Sacred Geometry Philosophy and Practice by Robert Lawlor

Author:Robert Lawlor [Lawlor, Robert]
Language: eng
Format: epub
Tags: Métaphysique, Mathématiques
Published: 2015-10-13T22:00:00+00:00

Workbook 5

The Golden Proportion

We begin our search for a geometric division which requires only two terms by using two geometric ideas which are already familiar: the right triangle inscribed in a semicircle (Theorem of Thales) and the √2 (Workbook 1) which in this case will be the'radius of this semicircle. As shown on p. 45, we can use √2 as radius to give a division of line segments, a, b, c into a three-term geometric proportion.

Drawing 5.1a. Taking square ABCD, project the inner surface divisions by circular arcs onto the square's linear base. From this base line we will derive proportional relationships. With C as centre and radius CA, project base line we will derive proportional relationships. With C as centre and radius CA, project base line EG. Project line CD in a similar manner, giving line DF. Using the geometric theorem that the angle inscribed in a semicircle (diameter EG) is a right angle, we join AE and AG and find three similar triangles:

ΔEDA ≈ ΔEAG

ΔEAG ≈ ΔADG

ΔADG ≈ ΔEDA

Therefore,

a:b :: b:c,

and if a/b = b/c then b2 = ac.

In this case,

c = 2b+a, and a:b::b:2b+a.

a/b : b/(2b+a)

as compared with

a/b:b/(b+a)

b 2b+a

a:b::b:c.

c = a+b

hence, a:b::b:a+b

We then have the values:

side of the square AB = b = 1

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