This is how it all began. I
received the line of numbers – 2193462 – in an early morning
dream transmission. I immediately wrote it down in the dream journal
that I keep next to my bed. Then I lay there puzzling over it. I had
never before dreamed of just numbers – no story, no images – just
that line of numbers crystal clear in my mind as I woke. I thought
perhaps it had something to do with my decades long interest in the
numerical basis of the I Ching. Or perhaps it reflected Carl Jung’s
idea that number is the purest essence of the archetypes. I am an
artist – a painter. Archetypes and their metaphors are essential
contents of my paintings. But numbers?

Two years later, in 2006, I dreamed that I should not regard geometric shapes as shape in two or three dimensions, but should learn to see them as frequencies of energy. I dutifully entered this in my dream journal.

By July of 2011, I had developed an interest in aperiodic (ie. not symmetrical) tiling patterns and even managed to produce a rather rudimentary one based on the square root of 2 and the octagon. A dream advised me to introduce the Golden Mean proportion into my tiling pattern via shapes derived from a pentagon. I worked on doing this – but without success.

In September of that year, I dreamed about the Eiffel Tower (and yes, Freud, sometimes a tower is just a tower). In the dream, I am somewhat lost, wandering about, but confident that I will find my way. I come to a large wide boulevard paved with huge slabs of stone. There are elaborate gardens on both sides, all laid out in a geometric design. In the far distance, straight ahead, I can see the Eiffel Tower. “Wow” I exclaim,” That architect really knew what he was doing”. I move forward and pass a gardener who says – “Use positive thinking.” I walk faster towards the tower, then break into a jog. In this dream, I was led to consider the geometric order of the gardens and the tower – the square base that then rises as an elegant elongated curved pyramid constructed of interconnected triangles.

And then in November of 2012, it happened again: Bam! Crystal clear - the line of numbers again, but in a somewhat truncated sequence – 46219. I dug out my old journal and made notes. I still didn’t know what to make of it all. It seemed to be referencing the I Ching and a shifting cycle of numbers that transform one into the other. Accordingly I constructed a grid that was six integers wide and eight integers high. The numbers 462193 appeared unexpectedly on the bottom row. The numbers 193462 appeared on the fifth row.

For the next two months, I receive a steady stream of dreams about numbers. Sometimes accompanied by enigmatic words such as “Cull out the watcher dolls and place them on the right”. And “The numbers are trying to get somewhere and encounter obstacles. They move in spiral cycles”. And “Our ego was so inflamed in a wall of squares”.

The entry for November 17th describes a profusion of geometric forms and the message: “These are treasures. Guard them well.” Two weeks later, I dreamed of eight numbers in four pairs: 6 and 3, 4 and 7, 1 and 8, 9 and 2. I was intrigued to note that this group of numbers matched a set of paired numbers essential to the I Ching. I drew more grids, discovered more patterns, all swirling and moving and transforming one number into another. I made another aperiodic tiling – a rather clumsy and ugly one. But did find a most interesting symmetrical pattern that exhibited optical illusions of depth. (fig. 1)

A year later, I dream of my aperiodic tiling triangles and the number .891. I am seriously perplexed, but later

that day when consulting my trigonometry tables, I discover that .891 is the cosine of angle 27 degrees and sine of angle 63 degrees. And they are pentagonal numbers.

The next morning, I dream that I am shuffling my cut-out tiling triangles around until they form a rectangle and then a square. Then other triangles give the same rectangle/square, only it is smaller. Upon waking, I find this smaller square embedded in the geometric drawing that I had recently made. That smaller square enabled me to discover some previously unseen triangles. Now I had three triangles in the ratios of the square root of 2

and the square root of 3 and the Golden Mean. To make the tiling, I needed a mirror image of two of those

triangles. That gave me a total of five usable triangles that I then combined to create the tiling pattern in fig. 2. Whew!

I regard this pattern as a matrix from which other patterns can be derived. Such as fig.3 – drawn by connecting the center point of each triangle around a common vertex. It has a rather organic appearance.

Fig. 4 is realized by bisecting the angles of each triangle. It exhibits optical illusions of space and depth.

Two years later, in 2006, I dreamed that I should not regard geometric shapes as shape in two or three dimensions, but should learn to see them as frequencies of energy. I dutifully entered this in my dream journal.

By July of 2011, I had developed an interest in aperiodic (ie. not symmetrical) tiling patterns and even managed to produce a rather rudimentary one based on the square root of 2 and the octagon. A dream advised me to introduce the Golden Mean proportion into my tiling pattern via shapes derived from a pentagon. I worked on doing this – but without success.

In September of that year, I dreamed about the Eiffel Tower (and yes, Freud, sometimes a tower is just a tower). In the dream, I am somewhat lost, wandering about, but confident that I will find my way. I come to a large wide boulevard paved with huge slabs of stone. There are elaborate gardens on both sides, all laid out in a geometric design. In the far distance, straight ahead, I can see the Eiffel Tower. “Wow” I exclaim,” That architect really knew what he was doing”. I move forward and pass a gardener who says – “Use positive thinking.” I walk faster towards the tower, then break into a jog. In this dream, I was led to consider the geometric order of the gardens and the tower – the square base that then rises as an elegant elongated curved pyramid constructed of interconnected triangles.

And then in November of 2012, it happened again: Bam! Crystal clear - the line of numbers again, but in a somewhat truncated sequence – 46219. I dug out my old journal and made notes. I still didn’t know what to make of it all. It seemed to be referencing the I Ching and a shifting cycle of numbers that transform one into the other. Accordingly I constructed a grid that was six integers wide and eight integers high. The numbers 462193 appeared unexpectedly on the bottom row. The numbers 193462 appeared on the fifth row.

For the next two months, I receive a steady stream of dreams about numbers. Sometimes accompanied by enigmatic words such as “Cull out the watcher dolls and place them on the right”. And “The numbers are trying to get somewhere and encounter obstacles. They move in spiral cycles”. And “Our ego was so inflamed in a wall of squares”.

The entry for November 17th describes a profusion of geometric forms and the message: “These are treasures. Guard them well.” Two weeks later, I dreamed of eight numbers in four pairs: 6 and 3, 4 and 7, 1 and 8, 9 and 2. I was intrigued to note that this group of numbers matched a set of paired numbers essential to the I Ching. I drew more grids, discovered more patterns, all swirling and moving and transforming one number into another. I made another aperiodic tiling – a rather clumsy and ugly one. But did find a most interesting symmetrical pattern that exhibited optical illusions of depth. (fig. 1)

A year later, I dream of my aperiodic tiling triangles and the number .891. I am seriously perplexed, but later

that day when consulting my trigonometry tables, I discover that .891 is the cosine of angle 27 degrees and sine of angle 63 degrees. And they are pentagonal numbers.

The next morning, I dream that I am shuffling my cut-out tiling triangles around until they form a rectangle and then a square. Then other triangles give the same rectangle/square, only it is smaller. Upon waking, I find this smaller square embedded in the geometric drawing that I had recently made. That smaller square enabled me to discover some previously unseen triangles. Now I had three triangles in the ratios of the square root of 2

and the square root of 3 and the Golden Mean. To make the tiling, I needed a mirror image of two of those

triangles. That gave me a total of five usable triangles that I then combined to create the tiling pattern in fig. 2. Whew!

I regard this pattern as a matrix from which other patterns can be derived. Such as fig.3 – drawn by connecting the center point of each triangle around a common vertex. It has a rather organic appearance.

Fig. 4 is realized by bisecting the angles of each triangle. It exhibits optical illusions of space and depth.

A successful tiling has to have some side lengths that match – some common ground. Since numerical ratios are mathematical metaphors: ie. 2 is to 4 as 4 is to 8, etc; my artist’s mind had linked my three incommensurable numbers (square root of 2, square root of 3, and the Golden Mean) by using the Golden Mean as the common ground. Three regular polygons – an equilateral triangle, a square, and pentagon were drawn in such a way that they all shared the same measure of side length. I then divided them up in various triangles.

And why was I, an artist, attracted to these numbers and these shapes? It is because of their archetypal qualities. The most fundamental of which is Expansion/Contraction. We usually think of these as a pair of opposites and may diagram them accordingly. Expansion is movement away from a center. Contraction is movement toward a center. Thus you will note in fig. 5

And why was I, an artist, attracted to these numbers and these shapes? It is because of their archetypal qualities. The most fundamental of which is Expansion/Contraction. We usually think of these as a pair of opposites and may diagram them accordingly. Expansion is movement away from a center. Contraction is movement toward a center. Thus you will note in fig. 5

that as Expansion moves away
from one center, it begins to move toward the other center and
becomes Contraction. They are not mutually exclusive as we usually
think of opposites but instead are constantly transformed one into
the other. Thus Expansion is simultaneously Contraction, ie they
exhibit complementarity. This idea is fundamental to the I Ching. And
it is reflected quite nicely in the numerical patterns that represent
the cycles of change of the 64 hexagrams. Let us examine how this
works. The numbers 5 and 0 are the two centers with four integers
between them as in fig.6:

This cycle is repeated again
after number 9, so that instead of number 10, you begin again with 0.
Acting on the premise that addition is Expansion and subtraction is
Contraction, we can then carry out simple + and - manipulations
of the numbers. Initially this is done in a symmetrical manner around
5 as the center. Add 1+9=10, 2+8=10, 3+7=10, 4+6=10. The numbers
added are equidistant from 5 as a center. The I Ching names this
pattern as The River Lo. Subtract 9-4=5, 8-3=5, 7-2=5, 6-1=5. Now we
have numbers that are equidistant from both centers, 0 and 5, in
turn. This pattern was called The Yellow River. Expansion and
Contraction. Having two centers beautifully represents the condition
of Duality in the physical plane, the yin and yang.

Moving symmetrically around the two centers always produces symmetrical results – the metaphysical Ideal, a state of perfect balance. But that is stasis, complete in itself. No further changes occur. Moving asymmetrically

around the two centers produces asymmetrical (ie, aperiodic) results – a condition of constant change.

Those are the conditions of life on planet Earth – constant change in search of balance. An aperiodic tiling is a visual analogy of this process. It has no symmetrical repetition in the tiling of the plane and can, apparently, continue in this manner into infinity. The tiling triangles combine and transform into larger shapes, and those can be combined into ever larger shapes. Thus there is repetition and transformation at various scales – one of the subjects which chaos theory so brilliantly investigates.

These ideas – repetition, transformation, balance, a center (or two centers) as focal point, ratio and scale, reconciliation of opposites (as in Expansion/Contraction and Symmetry/Asymmetry) and archetypal metaphor all serve as some of the bedrock structural symbols for all of the Arts. In their search for order, artist’s and mathematicians use the same processes. I feel that the Cosmos has truly gifted me with these insights by way of my dreams. And my paintings have begun to change accordingly.

This article appeared in the fall 2014 issue of Dream Time published by The International Association for the Study of Dreams

Moving symmetrically around the two centers always produces symmetrical results – the metaphysical Ideal, a state of perfect balance. But that is stasis, complete in itself. No further changes occur. Moving asymmetrically

around the two centers produces asymmetrical (ie, aperiodic) results – a condition of constant change.

Those are the conditions of life on planet Earth – constant change in search of balance. An aperiodic tiling is a visual analogy of this process. It has no symmetrical repetition in the tiling of the plane and can, apparently, continue in this manner into infinity. The tiling triangles combine and transform into larger shapes, and those can be combined into ever larger shapes. Thus there is repetition and transformation at various scales – one of the subjects which chaos theory so brilliantly investigates.

These ideas – repetition, transformation, balance, a center (or two centers) as focal point, ratio and scale, reconciliation of opposites (as in Expansion/Contraction and Symmetry/Asymmetry) and archetypal metaphor all serve as some of the bedrock structural symbols for all of the Arts. In their search for order, artist’s and mathematicians use the same processes. I feel that the Cosmos has truly gifted me with these insights by way of my dreams. And my paintings have begun to change accordingly.

This article appeared in the fall 2014 issue of Dream Time published by The International Association for the Study of Dreams