The document discusses the polygonal technique for deriving Islamic geometric patterns. It describes how patterns are created by placing a compass at key points on a tessellating polygonal grid. Various historical examples from the 11th-16th centuries demonstrating this technique are provided. Complex compound patterns combining different symmetrical grids, such as 10-fold and 12-fold, or 11-fold and 13-fold, are also examined.

1. The Polygonal Technique The traditional use of polygonal matrices in the design of Islamic geometric patterns. Jay Bonner Design Consultant Santa Fe, New Mexico USA Email: [email_address] Web Site: www.bonner-design.com

2. Pattern Derivation – Compass-Work: Great Mosque of Damascus (715 C.E.), Pierced Stone Screen

3. Pattern Derivation – Compass-Work: Patterns are created by placing the compass at key points upon a tessellating polygonal grid

4. Pattern Derivation – Compass-Work: Quran of Ibn al-Bawwab (1000-1001 C.E.)

5. Pattern Derivation – Graph Paper Technique:

6. Pattern Derivation - Parallel Radii Technique

7. Pattern Derivation - Parallel Radii Technique

8. Pattern Derivation – Parallel Radii Technique: Sultan Han, Aksaray, Turkey (1229 C.E.)

9. Pattern Derivation – Additive Technique

10. Pattern Derivation – Point Joining

11. Pattern Derivation – Point Joining

12. Pattern Derivation – Comparative Methods

13. Pattern Derivation – Polygonal Technique The Four Pattern Families

14. The Polygonal Technique: Historical Evidence Mamluk stone work: Sultan Hassan Madrasa, Cairo (1363 C.E.)

15. The Polygonal Technique: Historical Evidence Mamluk historical example with scribed “dead-lines” under the painting.

16. The Polygonal Technique: Historical Evidence 15 th Century Mughal Jali Screen, Probably Fatipur Sikri, Agra, India

17. The Polygonal Technique: Historical Evidence One of 24 examples of the Polygonal Technique from the Topkapi Scroll. Fifteenth or sixteenth century: Western Persia.

18. Polygonal Technique The Classic 5-Fold Acute Pattern

19. Polygonal Technique The Classic 5-Fold Middle Pattern

20. Polygonal Technique The Classic 5-Fold Obtuse Pattern

21. Polygonal Technique The Classic 5-Fold 2-Point Pattern

22. 5-Fold System – the Classic Acute Pattern: Qala-i Bust, Bust, Iran (c. 1000 C.E.)

23. 5-Fold System – the Classic 2-Point Pattern: Inlay Work, Egypt, 16 th Century C.E.

24. The Polygonal Technique A Single Pattern Can Sometimes Be Produced From More Than A One Sub-Grid. The Ring of Pentagons allows for tremendous design versatility.

25. The Polygonal Technique Variations to the Pattern Lines within the Ring of Pentagons

26. 3-Fold Sub-Grid 12 Non-Regular Pentagons around a Regular Dodecagon

27. 3-Fold Pattern with 12-Pointed Stars Pattern Derivation

28. 3-Fold Pattern: 12-Pointed Stars Sub-Grid Construction Note: Three Pentagons Together are Common in 3-Fold Sub-Grids

29. Compound Pattern – 10s and 12s:

30. Compound Pattern – 10s and 12s: Topkapi Scroll with Variation at the Three Touching Pentagons

31. Compound Pattern – 10s and 12s: Sub-Grid Note: the Proportion of the Rectangular Repeat Unit is a Direct Result of the Compounded 10-Fold and 12-Fold Symmetry

32. Compound Pattern – 10s and 12s: Pattern Derivation

33. Sub-Grid Construction: 10s and 12s: Note: The Problem of the Non-Aligning Edges is Overcome by the Ring-of-Pentagons

34. Compound Pattern – 11s and 13s: A particularly complex and beautiful compound pattern is found at the Tomb of Mu’minah Khatun, Nakhichevan, Azerbaijan.

35. Compound Pattern – 11s and 13s: Detail from the Tomb of Mu’minah Khatujn, Nakhichevan, Azerbaijan. This compound Islamic geometric pattern exemplifies the highest expression of this art form.

36. Compound Pattern – 11s and 13s: Once Again, Two Different Elongated Hexagons as the Repeat Unit Having Equally 11-Fold and 13-Fold Symmetry!

37. Compound Pattern – 11s and 13s: Pattern Derivation

38. Sub-Grid Construction: 11s and 13s: Polygonal Sub-Grid Construction using the Radii Matrix technique – Part One

39. Sub-Grid Construction: 11s and 13s: Polygonal Sub-Grid Construction using the Radii Matrix technique – Part Two

40. Sub-Grid Construction: 11s and 13s: Polygonal Sub-Grid Construction using the Radii Matrix technique – Part Three

41. Sub-Grid Construction: 11s and 13s: Polygonal Sub-Grid Construction using the Radii Matrix technique – Part Four

42. Completed Sub-Grid: 11s and 13s

43. Pattern Construction: 11s and 13s:

44. Pattern Construction: 11s and 13s:

45. Pattern Construction: 11s and 13s:

46. Pattern Construction: 11s and 13s:

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50. Pattern Construction: 11s and 13s:

51. Pattern Construction: 11s and 13s:

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53. Pattern Construction: 11s and 13s:

54. Pattern Construction: 11s and 13s:

55. Pattern Construction: 11s and 13s:

56. Pattern Construction: 11s and 13s:

57. Pattern Construction: 11s and 13s:

58. Pattern Construction: 11s and 13s:

59. Pattern Construction: 11s and 13s:

60. Pattern Construction: 11s and 13s:

61. Pattern Construction: 11s and 13s:

62. Pattern Construction: 11s and 13s:

63. Pattern Construction: 11s and 13s:

64. Pattern Construction: 11s and 13s:

65. Compound Pattern – 11s and 13s: Redrawn image of this exceptional Azerbaijani pattern

66. Allahu Akbar Thank You Jay Bonner Design Consultant Santa Fe, New Mexico USA Email: [email_address] Web Site: www.bonner-design.com