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

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

3. 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

4. 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:

5. Fig. 2 Definition of hexa-
Picture taken from

6. 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.

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

8. 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.

9. 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.

10. 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?

11. 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?

12. 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?

13. Fields of Mathematics Combinatorics Catalan numbers Probability
Algebra
Geometry

14. 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

15. 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

16. 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.

17. 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.

18. 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

19. 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.

20. 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.

21. 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.

22. 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

23. 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

24. 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

25. References
Anderson, T., McLean, T., Pajoohesh, H., & Smith, C. (2009). The combinatorics of all regular flexagons. European Journal of Combinatorics, 31,
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,  
Watt, S. (2013). What the hexaflexagon? Retrieved February 7, 2015, from