Transcript Sacred Geometry - Philosophic Musings

Sacred
Geometry
Dennis Blejer
Fall 2009
School of Practical Philosophy and Meditation
Outline
1. Introduction
1.
2.
3.
4.
What is Sacred Geometry?
Why study Sacred Geometry
Examples from architecture, art, and astronomy
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?
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.
Which Rectangle is Most Pleasing?
1
2
3
4
5
6
7
8
Histogram of Preferences
The Divine Proportion
1
Φ-1
Logarithmic Spiral
Golden Church
Pyramids of Giza
Parthenon
Roman Arch
Notre Dame
Rose Window
Strasbourg 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 Circle
The 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 β
β α
α β
β α
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
3/2
1
1
2
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)
Pythagorean Theorem
b
C
B
a
A
Golden Ratio Proportion
A AB

B
A
 2    1/ 4  5 / 4
(  1/ 2)  5 / 4
2
A
Define  
so
B
  1
1

 2   1
   1
2
  1/ 2  5 / 2
1 5

 1.618
2
Golden Function
Golden Function
Golden Function
10
2
1.5
8
1
0.5
F(phi)
F(phi)
6
4
0
-0.5
2
-1
0
-2
-4
-1.5
-2
0
2
4
-2
0
0.5
1
1.5
phi
phi
F ( )      1
1 5
F ( )  0   
2
2
2
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
√5/2
½
h = 1/√5/2
= 4√5/10
h
1
ℓ = 3√5/10
α
ℓ
√5/2 = 5√5/10
½
α
½
½
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 square
=1
• Radius of circle
=Φ
• Side of pentagon
= √(Φ+2)
• Side of dodecagon
=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