2-D Grids

Generation of the Two Dimentional Grids

In the initial section of this chapter, the circle, as symbol of Monadic unity, divided from within itself to create its reflection in the form of a second circle.  This second circle could only manifest itself by movement away from the primary circle.  Its movement was continued until the centers and boundaries of the two circles touched, creating the Vesica, from which the individual regular polygons were then given birth.  

If the movement of the second circle is continued until both circle boundaries are tangent at only one point, as in figure 1a, a new range of properties is generated.  

Critchlow says of this [IP, p15],“Here the connection between the two has been reduced to a point of contact, in its own way reflecting the source of the original circle.  It is from this point, strictly speaking, that measurement can be identified as it is the ‘between’ of the two primary unities and therefore represents the first measurable distance, i.e. from center to center.   Curiously, the figure can be read as another expression of eternity in the form of infinity as meniscus.  This can be seen to be relevant to the figure if the path is taken as a continuous flow, say from the top, although (as with the circle) the beginning can be anywhere, down right through the contact point, down further left around the bottom circle and back right up through contact point again left to top.  This is a symbol of a path equally traversing origin and manifest reflection – the invisible but implied inseparable relationship – as the boundaries of each  circle belong in this sense to both.”.  

This diagram of the primary and reflected circles can also be thought of as symbolizing the concept of the ‘same’ and the ‘different’ joined at the tangent point by ‘being’, as spoken of in Plato’s Timaeus [35]. 

The relationship of these circles may be said to be one dimensional, because a single line can join the two centers and the tangent point.  The indefinite quality of the Dyad is illustrated, in that the second or exterior circle may ‘roll’ freely around the primary or inner circle.  To achieve stability, or form, a third circle is necessary, figure 1b,.  

As Critchlow puts it [P, p17],  “To establish a finite primordial area we can polarize the exteriorized circle by a lateral expansion.  We will keep this expansion related to the origin by keeping all three peripheries in contact and thereby the expanding centers on an arc determined by the original circle.  The expansion between these polarizations of the lower circle will pass through a series of significant arcs on its way to a state of equilibrium and maximum expansion.  When maximum lower expansion has been established all three circles come to rest with their outermost points just touching; in this way the principle polygon, the triangle is established.”   

This process of expansion may be continued and according to Critchlow [IP, p17], figure 1c,“… each additional manifest circle can be expressed as a lateral polarization of its predecessor” ;  until [IP, p18], figure 1d,“…the sixth peripheral circle exactly contacts the first peripheral circle.  This establishes a primordiality of six possible outer reflections of a primary, or seventh central originating circle. … From the primary and principle unity, the circle, the second unity was the triangle and the third unity, if we may so use the phrase, was the ‘hexagonal’ group of seven circles.”.   

The only other regular polygon that completely fills two dimensional space on its own, in a regular manner is the square.  It can also be derived from the arrangement of seven circles as shown in figure 1e.  Thus, we have another kind of geometrical creation myth, in which the external six ‘perfectly’ surrounds the central Monadic one;  suggestive of the six days of creation and the seventh day of rest;  or the journey of the soul through the seven planetary heavens;  or the seven intervals of the musical octave

The triangle and the hexagon also have the property of ‘dualing’ each other, shown in figure 2.  If, for example, on the triangular grid, the centers of each triangle are connected, a hexagonal grid is produced and if the centers of the hexagons are similarly joined a triangular grid is produced.  The square duals itself. 

It is only the triangle, square and hexagon, the 3, the 4 and the 6, that fill 2-dimensional space uniformly and completely.  These are known as the regular grids because they use only one figure, and all vertices of the grid are identical regardless of the number of figures forming the vertex.  No other polygons have this property.  But there are eight possible combinations of the 3, 4, 6, 8, and 12 sided polygons that are known as the semi-regular grids.  ‘Semi’ because different figures are used in the grid but each vertex is still identical.

To create these grids, we return to the original Monadic circle, figure 3, and following Critchlow [IP, p20], “…examine a form of simultaneous expression of direction.  Each of the six possible directions from the center will manifest itself at the same time.  As before our starting point is the primordial circle – a symbol of simplicity and plentitude.  The first stage in the simultaneous expression is the establishment of three axes of expansion and six directions of departure or reflection [figure 3a].  As the circles differentiate from each other without departing in principle from the central prototype, they establish intersections of confirmation of the three axes.  The diagram  [figure 3b] serves in its own terms the energization of the externalizing of the center into six directions – both as fact and symbol.”   

Continuing the expansion outward, we reach the stage where the centers of the reflected circles fall on the circumference of the primary circle, figure 3c.  This is the position that creates the Vesica Pisces and here it creates six simultaneous Vesicas.  This is also the framework used to create the hexagon.  In this instance, it is the generator of four of the eight semi-regular grids, as shown in figures 4 & 5.  

The first two grids are combinations of triangles and hexagons.  #1 has two figures around each vertex, figure 4a.  #2 has five figures at each vertex, figure 4b.

Grids #3 and #4 are combinations of squares and triangles with five figures around each vertex. 

The  grid #5, figure 5a, is composed of the three polygons that form regular grids, the triangle, square, and hexagon.  It is generated by continuing the sixfold expansion until the reflected circles divide the primary circle into a peripheral division of twelve and an inner division of six.  Six, as noted in the Arithmetic chapter, is a ‘perfect’ number in that the sum of its factors is equal to the product of its factors, that is, 1+2+3=1x2x3=6.   

Grid #6, figure 5b, composed of the square, hexagon & dodecagon, is generated on the same circular expansion but the ‘interior’ hexagon is not used, allowing the dodecagon, which is present in grid number five, to be expressed. 

Grid #7, figure 5c, comprised of the triangle and the dodecagon, is generated by continuing the expansion of the six reflected circles until only one-twelveth of their circumferences are in contact with the primary circle.  

The last of the semi-regulars, #8, figure 5d, comprised of the octagon and square, is generated from the fully expanded group of seven circles, as was the square grid.  In this case, the square grid is rotated one-eighth turn, or 45o, to form the octagon.  The interstitial spaces are squares.

The three regular grids are each made up of one type of polygon and one type of vertex.  For example, in the hexagonal grid, each vertex is composed of three hexagons.  In the semi-regular grids two, three or four types of polygons occur but there is still only one type of vertex.  Grids in which there are more than one type of vertex are known as demi-regular.  There are fourteen of these.  For the sake of brevity they will not be described here.  The interested reader is referred to Critchlow’s book, Order in Space where they are thoroughly covered. 

Applied Examples

In the following section we look at some methods of developing the grids that have been handed down through the Islamic tradition.  In figure 6 ritchlow demonstrates how division of the sides of the regular polygons can be a key pattern generator.  Pat7terns based on the three regular grids are shown in figures 7, 8, 9, 10 & 11.  Patterns based on each of the semi-regular grids[except for some reason #7] are illustrated in figures 12 – 20.  In figure 21 Parman & El Said set the hexagon in a square.  In figures 22 and 23 the hexagon and pentagon are ‘unfolded’ to create revtangular ’tiles’.  In figures 23 and 25 Critchlow gives snother approach to the spacial expression of the pentagon.  In figure26 Critchlow presents a decorative pattern and a numerical-geometrical-cosmological ansalysis.  Figure 27 is his construction of a pattern starting with a triangle, proceeding through six, nine and twelve fols divisions.  These last two paatterns surprisingly include fivefold stars.  Lastly, figure 28 is a sample of actual Islamic work.  Figure 29 is a 12-fold pattern credited by Critchlow to Kepler.  Figure 30 is a version of the pattern called ‘the Inhalation and the Exhalation’.