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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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22:
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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,
361:
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
336:
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
348:
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
375:
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
249:
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.
396:, 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.
285:
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
352:
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.
337:
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:
60:). 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
99:
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
217:
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
159:
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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442:. These should be distinguished from the "ordinary" pentaflexagons and heptaflexagons described above, which are made out of
1604:
1143:
Hexaflexagons, Probability
Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games
415:
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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1107:
Hexaflexagons and Other
Mathematical Diversions: The First "Scientific American" Book of Puzzles and Games
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Hexaflexagons and Other
Mathematical Diversions: The First Scientific American Book of Puzzles and Games
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45:
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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509:, 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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1298: – 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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1287: – Robin Moseley's site has patterns for a large variety of flexagons.
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482:). 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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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
370:
25:
A hexaflexagon, shown with the same face in two configurations
1055:. Starmont Reader's Guide #20. Borgo Press. pp. 47–48.
1020:
Making Handmade Books: 100+ Bindings, Structures & Forms
711:
Rogers, Russell E.; Andrea, Leonard D. L. (April 21, 1959).
589:
1284:
484:
Making Handmade Books: 100+ Bindings, Structures and Forms
130:
in an article so well-received that it launched Gardner's
1327:"General Solution for Multiple Foldings of Hexaflexagons"
1295:
384:
In its flat state, the pentaflexagon looks much like the
694:"The Top 10 Martin Gardner Scientific American Articles"
555:
Oakley, C. O.; Wisner, R. J. (March 1957). "Flexagons".
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A high-order hexaflexagon was used as a plot element in
379:
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Flexagons are also a popular book structure used by
563:(3). Mathematical Association of America: 143–154.
419:
Nonplanar pentaflexagon and nonplanar heptaflexagon
402:
147:
120:Flexagons were introduced to the general public by
1251:Serious Fun with Flexagons, A Compendium and Guide
1094:
1017:
1160:Gardner, Martin (January 2012). "Hexaflexagons".
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152:In 1955, Russell Rogers and Leonard D'Andrea of
750:"Flexagon Discovery: The Shape-Shifting 12-Gon"
632:. Vol. 195, no. 6. pp. 162–168.
626:Gardner, Martin (December 1956). "Flexagons".
87:Discovery and introduction of the hexaflexagon
1363:
621:
619:
341:1 → 3 → 6 → 1 → 3 → 2 → 4 → 3 → 2 → 1 → 5 → 2
48:Flexagons are usually square or rectangular (
1281:– contains historical information and theory
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710:
592:"The combinatorics of all regular flexagons"
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713:"Changeable amusement devices and the like"
371:Right octaflexagon and right dodecaflexagon
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932:"Enneaflexagon: Isosceles Enneaflexagon"
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906:"Octaflexagon: Isosceles Octaflexagon"
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1109:. University of Chicago Press. 1988.
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226:and in the magic wallet trick or the
1605:Geometric Exercises in Paper Folding
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1145:. Cambridge University Press. 2008.
380:Pentaflexagon and right decaflexagon
244:
1626:A History of Folding in Mathematics
309:
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161:Herbick & Held Printing Company
13:
729:from the original on June 14, 2011
692:Mulcahy, Colm (October 21, 2014).
638:10.1038/scientificamerican1256-162
392:divided from the center into five
276:
14:
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596:European Journal of Combinatorics
557:The American Mathematical Monthly
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1329:IJPAM, Vol. 58, No. 1, 113–124.
1318:MathWorld entry on Hexaflexagons
1125:The Colossal Book of Mathematics
997:. Universidad AutĂłnoma de Puebla
967:. Universidad AutĂłnoma de Puebla
403:Generalized isosceles n-flexagon
264:
209:s is hidden inside the flexagon.
148:Attempted commercial development
1526:Alexandrov's uniqueness theorem
1184:10.4169/college.math.j.43.1.002
1176:10.4169/college.math.j.43.1.002
1163:The College Mathematics Journal
1127:. W. W. Norton & Co. 2001.
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671:. University of Chicago Press.
1234:. Cambridge University Press.
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124:in the December 1956 issue of
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1464:Regular paperfolding sequence
1101:. Simon & Schuster. 1959.
1049:Collings, Michael R. (1984).
538:
1612:Geometric Folding Algorithms
1379:Mathematics of paper folding
170:
154:Homestead Park, Pennsylvania
7:
516:
132:"Mathematical Games" column
10:
1799:
1662:Margherita Piazzola Beloch
1338:'s video on Hexaflexagons
345:And then back to 1 again.
302:whose single edge forms a
257:Hexatetraflexagon traverse
163:, the printing company in
81:
1649:
1596:
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1472:
1441:
1433:Yoshizawa–Randlett system
1385:
1016:Golden, Alisa J. (2011).
609:10.1016/j.ejc.2009.01.005
240:Tritetraflexagon traverse
1633:Origami Polyhedra Design
1211:Mitchell, David (2000).
1202:Jones, Madeline (1966).
1024:. Lark Crafts. pp.
667:Gardner, Martin (1988).
1331:"19 faces of Flexagons"
1275:My Flexagon Experiences
930:Sherman, Scott (2007).
904:Sherman, Scott (2007).
878:Sherman, Scott (2007).
852:Sherman, Scott (2007).
826:Sherman, Scott (2007).
800:Sherman, Scott (2007).
774:Sherman, Scott (2007).
722:. U.S. Patent 2883195.
1778:Geometric group theory
1423:Napkin folding problem
1308:, including three nets
1090:. It also appears in:
991:"Heptagonal Flexagons"
961:"Pentagonal Flexagons"
748:Schwartz, Ann (2005).
528:Geometric group theory
511:video on hexaflexagons
366:Higher order flexagons
330:
321:
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720:Freepatentsonline.com
450:be made to lie flat.
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95:, while a student at
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1583:Fold-and-cut theorem
1539:Steffen's polyhedron
1403:Huzita–Hatori axioms
1393:Big-little-big lemma
1232:Flexagons Inside Out
97:Princeton University
76:equivalence relation
1531:Flexible polyhedron
1285:The Flexagon Portal
1206:. Crown Publishers.
1088:Scientific American
987:McIntosh, Harold V.
959:(August 24, 2000).
957:McIntosh, Harold V.
698:Scientific American
629:Scientific American
444:isosceles triangles
394:isosceles triangles
357:Other hexaflexagons
222:children's toy, in
127:Scientific American
1768:Mechanical puzzles
1712:Toshikazu Kawasaki
1535:Bricard octahedron
1510:Yoshimura buckling
1408:Kawasaki's theorem
1279:Harold V. McIntosh
1249:Pook, Les (2009).
1230:Pook, Les (2006).
1084:Mathematical Games
989:(March 11, 2000).
472:Edward H. Hutchins
454:In popular culture
424:Harold V. McIntosh
331:
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1619:Geometric Origami
1490:Paper bag problem
1413:Maekawa's theorem
1260:978-90-481-2502-9
1152:978-0-521-73525-4
1035:978-1-60059-587-5
486:by Alisa Golden.
462:creators such as
245:Hexatetraflexagon
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1692:David A. Huffman
1657:Roger C. Alperin
1560:Source unfolding
1428:Pureland origami
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388:logo: a regular
310:Hexahexaflexagon
296:recycling symbol
181:Tritetraflexagon
102:Bryant Tuckerman
62:hexahexaflexagon
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298:. This makes a
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854:"Decaflexagon"
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776:"Octaflexagon"
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141:The Magic Show
122:Martin Gardner
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265:Hexaflexagons
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1747:Eve Torrence
1677:Erik Demaine
1638:
1631:
1624:
1617:
1610:
1603:
1597:Publications
1459:Möbius strip
1453:
1449:Dragon curve
1386:Flat folding
1253:. Springer.
1250:
1231:
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1074:Bibliography
1051:
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999:. Retrieved
995:Cinvestav.mx
994:
981:
969:. Retrieved
965:Cinvestav.mx
964:
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939:. Retrieved
935:
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913:. Retrieved
909:
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887:. Retrieved
883:
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861:. Retrieved
857:
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835:. Retrieved
831:
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783:. Retrieved
779:
769:
757:. Retrieved
753:
743:
731:. Retrieved
719:
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627:
602:(1): 72–80.
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533:Kaleidocycle
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300:Möbius strip
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18:
1732:KĹŤryĹŤ Miura
1727:Jun Maekawa
1702:KĂ´di Husimi
1418:Map folding
1304:'s page on
1215:. Tarquin.
1028:, 132–133.
1001:October 26,
971:October 26,
941:October 26,
915:October 26,
889:October 26,
863:October 26,
837:October 26,
811:October 26,
785:October 26,
759:October 26,
733:January 13,
523:Cayley tree
446:, and they
434:, and from
114:topological
16:Paper model
1783:Paper toys
1762:Categories
1722:Anna Lubiw
1555:Common net
1480:Miura fold
1170:(1): 2–5.
1086:column in
654:4657622161
539:References
468:Life Cycle
464:Julie Chen
411:-gon into
165:Pittsburgh
110:John Tukey
1640:Origamics
1519:Polyhedra
1312:Flexagons
1302:MathWorld
1296:Flexagons
1291:Flexagons
1192:218544330
936:Loki3.com
910:Loki3.com
884:Loki3.com
858:Loki3.com
832:Loki3.com
806:Loki3.com
780:Loki3.com
493:'s novel
436:heptagons
428:pentagons
171:Varieties
54:hexagonal
35:flexagons
1697:Tom Hull
1667:Yan Chen
1550:Blooming
1454:Flexagon
1325:(2010).
724:Archived
646:24941843
517:See also
390:pentagon
386:Chrysler
230:wallet.
31:geometry
1336:Vi Hart
577:2310544
507:Vi Hart
438:called
430:called
286:folded.
82:History
1650:People
1505:Sonobe
1343:part 2
1340:part 1
1257:
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470:) and
228:Himber
215:square
201:s and
108:, and
43:flexed
1188:S2CID
1180:JSTOR
727:(PDF)
716:(PDF)
642:JSTOR
573:JSTOR
476:Album
52:) or
1255:ISBN
1236:ISBN
1217:ISBN
1147:ISBN
1129:ISBN
1111:ISBN
1057:ISBN
1030:ISBN
1003:2012
973:2012
943:2012
917:2012
891:2012
865:2012
839:2012
813:2012
787:2012
761:2012
735:2011
673:ISBN
650:OCLC
478:and
69:pats
39:flat
37:are
1545:Net
1277:by
1172:doi
1026:130
634:doi
604:doi
565:doi
448:can
29:In
1764::
1537:,
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618:^
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498:0X
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413:n
409:n
207:C
203:B
199:A
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
