flex the hexahexaflexagon group 9-14 darrell goh (3s106) | tan yan wen (3s221)
TRANSCRIPT
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