Fig. 2: Fig. 2: The generation of the Golden Number. We can observe how, by means of the square, the form and harmony of all the mathematical derivations contained in this figure, symbol of matter, are generated. © Josep Gonzàlez

But

what

is

the

golden

number?

What

does

it

contribute

to

the

beauty

of

things,

to

their

form

and

growth,

and

to

the

organization of their internal structure?

We

could

describe

it

as

Euclid

did:

division

in

“Extreme

and

Mean

Ratio”

(Elements,

Book

VI,

def.

3):

“A

straight

line

is

said

to

have

been

cut

in

extreme

and

mean

ratio

when

the

[whole]

line

is

to

the

greater

segment

as

the

greater

segment

is

to

the

lesser”

(Fig.

2).

This

relation

between

segments

occurs

with

such

similarity

and

versatility

in

its

growth

that

it

is

able

to

create

the

family

of

the

five

bodies

or

“Platonic

solids”,

regular

polyhedra

constructed

and

analyzed

by

Theaetetus

and

with

which

Plato

studied

the

origin

and

structure

of

the

universe.

They

combined

Empedocles’

idea

of

the

existence

of

four

basic

elements

constituting

matter

with

Democritus’

atomic

theory.

And

the

fundamental

key

by

which

it

is

possible

to

unite

them

is

the

quintessence,

which

the

Pythagoreans

symbolized

with

the

five-

pointed

star,

a

polyhedral

figure

impossible

to

draw

if

the

“Golden”

division

is

unknown.

And

it

is

precisely

this

pentalpha

star

that

the

Pythagorean

fraternity

used

as

a

sign

of

recognition,

tracing

it

in

the

manner

of

the

master,

such

that

violation

of

the

supreme

secret,

the

golden

key,

meant

a

death

sentence

for

the

informer.

The

five

pre-Socratic

elements

of

creation:

earth,

water,

air,

fire

and

the

aether

or

universe,

were

represented

by

the

five

Platonic

solids.

Cube,

tetrahedron,

octahedron,

icosahedron

and

dodecahedron,

and

their

Pythagorean

equivalents

Gaia,

Heile,

Aer,

Hudor

and

Hieron

(Hyle),

were

generated

geometrically

by

inclusion

and

consecutive

tangency

of

each

in

the

next,

with

the

“Extreme

and

Mean

Ratio”

appearing

as

the

key

to

this

continuous

spatial

transformation

among

those

regular

polyhedra.

This

secret

key

would

be

inherited

by

all

the

initiatory

groups

derived

from

Pythagoreanism

and

Orphism,

being

kept

secret

(Fig.

3)

from

adept

to

adept,

or,

as

R.

Ambelain

would

say,

from

epopt

to

epopt,

until

at

the

dawn

of

the

Renaissance

it

came

to

be

revealed.

Its

geometric

formula

was

remembered

as

the

“figure

de

la

Victoire”

by

Philippe

Eléphant,

a

14th-century

sage

from

Toulouse

who

wrote

on

Euclid’s

Elementa.

Victory:

one

of

the

key

names

by

which

operative

masonry

and

the

compagnonnage

concealed the generating construction of the divine proportion.

.