Sacred Geometry

1. Sacred Geometry Dennis Blejer Fall 2009 School of Practical Philosophy and Meditation

2. Outline • Introduction • What is Sacred Geometry? • Why study Sacred Geometry • Examples from architecture, art, and astronomy • A few theorems from geometry and algebra • Equilateral Triangle, Regular Hexagon, and the Vesica Piscis • Square, Octagon, and the Golden Rectangle • Pentagon and Pentagram • Great Pyramid, Icosahedron, and Dodecahedraon

3. What is Sacred Geometry? • The study of the forms, proportions, and harmonies that underlie the growth and structure of things in the natural world, and in architecture, that glorifies the Divine • The tools of Sacred Geometry are the straight edge and compass, attention, creativity, and reason

4. Why Study Sacred Geometry? • “Let no one ignorant of Geometry enter herein” • Inscribed over the entrance to the Platonic Academy in Athens • Develops the higher faculties of man so that one becomes capable of contemplating and reflecting Truth itself (Platonic dialectic)

5. Why Study Sacred Geometry? Republic, Plato, Book VII You amuse me, you who seem worried that I impose impractical studies upon you. It does not only reside with mediocre minds, but all men have difficulty in persuading themselves that it is through these studies, as if with instruments, that one purifies the eye of the soul, and that one causes a new fire to burn in this organ which was obscured and as though extinguished by the shadows of the other sciences, an organ whose conservation is more important than ten thousand eyes, since it is by it alone that we contemplate the truth.

6. Which Rectangle is Most Pleasing? 1 2 3 4 5 6 7 8

7. Histogram of Preferences

8. 1 Φ-1 The Divine Proportion

9. Logarithmic Spiral

10. Golden Church

11. Pyramids of Giza

12. Parthenon

13. Roman Arch

14. Notre Dame

15. Rose WindowStrasbourg Cathedral, France

16. Mandala

17. Vesica Piscis

18. Vesica Piscis and Relationship to Great Pyramid

19. Vesica Piscis and Relationship to Gothic Arch

20. Vesica Piscis and the Hourglass Nebula

21. Stonehenge

22. Villa Emo

23. Waterperry House

24. Bronze and Geometry

25. Proportions of the Human Figure

26. Point, Line, Plane, and CircleThe Elements of Euclid • A point is that which has no part (dimensionless but defines a location) • A line is breadthless length (two points define a line; modern) • A plane surface is a surface which lies evenly with the straight lines on itself (Two intersecting lines define a plane; modern) • A circle is the locus of points equidistance from a central point (modern definition)

27. Sum of the Angles of a Triangle Equals 180 Degrees β α α φ β α β α φ β α β α α+β = 180° α+β+φ = 180°

28. All Triangles (inscribed) that have the Diagonal of a Circle as One Side are Right Triangles β α β α 2α + 2β = 180° α + β = 90°

29. Similar Triangles 3/2 2 1 1 3/2 2 • Corresponding angles are equal (AAA) • Corresponding sides are in proportion (SSS) • Two sides are in proportion and the included angles equal (SAS)

30. Pythagorean Theorem b C B a A

31. Golden Ratio Proportion

32. Golden Function

33. Constructing an Equilateral Triangle

34. Constructing a Regular Hexagon

35. Star of David

36. Circumscribe a Circle about an Equilateral Triangle

37. Vesica Piscis

38. Hexagonal Fleur de Li and the Vesica Piscis

39. Constructing a Square

40. Constructing a Regular Octagon

41. 3, 4, 5 Right Triangle √5/2 ½ h = 1/√5/2 = 4√5/10 ℓ = 3√5/10 √5/2 = 5√5/10 h 1 α ℓ ½ α ½ ½

42. Construction of the √2, √3, Double, and√5, Rectangles

43. Construction of the Golden Rectangle

44. Division of a Golden Rectangle into a Square and a Golden Rectangle

45. Golden Rectangle and Triangle

46. Golden Rectangle and the Pentagon

47. Pentagon and Golden Ratio • Side of square = 1 • Radius of circle = Φ • Side of pentagon = √(Φ+2) • Side of dodecagon = 1

48. Vesica Piscis as a Generating Figure

49. Pentagram

50. Pentagon and Pentagram