Sacred Geometry

Dennis BlejerFall 2009

School of Practical Philosophy and Meditation

Outline

1. Introduction1. What is Sacred Geometry?2. Why study Sacred Geometry3. Examples from architecture, art, and astronomy4. A few theorems from geometry and algebra

2. Equilateral Triangle, Regular Hexagon, and the Vesica Piscis

3. Square, Octagon, and the Golden Rectangle

4. Pentagon and Pentagram

5. Great Pyramid, Icosahedron, and Dodecahedraon

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

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)

Why Study Sacred Geometry?

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.

Republic, Plato, Book VII

Which Rectangle is Most Pleasing?

1 2 3 4

5 6 7 8

Histogram of Preferences

The Divine Proportion

Φ-11

Logarithmic Spiral

Golden Church

Pyramids of Giza

Parthenon

Roman Arch

Notre Dame

Rose WindowStrasbourg Cathedral, France

Mandala

Vesica Piscis

Vesica Piscis and Relationship to Great Pyramid

Vesica Piscis and Relationship to Gothic Arch

Vesica Piscis and the Hourglass Nebula

Stonehenge

Villa Emo

Waterperry House

Bronze and Geometry

Proportions of the Human Figure

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)

Sum of the Angles of a Triangle Equals 180 Degrees

α

α

α

α

β

β

β

β

α+β = 180°

α

α

β φ

φ

α+β+φ = 180°

All Triangles (inscribed) that have the Diagonal of a Circle as One

Side are Right Triangles

α

α

β

β 2α + 2β = 180°α + β = 90°

Similar Triangles

• Corresponding angles are equal (AAA)

• Corresponding sides are in proportion (SSS)

• Two sides are in proportion and the included angles equal (SAS)

1

1

3/2

3/2

2

2

Pythagorean Theorem

C

A

B

a

b

Golden Ratio Proportion

2

2

Define so

11 1

1

A A B

B A

A

B

2

2

1/ 4 5 / 4

( 1/ 2) 5 / 4

1/ 2 5 / 2

1 51.618

2

-4 -2 0 2 4-2

0

2

4

6

8

10Golden Function

phi

F(p

hi)

0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Golden Function

phi

F(p

hi)

Golden Function

1 5( ) 0

2F

2( ) 1F

Constructing an Equilateral Triangle

Constructing a Regular Hexagon

Star of David

Circumscribe a Circle about an Equilateral Triangle

Vesica Piscis

Hexagonal Fleur de Li and the Vesica Piscis

Constructing a Square

Constructing a Regular Octagon

3, 4, 5 Right Triangle

½

α

α

½

½

½

√5/2

1

h

ℓ

h = 1/√5/2 = 4√5/10

ℓ = 3√5/10

√5/2 = 5√5/10

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

Construction of the Golden Rectangle

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

Golden Rectangle and Triangle

Golden Rectangle and the Pentagon

Pentagon and Golden Ratio

• Side of squareSide of square = 1 = 1 • Radius of circle Radius of circle = = ΦΦ• Side of pentagonSide of pentagon = √( = √(ΦΦ+2)+2)• Side of dodecagonSide of dodecagon = 1 = 1

Vesica Piscis as a Generating Figure

Pentagram

Pentagon and Pentagram

Great Pyramid of Gizah

Platonic Solids

Golden Rectangular Solids

Icosahedron and Dodecahedron and Inscribed Golden Rectangles

Bibliography

• Sacred Geometry, Robert Lawlor, 1982, Thames and Hudson

• Geometry of Art and Life, Matila Ghyka, 1946, Dover