1.618

Title:  1.618...

Medium:  Oil on Canvas

Size:  34" Wide x 55" Tall

Completed:  APR 2009

Available through Fly Coop Studios, Asheville, North Carolina

Homage to Pythagoras

Painting Title: Homage to Pythagoras
Size: 34" wide x 55" tall
Medium: Acrylic on Board
2006  

No Longer Available

     Pythagoras (569-475 BC) was a Greek philosopher who made important discoveries in mathematics, astronomy and music theory.  He is best known for the mathematical theorem named for him.  The Pythagorean Theorem, written ALGEBRAICALLY, is

     It is depicted VISUALLY as the central motif, with the word PROOF alongside it.  The Babylonians knew about this theorem1000 years earlier, but Pythagoras proved it.

     He, or rather one of his followers, also proved that irrational numbers exist, as shown in the mathematical proof in the upper portion of the painting.  Phi, written 

is found in the relationships between the sides, the edges and the vertices of the Platonic Solids, and was considered by the Pythagoreans to be so significant that they were sworn to secrecy on the subject. When Hippasus of Metapontum (who is credited with discovering the dodecahedron) divulged the secret of the existence of the irrational, he was thrown in the river and drowned. Phi, expressed to about 20,000 places is printed to the surface in the painting.

     Pythagoras discovered the mathematics in music. By dividing a string into sections, so lengths have the ratios of 2:1, 3:2, 4:3, or 5:4 (octave, fifth, fourth, third), harmonic tunes result. He was quoted as having said, "There is geometry in the humming of the strings". In music theory, the diatonic scale, depicted on the left side of the theorem proof, is a 7-note musical scale with the pattern repeating at the octave (diatonic is translated by Greek, meaning literally "progressing through tones").

     Pythagoreans discovered the dodecahedron, the fifth regular/Platonic solid. Actually, Neolithic people of Scotland 2000 BC knew about these shapes, as evidenced by the stone carvings they left behind. However, the Pythagoreans were enamored and studied these shapes.
 

     If you fit any regular solid just inside a sphere, all the vertices touch the inside of the sphere. If you fit a sphere just inside any regular solid, it touches all the faces of that regular solid.

     The Pythagoreans believed that the universe consisted of a central, spherical earth surrounded by one of the five regular solids, in turn surrounded by a crystalline sphere surrounded by another regular solid, and so on until five spheres surrounded the earth, each circumscribed about a regular solid. The planets and the stars were attached and, as they rotated, created musical harmonies. The Pythagoreans believed most people couldn’t hear this "harmony of the spheres" because they had grown accustomed to it from birth, but that Pythagoras alone could hear it.

     The space is broken up into a Golden Section Proportion.  Click here to go to the Museum of the Golden Ratio

Six

2006

34" Wide x 55" Tall
Acrylic on Board

No Longer Available

This painting is on a board that has Golden Section Dimensions. The Space is broken up into the Golden Section Proportions.

The lower section is filled with numbers (Phi up to 20,000)that are printed onto the board (archival ink).

"Meia duzia," literally translated from Portuguese, means "half dozen."

The black bar with a dot over it is the Mayan glyph for the number Six.

In the upper right corner, two triangles represent 6.

The central motif is a circle with 6 circles around it. If this is about 6, why are there 7 circles? BECAUSE: If you take a circle of any given size and surround it by 6 more circles of the same size, EXACTLY 6 will fit around it, all touching.

Other interesting things about 6, some of which still shows through in the painting, but most of which was painted over:

*6 is the smallest perfect number (a number whose divisors add up to itself 1+2+3=6).

*6 cubed = 3 cubed + 4 cubed + 5 cubed.

*There are 6 strings on a standard guitar.

*Words that mean six: sextet, sestet, sextuplet, hexad, VI, sixer

*Carbon is # 6 on the Elemental Scale

*The honeycomb made by bees has six sides and exactly six of the six-sided shapes (a hexagon) fits around one.

Click here to go to the Museum of the Golden Ratio .

Nine

Nine

34" Wide x 55" Tall

Acrylic on Board

No Longer Available

In order to evoke the # 9, nine cats are stretched out around a spiral created using the Golden Section. The board is built to Golden Section dimensions, broken down into smaller and smaller parts:

34" x 55" board
34" x 34" square at the bottom
21 x 34" top section
21" x 21" top right square
13" x 21" top left section
13" x 13" top left square
8" x 13" section beneath that with
8" x 8" square and to the right of that,
5" x 8" section with
5" x 5" square and then
3" x 5" section with
3" x 3" square and
2" x 3" section with
(2) 1" x 1" squares

These numbers, 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55 are Fibonacci numbers, a sequence of numbers whose successive ratios approach the Golden Ratio, as they approach infinity. (The Golden Ratio, is an irrational number --- one that goes on to infinity without ever repeating itself.) For instance, take the 2 and the 3 in the Fibonacci series 3/2 = 1.5; 5/3 = 1.67; 8/5 = 1.6; 13/8 = 1.63; 21/13 = 1.62, etc.

This golden ratio measures the fraction of a turn between successive leaves on the stalk of a plant: 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc.

Words and Glyphs for the Number 9 are woven throughout the piece, with repetitions used to evoke the sense of patterns.

Click here to go to the Museum of the Golden Ratio .

Artist's Statement

I used to dance. There’s something exquisite about losing yourself in movement. Sometimes, when I’m painting, I’ll start to dance, especially if I’m listening to Reggae or Zydeco. I almost always paint to music. Being an idea person, an avid reader and researcher, my paintings manifest from contemplations on a variety of subjects --- Mathematics, The Golden Ratio, Quantum Physics, and Spiritual Matters. I glimpse connections and begin to draw thumbnail sketches around the edges of my notes. The internal dialogue is about the concept, then words gradually disappear and visual elements expand until only sketches are coming off the end of the pencil. Later, within the painting, words may reappear. I’ll have an entire body of work "idea", with dozens of sketches as potential paintings, but nothing set in stone, nothing certain. All the thumbnails represent a chaotic assortment of ideas, as if each one were a different perspective on the issue, full of the potential to explain it, but just a fragment. Only if a cluster of fragments is seen together, in a full body of work, can they explain the total concept, as if they were pottery shards that must be glued together in order to see the whole vessel. It reminds me of Quantum Physics which tells us that we are not solid, that we move and vibrate and are filled with the potential to manifest. We need only focus on our desire in order to make it real…solid. So that is how I approach the paintings. I focus on the desire to express the idea. I think about the idea…not the painting. I allow the painting to manifest itself. After working with the Golden Ratio for several years, I now have custom painting surfaces made that allow me to paint within that format. It sets the stage. It is the architecture, the structure on which I begin to build. I break up the space and then insert my one or two shapes, the initial idea, and then the dance begins. Getting lost in the painting process is the goal. It is the meditation, the prayer. It is timeless. There are no words. It is pure experience. And that experience is a joyful one. I sometimes have a tendency to tighten up in my work, to make it more accurate, precise, literal, but I recognize that as a state of distrust. I remind myself that the viewer (and the painter) CAN be trusted to figure it out. Allowing the mystery, the nuance, just the glimpse of the idea to exist in a more painterly realm makes the painting richer and more seductive. With that as my intention, the focus on giving up the inherent struggle between tight and painterly, I have recognized the duality within myself and allowed it to be. And so I paint the way I must. And I do the dance. Layer after layer of paint, I build up the surface, scraping/carving/writing, building it up like memory and digging into it in a search for the patterns and rhythms of Universal understanding. From subtle mark-making to deep cuts, glyphs/symbols, visual information--- anything I can use that will add depth---of spirit, of knowledge, of understanding, a sense of connectivity. We go back and forth, the painting and I, until the dance is complete.