The Spiral of Sevenfold Dynamic Systemization

THE SPIRAL OF SEVENFOLD

DYNAMIC SYSTEMIZATION

...the eternity of the DAO rests on the fact that all its move ments "return" into
itself. All opposites are eliminated by it by being balanced against one another,
so that every movement neces sarily turns into its opposite. --Wilhelm
[W533:19]

... the world materializes and man spiritualizes along the same spiral. It is the
breathing of the cosmos. --Purce [P538:11]

xxxx
Up until now we have allowed a certain ambiguity in our use of the circle
to go unexplained. On the one hand we have made much of the distinction
between the circumference (+) of a circle and its centre (-), with the former
representing reflection and its form (viz. the twelvefold form of compound
systemization) in one continuous dynamic unity and the latter representing
concrete experience, the resting-point of all reflective systemization. Yet on the
other hand, when it actually comes to mapping the twelve expressions onto the
circle, we have sometimes found it helpful to leave a gap at the top, in order to
represent the difference between the starting point and the end point of the
twelvefold progression. This gap implies that the dynamic circle is not a circle at
all, but a spiral. For if the end of the curve which is closest to the centre is
extended according to the same pattern (i.e. always leaving a gap when it reaches
the top again), then eventually the curve will come to rest at the centre of the
figure. As we shall see in 7.2-3, both these figures--i.e. both the complete circle
and the spiral which is generated out of the broken circle--are related to each
other by means of their common participation in the sphere. But before
examining how this is so [see 7.2], we should be clear about the ways in which
different sorts of two-dimensional, or "flat", spirals are relevant to the Geometry
of Logic.
Before a spiral can be used as a symbol for logical relations, we must
determine how to cope with its infinite character. For a true spiral, in the
geometrical sense, progresses infinitely around its centre, so that there is no
definite point at which it begins or ends. This infinite character makes the spiral a
profound symbol of the ultimate unity of analytic and synthetic logic, for they
both repeatedly "turn into" each other, ultimately merging in an indistinguishable
common point. But the operation of logical systemization always assumes some
definite perspective, which in turn requires limitations to be set up before a
concept or symbol can be clearly defined as a logical tool.
The most obvious limitation to put on the spiral is to specify a definite
number of revolutions on its path towards the centre. Positing three revolutions
not only enables us to represent the essential properties of the spiral in their
simplest and most elegant form, but also accords well with the higher degree of
logical systemization which the spiral can represent. For the dynamic circle starts
with threefold synthesis, and relates four such syntheses together in an analytic
pattern; so if we assume an alternating pattern, we must now return to three. As
such, the spiral represents the systematic synthesis of three systems of
twelvefold compound relations (i.e. "3.4.3 = 36"). An initial, "static"
representation of the flat spiral, therefore, is given in Figure 7.1a.@ Note that
the easiest way to construct this figure is to inscribe it in a rectilinear spiralling
maze [see Figure 7.1b]. The maze is constructed by lengthening each line by a
consistent length (say 1 cm), starting from the centre. The spiral can then be
constructed by positioning the fixed point of a compass either at the centre or at
the first corner in the maze (depending on which is

Figure 7.1: The Simple Threefold Spiral
(a) The Curvilinear Figure(b) The Rectilinear Matrix for 1)the Construction of
the Spiral

Figure 7.2: The Four Types of Regular Threefold Spiral, Mapped onto the Cross

central to a given side), and constructing successively connecting semicircles
within each half-square.
An important feature of this simple threefold spiral is that any straight line
which passes through its centre will intersect the curve exactly six times.
Including the tangential intersection of the curve at its centre (i.e. at its end or
starting point, as the case may be), this makes a total of seven intersections--yet
another example of the "6+1 = 7" pattern which is becoming more and more
prominent as we proceed. For now the simple recognition of the occurrence of
this pattern will have to suffice; but in 7.3-4 we will be adequately prepared to
give a full account of its significance.
In order to convert the static spiral into a dynamic spiral, we must
determine the direction of its logical development (i.e. either clockwise or
counterclockwise), as well as the direction towards which it is open (i.e. either to
the left or to the right). Taken together, these alternatives yield a total of four
ways of drawing the two-dimensional, three-revolution spiral mentioned above.
(Compare the four corresponding figures given in Figure 6.8.) By regarding the
former pair as presenting us with a first level analytic division, and the latter as
presenting us with a second level analytic division, and by correlating
"clockwise" and "right" with "+" and their opposites with "-", we can map all
four possibilities onto the standard cross as in Figure 7.2.@ The fact that the two
spirals which point outwards are at the "pure" points of the cross ("++" and "--
"), while those which point inwards are at the "mixed" points ("-+" and "+-") is
not accidental. For, as we noted in 2.2, the pure points are both strong or active
in relation to the mixed points, as represented by the arrows pointing down and
to the right. Accordingly, the evolution of the outward pointing spirals conveys a
definite sense of activity or control (cf. the perspective of analysis), while the
involution of the inward pointing spirals conveys the opposite sense of
possessing or being under the control of another (cf. the perspective of synthesis
[see 4.1].
The importance of recognizing which of the four spirals is appropriate to
a given material system cannot be over-stressed. For the way in which the twelve
expressions in compound systemization are mapped onto the dynamic circle is
entirely dependent upon the direction of developmentand the direction of
openness of the spiral of which it is a part. Indeed, we have already seen in
Figure 6.8 that, for example, a circle whose direction is counterclockwise and
flowing inwards (-+) is mapped quite differently than the standard logical circle
(+-). Both of these cases use the same rule for mapping each of the four stages
of synthetic integration: viz. they both assume that each simple synthetic
integration follows the direction of development, so that the expressions with 1's
in them suppress those with 0's. These two spirals therefore follow a synthetic
method of mapping. By contrast, the two clockwise spirals follow an analytic
method, in which the 0's suppress the 1's, and the order of the synthetic
progression is reversed [see 6.3]. Thus the analytic method views each spiral as
pushing its centre out towards the extremes, while the synthetic method views
each spiral as drawing its extremities in towards the centre.
Each of the four "regular" forms of the flat spiral given in Figure 7.2 is
partial in itself. One reason is that only a relatively small number of material
systems could actually be regarded as literally corresponding to the spiral in this
threefold form. An even more important reason is that the spiral implies a balance
of opposites, yet on its own each of the spirals considered above is imbalanced,
since the logical development progresses in only one direction in each case. The
imbalance was offset by the fact that we mapped the four spirals onto the
balanced poles of the analytic cross. But in order fully to rectify the situation, we
must suggest several ways in which the balance can be fully restored within a
single spiral figure as such. This can be accomplished either by altering the shape
of the standard figure, or by combining two or more of them, or by using both
methods.
The simplest way to alter the spiral is to change the number of
revolutions it makes. The main reason for making the threefold spiral the
standard is to enable us to draw some crucial systematic distinctions towards the
end of this chapter. If a particular system is better symbolized by a spiral with
two, or five, or any number of revolutions, then there is certainly no logical or
geometrical reason why it should not be employed.
Another modification would be to alter the number of curves, or
"spokes", which converge on or diverge from the centre. In particular, a two-
spoke spiral could be used to depict both the evolutionary and the involutionary
paths of logical development [see Figure 7.3].@ The ancient good luck sign
known as the swastika, incidentally, is a four-spoked, rectilinear spiral with only
one-half a revolution.
Any of the above variations can be duplicated by revolving the figure by
180\ on its outermost point and attaching it to another figure in the original
position. Thus, for example, the "--" and "+-" spirals in Figure 7.2 can be
combined to form a double spiral [Figure 7.4a]. The same can be done with the
"++" and "-+" spirals, or indeed, with all four flat spirals at once [Figure 7.4b].
(Spirals such as the one in Figure 7.3 can also be combined with their 180\
counterpart in the same way.) Such connections are possible because of the
spiral's inherent regularity. However, because of its inherent lack of symmetry,
a spiral can never be made to face the opposite direction merely by manipulating
it within a plane. Of course, if the "++" and "+-" spirals, for instance, are both
given, then the two can easily be combined into a double spiral, as in Figure
7.4c.@ But in order to convert the "++" spiral to the "--" spiral, we would have
to invert it by rotating it through a third dimension. Since a spiral in three
dimensions is based not so much on the circle as on the sphere, we shall turn
now to a discussion of the latter. Only after understanding the logical symbolism
of its geometrical structure will we be adequately equipped to consider the
spherical spiral.

Figure 7.3: The Two-Spoke Flat Spiral

Figure 7.4: The Multiple Flat Spiral
(a) The "--" and "+-" Spirals(c) The "++" and "+-" Spirals
(b) All Four Flat Spirals

7.2 The Sphere as a Sixty-Four-Fold System
The solid figure whose radical simplicity is the three-dimensional
equivalent of the two-dimensional simplicity of the circle is the sphere. Plato,
like many others, regarded the perfect symmetry of the sphere as symbolic of the
perfect form upon which the universe itself is structured [P494:33B].
Commenting on this symbolism, C495:54 explains:

The sphere is the most uniform of all solid figures, and the only one which, by
rotating on its axis, can move within its own limits without change of place. This
axial rotation symbolizes the movement of Reason and is superior to all
rectilinear motions.

.... [Moreover,] the sphere has a greater volume than any other solid figure with
plane sides, having the same perimeter.

Like the spiral, the infinite character of the sphere (in this case, its infinite
symmetry) must be limited by some defining rules before it can be a useful tool
in the Geometry of Logic. Just as the circle is defined either in terms of its
circumference or of its area [see e.g. note 6.18], so also the logical definition of
the sphere can refer either to its surface or to its volume. Because of its perfect
symmetry, there is an infinite number of ways to draw limitations of either sort.
But we shall limit our discussion to one example of each. We can then return to
our discussion of the spiral by mapping it onto the sphere in 7.3.
The area of the sphere can be divided into eight equal sections by
inscribing it within the "three-dimensional cross", mentioned briefly in 2.2 [see
Figure 2.21]. This tames the sphere to some extent, and enables the logician to
use it as a map for a number of different sorts of logical relation. The three
planes intersect within the sphere to form three axes (and hence the six
directions: right, left, up, down, front, back); and these combine to form three
distinct ("internal") two-dimensional crosses [see Figure 7.5a]. The edges of
these planes extend beyond the sphere to form six additional ("external") two-
dimensional crosses at the points where they would intersect the planes of a
concentric cube [see Figure 7.5b].@
The rich parallelism which can be drawn between the geometrical figure
of the double cross and the logical distinctions we have introduced is
noteworthy, to say the least. We already saw in 2.3 how the eight spaces defined
by the three internal double crosses (xy, xz, yz) can be correlated with third level
(eightfold) analytic division [Figure 2.21], and how the cube is actually the
inversion of the three-dimensional cross [Figure 2.20]. We can now point out
the additional fact that the figure contains twelve external corners, or vertices,
twelve sections of plane, and twelve external axes (two for each external two-
dimensional cross). The twelve expressions which define compound
systemization could be mapped onto any of these aspects of the figure. But the
sections of plane would be most appropriate, since the external axes (and so also
the corners) are better reserved for use in combination with the three internal axes
(6+3 = 9) to represent the logical relation between first and second levels of
synthetic integration. For the three crosses which are parallel to each other in
each of the three directions (e.g. x, x, x) could represent a simple synthetic
relation, while the three sets of three in conjunction (i.e. the x's, y's and z's)
could represent synthetic integration on the more complex, second level.
Assigning material elements to the four poles of each cross in these three
sets of three crosses would yield a twelvefold system of

Figure 7.5: The Sphere and the Three-Dimensional Cross
(a) The Three Internal Crosses
(b) The Six External Crosses

relations proceeding in each direction; thus the three-dimensional cross can be
used to represent a thirty-six-fold relation (3.3.4 = 36), much as the standard
spiral represented in 7.1. A possible source of confusion here is that each of the
corners, along with the centres of the six external crosses, could be labelled
twice [cf. 4.2]; that is, each of the eighteen geometrical elements would have to
be viewed from two different perspectives (one from each of the two dimensions
in the plane in which the element in question is placed) in order to yield a total of
thirty-six logical elements. A final point of interest is that there are six distinct
centres for the external crosses, while the internal crosses share a common centre
(6+1 = 7). Each of the former plays an active role in the thirty-six-fold system
mentioned above, while the latter does not, since it is never the end point for a
cross, viewed from any of the dimensional perspectives. Once again, therefore,
we have a graphical illustration not only of the interesting relation between "7"
and "12", but also of the logical principle that the sixfold, perspective-bound
elements around which compound systemization revolves come to rest in a
seventh, perspective-free (or perhaps, omni-perspectival) element.
Since each of the eight divisions of the area of the three-dimensional
cross is itself cube-shaped like the original, it too can be divided into eight parts
(e.g. by its eight corners), thus yielding a total of sixty-four elements (8.8 = 64).
This way of dividing the solid cross would not apply to the smaller area which is
enclosed by the sphere. However, the sphere's surface does allow for such a
division. For the three-dimensional cross cuts the sphere's surface with twelve
arcs into eight parts, each of which can be divided into eight distinct curved
triangles [see Figures 7.6a and b].@ The sixty-four surface sections of the
sphere can also be described in terms of the analytical relation between its three
internal crosses, since 4.4.4 = 64. As such the sphere represents sixth level
analytic

Figure 7.6: The Sixty-Four-Fold Division of the Sphere
(a) A One-Eighth Segment(b) The Solid
(c) The Sphere's Projection

division (2 = (2) = 64). Yet analytic division on this level also evinces some of
the patterns of compound systemization. By projecting the sphere onto a plane
[see Figure 7.6c], the similarity between it and the star figures inscribed into the
circle in 6.3 becomes immediately apparent. The points of the stars are created by
various combinations of the "3" triangles in the "3+1 = 4" pattern on either side
of each of the eight segments (as shown in Figure 7.6a). Moreover, the two
middle rings in Figure 7.6c each consist of twenty-four triangles (12.2), while
the outer and inner rings each consist of only eight triangles (8+12+12+12+12+8
= 64).
An obvious application of this way of dividing the sphere should be
mentioned at this point, since it is not directly related to the applications of the
spherical spiral which will be given in 7.4. The sixty-four-fold division of the
sphere is a geometrical model for what is undoubtedly the oldest logical system
known to modern man. For the ancient Chinese developed in elaborate detail the
logical relations between the elements in a sixth level analytic division. The fact
that this logical system, expounded most thoroughly in the I Ching (Book of
Changes) [W509], was used largely as a canon for divination by its Chinese
advocates should not prevent us from recognizing its thoroughly logical, and
thoroughly systematic character. We shall conclude our examination of the
sphere, therefore, by showing how the elements of the I Ching can be mapped
onto its sixty-four segments.
Sixth level analytic division requires six positive or negative terms in
each expression in order to produce its requisite sixty-four expressions. The
Chinese, rather appropriately, categorized the six terms in each expression into
two "trigrams", thus giving each expression (each "hexagram") the form of a
double synthetic relation (i.e. "2.3 = 6"). At the level of the individual
expression, this type of logical relation has a clearly synthetic structure (as
reflected by the use of triangles in Figure 7.6), even though the general structure
of the overall system is best regarded as analytic. One way of reflecting this
difference in one's interpretation of the system is to arrange the six terms in three
rows of two, so that each hexagram could be read either vertically, as three
single level analytic relations, or horizontally, as two simple synthetic relations,
or (best of all) as a combination of both. The traditional Chinese notation [see
Figure 7.7a] uses solid and broken horizontal lines stacked up in vertical
columns of six, which makes it unfortunately difficult to determine the analytic
relations at a glance. This is rectified, and the systematic coherence of sixty-four-
fold forms of relation is established, in Figure 7.7b, where the same sixty-four
expressions are mapped onto the projection of the sphere, using the same rules
we have been using all along to guide the arrangement of the expressions on the
map.@
Any suggestion that systems such as the I Ching in the East, or the
zodiacal astrology in the West, might have some firm foundation in formal logic
is likely to be met with a good deal of scepticism (perhaps even some laughter)
by the average post-Enlightenment thinker. The reason, no doubt, is that we in
the West tend to view logic and logical principles exclusively from what I would
call the "analytic a priori", or "logical" perspective, whereas these systems have
traditionally employed logical principles from what I would call the "analytic a
posteriori", or "practical" perspective. The former emphasizes the rules being
followed and the rigorous argumentation upon which each step in a system is
based; the latter emphasizes the way the system is to be applied to one's
everyday life. The latter systems are not illogical merely because they choose not
to adopt a logical perspective. On the contrary, they are rooted in the fundamental
principles of formal logic.
The difficulty in accepting this, for the Western man, is only made worse
by the fact that virtually all who expound the Eastern approach do so without
giving clearly explained and cogently argued

Figure 7.7: The Arrangement of the I Ching Hexagrams
(a) Using the Traditional Notation
(b) Using the "+" and "-" Notation, Mapped onto the Sphere's Projection

reasons for making the distinctions and applications they make. This is perhaps
not surprising, since the need for such a defence arises only when the opposite
perspective is assumed. Yet this is not a sufficient excuse, for the two
perspectives are at least potentially complementary. The apparent flights of fancy
which characterize the works of many defenders of the practical perspective are
bound to grate against the mind of one who believes exclusively in the validity of
an analytic a priori approach to logic. But there is no reason why one cannot use
both perspectives, each to fill the gap left by the other, as I have been attempting
to do in this book. When this affinity begins to be recognized to the extent that it
can be incorporated into the way we naturally think and act, then the alienated
siblings of Western and Eastern modes of thought will finally begin to be
reunited in the common unity in which both have their roots.

7.3 The Spherical Spiral as a Sevenfold Process
The main reason for considering the symbolic value of the sphere in 7.3
has been to set the scence for an examination of the spiral in its various three-
dimensional forms. The simple threefold spiral can be transferred to a three-
dimensional figure with minimal effort of the imagination simply by grabbing
hold of the centre of the flat spiral with one hand and the outermost point with
the other hand, and pulling them apart like a spring. On its own the resulting
figure would represent the same thirty-six-fold (12.3) systematic relations as the
flat spiral does, so the extra dimension would be superfluous. But the expanded
version is indispensable when it comes to combining spirals into larger
systematic units.
The most important example of a compound spiral in three dimensions is
the one which can be constructed by mapping two expanded flat spirals onto the
surface of a sphere. The result [see Figure 7.8a] is a geometrical representation
of the logical integration of a set of systems in a higher System.@ This standard
model of the spherical spiral represents the ultimate integration of the static and
the dynamic, as well as of the analytic and the synthetic, aspects of formal logic.
Its cyclical pattern clearly implies a continuous dynamic flow of relations
between opposing yet integrated elements of all sorts. The six revolutions
represent the combination of two opposing spirals (either the "++" and "+-"
spirals or the "--" and "-+" spirals in Figure 7.2), and the line down the centre of
the sphere unites the starting point of one with the end point of another in a
common seventh point of rest (i.e. non-revolution). When this spherical spiral is
viewed from the top [see Figure 7.8b], it looks just like two flat spirals
overlapping: the seventh "revolution appears as a single point.
An awareness of the dynamic logical development inherent in the
spherical spiral is essential for anyone who wishes to grasp its symbolic
meaning. But no less important is an awareness of the static stages through
which this process is passing as it flows through its cyclical pattern. This pattern
can be broken down into stages in a number of ways without altering the
systematic level at which we are working. (At lower levels, of course, this same
system can be broken down into a multitude of particular elements or relations
between elements.) One way would be to regard the extremities of the widest
loop of the spiral as representing two opposing stages, and the extremities of top
and bottom (i.e. the endpoints of the two smallest loops) as representing two
others. By connecting these points with straight lines within the spiral and
labelling the four points with "0", "+", "-", and "1", we can depict the
extremities of the spiral as united by a simple synthetic relation, pictured in
Figure 7.9 in its expanded (analytic) form (assuming a downward movement
from top to bottom).@ The "0" and "1" could also be depicted in their synthetic
unity by mapping this same spiral onto a solid ring (i.e. one with no hole in the
centre [cf. Figure 7.13]), so that the spiral actually returns to the same point

Figure 7.8: The Spherical Spiral
(a) Side View(b) Top View

Figure 7.9: The Simple Synthetic Integration of the Spherical Spiral

from which it starts, without the need for the centre line. In either case the three
synthetic stages outlined here are those of outward movement (or evolution),
inward movement (or involution), and return to the state of no movement (or
rest).
The other obvious way of distinguishing the static elements, or stages,
through which the standard spherical spiral passes is to count each loop as a
stage. This yields the same type of sevenfold distinction (here constituted by the
six loops plus the central line) which we have noted on several occasions so far
without elaborating on its symbolic significance. We have now finally reached
the point where it is possible to dwell on this important type of distinction in
more detail. The seven stages represent an elaboration of simple threefold
synthesis when the latter is applied in higher level systematic contexts. The best
way of clarifying the nuances of this sevenfold development of logical relations
implicit in the spherical spiral will be to relate each loop to one of the standard
static geometrical figures we have used in preceeding chapters. (We could use
the seven figures of the heptagon described in 5.3; but these have not had
enough use to merit standardization at such a crucial point in our inquiry.)
Among the most commonly employed figures in the Geometry of Logic are the
cross and its inversion, the square, in analytic contexts, and the circle and its
inversion, the point, in synthetic contexts. (Note that the solid cross, the sphere,
and the cube are simply three-dimensional versions of these two-dimensional
figures. As such they could just as well have been used as standard symbols.
The triangle could have been chosen here instead of the circle; but its inversion
has not played an important part in our System, so it would not make such a
good parallel to the square and cross.) The successive systematic importance of
these figures can be depicted either by mapping them onto the cross [Figure
7.10a], or by combining them in a single, concentric series [Figure 7.10b].@ By
relating these figures to the loops of the spiral, not

Figure 7.10: Four Standard Figures
(a) Mapped onto the Cross (b) Mapped Concentrically

Figure 7.11: Analysis of the Sevenfold Spiral Process

only will our previous discussion serve to shed light on the nature of the spiral,
but the systematic order of the spiral will serve to integrate the Geometry of
Logic in a higher level of systematic coherence than what has been attained so
far.
The obvious choice of a geometrical figure to represent both the first and
last bits of the sevenfold dynamic process is the point. For it is from it that all
other figures are constructed [see 1.1] and to it that all such figures can be
reduced. The widest loop should be represented by the figure which has tended
throughout our discussion to be used as a symbol of full extension, or even
transcendence, the square [see e.g. 1.2, 2.2, 3.4]. On either side of the "square
loop" are two sets of loops of equal sizes but opposite shapes (i.e. mirror
images). Those closest to the square can be represented by the symbol for
wholeness, the circle. The first could be depicted as rotating clockwise (the
direction of time) to represent experience, and the second as rotating
counterclockwise (the direction of the stars) to represent reflection upon
experience. And the pair of opposites closest to the two points could be
represented as crosses, the former pointing outward in all directions (reaching
out through experience) and the latter pointing inward in all directions (in
introspection and reflection). This thrusting action, in contrast to the stable
revolution of the circle, is necessary to compensate for the steeper spirals closer
to the centre, much as a spaceship depends on special booster rockets to push it
up into orbit or direct it back to earth [see Figure 7.11].@ These seven figures
can be reduced to two essential figures (the point and the cross) and their
inversions (the circle and the square). Hence the symbolism suggests an
alternating pattern between synthetic and analytic symbols within the spiral.
Finally, it should be noted that the first and last points are best viewed as if they
were the same point: if the figures were arranged in a circle rather than a line,
these two points would coincide. The seventh element can therefore be viewed as
the empty space which provides the context for their union. The sevenfold
process of transformation which this point undergoes is symbolic of a renewed
life, or even rebirth into a new state.
To represent the spiral in terms of seven successive and discrete
geometrical figures is to view a predominantly synthetic diagram from an analytic
perspective (i.e. to analyse it). Yet these two methods of symbolizing synthetic
relations are saying precisely the same thing. For in both cases a fundamental
perspectival shift is represented as a change in the direction of development.
When the spherical spiral reaches its widest point and begins to contract, the
rules for mapping actually change, since one half of the figure represents either
the "++" or the "-+" spiral, as given in Figure 7.2. (Figure 6.8 explicates the
differences in mapping which must hold for the twelve elements on each loop in
these four spirals.) The change which is represented in the spherical spiral as a
change from evolution to involution, or vice versa, is represented in two similar
ways by the seven figures in Figure 7.11. It is represented first by the change in
the circle's direction of revolution (compare the third and fifth figures), and then
by the change from an outward pointing cross to an inward pointing cross.
Mapping the logical relations between the elements of a material system onto
such geometrical models helps an interpeter to recognize fundamental changes in
perspective and to adapt his interpretive rules accordingly. Incidentally, Figure
7.8a was purposefully drawn without any dotted line in order to provide the
reader with a geometrical example of a perspectival shift. Is the figure drawn
with the top tilted towards the viewer, or away from the viewer? The decision is
crucial, since it will determine whether the direction of logical development is
clockwise or counterclockwise, and so also whether the pair of threefold spirals
which are being combined are the "++", "+-" pair or the "--", "-+" pair. The best
answer is that it is both: the diagram as it stands continually shifts from one
perspective to the other, just as do the elements of many real systems (e.g. light
as wave and particle).
Three-dimensional spirals can also take numerous forms other than the
standard spherical form we have been considering. A few examples will suffice
to give an indication of some of the interesting alternatives. First, the spherical
spiral can be inverted, so that the widest loop becomes a point, while the top and
bottom points become wide loops [see Figure 7.12a]. The analytic model of this
spiral depicts the same symbols arranged in an inverse order [cf. Figures 7.11
and 7.12b].@ Such a spiral could then be extended in both directions
indefinitely, indicating the eternal development of opposite forces. Thus, for
instance, by altering the number of revolutions, the central point could represent
the ninth (central) square in the eightfold double cross, or the thirteenth (central)
point in a twelvefold circle, etc. For such applications, of course, the analytic
model would have to be varied by supplying new figures to participate in the
progression [cf. 4.2].
Another spiral with potential applications for the Geometry of Logic is
the one which can be mapped on to a solid ring. We have already mentioned one
way of doing so, by compressing the two poles of the sphere until they meet at
the centre. A variation--one which is considerably easier to draw on a flat
surface--is to regard the spiral as a series of vertical, rather than horizontal, loops
[as in Figure 7.13].@ The number of loops is flexible (though twelve would
serve as the best standard number for this circular spiral),so this figure would
make an appropriate model for a System containing a large or irregular number
of systems, all of which are more or less the same in form and relational value.
Perhaps the most significant variation of the standard spherical spiral is
the dual spherical spiral, which can be constructed either by joining two spherical
spirals by lines at their tops and bottoms, or by inscribing the standard spherical
spiral with its inversion [see Figures 7.14a and b]. In both cases the logical
development continuously

Figure 7.12: The Inverted Spherical Spiral
(a) The Synthetic Model (b) The Analytic Model

Figure 7.13: The Vertical, Solid Ring Spiral

Figure 7.14: The Dual Spherical Spiral
(a) As Two Spherical Spirals(b) As a Spherical Spiral 1+Inscribed by its
Inversion

alternates between the two opposing, yet connected, figures.@ The significance
of these figures is that they are both constructed out of all four types of flat
spiral, as depicted in Figure 7.2 in the form of a perfect second level analytic
relation, expanding them into their three-dimensional equivalents, and integrating
them in a single System of systematic forms of relation. They indicate, therefore,
that with the spiral the Geometry of Logic has come full circle. For the simple
flat spiral has fulfilled the same role in this chapter as the undifferentiated point
(unity) played in the first chapter.
The difference is that we have now passed through a transforming
process of discovering its logical significance, so we stand at the same point with
a higher level of awareness. Likewise, the standard spherical spiral can be seen
as a way of splitting this unity into a "first level" duality between a "+" (the top
simple spiral) and a "-" (the bottom simple spiral). And now we have taken this
to the second level with the dual spherical spiral.
From here we could go on to construct yet a higher level of spirals, in
which each element is itself a spiral. Just as the earth is a planetary system in its
own right, yet combines with the other planets and the sun to constitute a "solar
system", so also systematically related sets of logical relations can be combined
at a number of distinct systematic levels. In this seventh chapter we have in a
sense completed our construction of the formal System which has served as the
pattern for organizing our treatment of the Geometry of Logic in this book. Yet
we must recognize that this pattern actually continues to develop, ad infinitum. In
so doing, it always follows the same pattern of patterns; the same perspectival
shifts continually arise, only in new forms. For logical systems are like natural
systems: they begin with a seed ( ); the seed sprouts ( ); it grows to maturity ( );
it bears fruit ( ); the fruit ripens ( ); it falls to the ground ( ); the seed dies and so
becomes the source of a new life ( ). But the more abstract the formal relations
become, the more difficult it is to apply them to concrete situations. Our final
task, therefore, must be to sample some of the applications which can be made of
the logical relations which we have symbolized with the help of various forms of
the spiral.

7.4 Applications: Learning, Wandering
and Worshipping
By now it should be obvious enough that the spiral represents a
"quantum leap" to a higher level of systematic unity than what was previously
possible in the Geometry of Logic. The suitability of using it in this way is
confirmed by the fact that this same symbol, as the geometrical representation of
the cyclical powers of Nature, manifested in tornados, whirlpools, columns of
smoke, etc., has traditionally been used to represent the ultimate balance of
opposites within oneself, the final integration (or "enlightenment") of all one's
faculties into a single unified Person [see e.g. P538:7-8]. Its self-connected
character symbolizes eternity in time and the wholeness of the universe in space.
Moreover, the spiral is the principle which governs many forms of growth,
whether it be the growth of the universe from a Big Bang, or of a person from a
human embryo, or even of one of the many plants, such as the mushroom,
which grow in spiral shapes. "The entire universe", as Purce puts it, "is but the
spiral manifestation of the still centre; as it contracts and disappears to the source
whence it came" [18]. As a symbol the spiral represents growth towards
personal and/or intellectual wholeness and integrity through an integration of
experience and reflection. This union of analysis and synthesis is perfectly
symbolized by the Caduceus [see cover drawing], the ancient symbol of healing,
composed of two spiralling serpants intertwining around a physician's staff, out
of which sprout the wings of a dove, symbolizing "dynamic equilibrium, the
ecstatic union of these currents" [P538:25]. In this concluding section I will
discuss three categories of examples of how such Systematic wholeness
manifests itself in real situations: in epistemology (or the learning process), in the
specific experiences of inward and outward religious wandering, and in religious
symbolism in general.
The sevenfold process represented by the standard spherical spiral
[Figure 7.12] manifests itself epistemologically whenever a person seriously
undertakes the task of understanding something which he has not previously
understood, whether it be the meaning of a word, the content (as opposed to the
validity) of an argument, the proper interpretation of a book, or any other set of
conceptual objects. In each case the process of learning can be described in terms
of the same seven stages: (1) one begins with the recognition of ignorance and
the consequent decision to learn ( ); (2) this is followed by a period of reaching
out towards the object of one's interest in a more or less haphazard way ( ); (3)
as the standpoint of the Other (in this case the desired object of knowledge)
begins to be understood, its content is put into some kind of tentative systematic
order ( ); and (4) if the learner presses on, there eventually comes a point at
which his previous perspective is overcome by that of the Other ( ). The first
stage is now complete and in a sense the subject has come to know the object; yet
this extreme state of captivity cannot last if true learning is to take place. Instead,
the second stage enables the learner to regain his balance: (5) the learner begins
to reflect on the application of this knowledge to his own perspective ( ); (6) he
then makes an active attempt to reclaim his own perspective as his own ( ); and
finally (7) he establishes a balance which includes both his previous standpoint
and the newfound knowledge, synthesized in a higher state of awareness ( ). To
progress through these seven stages in one's grappling with the Unknown is to
learn. And to learn is to experience the power of logic.
Given the widespread assumption that the individual man is a microcosm
reflecting the structure of the macrocosm in which he partakes, we can expect
that a fully integrated System of metaphysics would contain a sub-system
corresponding to each of the seven stages in the learning process (metaphysics
itself being a macrocosmic model of the microcosmic learning experience of each
individual). This is no place even to begin the task of constructing such a
System. Nevertheless it will be interesting to draw attention to one implication
this view of metaphysics has on the history of philosophy. In 6.4 we gave a
summary of the twelvefold structure of Kant's Critique of Pure Reason. We can
now add to that summary the significant fact that Kant included three such
Critical systems in the structure of his overall philosophical System. These three
systems can be mapped straightforwardly onto the three revolutions of the flat
spiral [see P1:9.4]. If Kant's System is taken together with an opposing (though
complementary) System such as Hegel's, it may even be possible to map the
combination onto the more complete spherical spiral. Hegel's System would be
particularly appropriate for such a task since it uses many of the same terms as
Kant's System, and in many respects presupposes it as a basis, yet proceeds
(synthetically) from the concrete ("this" and "that") to the abstract (the
"Absolute") rather than (analytically) from the abstract (the "thing in itself") to
the concrete (real objects of knowledge and experience), as in Kant's System.
Thus, if Hegel's System is interpreted as a three-system complement of Kant's
three Critiques, then the sevenfold spiral would be completed by any third party
who can hold both Systems simultaneously in a "higher unity" of his own.
The inversion of the spherical spiral [see Figure 7.14] applies as a
symbol for self-integration not through an intellectual grasp of an Object or
rational System, but through the process of being grasped by meaningful
personal experiences. Many types of experience follow the pattern of "departure
and return", from the mundane activities which characterize our daily routine to
life-changing experiences such as "falling in love" or religious "conversion". We
shall concentrate, however, on two complementary examples of this category of
applications for the spiral: viz. the dynamic processes which characterize the
systematic wanderings of the spiritual quest in its inward and outward forms.
The journey of the mystic through the stages of his own soul as he
experiences union with God tends to follow the same orderly pattern, no matter
which religious tradition one uses as the point of departure. (The mystical
journey appears to the mystic to be an experience of wandering as he is going
through it; only afterwards does it become clear that he has been guided along an
orderly path.) "Mysticism" has nothing to do with the confusion or ambiguity
which the term connotes for many people; rather those who have had mystical
experiences invariably describe them in terms of "the clarity gained by such
ordering". They enable a person to see his life as a systematically interrelated
whole, each part interwoven with all the others, like a massive spiral
configuration of spirals, and to work towards the goal of perfection--a perfection
which is traditionally symbolized by the diamond [see e.g. P538:16]. The
diamond is usually regarded as an appropriate symbol for the mystical experience
of order and clarity because of its hardness (able to withstand great pressures
without sacrificing its integrity) and its brilliance (able to reflect the beauty of
light when cut in orderly patters). The Geometry of Logic confirms the
appropriateness of the diamond as a symbol for perfection in yet another way;
for the rock in its uncut form usually has an octahedral shape: eight triangular
faces arranged in fours on either side of a square base. This naturally stable
object therefore has the geometrical shape which symbolizes the logical integrity
of any system which employs synthetic (triangular) relations on an analytic
(square) basis. Perhaps it was this alternative symbolic quality which led Lao
Tzu to proclaim: "Do not desire the glitter of the jewel but the rawness of the
stone" [L532:39].
The inward mystical experience, which by its very nature tends to be
experienced only by a small proportion of religious people in most traditions, has
a direct correlate in the outward experience of pilgrimage, which is experienced
by a much larger proportion of the members of most religions.17 Just as the
mystic leaves the ordinary levels of everyday inward experience ( ) and through a
systematic series of disciplines or exercises ( ) ventures deep into the heart of his
soul ( ) in hopes of finding its true Centre ( ), so also the pilgrim leaves the
familiar surroundings of his homeland ( ) and travels over long distances ( ) in
his trek to the Sacred City ( ), the Centre of his religious tradition ( ). Both
wayfarers are likely to undergo considerable hardships in their quest for a vision
of the dwelling-place of God. And in both cases the journey is not complete until
the traveller leaves his long-awaited destination ( ) and makes the long journey
home ( ), where it will be his task to live according to his newfound Way in the
same old surroundings ( ). The sevenfold spherical spiral symbolizes the way in
which the second stage of such cyclical experience (i.e. the threefold return
journey) is to be regarded not as backtracking, but as pressing on to the future
which awaits upon one"s return to the point of departure:

Pay attention to the end as much as to the beginning: then nothing will be
spoiled. [L532:64]

Symbolism relevant to the application of the Geometry of Logic abounds
whenever a true pilgrimage takes place, and is therefore reflected in literary
descriptions of pilgrimages as well. Christian's spiral trek to the Celestial City in
Pilgrim's Progress, complete with its seven points of temptation along the way,
and Dante's spiral descent into Hell and ascent up Mount Purgatory in the Divine
Comedy are two obvious examples. Such symbols are by no means unique to
literary accounts of pilgrimages. For instance, the climax of the Muslim's ritual
pilgrimage to Mecca is his seven circumambulations around a cubical stone called
the "Ka'aba" [P538:31]. A thorough study of the pilgrimages of history and
literature would no doubt reveal the common use of such symbols.
The symbolism we have developed is also in agreement with the vast
majority of religious symbolism in general. The simultaneous immanence and
transcendence of God which is recognized not only by the spiritually adept, but
even by many theologians, is symbolized as accurately as is possible by the
spiral, especially in the standard form given in Figure 7.8. The spiral shape of
perfect Systems could also help to explain the recurrence in religious traditions of
doctrines of salvation through a transformed mode of existence, whether it be
through reincarnation, through a heavenly afterlife, or through some method of
transcendence in the "now" (such as the Buddhist's nirvana, or rather
differently, Neitzche's essentially religious doctrine of the "eternal return" of
each moment).
Not only doctrinal symbolism, but also the symbolism of sacramental
activities can be clearly understood on the basis of the tools for interpretation
provided by the Geometry of Logic. One good example is the dance, which
plays a central role in many religions, and is ignored only at the expense of the
worshippers of any religion. Purce's comments on the dance in P538:30 are
valuable:

By dancing and emulating the macrocosmic creative dance of Siva, the
whirling of the planets or the dance of the atoms, man active ly incorporates the
creative vibrations and ordering movements of the cosmos. His body becomes
the universe, his movements its movements, and when these are harmonious,
then he is not only in harmony with himself, but with the universe which he has
become.

When used in religious contexts as a map or a guide to spiritual
consciousness, the spiral, or more often the circle, is referred to as a "mandala"
[see note 6.9]. The doctrine of the seven "chakras" (= "wheels"), or energy
centres, which pass through the body of the meditating Hindu, are represented
either as mandalas or as lotuses. The seventh chakra is the point on the crown of
the head from which the hair grows out in a spiral pattern, and the form of which
determines the general way a person's hair will cover his head when left in its
natural state. The Hindus describe this point on the crown of the head as the
"Thousand Petalled Lotus through which cosmic light can enter the body"
[P538:127]. The seven chakras "unfold like a flower, spirally from within; all
flower symbolism, particularly that of the rose, is an expression of growth and
the unfolding spirit" [P538:22].
This same use of the rose as a symbol of the peace and rest at the heart of
all Systems is used in the poetry of T.S. Eliot, for whom "the end of all our
seeking [is] a state of being where fire and rose are one" [V531:224]. And this is
precisely the point being made by all the applications of the spiral given in this
section. Thus, it implies that "man at his best is outward bound only in order to
return across the seven oceans the inward way from where he started, and to see
the place for the first time" [187].

NOTES TO CHAPTERN SEVEN

1. Compare the endless repetition of periodic functions such as the sine wave.
Doczi uses such periodic waves throughout D491 (see esp. pp. 86-124) as a
geometrical tool for representing the logical proportions discernable throughout
nature, art and architecture. (Incidentally, the central dividing curve of the t'ai chi
symbol [see Figures 1.7, 2.6] can be regarded as a single revolution of a sine
wave.)

2. Many of the spirals which are traditionally used in religious and mystical art
and architecture have three and a half revolutions [see P538:24,70,101], so that
when two spirals are combined a total of seven revolutions results. However,
this destroys the regularity of the spiral, since some cross sections would then
cut the curve in eight places and others in seven. Moreover, we shall see below
that the double spiral evinces the "6+1 = 7" pattern even better when it has only
six revolutions.

3. The ancient Egyptians devised a configuration of flat spirals which they
believed provided the "key to alchemy" [P538:75]. ("Alcemon", incidentally,
was a pupil of Pythagoras (the Father of ancient Greek geometry!); he "thought
that men die because they cannot join their beginning to their end" [Yeats, A
Vision, a.q.i. P538:25].) The pattern looked something like Figure (a) on the
following page.@ This figure can actually be broken down into spirals
resembling each of the four standard flat spirals in Figure 7.2, as I have done in
Figure (b) on the following page. The disadvantage of the figure, however, is
that the curve intersects itself on four different occasions. Expanding the figure
into three dimensions would solve this problem; but in three dimensions, as we
shall see in 7.3, we have much more suggestivefigures to choose from.

Figures for Note 7.3
(a) The Spiral of Egyptian Alchemy
(b) An Analysis of the Spiral

4. The ancient Greeks used the term "spherical number" to refer to any number
whose cube and square both end with the same number as does the original
number [H488: .291]. As such, only multiples of "5" or "6" are spherical
numbers. (Consider the two series: "5, 25, 125, 625, ..." and "6, 36, 216,
1296, ...") However, on this basis the number "4" should at least be considered
as semi-spherical, since it ends in a "4" whenever it is raised to an odd power,
and a "6" when raised to an even power. (Consider the series: "4, 16, 64, 256,
1024, ...") Thus, "4" is spherical in its cube (and all subsequent odd powers),
though it is not spherical in its square (and all subsequent even powers).

5. The segments of the outer ring do not appear to be triangles in Figure 7.6c,
but they are. For the dotted line around the perimeter is meant to indicate that the
eight outer "spokes" are actually connected at the bottom of the sphere, just as
they are at the top.

6. Two other facts make a similar point without involving the sphere. The first is
quite simply that "64" is to "12" as multiplication is to addition, since "4.4.4 =
64", while "4+4+4 = 12" (cf. 5.3, where a similar correlation is drawn between
"7" and "12"). The second fact is that if a cube, with its eight vertices, is divided
according to a third level synthetic relation--that is, divided into twenty-seven
subcubes (as it would be, for example, if a solid double cross were inscribed in
it)--then there will be sixty-four discrete vertices in the entire figure (four planes
with sixteen points on each).

7. The Tai Shuan Ching has a rather more dominant synthetic structure, for it
uses three types of line, and arranges them in four term expressions (tetragrams),
thus providing for a total of eighty-one logical expressions [S493:178-9]. It
would be interesting to make a detailed comparison between this 3 structure and
the 4 structure of the I Ching. We can simply note in passing, however, that both
of the combinations total 12 if the numbers are multiplied rather than raised to a
power (i.e. "3.4 = 4.3 = 12").

8. Note that the segments on the periphery of the projected sphere refer to the top
of the solid; those in the centre refer to the bottom; and the middle of the three
solid circles refers to the sphere's equator. Each expression should be read in
rows from left to right, starting with the top row. The top row of three terms
remains the same throughout a given "slice of the pie" (i.e. 1/8 of the sphere),
while the bottom row either stays the same or follows the same pattern within a
given ring. When mapped onto a sphere in this order, every expression has its
exact opposite at the space 180\ on the opposite side of the sphere. The values in
brackets at the four extremities of the figure indicate that each quadrant begins
with the two values appropriate to the inverted form of second level analytic
division (viz. "++", "+-", "--", "-+", progressing clockwise). Eastern systems
tend to follow this inverted form of mapping opposites, rather than the standard
form introduced in 2.2.

9. S493, for example, is a masterpiece of scholarship on Chinese astrology, yet
the authors continually fail to explain why they make certain distinctions in
expounding the systems they deal with, or why they make the empirical
applications they do. As a rule the only justification they give is that this is the
way correlations have been made for 4000 years.

10. The resulting diamond shape which can be inscribed in the sphere [see
Figure 7.9] can be regarded as the geometrical representation of "the
Philosopher's Stone', which is described by Purce in terms of "the fourfold
sphere of wholeness, the goal of death and rebirth [P538:78].

11. Euclid constructs four diagrams using these concentric figures (excluding the
point) in E487:4.6-9, where he is establishing the geometrical relationship
between the circle and the square. Each of four successive propositions states:
"Inside/outside a given circle/square, construct a given square/circle."
Interestingly, if "inside a ..." and "construct a square" are represented by "+"
and "outside a ..." and "construct a circle" by "-", then his four diagrams follow
the order of second level analytic division (++, -+, +-, --). Each diagram
includes an inscribed cross as well [see Figures below]. (He follows the same
fourfold pattern in E487:4.11-14 with respect to the circle and the pentagon.)
6. (++)7. (-+)
8. (+-)9. (--)

12. Compare the hidden heptagon at the centre of Figure 5.5 with its transformed
state as a heptagram extending beyond the limits of the original figure.

13. This is precisely how Yeats uses the inverted twelvefold spiral in A Vision
[see P538:21].

14. If, for example, each loop of the standard sevenfold spherical spiral is
assumed to symbolize one year in a person's life (a year being a system of
twelve months), then each spherical spiral would have to combine with others in
a higher pattern in order to symbolize an entire lifetime. This corresponds, in
fact, to some astrological systems, such as that elaborated by Yeats in A Vision
[cf. P538:21,118-9], in which an average lifespan of eighty-four years (the
length of one orbit of Uranus around the sun) is divided first into three periods
of twenty-eight years (cf. the length of the lunar cycle in days), and then each of
these is divided into four cycles of seven years (cf. the number of days in a
week), thus yielding a total of twelve such cycles of seven years each (7.4.3 =
84). Such traditions generally associate "wisdom" with the planet Uranus, and
therefore with a single lifespan, and "knowledge" with Pluto, which "takes about
three lifetimes to complete one cycle" [23]. Hence the twelve spherical spirals
which make up a lifetime are themselves regarded as part of a threefold (flat
spiral) system, and these can be combined in still higher levels of systematic
patterns! On the analytic a posteriori logic which guides the application of such
systems, see 7.2.

15. Everyone is familiar nowadays with the astronomer's belief in the spiral
shape of our own and of other galaxies. But what is not often taken into
consideration is that the circuit followed by our solar system is a spiral when the
galaxy's movement is taken into account; likewise the orbits of the planets are
spirals when the movement of the solar system is taken into account. On this
basis it seems reasonable to conjecture that the universe itself in some sense
develops in a spiral pattern, at least as viewed from the perspective of someone
outside the universe who can view it as a whole from some higher level.

16.Consider the saying of Lao Tzu in L532:16:
Things in all their multitude:
each one returns to its root.
Return to the root means stillness.
Stillness means return to fate.
Return to fate means eternity.
Cognition of eternity means clarity.
If one does not recognize the eternal
one falls into confusion and sin.

17. See B528:97. The love of tourism which goes hand in hand with the
secularization of a society is not accidental: it stems from an unconscious need to
participate in this essentially religious pattern. The tragedy is that tourism is
almost always a perversion of the sevenfold spiral, since it is usually employed
as a (temporary) means of escaping from reality rather than a (permanent) means
of instilling reality with spiritual life. The typical tourist is always pressing
forward to see new sights, but never arriving at the destination, always planning
ahead, but never really returning home. Thus pilgrimage (and tourism) can be
compared to a maze or a puzzle: "it reveals and conceals. It is cosmos [systematic
order] to those who know the way, and chaos to those who lose it" [P538:29].

18. V531:263 writes from within the Tantric tradition that "in this pattern of
departure and return ... all our separation and traffic and travail in-between our
several beginnings serves only to enable us to see with a clarity we could never
have achieved any other way how departure and return are one and our home is
where we started from."

19. See P538:64,114 for good illustrations depicting these pilgrimages. She also
mentions [18] the symbolic importance of "the seven [steps] of the Persian
Mithraic initiate, the seven steps of Buddha, the seven notches on the Siberian
shamanic tree [and] the seven steps of the Babylonian ziggurat." To this could be
added numerous examples from the Bible, such as the seven days of Creation,
the seven revolutions of the Children of Israel around the wall of Jerico, or the
corresponding destruction announced by the seven trumpets in the Revelation of
St. John, to name just a random selection of the many which could be cited from
the Jewish and Christian traditions.