Curvatures 01 02

Lao Tsu, the philosopher and poet who lived in the 6th century B.C. China, described the "genius in seeing things in the seed." One botanical seed that has always intrigued me is that of the Vine Maple Tree (Aceraceae circinatum), which is indigenous to the Pacific Northwest where these paintings were created.

As many of us have seen as children, the Vine Maple Tree produces seed pods with propeller-like wings allowing them to fall through the air like a helicopter. Early in the seed development, the two propeller wings are fused together and slowly open in preparation for flight. This fugitive moment of separating is visually analogous to progressing from a 2-cusped to a 3-cusped hypocycloid.

The Persian astronomer and mathematician Nasir Al-Din al-Tusi (1201 - 1274) studied the 2-cusped hypocycloid. Called the Tusi couple, this line segment results from rolling a circle of radius b inside a circle of radius 2b. The ratio of inside circle to outside circle is

a/b=2

A 3-cusped hypocycloid, called a deltoid, has a ratio

a/b = 3

These paintings chronicle my thoughts on the visually curious and precarious moment just before two tangent curves separate, when the Tusi couple becomes a deltoid, and the maple seed opens.

Ten Equal Tangents 01 02 03

I sit at my studio window and look out across the Puget Sound to the Olympic Mountain. On one blustery day, a flock of ten birds issued from the top of these snow-capped peaks. At first, they appeared just like flecks of pepper but, as they approached, they became more like a small storm cloud. Breaking off and joining up, their cloud-like form soared and twisted and drove through the street in front of my window, scattering their collective and then, just as easily, picking up where they left off, a forgotten bed sheet in the wind.

Mesmerized by their synchronity, I found myself lost in thought while they banked to the left, twirled to the right, and folded onto themselves like a sandwich. Unpredictable tempests and sudden, they formed a perfect monkey-saddle shape before swarming over my studio and out of sight.

I blinked in disbelief, and looked out my window: nothing but water, mountains, sky. What remained were a set of doodles I had made in my notebook while watching the group of ten birds fly like one. The doodles evolved into this painting titled "Ten Equal Tangents" based on the observation that when all ten birds shift simultaneously in mid-air, the angles of intersections (Ψ's) of the curves of their wings are all equal: Ten Equal Tangents of Ψ, given by the equation:

tan Ψ1 = tan Ψ1+1
tan Ψ1 = {tan Ψ1 - tan Ψ2}/{1 + tan Ψ1 tan Ψ2}, I = 1, ., 9

At the end of the day I sat down again at the window. There were no birds. There were no doodles. There was only this painting hanging on the wall.

Paraboloids in Blue 01 02 03

This triptych developed from a poem, a song, and the shadow that formed in my mind while thinking about an equation for a quadratic surface.

In "Narrow Road to the Interior," the poet Matsui Basho reveals that the journey itself is the destination. "A lifetime adrift in a boat," Basho writes, "or in old age leading a tired horse into the years, every day is a journey, and the journey itself is home." This drifting boat, as I imagine it, floats across the painting.

Kohachiro Miyata, Japan's leading player of shakuhachi, the bamboo flute, performs with the Ensemble Nipponia in Japan. One of his songs, "Honshirabe," is gradual and rhythmic, bringing to my mind the image of a boat's oar skimming the water in a dense fog.

A Paraboloid is a geometric surface, a revolving of a parabola, in which intersections with planes produce parabolas, ellipses, and hyperbolas. It is a quadratic surface specified by the Cartesian equation

z = b(x2 = y2 ).

The negative space at the base of the paraboloid is the shape that's formed by the interstice and its shadow. This shape is the drifting boat, the x-axis, the oar skimming the water's surface.

Cycloids 01

A cycloid is a geometric curve formed by a point on the circumference of a circle that rolls along a straight line. If a circle has radius a then the cycloid is described by the parametric equation

X=a(t-sin t)

Y= a(1-cos t).

These Cycloid paintings were inspired by Galileo Galilei's study of the cycloid in 1599, inventions by Qi Jigunag, and their subsequent application to contemporary architecture by Wallace K. Harrison. In 1970, Harrison designed the National Academies Auditorium using the cycloidal curve, which turns a coordinate system based on the cycloid into the acoustically perfect interior space of this room.

At approximately the same time that Galileo was studying the cycloid, the Chinese military strategist and inventor Qi Jiguang was defending the Great Wall of China during the Ming Dynasty. Employing his own inventions on the battlefield, Qi Jiguang invented a wheel and flint apparatus that produced sparks from the rim when rolled.

The sparks from General Qi Jiguang's invention formed the locus of a cycloid, and can be seen in red throughout these paintings. The orange and brown palette of these works was also influenced by the official portrait of the General.

Paracycles

My grandmother once planted a lilac tree (syringe vulgaris) in the backyard where I grew up and every year, just before the petals on the blossoms would open to expose the yellow anthers, the interstices of the petals formed the shape of a paracycle.

Known also as an asteroid or cubocycloid, the paracycle is a four-cusped hypocycloid drawn with an outside circle radius that is four times as large as the inside circle radius. Johann Bernoulli first studied this shape in 1691.

This shape can evolve from an ellipse's envelope, an ellipse being "squished" from top to bottom, sliding down like a glissette.

Hypocycloids 01 02 03 04

Measuring 3 feet high and 12 feet across, these paintings are the largest in the exhibition for the National Academies and the National Academy of Sciences. They are a tribute to the importance of being carried away by imagination, and the swift adventurous journey into creativity.

When I first entered the Auditorium at the National Academy of Sciences, I felt like I had stepped into a 3-dimensional rendering of my childhood Spirograph toy. This toy allowed me to create all kinds of shapes on paper by rolling a circle inside of another circle, and it provided me with endless hours of geometric discovery.

The shape my toy made is a hypocycloid, a unique plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. Roemer first discovered hypocycloids in 1674 while he was studying geometric forms to create gear teeth. The hypocycloid is unique in that the ratio if the radius of the larger circle to the radius of the smaller circle determines the number of cusps of the curve.

In philosopher Gao Xingjian's book Soul Mountain, he writes," What is crucial is this image in your mind." I took the image of the Spirograph orbiting in my mind and the equations to derive the hypocycloid, and re-entered the world of childlike discovery in this painting.

Autocorrelation 06

Correlation is the degree to which two or more quantities are linearly associated. Autocorrelation refers to the dependence of current data on previous data points, and can be used to describe this relationship over time, with multiple observations, such as in a time-series analysis.

After first laying out the equations on the canvas, it occurred to me that the elements in a correlation matrix could become active, like performers on a theater stage, and act out their parts in a 3-dimensional scene. Layering these scenes one on top of each other provides for "looking through" their performance, as in panel data over time.

The palette for this painting is based on Rembrandt's Nightwatch, which has always appeared as a theatrical set, with characters circling in correlated fashion.

Parabolic Symmetries

The paraboloid is the most frequently-drawn shape in my sketchbook because of its inherent symmetry which is the archetypal form of a bowl, basin, and goblet. It shows up unannounced and freely, sometimes wet and filled with water, and at other times, empty except for a dusty bee.

A parabola, on which it is based, is the geometric curve formed by the intersection of a cone with a plane parallel to its side. A paraboloid is a geometric surface, a revolution of the parabola, in which intersections with planes produce parabolas, ellipses, or hyperbolas. It is a quadratic surface specified by the Carterian equation

z = b(x2 + y2 ).

This very common shape in every day things, is a shape that is always in my thoughts when I paint; it is convex and concave, inverting and rolling around the floor after the milk and cereal have all spilled out, and its perfect symmetry continues to inspire many of my ideas.

Riddle in Gold 01 02

This painting is my answer to the following riddle:

Three people, A, B, and C, enter a room and sit opposite each other at a table. They are each given a number written on a card that is attached to their forehead. Each person can see the other two numbers, but not their own. The numbers are Real, positive integers, and two of the numbers add up to the third. Starting with Person A, and going around the table as many times as necessary, they attempt to discover their numbers.

Person A says, "I don't know what my number is." Person B then responds with, "I don't know what my number is." Person C replies, "I don't know what my number is."

Based on this, Person A deduces: "Oh! My number is 50."

How did Person A know this, and what numbers do person B and C have?

Centroids Involutus

Recurring themes throughout this series of paintings include optimization, identifying a state of equilibrium, and where the center or centroid is located. Artists like Robert Smithson, who created the large-scale "Spiral Jetty" earthwork that curves around itself as it lies embedded in the floor of the ocean, have given thought to the center of things. The process of finding these centers, whether they are centroid of an arc or the volume of something, can be an elusive but rewarding task.

The centroid is considered the center of mass for a two-dimensional planar closed surface, known as a lamina, or the center of gravity for an object such as a paraboloid. The centroid of a lamina is the point on which it would balance when placed on a needle.

The involute of a circle is a curve traced by the end of a taut thread that cannot be extended as it is unwound from the surface of a curve. The locus of points traced out by the end of the string is called the involute of the original curve, and the original curve is called the evolute of its involute. The Dutch physicist Huygens first studied the involute of a circle during his research on pendula in the 17th century.

In this painting, the centroid of a boat-like shape travels in my mind around the Robert Smithson's Spiral Jetty earthwork, the wake forming an involute. In the waters below, sea shells also have the same spiral shape.

Convexity Involutus 01

The Möbius strip has fascinated me since I first saw it in a print by M.C. Escher. Invented by the German astronomer and mathematician August Ferdinand Möbius in 1858, the concept named after him is a one-sided non-orientable surface obtained by taking a long strip of tape and half twisting it before attaching the two ends.

The involute of a circle is a curve traced by the end of a taut thread that cannot be extended as it is unwound from the surface of a curve. The locus of points traced out by the end of the string is called the involute of the original curve, and the original curve is called the evolute of its involute. The Dutch physicist Huygens first studied the involute of the circle during his research on pendula in the 17th Century.

This painting explores my own abstract visualization of an involute of a Möbius strip, and the resulting question as to whether there is a convex set of line segments connecting any pair of points on that strip and what it would look like. Here, I imagined a non-orientable tape used as a kite string that wraps around the world with just one very gradual half turn.