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

Introduction A flexagon

◦ Is a flexible hexagon◦ can be folded to reveal multiple patterns

A hexaflexagon◦ Is a hexagonal flexagon

Picture taken fromhttp://38.media.tumblr.com/tumblr_mbnwg3ex1L1qhd8sao1_500.gif

Fig. 1 The Hexahexaflexagon

History 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

INTRODUCTION

Picture taken from:http://i00.i.aliimg.com/img/pb/146/476/378/378476146_255.jpg

EnglishBinder American

Paper

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

Definition A hexahexaflexagon is

◦an advanced version of a hexaflexagon

◦all 6 of its sides will reveal different patterns when folded.

INTRODUCTION

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

Feynman Diagram Tuckerman (1939) discovered the Tuckerman Traverse

◦ the simplest way to bring out all the faces of a hexaflexagon

LITERATURE REVIEW

Picture taken from Hexaflexagons and other mathematical diversionsFig 3. Tuckerman Traverse

Feynman Diagram◦ A visual representation of the Tuckerman Traverse (Feynman, 1948).

◦ Allows us to better perceive flexagons and the order in which the patterns are revealed.

LITERATURE REVIEW

Picture taken fromhttp://www.explorecuriocity.org/Portals/2/article%20images/feyman%20diagram.png

Fig. 4 Feynman Diagram

Combinatorics and Catalan Numbers

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.

LITERATURE REVIEW

Flexes of a Flexagon

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.

LITERATURE REVIEW

Flexes of a Flexagon Pinch Flex

V-Flex

Identity Flex

Pyramid Shuffle Flex

Flip Flex

Silver Tetra Flex

Pocket Flex

Slot-tuck Flex

Ticket flex

Slot Flex

LITERATURE REVIEW

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.

TimelineWeek 9-10Term 1

• Complete Project Proposal• Start on PowerPoint slides

16-22 MarchMarch Holidays• Complete PowerPoint slides

Week 1-3Term 2• Project Rehearsals• Final Preparations for Prelims

6 AprilPrelims Judging• Reflection on performance

TimelineWeek 4-10Term 2

• Start on web report• Continuing of project in view of semis

June Holidays• Finishing up of project• Prepare updated PowerPoint slides

Week 1-2Term 3• Project Rehearsals• Semi-finals preparation• Complete updated PowerPoint slides

9 July 2015Semis Judging• Reflections on performance

TimelineWeek 3-8Term 3

•Completion of project•Finalise results and update PowerPoint slides•Completion of web report

21 August 2015Finals Judging

•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