Mathematics, Machines, and Drawing


Using smooth synthetic polypropylene paper and modified technical pens that ‘paint’ with diluted layers of india ink, I created this series of mathematical drawings. Firstly they are geometric conceptions, and then labored products of code, computational objects. As well, they are the aesthetic gems of long exploratory searches in vast parameter spaces. These process geometries are then mechanically drawn with diluted india ink, for 2 to 12 hours at a time, in order to capture the actual processes represented by a numerical sequence (such as a biometric signal or algebraic function). An emergent surface of layered india ink is produced from a line geometry map of numerical patterns. Thousands of lines are drawn out in time, each representing a moment of change in the numerical sequence. Together they aggregate pigment, ‘painting’ mathematically precise washed gradients of tones. The imperfections of the actual process of drawing brings the number forms to life and translates them into a material object to contemplate. Further details of the geometry are discussed in my webpage on Process Geometry.

For inquiries about purchasing any of these drawings, or commissions, please use my contact form: http://christopherwalsh.cc/contact/ 

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Art and mathematics are intertwined processes. When I make art– drawings, dance, music– there is always an underlying mathematical pulse of ideas, which helps organize and also catapult the work into different directions. Further, this pulse leads to precise questions, and can become a journey in itself. So began my project to create a geometry of process: what is the shape of musical tones, color harmonies, or number sequences? This question led to the experimental study of rotational symmetry in circle maps and their extensive gauge-planar walks. All the while, as I learn new mathematical ideas I am inspired in unforeseeable ways.

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Dynamic Flight of the Spiral Model:
Radius(mod(16364),mult(32000)),Angle(mod(4063), mult(5417))
8×8″, 2 hours
India ink, polypropylene paper, drafting plotter
2016
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Process Caustics
8 x 8″, 6 hours
India Ink, polypropylene paper, HP 7550a drafting plotter
2017
A .3mm technical pen draws a continuous line for over 6 hours with a measured dilution of india ink, which lets pigment aggregate along the envelopes of curves and line-intercept areas. This ink caustic effect visualizes the essential geometric features of a process, letting the primitive elements of points and lines recede from view. The fine, organic quality of line and pigment give an immanence to the mathematical equation; it becomes a material object drawn out in time. The function is a sum of four circular coordinates, computed as a set of interlinked recurrence relations (+,x), with parameters found in a linear search space.

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Gauge-planar Walk for 49146-gonal Numbers
8×10″, 6 hours
India Ink, polypropylene paper, HP 7550a drafting plotter
2017
A gauge-planar walk is the set of partial sums of a sequence of vectors. Each of the steps is a mapping from consecutive differences in a numerical sequence to radial angles of direction in the unit circle. The mapping begins with a circle divided into a number of degrees equal to a modulus, in this case it is 49146 degrees. Each point on the circle is a degree of freedom, and so defines a finite field of motion. The sequence used in this map is the set of polygonal numbers for the 49146-gon. Although the polygonal number sequence is discrete, the circle map f(n)→(nk2π)/mod(p) | k∈R is continuous. Here I modulated (k) to extrude a single stack of loxodromic spirals into the chain of eight vortical funnels which exhibit chaotic stability.
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The Complex Wave Function No.1
8×10″, 12 hours
India ink, polypropylene paper, drafting plotter
2017
This figure is the resultant motion of the gauge-planar model. As each of its component vectors rotate, the endpoint of its motion graphs a complex waveform. A single rotating vector traces out the path of a circle. When the vector endpoint along this circle is measured by its vertical motion, we get the successive heights of a sine wave. If a second vector is rotated around the endpoint of the first, two sine wave motions are added together into a composite wave, since the first produces a wave motion, and the second a wave upon this wave. In this way, a whole chain of vectors when each is individually rotated will produce a single wave motion built up from all the rotating vectors, each moving relative to the previous one. A complimentary wave is found if instead we measure the horizontal motions of the vector endpoint rotating around the circle. Yet together they form a new whole, called the complex wave, where the resultant motion of all the rotations of vectors in a chain result in a composite circular motion. The key insight is that this spiral so produced if we extend it along an axis of time, is a sine wave when viewed on the vertical axis, and a cosine wave when viewed on the horizontal axis, as time proceeds along a Z axis.

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Emerging Rose of the Spiral Model
8×8″, 2 hours
India ink, polypropylene paper, drafting plotter
2017

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Untitled Spiral Model #1
8×8″, 2.5 hours
India ink, polypropylene paper, drafting plotter
2017

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Untitled Spiral Model #2
8×8″, 2.5 hours
India ink, polypropylene paper, drafting plotter
2017

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Quadrilateral Twists
8×8″, 2.5 hours
India ink, polypropylene paper, drafting plotter
2017
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Heartbeat, Respiration, and Blood Oxygen Levels Correlation
8×8″, 6.5 hours
India ink, polypropylene paper, drafting plotter
2017
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Hexagonal Vortex
8×8″, 3 hours
India ink, polypropylene paper, drafting plotter
2016
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Cardioid Permutation
8×8″, 2 hours
India ink, polypropylene paper, drafting plotter
2017
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