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gray, and yellow. Note the different patterns used for the colors on the two sides. Figure 3 shows the first fold, and figure 4 the result of the first nine folds, which form a spiral. Figures 5-6 show the final folding of the spiral to make a hexagon; in 5, two red faces have been hidden by a valley fold, and in 6, two red faces on the bottom side have been hidden by a mountain fold. After figure 6, the final loose triangle is folded over and attached to the other end of the original strip so that one side is all blue, and the other all orange. Photos 7 and 8 show the process of everting the hexaflexagon to show the formerly hidden red triangles. By further manipulations, all six colors can be exposed.
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triangles with different colors, and they can be seen by flexing the hexahexaflexagon in all possible ways in theory, but only 15 can be flexed by the ordinary hexahexaflexagon. The 3 extra configurations are impossible due to the arrangement of the 4, 5, and 6 tiles at the back flap. (The 60-degree angles in the rhombi formed by the adjacent 4, 5, or 6 tiles will only appear on the sides and never will appear at the center because it would require one to cut the strip, which is topologically forbidden.)
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Figures 1-6 show the construction of a hexaflexagon made out of cardboard triangles on a backing made from a strip of cloth. It has been decorated in six colours; orange, blue, and red in figure 1 correspond to 1, 2, and 3 in the diagram above. The opposite side, figure 2, is decorated with purple,
372:
While the most commonly seen hexaflexagons have either three or six faces, variations exist with any number of faces. Straight strips produce hexaflexagons with a multiple of three number of faces. Other numbers are obtained from nonstraight strips, that are just straight strips with some joints
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An easy way to expose all six faces is using the
Tuckerman traverse, named after Bryant Tuckerman, one of the first to investigate the properties of hexaflexagons. The Tuckerman traverse involves the repeated flexing by pinching one corner and flex from exactly the same corner every time. If the
359:
Each color/face can also be exposed in more than one way. In figure 6, for example, each blue triangle has at the center its corner decorated with a wedge, but it is also possible, for example, to make the ones decorated with Y's come to the center. There are 18 such possible configurations for
386:
In these more recently discovered flexagons, each square or equilateral triangular face of a conventional flexagon is further divided into two right triangles, permitting additional flexing modes. The division of the square faces of tetraflexagons into right isosceles triangles yields the
260:
A more complicated cyclic hexatetraflexagon requires no gluing. A cyclic hexatetraflexagon does not have any "dead ends", but the person making it can keep folding it until they reach the starting position. If the sides are colored in the process, the states can be seen more clearly.
407:, with angles 72–54–54. Because of its fivefold symmetry, the pentaflexagon cannot be folded in half. However, a complex series of flexes results in its transformation from displaying sides one and two on the front and back, to displaying its previously hidden sides three and four.
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This trihexaflexagon template shows 3 colors of 9 triangles, printed on one side, and folded to be colored on both sides. The two yellow triangles on the ends will end up taped together. The red and blue arcs are seen as full circles on the inside of one side or the other when
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Hexahexaflexagons can be constructed from different shaped nets of eighteen equilateral triangles. One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version.
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A hexaflexagon with three faces is the simplest of the hexaflexagons to make and to manage, and is made from a single strip of paper, divided into nine equilateral triangles. (Some patterns provide ten triangles, two of which are glued together in the final assembly.)
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method, called the
Tuckerman traverse, for revealing all the faces of a flexagon. Tuckerman traverses are shown as a diagram that maps each face of the flexagon to each other face. In doing so, he realized that each face does not always appear in the same state.
348:
corner refuses to open, move to an adjacent corner and keep flexing. This procedure brings you to a 12-face cycle. During this procedure, however, 1, 2, and 3 show up three times as frequently as 4, 5, and 6. The cycle proceeds as follows:
71:). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a
110:
in the United States in 1939. His new
American paper would not fit in his English binder so he cut off the ends of the paper and began folding them into different shapes. One of these formed a trihexaflexagon. Stone's colleagues
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sides). The "tri" in the name means it has three faces, two of which are visible at any given time if the flexagon is pressed flat. The construction of the tritetraflexagon is similar to the mechanism used in the traditional
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Their patent imagined possible applications of the device "as a toy, as an advertising display device, or as an educational geometric device." A few such novelties were produced by the
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applied for a patent, and in 1959 they were granted U.S. Patent number 2,883,195 for the hexahexaflexagon, under the title "Changeable
Amusement Devices and the Like."
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The tritetraflexagon has two dead ends, where you cannot flex forward. To get to another face you must either flex backwards or flip the flexagon over.
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Hexaflexagons come in great variety, distinguished by the number of faces that can be achieved by flexing the assembled figure. (Note that the word
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By further dividing the 72-54-54 triangles of the pentaflexagon into 36-54-90 right triangles produces one variation of the 10-sided decaflexagon.
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folded, eliminating some faces. Many strips can be folded in different ways, producing different hexaflexagons, with different folding maps.
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To assemble, the strip is folded every third triangle, connecting back to itself after three inversions in the manner of the international
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octaflexagons, and the division of the triangular faces of the hexaflexagons into 30-60-90 right triangles yields the dodecaflexagons.
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Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an
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453:. These should be distinguished from the "ordinary" pentaflexagons and heptaflexagons described above, which are made out of
1615:
1154:
Hexaflexagons, Probability
Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games
426:
isosceles triangles. Other flexagons include the heptaflexagon, the isosceles octaflexagon, the enneaflexagon, and others.
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Contrary to the tritetraflexagon, the hexatetraflexagon has no dead ends, and does not ever need to be flexed backwards.
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In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of
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also describes "nonplanar" flexagons (i.e., ones which cannot be flexed so they lie flat); ones folded from
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Hexaflexagons and Other
Mathematical Diversions: The First "Scientific American" Book of Puzzles and Games
1778:
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Hexaflexagons and Other
Mathematical Diversions: The First Scientific American Book of Puzzles and Games
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1001:
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or folded in certain ways to reveal faces besides the two that were originally on the back and front.
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This hexaflexagon has six faces. It is made up of nineteen triangles folded from a strip of paper.
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became interested in the idea and formed the
Princeton Flexagon Committee. Tuckerman worked out a
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included a construct-your-own hexaflexagon with the original cast recording of his
Broadway show
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The discovery of the first flexagon, a trihexaflexagon, is credited to the
British mathematician
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can sometimes refer to an ordinary hexahexaflexagon, with six sides instead of other numbers.)
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The pentaflexagon is one of an infinite sequence of flexagons based on dividing a regular
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520:, a well-known recreational mathematician and public educator, gained attention for her
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The Mysterious Flexagons: An Introduction to a Fascinating New Concept in Paper Folding
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1309: – Scott Sherman's site, with variety of flexagons of different shapes.
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Anderson, Thomas; McLean, T. Bruce; Pajoohesh, Homeira; Smith, Chasen (January 2010).
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which then ran in that magazine for the next twenty-five years. In 1974, the magician
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1298: – Robin Moseley's site has patterns for a large variety of flexagons.
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493:). Instructions for making tetra-tetra-flexagon and cross-flexagons are included in
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The issue also contains another article by Pook, and one by Iacob, McLean, and Hua.
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where Rogers worked, but the device, marketed as the "Hexmo", failed to catch on.
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1325: – 1962 paper by Antony S. Conrad and Daniel K. Hartline (RIAS)
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Once folded, faces 1, 2, and 3 are easier to find than faces 4, 5, and 6.
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wrote an excellent introduction to hexaflexagons in the December 1956
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The "Scientific American" Book of Mathematical Puzzles and Diversions
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models, usually constructed by folding strips of paper, that can be
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The tritetraflexagon is the simplest tetraflexagon (flexagon with
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This figure has two faces visible, built of squares marked with
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A tritetraflexagon can be folded from a strip of paper as shown.
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The Magic of Flexagons – Paper curiosities to cut out and make
381:
36:
A hexaflexagon, shown with the same face in two configurations
1066:. Starmont Reader's Guide #20. Borgo Press. pp. 47–48.
1031:
Making Handmade Books: 100+ Bindings, Structures & Forms
722:
Rogers, Russell E.; Andrea, Leonard D. L. (April 21, 1959).
600:
1295:
495:
Making Handmade Books: 100+ Bindings, Structures and Forms
141:
in an article so well-received that it launched Gardner's
1338:"General Solution for Multiple Foldings of Hexaflexagons"
1306:
395:
In its flat state, the pentaflexagon looks much like the
705:"The Top 10 Martin Gardner Scientific American Articles"
566:
Oakley, C. O.; Wisner, R. J. (March 1957). "Flexagons".
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A high-order hexaflexagon was used as a plot element in
390:
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Flexagons are also a popular book structure used by
574:(3). Mathematical Association of America: 143–154.
430:
Nonplanar pentaflexagon and nonplanar heptaflexagon
413:
158:
131:Flexagons were introduced to the general public by
1262:Serious Fun with Flexagons, A Compendium and Guide
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1028:
1171:Gardner, Martin (January 2012). "Hexaflexagons".
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163:In 1955, Russell Rogers and Leonard D'Andrea of
761:"Flexagon Discovery: The Shape-Shifting 12-Gon"
643:. Vol. 195, no. 6. pp. 162–168.
637:Gardner, Martin (December 1956). "Flexagons".
98:Discovery and introduction of the hexaflexagon
1374:
632:
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352:1 → 3 → 6 → 1 → 3 → 2 → 4 → 3 → 2 → 1 → 5 → 2
59:Flexagons are usually square or rectangular (
1292:– contains historical information and theory
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603:"The combinatorics of all regular flexagons"
565:
724:"Changeable amusement devices and the like"
382:Right octaflexagon and right dodecaflexagon
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943:"Enneaflexagon: Isosceles Enneaflexagon"
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917:"Octaflexagon: Isosceles Octaflexagon"
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1120:. University of Chicago Press. 1988.
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237:and in the magic wallet trick or the
1616:Geometric Exercises in Paper Folding
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1156:. Cambridge University Press. 2008.
391:Pentaflexagon and right decaflexagon
255:
1637:A History of Folding in Mathematics
320:
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172:Herbick & Held Printing Company
24:
740:from the original on June 14, 2011
703:Mulcahy, Colm (October 21, 2014).
649:10.1038/scientificamerican1256-162
403:divided from the center into five
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25:
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1279:
607:European Journal of Combinatorics
568:The American Mathematical Monthly
186:
1340:IJPAM, Vol. 58, No. 1, 113–124.
1329:MathWorld entry on Hexaflexagons
1136:The Colossal Book of Mathematics
1008:. Universidad AutĂłnoma de Puebla
978:. Universidad AutĂłnoma de Puebla
414:Generalized isosceles n-flexagon
275:
220:s is hidden inside the flexagon.
159:Attempted commercial development
1537:Alexandrov's uniqueness theorem
1195:10.4169/college.math.j.43.1.002
1187:10.4169/college.math.j.43.1.002
1174:The College Mathematics Journal
1138:. W. W. Norton & Co. 2001.
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682:. University of Chicago Press.
1245:. Cambridge University Press.
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135:in the December 1956 issue of
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1:
1475:Regular paperfolding sequence
1112:. Simon & Schuster. 1959.
1060:Collings, Michael R. (1984).
549:
1623:Geometric Folding Algorithms
1390:Mathematics of paper folding
181:
165:Homestead Park, Pennsylvania
7:
527:
143:"Mathematical Games" column
10:
1810:
1673:Margherita Piazzola Beloch
1349:'s video on Hexaflexagons
356:And then back to 1 again.
313:whose single edge forms a
268:Hexatetraflexagon traverse
174:, the printing company in
92:
1660:
1607:
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1529:
1483:
1452:
1444:Yoshizawa–Randlett system
1396:
1027:Golden, Alisa J. (2011).
620:10.1016/j.ejc.2009.01.005
251:Tritetraflexagon traverse
1644:Origami Polyhedra Design
1222:Mitchell, David (2000).
1213:Jones, Madeline (1966).
1035:. Lark Crafts. pp.
678:Gardner, Martin (1988).
1342:"19 faces of Flexagons"
1286:My Flexagon Experiences
941:Sherman, Scott (2007).
915:Sherman, Scott (2007).
889:Sherman, Scott (2007).
863:Sherman, Scott (2007).
837:Sherman, Scott (2007).
811:Sherman, Scott (2007).
785:Sherman, Scott (2007).
733:. U.S. Patent 2883195.
1789:Geometric group theory
1434:Napkin folding problem
1319:, including three nets
1101:. It also appears in:
1002:"Heptagonal Flexagons"
972:"Pentagonal Flexagons"
759:Schwartz, Ann (2005).
539:Geometric group theory
522:video on hexaflexagons
377:Higher order flexagons
341:
332:
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731:Freepatentsonline.com
461:be made to lie flat.
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106:, while a student at
35:
1594:Fold-and-cut theorem
1550:Steffen's polyhedron
1414:Huzita–Hatori axioms
1404:Big-little-big lemma
1243:Flexagons Inside Out
108:Princeton University
87:equivalence relation
1542:Flexible polyhedron
1296:The Flexagon Portal
1217:. Crown Publishers.
1099:Scientific American
998:McIntosh, Harold V.
970:(August 24, 2000).
968:McIntosh, Harold V.
709:Scientific American
640:Scientific American
455:isosceles triangles
405:isosceles triangles
368:Other hexaflexagons
233:children's toy, in
138:Scientific American
1779:Mechanical puzzles
1723:Toshikazu Kawasaki
1546:Bricard octahedron
1521:Yoshimura buckling
1419:Kawasaki's theorem
1290:Harold V. McIntosh
1260:Pook, Les (2009).
1241:Pook, Les (2006).
1095:Mathematical Games
1000:(March 11, 2000).
483:Edward H. Hutchins
465:In popular culture
435:Harold V. McIntosh
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1630:Geometric Origami
1501:Paper bag problem
1424:Maekawa's theorem
1271:978-90-481-2502-9
1163:978-0-521-73525-4
1046:978-1-60059-587-5
497:by Alisa Golden.
473:creators such as
256:Hexatetraflexagon
16:(Redirected from
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1703:David A. Huffman
1668:Roger C. Alperin
1571:Source unfolding
1439:Pureland origami
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399:logo: a regular
321:Hexahexaflexagon
307:recycling symbol
192:Tritetraflexagon
113:Bryant Tuckerman
73:hexahexaflexagon
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891:"Heptaflexagon"
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839:"Pentaflexagon"
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309:. This makes a
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288:Trihexaflexagon
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216:s. The face of
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117:Richard Feynman
104:Arthur H. Stone
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1280:External links
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865:"Decaflexagon"
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787:"Octaflexagon"
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443:pentaflexagons
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231:Jacob's Ladder
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187:Tetraflexagons
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276:Hexaflexagons
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1758:Eve Torrence
1688:Erik Demaine
1649:
1642:
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1628:
1621:
1614:
1608:Publications
1470:Möbius strip
1464:
1460:Dragon curve
1397:Flat folding
1264:. Springer.
1261:
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1085:Bibliography
1062:
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1022:
1010:. Retrieved
1006:Cinvestav.mx
1005:
992:
980:. Retrieved
976:Cinvestav.mx
975:
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950:. Retrieved
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924:. Retrieved
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898:. Retrieved
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872:. Retrieved
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846:. Retrieved
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790:
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768:. Retrieved
764:
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742:. Retrieved
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613:(1): 72–80.
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544:Kaleidocycle
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311:Möbius strip
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29:
1743:KĹŤryĹŤ Miura
1738:Jun Maekawa
1713:KĂ´di Husimi
1429:Map folding
1315:'s page on
1226:. Tarquin.
1039:, 132–133.
1012:October 26,
982:October 26,
952:October 26,
926:October 26,
900:October 26,
874:October 26,
848:October 26,
822:October 26,
796:October 26,
770:October 26,
744:January 13,
534:Cayley tree
457:, and they
445:, and from
125:topological
27:Paper model
1794:Paper toys
1773:Categories
1733:Anna Lubiw
1566:Common net
1491:Miura fold
1181:(1): 2–5.
1097:column in
665:4657622161
550:References
479:Life Cycle
475:Julie Chen
422:-gon into
176:Pittsburgh
121:John Tukey
1651:Origamics
1530:Polyhedra
1323:Flexagons
1313:MathWorld
1307:Flexagons
1302:Flexagons
1203:218544330
947:Loki3.com
921:Loki3.com
895:Loki3.com
869:Loki3.com
843:Loki3.com
817:Loki3.com
791:Loki3.com
504:'s novel
447:heptagons
439:pentagons
182:Varieties
65:hexagonal
46:flexagons
1708:Tom Hull
1678:Yan Chen
1561:Blooming
1465:Flexagon
1336:(2010).
735:Archived
657:24941843
528:See also
401:pentagon
397:Chrysler
241:wallet.
42:geometry
1347:Vi Hart
588:2310544
518:Vi Hart
449:called
441:called
297:folded.
93:History
1661:People
1516:Sonobe
1354:part 2
1351:part 1
1268:
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481:) and
239:Himber
226:square
212:s and
119:, and
54:flexed
1199:S2CID
1191:JSTOR
738:(PDF)
727:(PDF)
653:JSTOR
584:JSTOR
487:Album
63:) or
1266:ISBN
1247:ISBN
1228:ISBN
1158:ISBN
1140:ISBN
1122:ISBN
1068:ISBN
1041:ISBN
1014:2012
984:2012
954:2012
928:2012
902:2012
876:2012
850:2012
824:2012
798:2012
772:2012
746:2011
684:ISBN
661:OCLC
489:and
80:pats
50:flat
48:are
1556:Net
1288:by
1183:doi
1037:130
645:doi
615:doi
576:doi
459:can
40:In
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558:^
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509:0X
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485:(
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424:n
420:n
218:C
214:B
210:A
67:(
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
