Sacred Geometry Dennis Blejer Fall 2009 School of Practical Philosophy and Meditation.
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Transcript of Sacred Geometry Dennis Blejer Fall 2009 School of Practical Philosophy and Meditation.
![Page 1: Sacred Geometry Dennis Blejer Fall 2009 School of Practical Philosophy and Meditation.](https://reader035.fdocuments.net/reader035/viewer/2022062221/56649f325503460f94c4ee46/html5/thumbnails/1.jpg)
Sacred Geometry
Dennis BlejerFall 2009
School of Practical Philosophy and Meditation
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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
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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
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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)
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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
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Which Rectangle is Most Pleasing?
1 2 3 4
5 6 7 8
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Histogram of Preferences
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The Divine Proportion
Φ-11
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Logarithmic Spiral
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Golden Church
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Pyramids of Giza
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Parthenon
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Roman Arch
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Rose WindowStrasbourg Cathedral, France
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Mandala
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Vesica Piscis
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Vesica Piscis and Relationship to Great Pyramid
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Vesica Piscis and Relationship to Gothic Arch
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Vesica Piscis and the Hourglass Nebula
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Stonehenge
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Villa Emo
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Waterperry House
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Bronze and Geometry
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Proportions of the Human Figure
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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)
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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°
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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°
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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
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Pythagorean Theorem
C
A
B
a
b
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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
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-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
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Constructing an Equilateral Triangle
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Constructing a Regular Hexagon
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Star of David
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Circumscribe a Circle about an Equilateral Triangle
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Vesica Piscis
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Hexagonal Fleur de Li and the Vesica Piscis
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Constructing a Square
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Constructing a Regular Octagon
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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
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Construction of the √2, √3, Double, and√5, Rectangles
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Construction of the Golden Rectangle
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Division of a Golden Rectangle into a Square and a Golden Rectangle
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Golden Rectangle and Triangle
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Golden Rectangle and the Pentagon
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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
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Vesica Piscis as a Generating Figure
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Pentagram
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Pentagon and Pentagram
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Great Pyramid of Gizah
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Platonic Solids
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Golden Rectangular Solids
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Icosahedron and Dodecahedron and Inscribed Golden Rectangles
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Bibliography
• Sacred Geometry, Robert Lawlor, 1982, Thames and Hudson
• Geometry of Art and Life, Matila Ghyka, 1946, Dover