton (rotatons and reflectons), then the resultng tle generates the hnge pattern requred to fold a polycube wth the Cube Extruson Algorthm. Ths proposton s nontrval as t s possble that some combnaton of cube gadgets would create new tles that are not present n any sngle gadget whch are thus not n the tle set. Proof: We prove that no other hnges are needed beyond those found n the constructed generator tle. There are two types of folded tles used by cube gadgets to make polycubes as descrbed n the proposton: nner tles whch make up the nonvsble parts nsde the cubes and pleat tles that make up the non-vsble part of the pleats (outsde of the cube). Inner tles are never folded once they are made part of a cube as our foldngs never modfy the nner structure of an exstng cube, so they do not requre any addtonal hnges beyond those n the tle set. Pleat tles may be folded agan but they are all half-square pleats, whch means that they smply reflect a crease along the mdlne of the tle but each of the reflected halves would already have exsted n reflectons of that tle of the cube gadget, so ths also does not create addtonal hnges. 4.2 Paper Dmensons We now show the specfc bounds on the dmensons of the requred rectangle of paper for each of the three cube gadgets consdered n ths paper. Corollary 4 (arctan 1 2 Unversalty Lemma) Any polycube of N cubes can be folded wth all faces seamless from an arctan 1 2 hnge pattern on a (2N +1) (2N +2) rectangle of paper usng the Cube Extruson Algorthm. Proof: The arctan 1 2 gadget has r = 1 and c = 1. Pluggng these constants nto Theorem 2 yelds a process for foldng the polycube from a rectangle of paper of sze (2N + 1) (2N + 2). Corollary 5 (Slt Unversalty Lemma) Any polycube of N cubes can be folded wth all faces seamless from a slt hnge pattern on a (2N + 1) (2N + 2) rectangle of paper usng the Cube Extruson Algorthm. Proof: Same as Corollary 4. The prevously dscussed gadgets create foldngs from a rectangle of paper that s wthn an addtve constant of beng square. As we show 11

12 now, drectly applyng the Cube Extruson Algorthm wth the tetraks cube gadget generates a foldng wth a rato wthn a constant of 1 2, but a slght modfcaton allows us to use an approxmately square sheet of paper. Corollary 6 (Box-Pleated Unversalty Lemma) Any polycube P of N cubes can be folded wth all faces seamless from a box-pleated (tetraks) hnge pattern on a (2N +1) (4N +2) rectangle of paper usng the Cube Extruson Algorthm. A slght modfcaton of the Cube Extruson Algorthm uses a (3n+1) (3n+2) rectangle of paper for even N and a (3N) (3N +3) rectangle of paper for odd N. Proof: The box-pleated gadget has r = 1 and c = 2. Pluggng these constants nto Theorem 2 yelds a process for foldng P from a rectangle of paper of sze (2N + 1) (4N + 2). Now we descrbe the modfed approach. Defne the transpose of a cube gadget to be the cube gadget wth r and c nterchanged, so that now we nsert r columns to the left and rght of the column and c rows below and above the specfed row. We alternate the box-pleated gadget and ts transpose for a polycube of N cubes such that the box-pleated gadget s appled N/2 tmes and ts transpose s appled N/2 tmes. Ths yelds a fnal folded state F n wth a startng sheet of sze (2r N 2 + 2c N 2 + 1) (2c N 2 + 2r N 2 + 2) whch smplfes slghtly to (2 N N 2 + 1) (4 N N 2 + 2). For even N, ths bound s (3N + 1) (3N + 2), and for odd N, t s (3N) (3N + 3). 4.3 Number of Layers Proposton 7 For any polycube of N cubes, the Cube Extruson Algorthm produces a foldng that uses O(N 2 ) layers. Proof: The folded state produced by Theorem 2 has a startng sheet of sze (2rN + 1) (2cN + 2), whch s clearly O(N 2 ) for constants r and c. And because each square of a hnge pattern contans O(1) hnges (by defnton), there can be at most O(N 2 ) layers n the foldng (even f we folded the paper up nto the smallest unt of area allowable by the hnge pattern). Unfortunately, ths quadratc bound on the number of layers s tght n the worst case: Proposton 8 For any N and any cube gadget G, there exsts a polycube of N cubes for whch the Cube Extruson Algorthm yelds a foldng requrng Ω(N 2 ) layers. 12

13 f b f b Fgure 7: Ths horzontal L-shaped polycube uses Ω(N 2 ) layers when folded by the Cube Extruson Algorthm. Proof: Wthout loss of generalty, assume that N s odd. The example we use s a horzontal L-shaped polycube, as shown n Fgure 7. To construct t, take a sngle cube b and two faces whch share an edge of the cube. Extrude (N 1)/2 cubes from each of the faces. (If N were even, we would extrude (N/2) 1 cubes from one of the faces and N/2 cubes from the other face). We now show that our foldng algorthm would construct a foldng havng Ω(N 2 ) layers. Let the bottom cube of the foldng algorthm be b, and take the bottom face f b to be one of the faces parallel the plane spanned by the legs of the L. Now consder the face n the resultng folded state opposte to f b : t s seamless, but hdden beneath t are pleats from prsms of both legs of the L. There are Ω(N) pleats from each leg, and the pleats are orthogonal to each other. Ths results n Ω(N 2 ) layers. 5 Implementaton A smplfcaton of ths algorthm was mplemented n Ruby. It allows the user to defne a polycube through extrusons and dsplays the correspondng foldng sequence to the screen as a seres of angle patterns, but t does not generate a fnal folded state. It can also show a smplfed composton of the foldng sequence whch can be saved to a PostScrpt fle. The smplfed composton just shows the orgnal angle pattern for each square tle on the full sheet of paper (that s, the angle pattern from the step when the 13

14 1.68 n Abstract Polycube Ruby Implementaton PostScrpt Cuttng Plotter (Graphtec/CraftROBO) Folded Model Fgure 8: The process by whch paper orgam can be constructed usng the descrbed algorthms. square was nserted as part of a row or column), and does not take nto account any non-orthogonal creases made as a result of later steps. The PostScrpt fle can be sent to a cuttng plotter, whch can etch the smplfed composton onto a sheet of paper. Alternatvely, one can use a laser cutter to score the paper. See Fgure 8. Only a sngle cube gadget s foldable at each step when foldng along vsble etchngs of the smplfed composton, so one can smply fold cube gadgets untl there are no vsble etchngs. If the foldngs generated by the Cube Extruson Algorthm are appled to a larger sheet of paper, t appears that the polycubes are lyng of top of or extruded from the sheet. Artstcally ths effect s generally preferable to havng a polycube closed off by a small flap. In partcular, t prevents the pleats from expandng as much, and n some cases enables multple polycubes to be extruded from a sngle sheet. 6 Rgd Foldablty and Self-Foldng Sheets The Cube Extruson Algorthm can be used to make artstc paper orgam, but to get from one folded state to the next may requre curvng the paper or ntroducng temporary nonhnge creases. Ths becomes an ssue for controllng a self-foldng programmable matter sheet, where the polygons of a hnge pattern are (nearly) rgd. It s an open queston whch polycubes are rgdly foldable from a partcular cube gadget, though t seems through smple emprcal testng that our cube gadgets fold rgdly n solaton (when makng one-cube polycubes). Polycubes wth more than one cube may nonetheless not be rgdly foldable, and polycubes wth ntermedary folded states that self-ntersect are almost certanly not rgdly foldable. Our ntuton suggests that the slt pattern has the broadest potental for programmable matter. 14

15 Acknowledgments References A. Ovadya would lke to thank a host of frends who have not complaned about hs ncessant foldng of cubes, and n partcular Smone Agha, Tucker Chan, Lyla Fscher, Andrea Hawksley, Robert Johnson, and Mara Monks for beng soundng boards, foldng, and/or commentng. E. Demane thanks the Huffman famly Else, Lnda, and Marlyn for access to Davd A. Huffman s notes, whch nclude the arctan 1 2 cube gadget. We also thank Jason Ku and Scott Macr for assstng wth Fgure 3 and Fgure 4 respectvely. [DDM00] Erk D. Demane, Martn L. Demane, and Joseph S. B. Mtchell. Foldng flat slhouettes and wrappng polyhedral packages: New results n computatonal orgam. Computatonal Geometry: Theory and Applcatons, 16(1):3 21, [DDMO04] Erk D. Demane, Satyan L. Devadoss, Joseph S. B. Mtchell, and Joseph O Rourke. Contnuous foldablty of polygonal paper. In Proceedngs of the 16th Canadan Conference on Computatonal Geometry, pages 64 67, Montréal, Canada, August [DO07] Erk D. Demane and Joseph O Rourke. Geometrc Foldng Algorthms: Lnkages, Orgam, Polyhedra. Cambrdge Unversty Press, July [DT10] Erk D. Demane and Tomohro Tach. Orgamzer: A practcal algorthm for foldng any polyhedron. Manuscrpt, [HAB + 10] E. Hawkes, B. An, N. M. Benbernou, H. Tanaka, S. Km, E. D. Demane, D. Rus, and R. J. Wood. Programmable matter by foldng. Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, To appear. pnas.org/cg/do/ /pnas