The work above is an exercise and exploration that I created to better understand certain techniques in geometric analysis based on a reading of the book "The Painter's Secret Geometry" by Charles Bouleau. It is by no means perfect or even necessarily complete. I did the work quickly and it contains many imperfections. Nonetheless, it still illustrates many key principles in geometry as an art form and will certainly be the basis for future serious works.

Artists from Medieval times and into the Renaissance period were trained using geometry as a tool in the composition of paintings. Some artists today continue to use these basic techniques in the composition of their paintings. In earlier times, however, geometry and mathematics were not disassociated from art as they are in today’s world. The two fields were part of understanding the overall order, unity, and beauty of the world. Geometry was used not only to organize the space but also to achieve an overall harmony within the painting. The term “concinnitas” was revered, defined as the intellectual harmony born of a right ratio of numbers. It was felt that greater beauty was achieved when the work of art was in alignment with predetermined geometric harmonies.

While medieval painters mostly stuck to fairly simple ratios in their compositions, Renaissance painters and architects branched out and began to use more complex ratios. While a medieval artist might choose a basic rectangle with width of 2 and length of 3 (the length being exactly 1-1/2 or 3/2 times longer than the width or short side), the Renaissance artists often worked with some more complex dimensions. They might, for example, choose a canvas with a ratio between sides of 2.25, 1.77, etc. But there was always a distinct logic and harmony to each of the rectangles chosen.

The dimensions of a rectangle were designed to emulate musical consonances. It was Pythagoras, way back in the 6th century BCE, that first discovered that number and geometry was the basis of musical harmony. He discovered that if one divides the string exactly in half, that a note one octave up from the original note is heard. If a string is shortened by exactly 1/3, when one plucks the string, one hears the most basic interval in music – the perfect fifth. The perfect fifth, in other words, is achieved through creating a ratio of 2 to 3, the remaining string length relative to the original. When one shortens the string by ¼, leaving a string length ¾ of the length of the original, one hears another basic musical interval – the perfect fourth.

Medieval Platonic thinking held ratios between small whole numbers as divine and archetypal – 1:2, 2:3, 3:4, in particular. This thinking hailed back to Pythagoras, who chose the symbol of the “tetraktys”, a triangular figure with 1 dot in the first row, 2 in the second, 3 in the 3rd, and 4 in the 4th. All musical harmonies could be found symbolized within this triangle and it was the basis for what we now call the “Pythagorean scale.”

A rectangle with a ratio between the sides of 2/3 (width 2 and length 3) was known as a Diapente, the same term used for the perfect fifth in music. For example, if one had a square with the length of the side of 2, multiplying that length by 3/2 to get a Diapente would give you the length of the other side of 3. In turn, a ratio with the ratio between sides of 3 to 4 was known as the Diatessaron, again, the same term used for the perfect fourth in music. The Diatessaron was formed by multiplying the short side by a ratio of 4:3.

This piece is based upon a rectangle known as the Double Diapente. What does this mean? This rectangle is based upon a ratio of 4 to 9. It is sometimes called the 4/6/9 rectangle. As with most geometry, one starts with unity, in this case, symbolized by the square. The square is the most basic rectangle of all. The reason a Double Diapente rectangle is so called is that one multiplies twice by 3/2 to get the eventual dimensions of the rectangle. Let’s assume that one starts with a square with the length of the sides being 4. When one multiplies 4 times 3/2, one gets 6. And if one multiplies 6 by 3/2, one gets 9 – 4/6/9!

There are certain basic techniques that one uses in geometry to divide any given length in 3 – this division is found where the diagonal of the rectangle meets the half-diagonal. Magic, right? What makes this particularly interesting is that if starts off with a square and you inscribe a circle within the square, then create another circle by placing the point of the compass on the circumference of the first circle, you create a diagram known as the “*vesica pisces*”, the perfect intersection of two circles in a perfect almond shape, often known as a "*mandorla*". In other words, there is a coincidence between the side of the second circle of the vesica pisces and multiplying the original square times 3/2. The two smaller circles within my own painting are fully contained within a Diapente rectangle (short side 4, long side 6.) Thus, there is a perfect connection between the highly symbolic *vesica pisces*, the Diapente rectangle, and the perfect fifth in music.

To continue, if I, in turn, multiply the first length of 6 by 3/2 again, I get a length of 9. The divisions of the long side are, therefore, 4, 6, and 9 and the overall rectangle has a dimension of 9:4. This is the Double Diapente rectangle. It is created by adding a length of 3 onto the original 4:6 Diapente rectangle.

If, however, I expanded the length of the short side of my original triangle from 4 to 6, creating a rectangle with the original short side having the length of 6 rather than 4, multiplying that by 3/2 would give you a rectangle with the ratio of 6:9 or in reduced terms, 2:3, a Diapente. The interesting point is that if one takes 2/3 of the big 6:9 Diapente rectangle, one gets the Double Diapente. By applying the techniques briefly referred to above in dividing the rectangle into thirds by following the intersection of diagonals and half-diagonals, one can easily find a geometric distance which is 2/3 of the side of the bigger square.

A basic analytical technique in geometric analysis of art is to first find the “armature” of the frame in question. This is done by taking a series of simple diagonals and half-diagonals in the painting and then creating a grid of horizontal and vertical lines based on the harmonious points of intersection. In my painting, I started with the large rectangle being a Diapente. Within that Diapente, there is a Double Diapente, and within that, there is yet again another Diapente and another Double Diapente. I used the color green in the colors of the three Diapente triangles contained within the piece. A purplish hue was used to denote the top corners of the 2 Double Diapente rectangles. I then drew in the diagonals for the big Diapente and the big Double Diapente. Interestingly, the diagonal for the smallest Double Diapente runs exactly parallel to the big Diapente. This is not surprising and it is what is known in geometric art as “the power of the diagonal”. One can think of the diagonal as pulling the two sides together in a very specific way, thereby creating the dominant energy of the rectangle.

The intersection of the parallel lines of the Diapentes and the Double Diapentes within the picture forms the wonderful cross at the center of my painting. It creates a feeling of ascent and jubilation in the overall picture. The half-diagonals of the large Double Diapente create a triangular shape that is colored yellow to denote illumination and light.

This picture is inspired by, but not strictly based upon a painting called “Transfiguration” by Raphael (1516-1520). In Raphael’s painting, one can clearly see a square at the top of the painting and a circle inscribed within the square where the rising Jesus is located. The yellow in my painting is roughly where the Jesus figure would have been in Raphael’s painting.

Knowing what is behind a drawing can help deepen the understanding of what the artist had in mind. Or one can simply look at the drawing and feel its beauty.

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