Flex the Hexahexaflexagon Group 9-14 Darrell Goh (3S106) | Tan Yan Wen (3S221)

Contents Introduction Objectives Research Problems Fields of Mathematics Literature Review Methodology Timeline References

Fig. 1 The Hexahexaflexagon Introduction A flexagon Is a flexible hexagon can be folded to reveal multiple patterns A hexaflexagon Is a hexagonal flexagon Fig. 1 The Hexahexaflexagon Picture taken from http://38.media.tumblr.com/tumblr_mbnwg3ex1L1qhd8sao1_500.gif

History INTRODUCTION Traces back to 1939 British Student Arthur H. Stone discovered the first flexagon, the trihexaflexagon Discovered when Stone could not fit an American paper in his English binder and thus cut and folded the extra part English Binder American Paper Picture taken from: http://i00.i.aliimg.com/img/pb/146/476/378/378476146_255.jpg

Fig. 2 Definition of hexa- Picture taken from http://dictionary.reference.com/browse/hexa

Definition A hexahexaflexagon is an advanced version of a hexaflexagon INTRODUCTION A hexahexaflexagon is an advanced version of a hexaflexagon all 6 of its sides will reveal different patterns when folded.

Objectives To create a Feynman diagram illustrating Hexahexaflexagons To find out all the possible combinations of folding Hexahexaflexagons to reveal all the patterns.

Objectives To find the general solution/formula to the number of possible combinations of the Hexahexaflexagon. To explore the combinatorics properties and relations with Catalan numbers of Hexahexaflexagons. To find the ideal flex to flex a Hexahexaflexagon.

Objectives To create a Feynman diagram illustrating Hexahexaflexagons To find out all the possible combinations of folding Hexahexaflexagons to reveal all the patterns. To find the general solution/formula to the number of possible combinations of the Hexahexaflexagon. To explore the combinatorics properties and Catalan numbers of Hexahexaflexagons. To find the ideal flex to flex a Hexahexaflexagon.

Research Problems What is the formula to find all the possible combinations folding hexahexaflexagons to reveal all the possible patterns? Is the formula applicable in every scenario to reveal all the possible patterns in the folding of hexahexaflexagons?

Research Problems What is the relationship between Catalan numbers and hexahexaflexagons? Can this relationship be used to create another formula to reveal all the possible patterns in the hexahexaflexagon? What is the ideal flex of a Hexahexaflexagon that can reveal every pattern in the most efficient way?

Research Problems What is the formula to find all the possible combinations folding hexahexaflexagons to reveal all the possible patterns? Is the formula applicable in every scenario to reveal all the possible patterns in the folding of hexahexaflexagons? What is the relationship between Catalan numbers and hexahexaflexagons? Can this relationship be used to create another formula to reveal all the possible patterns in the hexahexaflexagon? What is the ideal flex of a Hexahexaflexagon that can reveal every pattern in the most efficient way?

Fields of Mathematics Combinatorics Catalan numbers Probability Algebra Geometry

Fig 3. Tuckerman Traverse Feynman Diagram LITERATURE REVIEW Tuckerman (1939) discovered the Tuckerman Traverse the simplest way to bring out all the faces of a hexaflexagon Fig 3. Tuckerman Traverse Picture taken from Hexaflexagons and other mathematical diversions

Feynman Diagram A visual representation of LITERATURE REVIEW A visual representation of the Tuckerman Traverse (Feynman, 1948). Allows us to better perceive flexagons and the order in which the patterns are revealed. Fig. 4 Feynman Diagram Picture taken from http://www.explorecuriocity.org/Portals/2/article%20images/feyman%20diagram.png

Combinatorics and Catalan Numbers LITERATURE REVIEW Each triangular region of the hexahexaflexagon is called a pat. Each pat has a thickness, i.e., the number of triangles. This number is the degree of the pat. Anderson (2009) showed that there is also a pair of recursive relations for the number of pat classes with a given degree.

Flexes of a Flexagon Flex LITERATURE REVIEW Flex A series of modifications to a flexagon that takes it from one valid state to another, where the modifications consist of folding together, unfolding, and sliding pats.

Flexes of a Flexagon Pinch Flex Silver Tetra Flex V-Flex Pocket Flex LITERATURE REVIEW Pinch Flex Silver Tetra Flex V-Flex Pocket Flex Identity Flex Slot-tuck Flex Pyramid Shuffle Flex Ticket flex Flip Flex Slot Flex

Methodology To research on resources and related methods regarding Hexahexaflexagons such as the Tuckerman Traverse and Feynman diagrams. To manually fold and find all the possible combinations of folding Hexahexaflexagons to reveal all the patterns.

Methodology To analyse data and find trends in order to obtain Combinatorics properties To calculate the formula of the Hexahexaflexagon’s problem by analysing the data and use C++ programming to if the numbers get too big and complicated. To experiment folding the Hexahexaflexagon using the different flexes to find the ideal flex.

Methodology To research on resources and related methods regarding Hexahexaflexagons such as the Tuckerman Traverse and Feynman diagrams. To manually fold and find all the possible combinations of folding Hexahexaflexagons to reveal all the patterns. To analyse data and find trends in order to obtain Combinatorics properties To calculate the formula of the Hexahexaflexagon’s problem by analysing the data and use C++ programming to if the numbers get too big and complicated. To experiment folding the Hexahexaflexagon using the different flexes to find the ideal flex.

Timeline Week 9-10 Term 1 16-22 March March Holidays Week 1-3 Term 2 Complete Project Proposal Start on PowerPoint slides 16-22 March March Holidays Complete PowerPoint slides Week 1-3 Term 2 Project Rehearsals Final Preparations for Prelims 6 April Prelims Judging Reflection on performance

Timeline Week 4-10 Term 2 June Holidays Week 1-2 Term 3 9 July 2015 Start on web report Continuing of project in view of semis June Holidays Finishing up of project Prepare updated PowerPoint slides Week 1-2 Term 3 Project Rehearsals Semi-finals preparation Complete updated PowerPoint slides 9 July 2015 Semis Judging Reflections on performance

Timeline Term 3 Week 3-8 Completion of project Finalise results and update PowerPoint slides Completion of web report Finals Judging 21 August 2015 Reflections on performance After Finals Submission of research papers

References Anderson, T., McLean, T., Pajoohesh, H., & Smith, C. (2009). The combinatorics of all regular flexagons. European Journal of Combinatorics, 31, 72-80. Gardner, M. (1988). Hexaflexagons. In Hexaflexagons and Other Mathematical Diversions: The 1st Scientific American Book of Puzzles & Games. Chicago: University of Chicago Press. Iacob, I., McLean, T., & Wang, H. (2011). The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons. The College Mathematics Journal, 6-10.  Watt, S. (2013). What the hexaflexagon? Retrieved February 7, 2015, from http://www.explorecuriocity.org/Content.aspx?contentid=2648