Pook, L: Serious Fun with Flexagons
(Sprache: Englisch)
This profusely illustrated book on the geometry of flexagons is written to accommodate both recreational and serious mathematical study. It includes recent geometric analysis and over 700 nets and instructions for making paper models of various flexagons.
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Produktdetails
Produktinformationen zu „Pook, L: Serious Fun with Flexagons “
This profusely illustrated book on the geometry of flexagons is written to accommodate both recreational and serious mathematical study. It includes recent geometric analysis and over 700 nets and instructions for making paper models of various flexagons.
Klappentext zu „Pook, L: Serious Fun with Flexagons “
A flexagon is a motion structure that has the appearance of a ring of hinged polygons. It can be flexed to display different pairs of faces, usually in cyclic order. Flexagons can be appreciated as toys or puzzles, as a recreational mathematics topic, and as the subject of serious mathematical study. Workable paper models of flexagons are easy to make and entertaining to manipulate. The mathematics of flexagons is complex, and how a flexagon works is not immediately obvious on examination of a paper model. Recent geometric analysis, included in the book, has improved theoretical understanding of flexagons, especially relationships between different types.
Inhaltsverzeichnis zu „Pook, L: Serious Fun with Flexagons “
1 Introduction1.1 General Features
1.2 Terminology
1.3 Outline of Book
1.4 Making Flexagons
1.4.1 General Assembly Instructions
1.4.2 The Two Sector First Order Fundamental Square Even Edge Flexagon
2 Polygon Rings
2.1 Introduction
2.1.1 Multiple Polygons
2.1.2 Combinations
2.2 Edge Rings of Regular Polygons
2.2.1 General Properties
2.2.2 Regular Even Edge Rings
2.2.3 Regular Odd Edge Rings
2.2.4 Compound Edge Rings
2.2.5 Irregular Edge Rings
2.3 Edge Rings of Irregular Polygons
2.3.1 General Properties
2.3.2 Even Edge Rings of Silver and Bronze Triangles
2.4 Vertex Rings
2.4.1 General Properties
2.4.2 Vertex Rings of Squares
3 Fundamental Nets
3.1 Introduction
3.2 First Order Fundamental Edge Nets
3.3 Second Order Fundamental Edge Nets
3.4 Fundamental Vertex Nets
3.5 Fundamental Silver and Bronze Edge Nets
4 Fundamental Edge Flexagons
4.1 Introduction
4.1.1 Standard Face Numbering Sequences
4.1.2 Truncated Flexagons
4.2 First Order Fundamental Even Edge Flexagons
4.2.1 General Properties
4.2.2 Ring Even Edge Flexagons
4.2.3 First Order Fundamental Triangle Even Edge Flexagons
4.2.4 First Order Fundamental Square Even Edge Flexagons
4.2.5 Detailed Analysis of Flexagons
4.2.6 First Order Fundamental Pentagon Even Edge Flexagons
4.2.7 First Order Fundamental Hexagon Even Edge Flexagons
4.2.8 First Order Fundamental Octagon Even Edge Flexagons
4.2.9 First Order Fundamental Dodecagon Even Edge Flexagons
4.3 Second Order Fundamental Odd Edge Flexagons
4.3.1 General Properties
4.3.2 Second Order Fundamental Triangle Odd Edge Flexagons
4.3.3 A Second Order Fundamental Square Odd Edge Flexagon
4.3.4 A Second Order Fundamental 20-gon Odd Edge Flexagon
5 Fundamental Skeletal and Point Flexagons
5.1 Introduction
5.2 First Order Fundamental Even Skeletal Flexagons
5.2.1 General Properties
5.2.2 First Order Fundamental Triangle Even Skeletal Flexagons
5.2.3 A First Order Fundamental Square Even Skeletal Flexagon
5.3
... mehr
Fundamental Point Flexagons
5.3.1 General Properties and Unagons
5.3.2 The Fundamental Triangle Point Flexagon
5.3.3 The Fundamental Square Point Flexagon
5.3.4 Fundamental Pentagon Point Flexagons
5.3.5 The Fundamental Hexagon Point Flexagon
5.4 Interleaved Fundamental Point Flexagons
5.4.1 General Properties
5.4.2 The Interleaved Fundamental Pentagon Point Flexagon
5.4.3 An Interleaved Fundamental Enneagon Point Flexagon.
5.5 Augmented Fundamental Point Flexagons
5.5.1 General Properties
5.5.2 An Augmented Fundamental Triangle Point Flexagon
5.5.3 An Augmented Fundamental Square Point Flexagon
5.6 Augmented Interleaved Fundamental Point Flexagons
5.6.1 General Properties
5.6.2 Augmented Interleaved Fundamental Triangle Point Flexagons
5.6.3 An Augmented Interleaved Fundamental Square Point Flexagon
6 Fundamental Compound Edge Flexagons
6.1 Introduction
6.2 General Properties
6.3 Triangular Fundamental Compound Edge Flexagons
6.3.1 Some Properties
6.3.2 A Fundamental Square Compound Edge Flexagon
6.3.3 A Fundamental Pentagon Compound Edge Flexagon
6.4 A Square-Like Fundamental Compound Edge Flexagon
6.4.1 Some Properties
6.4.2 A fundamental hexagon compound edge flexagon
6.5 Pentagonal Fundamental Compound Edge Flexagons
6.5.1 Some Properties
6.5.2 A Fundamental Square Compound Edge Flexagon
6.5.3 A Fundamental Hexagon Compound Edge Flexagon
6.6 A Hexagonal Fundamental Compound Edge Flexagon
6.6.1 Some Properties
6.6.2 A Fundamental Octagon Compound Edge Flexagon
6.7 Heptagonal Fundamental Compound Edge Flexagons
6.7.1 Some Properties
6.7.2 A Fundamental Hexagon Compound Edge Flexagon
6.7.3 A Fundamental Decagon Compound Edge Flexagon
7 Irregular Cycle Flexagons
7.1 Introduction
7.2 Irregular Cycle Even Edge Flexagons
7.2.1 General Properties
7.2.2 Derivation of Nets
7.2.3 The Irregular Cycle Square Even Edge Flexagon
7.2.4 An Irregular Cycle Pentagon Even Edge Flexagon
7.2.5 Irregular Cycle Hexagon Even Edge Flexagons
7.3 Irregular Cycle Interleaved Point Flexagons
7.3.1 General Properties
7.3.2 Interleaf Flexes
7.3.3 The Irregular Cycle Interleaved Square Point; Flexagon
7.3.4 Irregular Cycle Interleaved Pentagon Point Flexagons
7.3.5 Irregular Cycle Interleaved Hexagon Point Flexagons
7.3.6 Augmented Irregular Cycle Interleaved Triangle Point Flexagons
7.4 Distinct Face Numbering Sequences
7.5 Irregular Cycle Non Interleaved Point Flexagons
7.5.1 General Properties
7.5.2 The Irregular Cycle Non Interleaved Square; Point Flexagon
7.5.3 An Irregular Cycle Non Interleaved Pentagon Point Flexagon
8 Degenerate flexagons
8.1 Introduction
8.2 Degenerate Even Edge Flexagons
8.2.1 General Properties
8.2.2 A Degenerate Square Even Edge Flexagon
8.2.3 Degenerate Pentagon Even Edge Flexagons
8.2.4 Degenerate Hexagon Even Edge Flexagons
8.2.5 Degenerate Octagon Even Edge Flexagons
8.2.6 A Degenerate Dodecagon Even Edge Flexagon
8.3 Degenerate Non Interleaved Point Flexagons
8.3.1 General Properties
8.3.2 The Degenerate Non Interleaved Square Point Flexagon
8.3.3 Degenerate Non Interleaved Pentagon Point Flexagons
8.4 Degenerate Irregular Cycle Interleaved Point Flexagons
8.4.1 General Properties
8.4.2 Degenerate Interleaved Triangle Point Flexagons
8.5 Degenerate Compound Edge Flexagons
8.5.1 General Properties
8.5.2 The Degenerate Square-Like Hexagon Compound Edge Flexagon
8.5.3 Degenerate Pentagonal Compound Edge Flexagons; 9 Irregular Ring Even Edge Flexagons
9.1 Introduction
9.1.1 Fundamental Irregular Ring Even Edge Flexagons
9.2 Irregular Ring Triangle Even Edge Flexagons
9.2.1 General Properties
9.2.2 An Irregular Ring 12 Triangle Even Edge Flexagon
9.2.3 An Irregular Ring 8 Triangle Even Edge Flexagon
9.2.4 An Irregular Ring 16 Triangle Even Edge Flexagon
9.3 Irregular Ring Square Even Edge Flexagons
9.3.1 General Properties
9.3.2 Irregular Ring 6 Square Even Edge Flexagons
9.3.3 Irregular Ring 8 Square Even Edge Flexagons
9.4 Irregular Ring Pentagon Even Edge Flexagons
9.4.1 General Properties
9.4.2 An Irregular Ring 6 Pentagon Even Edge Flexagon
9.4.3 An Irregular Ring 8 Pentagon Even Edge Flexagon
9.5 Irregular Ring Hexagon Even Edge Flexagons
9.5.1 General Properties
9.5.2 An Irregular Ring 6 Hexagon Even Edge Flexagon
9.5.3 Irregular Ring 8 Hexagon Even Edge Flexagons
9.6 Irregular Ring Dodecagon Even Edge Flexagons
9.6.1 General Properties
9.6.2 Irregular Ring 8 Dodecagon Even Edge Flexagons
10 Irregular Polygon Edge Flexagons
10.1 Introduction
10.1.1 Transformation of Polygons
10.1.2 Stretch Flexagons and Stretch Polygon Rings
10.2 Irregular Triangle Edge Flexagons
10.2.1 Irregular Triangles
10.2.2 Isosceles Triangle Edge Rings
10.2.3 Bronze Even Edge Rings
10.2.4 Isosceles Triangle Even Edge Flexagons
10.2.5 Isosceles Triangle Odd Edge Flexagons
10.2.6 Scalene Triangle Even Edge Flexagons
10.2.7 A Partial Overlap Bronze Even Edge Flexagon
10.2.8 A Scalene Triangle Even Edge Flexagon
10.3 Irregular Quadrilateral Even Edge Flexagons
10.3.1 Irregular Quadrilaterals
10.3.2 Irregular Quadrilateral Even Edge Rings
10.3.3 A Rectangle Even Edge Flexagon
10.3.4 Rhombus Even Edge Flexagons
10.3.5 A Trapezium Even Edge Flexagon
10.4 Irregular Pentagon Even Edge Flexagons
10.4.1 Irregular Pentagons
10.4.2 Irregular Pentagon Even Edge Rings
10.4.3 An Equiangular Irregular Pentagon Even Edge Flexagon
10.4.4 An Irregular Ring 8 Irregular Pentagon Even Edge Flexagon
11 Complex Flexagons
11.1 Introduction
11.2 Linked Even Edge Flexagons
11.2.1 Methods of Linking
11.2.2 Linked Hexaflexagons
11.2.3 Linked Square Even Edge Flexagons
11.2.4 Linked Pentagon Even Edge Flexagons
11.2.5 Linked Silver Even Edge Flexagons
11.2.6 Linked Bronze Even Edge Flexagons
11.3 Linked Point Flexagons
11.3.1 Methods of Linking
11.3.2 Linked Triangle Point Flexagons
11.3.3 Linked Square Point Flexagons
11.4 Conjoined Point Flexagons
11.4.1 General Properties
11.4.2 Conjoined Triangle Point Flexagons
11.4.3 A Conjoined Pentagon Point Flexagon
11.5 Bundled Odd Edge Flexagons
11.5.1 General Properties
11.5.2 A Five Sector Bundled Triangle Odd Edge Flexagon
11.5.3 The Seven and Six Flexagon
11.6 Slipagons
11.6.1 General Properties
11.6.2 A Trihexaflexagon Slipagon
11.6.3 A Partial Overlap Silver Even Edge Slipagon
11.7 Coupled Point Flexagons
11.7.1 General Properties
11.7.2 A Coupled Triangle Point Flexagon
11.7.3 A Coupled Square Point Flexagon
12 Miscellaneous Flexagons
12.1 Introduction
12.2 Three Sector Odd Flexagons
12.2.1 General Properties
12.2.2 The Three Sector Fundamental Dodecagon Odd Edge Flexagon
12.2.3 A Three Sector Isosceles Triangle Odd Edge Flexagon
12.2.4 A Three Sector Hexagon Odd Skeletal Flexagon
12.3 Degenerate Odd Edge Flexagons
12.3.1 General Properties
12.3.2 A Degenerate Square Odd Edge Flexagon
12.4 Alternating Odd Edge Flexagons
12.4.1 General Properties
12.4.2 A Square Alternating Odd Edge Flexagon
12.5 Flapagons
12.5.1 General Properties
12.5.2 The Fundamental Square Duplex Flapagon
12.5.3 A Square Flapagon-Flexagon Hybrid
12.5.4 The Fundamental Isosceles Triangle Triplex Flapagon
12.6 Multiplex Edge Flexagons
12.6.1 General Properties
12.6.2 A Square Duplex Edge Flexagon
12.6.3 The Thrice Three-Fold Flexagon
12.7 A Hooke's Joint Flexagon
Indext Flexagon
12 Miscellaneous Flexagons
12.1 Introduction
12.2 Three Sector Odd Flexagons
12.2.1 General Properties
12.2.2 The Three Sector Fundamental Dodecagon Odd Edge Flexagon
12.2.3 A Three Sector Isosceles Triangle Odd Edge Flexagon
12.2.4 A Three Sector Hexagon Odd Skeletal Flexagon
12.3 Degenerate Odd Edge Flexagons
12.3.1 General Properties
12.3.2 A Degenerate Square Odd Edge Flexagon
12.4 Alternating Odd Edge Flexagons
12.4.1 General Pro
5.3.1 General Properties and Unagons
5.3.2 The Fundamental Triangle Point Flexagon
5.3.3 The Fundamental Square Point Flexagon
5.3.4 Fundamental Pentagon Point Flexagons
5.3.5 The Fundamental Hexagon Point Flexagon
5.4 Interleaved Fundamental Point Flexagons
5.4.1 General Properties
5.4.2 The Interleaved Fundamental Pentagon Point Flexagon
5.4.3 An Interleaved Fundamental Enneagon Point Flexagon.
5.5 Augmented Fundamental Point Flexagons
5.5.1 General Properties
5.5.2 An Augmented Fundamental Triangle Point Flexagon
5.5.3 An Augmented Fundamental Square Point Flexagon
5.6 Augmented Interleaved Fundamental Point Flexagons
5.6.1 General Properties
5.6.2 Augmented Interleaved Fundamental Triangle Point Flexagons
5.6.3 An Augmented Interleaved Fundamental Square Point Flexagon
6 Fundamental Compound Edge Flexagons
6.1 Introduction
6.2 General Properties
6.3 Triangular Fundamental Compound Edge Flexagons
6.3.1 Some Properties
6.3.2 A Fundamental Square Compound Edge Flexagon
6.3.3 A Fundamental Pentagon Compound Edge Flexagon
6.4 A Square-Like Fundamental Compound Edge Flexagon
6.4.1 Some Properties
6.4.2 A fundamental hexagon compound edge flexagon
6.5 Pentagonal Fundamental Compound Edge Flexagons
6.5.1 Some Properties
6.5.2 A Fundamental Square Compound Edge Flexagon
6.5.3 A Fundamental Hexagon Compound Edge Flexagon
6.6 A Hexagonal Fundamental Compound Edge Flexagon
6.6.1 Some Properties
6.6.2 A Fundamental Octagon Compound Edge Flexagon
6.7 Heptagonal Fundamental Compound Edge Flexagons
6.7.1 Some Properties
6.7.2 A Fundamental Hexagon Compound Edge Flexagon
6.7.3 A Fundamental Decagon Compound Edge Flexagon
7 Irregular Cycle Flexagons
7.1 Introduction
7.2 Irregular Cycle Even Edge Flexagons
7.2.1 General Properties
7.2.2 Derivation of Nets
7.2.3 The Irregular Cycle Square Even Edge Flexagon
7.2.4 An Irregular Cycle Pentagon Even Edge Flexagon
7.2.5 Irregular Cycle Hexagon Even Edge Flexagons
7.3 Irregular Cycle Interleaved Point Flexagons
7.3.1 General Properties
7.3.2 Interleaf Flexes
7.3.3 The Irregular Cycle Interleaved Square Point; Flexagon
7.3.4 Irregular Cycle Interleaved Pentagon Point Flexagons
7.3.5 Irregular Cycle Interleaved Hexagon Point Flexagons
7.3.6 Augmented Irregular Cycle Interleaved Triangle Point Flexagons
7.4 Distinct Face Numbering Sequences
7.5 Irregular Cycle Non Interleaved Point Flexagons
7.5.1 General Properties
7.5.2 The Irregular Cycle Non Interleaved Square; Point Flexagon
7.5.3 An Irregular Cycle Non Interleaved Pentagon Point Flexagon
8 Degenerate flexagons
8.1 Introduction
8.2 Degenerate Even Edge Flexagons
8.2.1 General Properties
8.2.2 A Degenerate Square Even Edge Flexagon
8.2.3 Degenerate Pentagon Even Edge Flexagons
8.2.4 Degenerate Hexagon Even Edge Flexagons
8.2.5 Degenerate Octagon Even Edge Flexagons
8.2.6 A Degenerate Dodecagon Even Edge Flexagon
8.3 Degenerate Non Interleaved Point Flexagons
8.3.1 General Properties
8.3.2 The Degenerate Non Interleaved Square Point Flexagon
8.3.3 Degenerate Non Interleaved Pentagon Point Flexagons
8.4 Degenerate Irregular Cycle Interleaved Point Flexagons
8.4.1 General Properties
8.4.2 Degenerate Interleaved Triangle Point Flexagons
8.5 Degenerate Compound Edge Flexagons
8.5.1 General Properties
8.5.2 The Degenerate Square-Like Hexagon Compound Edge Flexagon
8.5.3 Degenerate Pentagonal Compound Edge Flexagons; 9 Irregular Ring Even Edge Flexagons
9.1 Introduction
9.1.1 Fundamental Irregular Ring Even Edge Flexagons
9.2 Irregular Ring Triangle Even Edge Flexagons
9.2.1 General Properties
9.2.2 An Irregular Ring 12 Triangle Even Edge Flexagon
9.2.3 An Irregular Ring 8 Triangle Even Edge Flexagon
9.2.4 An Irregular Ring 16 Triangle Even Edge Flexagon
9.3 Irregular Ring Square Even Edge Flexagons
9.3.1 General Properties
9.3.2 Irregular Ring 6 Square Even Edge Flexagons
9.3.3 Irregular Ring 8 Square Even Edge Flexagons
9.4 Irregular Ring Pentagon Even Edge Flexagons
9.4.1 General Properties
9.4.2 An Irregular Ring 6 Pentagon Even Edge Flexagon
9.4.3 An Irregular Ring 8 Pentagon Even Edge Flexagon
9.5 Irregular Ring Hexagon Even Edge Flexagons
9.5.1 General Properties
9.5.2 An Irregular Ring 6 Hexagon Even Edge Flexagon
9.5.3 Irregular Ring 8 Hexagon Even Edge Flexagons
9.6 Irregular Ring Dodecagon Even Edge Flexagons
9.6.1 General Properties
9.6.2 Irregular Ring 8 Dodecagon Even Edge Flexagons
10 Irregular Polygon Edge Flexagons
10.1 Introduction
10.1.1 Transformation of Polygons
10.1.2 Stretch Flexagons and Stretch Polygon Rings
10.2 Irregular Triangle Edge Flexagons
10.2.1 Irregular Triangles
10.2.2 Isosceles Triangle Edge Rings
10.2.3 Bronze Even Edge Rings
10.2.4 Isosceles Triangle Even Edge Flexagons
10.2.5 Isosceles Triangle Odd Edge Flexagons
10.2.6 Scalene Triangle Even Edge Flexagons
10.2.7 A Partial Overlap Bronze Even Edge Flexagon
10.2.8 A Scalene Triangle Even Edge Flexagon
10.3 Irregular Quadrilateral Even Edge Flexagons
10.3.1 Irregular Quadrilaterals
10.3.2 Irregular Quadrilateral Even Edge Rings
10.3.3 A Rectangle Even Edge Flexagon
10.3.4 Rhombus Even Edge Flexagons
10.3.5 A Trapezium Even Edge Flexagon
10.4 Irregular Pentagon Even Edge Flexagons
10.4.1 Irregular Pentagons
10.4.2 Irregular Pentagon Even Edge Rings
10.4.3 An Equiangular Irregular Pentagon Even Edge Flexagon
10.4.4 An Irregular Ring 8 Irregular Pentagon Even Edge Flexagon
11 Complex Flexagons
11.1 Introduction
11.2 Linked Even Edge Flexagons
11.2.1 Methods of Linking
11.2.2 Linked Hexaflexagons
11.2.3 Linked Square Even Edge Flexagons
11.2.4 Linked Pentagon Even Edge Flexagons
11.2.5 Linked Silver Even Edge Flexagons
11.2.6 Linked Bronze Even Edge Flexagons
11.3 Linked Point Flexagons
11.3.1 Methods of Linking
11.3.2 Linked Triangle Point Flexagons
11.3.3 Linked Square Point Flexagons
11.4 Conjoined Point Flexagons
11.4.1 General Properties
11.4.2 Conjoined Triangle Point Flexagons
11.4.3 A Conjoined Pentagon Point Flexagon
11.5 Bundled Odd Edge Flexagons
11.5.1 General Properties
11.5.2 A Five Sector Bundled Triangle Odd Edge Flexagon
11.5.3 The Seven and Six Flexagon
11.6 Slipagons
11.6.1 General Properties
11.6.2 A Trihexaflexagon Slipagon
11.6.3 A Partial Overlap Silver Even Edge Slipagon
11.7 Coupled Point Flexagons
11.7.1 General Properties
11.7.2 A Coupled Triangle Point Flexagon
11.7.3 A Coupled Square Point Flexagon
12 Miscellaneous Flexagons
12.1 Introduction
12.2 Three Sector Odd Flexagons
12.2.1 General Properties
12.2.2 The Three Sector Fundamental Dodecagon Odd Edge Flexagon
12.2.3 A Three Sector Isosceles Triangle Odd Edge Flexagon
12.2.4 A Three Sector Hexagon Odd Skeletal Flexagon
12.3 Degenerate Odd Edge Flexagons
12.3.1 General Properties
12.3.2 A Degenerate Square Odd Edge Flexagon
12.4 Alternating Odd Edge Flexagons
12.4.1 General Properties
12.4.2 A Square Alternating Odd Edge Flexagon
12.5 Flapagons
12.5.1 General Properties
12.5.2 The Fundamental Square Duplex Flapagon
12.5.3 A Square Flapagon-Flexagon Hybrid
12.5.4 The Fundamental Isosceles Triangle Triplex Flapagon
12.6 Multiplex Edge Flexagons
12.6.1 General Properties
12.6.2 A Square Duplex Edge Flexagon
12.6.3 The Thrice Three-Fold Flexagon
12.7 A Hooke's Joint Flexagon
Indext Flexagon
12 Miscellaneous Flexagons
12.1 Introduction
12.2 Three Sector Odd Flexagons
12.2.1 General Properties
12.2.2 The Three Sector Fundamental Dodecagon Odd Edge Flexagon
12.2.3 A Three Sector Isosceles Triangle Odd Edge Flexagon
12.2.4 A Three Sector Hexagon Odd Skeletal Flexagon
12.3 Degenerate Odd Edge Flexagons
12.3.1 General Properties
12.3.2 A Degenerate Square Odd Edge Flexagon
12.4 Alternating Odd Edge Flexagons
12.4.1 General Pro
... weniger
Autoren-Porträt von Les Pook
Since my formal retirement in 1998 I have been a Visiting Professor in the Department of Mechanical Engineering at University College London, where I do some part time teaching in the fields of metal fatigue and fracture mechanics. I am a senior official of the European Structural Integrity Society, and have been involved with the organisation of two of their conferences (FCP 2003 and CP 2006), including editing conference proceedings and associated special issues of journals.
Bibliographische Angaben
- Autor: Les Pook
- XV, 329 Seiten, Maße: 16,5 x 24,2 cm, Gebunden, Englisch
- Verlag: Springer Netherland
- ISBN-10: 9048125022
- ISBN-13: 9789048125029
- Erscheinungsdatum: 27.08.2009
Sprache:
Englisch
Rezension zu „Pook, L: Serious Fun with Flexagons “
From the reviews:"This book comes across as a sort of encyclopedia or a classification of flexagons. ... The more serious researcher ... will find this book an amazing resource for various kinds of flexagons. ... I'm sure this book will remain a reference book of choice in the subject of flexagons for a long time. ... If you are looking for ... 'Encyclopedia of Flexagons' this flexagon book appears to be the best on the market." (Collin Carbno, The Mathematical Association of America, December, 2009)
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