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19T-h-i-c-k-e-n-i-n-g

**Joseph Choma**

*is an architect and designer, and founder of Design Topology Lab, a research platform dedicated to the ontology of space defined by mathematics. He is writing the first pedagogical guide into trigonometric transformations. Choma completed his graduate studies in design and computation at Massachusetts Institute of Technology, and is currently an assistant professor at Southern Polytechnic State University, where he directs the Digital Fabrication Lab.*

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Two seemingly identical lines—each defined by a unique parametric equation—are different, because the location of an object in space is part of a shape’s mathematical DNA: the *x*and

*y*values define the location of points in a two-dimensional Cartesian coordinate system.

*x = u*

*y =*π / 3

for all

*u*є [0, 2π]

*x = u*

*y = -*π / 3

for all

*u*є [0, 2π]

If we imagine that this is a section cut through two surfaces in space, this shape acquires thickness. If this example were to be a three-dimensional shape, each surface would be defined by the parameters:

*u*and

*v*. Within digital software, the distance or thickness between two surfaces is considered to be a separate parameter:

*w*. However,

*w*does not always operate separately. For example, a sphere with an offset thickness would require two parametric equations, defining two different spheres in space. Although the parameter

*w*is oriented perpendicular to the surface normal, it is not used to define the inherent geometry or “DNA” of either shape.

Within typical digital software environments, we do not have a means to manipulate the

*w*parameter—that which is not part of the shape’s “DNA.” However, if we begin to reconsider

*w*as part of the inherent DNA of the shape, the possibility of creating a “thick shape” with one parametric equation arises.

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A simple sine curve mediates between two boundary conditions. Based on our new definition, this could be considered a “thick shape,” although the curve still reads as a single line. It is only after the sine curve’s frequency increases to a certain threshold that a zero-thickness shape begins to have the illusion of thickness. Eventually the undulations visually read as a solid “thick” black rectangle: a thick shape. *x = u*

*y =*sin(25

*u*)

for all

*u*є [0, 2π]

Accepting this, a question arises: Is it possible to define more than one boundary condition with a single parametric equation? In the example below, a curve is embedded inside another curve, such that the curve modulates between four boundaries. Mathematically, it is possible to combine parts of shapes to create a different singular shape. This type of trigonometric transformation might be called

*texturing*: shapes of different frequencies combine; the lower-frequency shape becomes the receiver for the higher-frequency shape. Here, the curve of frequency 25 is “textured” by the curve of frequency 50, which is placed inside it.[1]

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Like constructive interference in sound waves, as waves combine the increase in amplitude is determined by the collision of the apexes.[2]*x = u*

*y =*(sin(50

*u*) + sin(25

*u*)) / 1.75

for all

*u*є [0, 2π]

By introducing an additional parameter to the parametric equation, we are able to see the shape grow in another manner. In this simplified two-dimensional example, by adding the parameter,

*v*, the previous curve transforms into a surface.

*x = u*

*y = v*+ (sin(50

*u*) + sin(25

*u*)) / 1.75

for all

*u*є [0, 2π],

*v*є [–π / 8, π / 8]

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In some ways, this is similar to the three-dimensional example in the pages that follow. However, in the those examples, where parameter *w*is introduced, the parametric equation needs to be defined by a second subset of

*x*,

*y*, and

*z*. This subset allows a single geometry to modulate infinitely between multiple three-dimensional boundary conditions. The depth at which the geometry modulates between these boundaries is controlled by the parameter

*w*. This type of trigonometric transformation could be called

*thickening*.

After a shape has been defined by the second subset of

*x*,

*y*, and

*z*, a whole new range of geometric freedom is introduced. In a more complex trigonometric transformation (called

*containing*), the boundary of one three-dimensional shape can be placed inside the boundary of another three-dimensional shape. Finally, the complete geometric hierarchy of a shape can be altered in a single transformation, within one parametric equation. As the sphere thickens over a series of recursions, its geometry begins to mediate between multiple envelopes. The sphere no longer has one boundary but rather multiple boundaries, while the drawing on its surface is perceived as a thickening; from an object state to that of an atmosphere.

*All images by Joseph Choma, 2012.*

**Endnotes**

1. Note that these two parts in

*y*are scaled by 1.75.

2. The scaling function in this example is used to keep the amplitude (or thickness) consistent with the examples above.

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