PHI & Five
PHIvesided Symmetry
The easiest way to figure PHI is with a simple pentagon,
the geometric figure with five equal sides
meeting at five equal angles.
This fivesided figure embodies PHI. Most simply and precisely,
PHI is the ratio of any diagonal
to any side of a pentagon.
Draw a regular pentagon ABCDE,
with equal sides and equal angles.
Then draw a diagonal as line AC (red)
connecting any two vertexes of the pentagon.
Divide the length of a diagonal AC
by the length of a side AB
This ratio is an accurate value for PHI.
You can install five such diagonal lines,
and they are all of equal length.
Here we encounter PHI's first curious quirk.
Draw a second diagonal BD inside the pentagon
so that this new line crosses the first diagonal at point O.
Each diagonal is divided into two parts,
and each part is in PHI ratio to the other,
and to the whole diagonal.
Thus, the PHI ratio reoccursÑor "recurs"Ñ
every time any diagonal is divided by another diagonal.
In the next dimension, you see how valuable this recurrence is.
When you draw all five pentagon diagonals,
they form a fivepoint star:
a Pentangle.
Inside this star is a smaller, inverted pentagon (green).
Each diagonal is crossed by two other diagonals,
and each segment is in PHI ratio to the larger segments
and to the whole.
Thus, the inverted inner pentagon
is in PHI ratio to the initial outer pentagon.
Thus, PHI is the ratio of fivesided symmetry.
Inscribe the Pentangle star inside a Pentagon
and you have the Pentagram,
symbol of the ancient Greek School of Mathematics
founded by PythagorusÑ
solid evidence the ancient Mystery Schools knew about PHI
and appreciated the Divine Ratio's multitude of uses
to form our physical and biological worlds.
While the pentagon makes it very easy and simple to derive and describe PHI, this ratio has several remarkable properties that are much more difficult to understand, yet they allow this ratio to find numerous profound applications to create our biological and cosmological realities.
Extreme & Mean
Resonant Recursion
PHI is also found in another much more obscure and obtuse relationship.
The Extreme and Mean Proportion reveals more insight
into PHI's unique and useful properties.
In simple language, this mathematical mindtwister says:
One unique point exists
that divides a line into two unequal segments
so that the whole is to the greater
as the greater is to the lesser


In simple mathematical shorthand:
whole
greater 
= 
greater
lesser



In a simple geometric line diagram:
where:
This odd, obscure ratio equals PHI.
The value of this isn't immediately obvious,
but becomes much more apparent when we divide the long segment
by the short segment
to locate a new point D.
The simple way to accomplish this
is to set a compass to line segment BC,
then rotate around point C
to mark a new point D on segment AC.
ACÑthe old long segmentÑ
is now the whole.
CD is the new long segment.
AD is the new short segment.
AC
CD 
= 
CD
AD 
= 1.618 = PHI



All three lines are in the same PHI ratio relationship.
Repeat this operation again.
Set the compass to line AD and rotate around point D to locate new point E on the long segment CD.
Line CD is thus divided
into two unequal segments CE and DE.
The three new line segments are again in PHI ratio:
CD
DE 
= 
DE
CE 
= 1.618 = PHI



Repeatedly dividing each long segment
by the short segment
will create new line segments
that are also in PHI ratio at each smaller scale, no matter how many times this division is repeated.
DE
EF 
= 
EF
DF 
= 1.618 = PHI



Thus, in Extreme & Mean Proportion, PHI recurs again and again. Every time the lesser is used to divide the greater into two shorter, unequal segments, the resulting line segments are in PHI ratio.
From these simple operations on a line, you can see that PHI is a resonant ratioÑa geometric echo. Once established, this ratio recreates itself through successive operations.
Recursion is a very useful property, and a key characteristic in the design of forms and flow systems that are stable at multiple levels of scale. Recursion is also a mechanism to pass information as ratio through successive layers of scale.
PHI provides the recursion required for regeneration
Ñthe resonant ratio to govern and guide growth.
Square & Circle
Embodiment
PI defines the geometric dance of the line and with the circle, while PHI describes the union of the square with the circle. PHI defines the squaring of a circle. Stated in mathematical terms, this says:
Given a square of known perimeter,
create a circle of equal circumference 

In Sacred Geometry, the Square represents the material worldÑ the Four Directions of space. The Circle is ZerOÑsymbol of The VoidÑdoorway to emptinessÑgateway to the dimensions of spiritÑthe Original Mystery.
Our numerical symbol for "Five" encodes this metamath mystery of mystic union:
above: two sides of a square
below: an open, partial circle 

Five unites spirit with matter to capture life in physical form. In its essence, PHI is embodimentÑhow to enclose spirit in a physical form.
In ancient Egypt, this mathematical mystery was encoded in the dimensions of The Great Pyramid. If the base of the Pyramid is the square, then the height of the Pyramid ("apotheom") is the radius of the circle.
Further, in The Great Pyramid, PHI is also encoded in the proportion of its base and height. If the base of the Pyramid is two units, then the height from apex down the center of each face to the edge of the base is PHI units.
In human anatomy, this is embodied most simply as four fingers and an opposable thumb.
Careful analysis of our human dimensions reveals many PHI ratios. Indeed, the great Italian Renaissance artist and inventor Leonardo daVinci made a classic study of the proportions of the human form embedded in a square, circle and pentagon.
Ñ
