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

Zero

Painting Title: Zero
Size: 34" wide x 55" tall
Medium: Acrylic on Board

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

No longer available

Zero has two uses…both important, but distinctly different from each other.

One use is as an empty place indicator in our place-value number system. For instance, in the number 102, the zero is used simply to put the 1 and 2 in the correct "place". When the place-value use of zero was introduced, a simple dot was used…note the diamond-shaped dot…the ancient Hindu form called the ‘bindu’.

Secondly, the zero is used as a number itself, meaning "none", "empty" or "nothing". Indians used the Glyph "0" as early as 640 BC. Fibonacci, the Italian mathematician who introduced the Hindu-Arabic number system to Europe around 1200, called it a "sign" rather than a number. His name for zero was "Zefiro"---from the Semitic sifr, which meant "empty".

In this painting, the most prominent figure is of the Glyph (symbol) we westerners use for the number zero. The glyphs in the upper right and left corners are ancient Sanskrit. Other words which mean zero are in the painting: nul, null, nol, nula, cipher, etc. Meden agan ……Greek for "nothing in excess"…the wise words of the Greek oracle at Delphi.

At the bottom right, the Greek Gnothi Seauton, means Know Thyself…it is the motto over the entrance to the temple of Apollo at Delphi.

Bottom left: In Quantum Physics, everything is in flux, a world of constant movement, a world of endless possibilities, so I have Zero’s in all sizes and shapes moving through that space…vibrating, swimming in the Akashic Field…ready to spring forth and become…what? A quantity…zero for now, but prepared to manifest. The Zero was, conceptually, the most difficult of all these paintings because as soon as a single mark was made, there was one, a mark…the only real Zero painting would have to be blank. Pulling Zero out of that blankness, manifesting it into solid form, through focus and intention tells a story of Quantum Physics.

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 .