Article written during an artist residency at Autodesk/Pier9. Originall published on Instructables.com, as part of a series of generative form and fabrication projects.
Phi and The Golden Proportion
The process of creation typically does not depend on a singular methodology when generating, perceiving and manipulating a form. Of the many ways to make a "thing", the observation and incorporation of natural phenomena is commonly used. However, in the process of observing and taking creative cues from the universe, one finds an enormous variety of phenomena to chose from. The beauty, order and variety in nature is readily visible. The golden proportion, or golden ratio is a classic pattern you've most likely seen that occurs in nature and demonstrates very interesting commonalities and relationships between various areas of physical, biological worlds. At its simpliest, it is a mathematical relationship between two quantities. By definition, two quantities are in the golden proportion if their ratio is the same as the ratio of their sum to the larger of the two quantities.
the ratio of A + B
is to A as A is to B
In math, the ratio is commonly referred to as the greek letter Phi , As a decimal, it is represented as 1.6180339887498948482... It is an irrational number, ie, has an infinite number of decimal places that never repeat, like e or PI.
The proportion is useful in the description of symmetry, as well as descriptions of polyhedra and polygons, as the ratio appears as a fundamental building block to base geometric solids. Mathematicians, designers and artists throughout history have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and the golden rectangle, which can be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems.
This article documents some distinct methods I've found for deriving the ratio with geometry. If you know of others, please comment and I'll add them.
Required Materials for doing it by hand
With a computer, you can use any drawing program that can do lines and arcs. Snapping helps as well.
The Golden Rectangle
Possibly the most common and widely used method to find the golden ratio is within the golden rectangle. The ratio of the short side of the rectangle to the long side is Phi. This is commonly seen when describing the Fibonacci sequence as a series of golden rectangles embedded within each other.
Draw a square that has sides that are 1 unit long
Place a compass point at 1/2 of the bottom side of the square
Rotate the compass clockwise it is on the same horizontal plane as the base of the square.
Where there compass arc and the base of the rectangle meet is Phi
Single Known Length
The simplest way to find Phi that I know of is not the traditional Golden Rectangle method. We can actually derive Phi from a single known length. In this method, we have a line that has a known length, in this case it has a length of 1.
Align the known length vertically (A)
Align another line of the same length to one end to the midpoint of the vertical length, and the other end to the horizontal plane. (B)
With a third duplicate length, align one end to the midpoint of the second length, and then other end to the horizontal plane. (G)
The ratio of (AB) to (AG) is Phi
Triangle in a Circle
We can find the golden ratio when inscribing a triangle inside a circle. This step assumes you can draw a perfect equilateral triangle within a circle. There are a few ways to draw a triangle without a computer or other devices that I listed in the requirements, such as this article on drawing an Equilateral Triangle
Draw a circle.
Align an equilateral triangle so it fits within the circle bounds.
Draw a horizontal line that starts from the 1/2 point of the left edge of the triangle (A)
Continue the line to the other side of the triangle (B) until it intersects with the edge of the circle (G)
Square in a Circle
The square-in-circle method is similar to the triangle, and related. It's interesting to see the pattern and relationships between these shapes emerge.
Draw a circle
Divide the circle in half with a horizontal line
Draw a square that rests on the horizontal line, and is scaled vertically so its two top corners intersect the edge of the circle.
The length one side of the square (AB) to the distance to the right or left edge of the circumference of the circle (AG) is Phi
Pentagon in a Circle
The pentagon in a circle method is similar to triangle and square.
Draw a circle
Within the bounds of the circle, draw a pentagon, and intersect the vertices with the edge of the circle.
Draw two lines that emanate from the bottom vertices of the pentagon, and intersect at the top vertex of the pentagon.
Draw a horizontal line from the leftmost vertex (A) of the pentagon to the rightmost vertex (B) of the pentagon.
The ratio of (AB) to (AG) is Phi
Pentagon in a Circle 2
If you've already inscribed a pentagon in a circle, the ratio actually exists in another measurable place.
Connect all the vertices into a 5 pointed star.
If the edge of the pentagon is 1, then the length of the connecting star lines in Phi
Additionally, the length of the smaller lines within the star are also related. They are 1 / Phi
The 1/2 square method
Draw a side of a rectangle with length 1 on the horizontal, and a height of 1/2 on the vertical.
Draw a diagonal line within the rectangle (xy)
Place your compass point at (y) and set it to length (yz)
Inscribe an arc until it intersects with the diagonal line. (w)
Place your compass point at (x) and set it to length (xw)
Inscribe an arc that starts at (w), and intersects with the bottom of the rectangle (u)
(xu) is 1/Phi
Below are the following reference files for all methods as AI, DXF, Solidworks, and PDF
laser-cut plywood, grasshopper, touchdesigner, python
A series of algorhythmically generated projection-mapped truncated tetrahedrons laser-cut from 10mm ply. All vary in size, orientation, truncation, and all have a boolean intersection with the wall as if each object is embedded within it. After manual assembly, the generated geometry is utilized as a base for projection mapping.
This project was realized as an experiment in creating a fast, efficient solution for a tightly-linked fabrication and projection-mapping workflow. Sculptural form, build plans, material cut files, and projected graphics are all generated from software and parametrically linked.
Geometry was created in Grasshopper/Rhino via the gHowl LunchBox toolkit and GhPython.
Gometry and supports layed out for cutting.
Forms were cut with a LaserSaur laser cutter from sheets of lightweight, low-grade 10mm ply.
Panels are superglued together by hand. The interior support structure relies on two angle wedges, which are attached on the interior of the forms.
After glue-up multiple coats of white primer is applied. I did a light hand sanding of the laser-cut plywood edges, as the laser-braze is a tough surface to get paint to adhere to, and therefor coat well enough so it appears completely white. Grey would have looked nice and worked well.
Ultra strong neodynium magnets attracted to large-headed nails or screws into the wall were used as a mounting solution.
Geometry is piped into TouchDesigner via OSC
Once geometry is generated in TouchDesigner, the projected meshes can be aligned onto the sculpture.
Shaders applied to the mesh faces. This installation was part of a group show, and I used the audio-input from the sound generated from Sasha's Drum Piece.
This project did not attempt to solve the task of of aligning projections to the physical objects. Because of their simple geometric shape, I had planned to first install them on a surface and register the projections manually. For future iterations, I'd am considering ideas that include embedded fiber-optics in the verticies that would be tracked with an IR camera for a precise, real-time alignment. A huge challenge involved in projection mapping objects is the sheer amount of time it takes to align projections, and creating a real-time mapping solution allows for articulated and or moving installations.
This iteration does not include a solution mitered corners, due to the physical tendancies of the laser cutter. However, the lack of mitres simplified the fabrication process considerably and was a benifit -- as this design reveals a channel between faces that makes a perfect area highlight the polygon edge when mapping.
A proof-of-concept collection of patches that shares polygon mesh geometry from Grasshopper to TouchDesigner in realtime via OSC. The ability to share mesh data between Grasshopper and TouchDesigner allows for a streamlined workflow when projection-mapping geometry. It eliminates a mesh export/import step, and doesn't rely on TouchDesigner's mesh editing tools. It streamlines the process of fabricating objects, as it closely links the fabrication process of the form to the projection mapping process. Grasshopper, a parametric geometry environment, allows for great control over the verticies, planes, and normals of a 3D mesh, which gets sent to the TouchDesigner graphics environment. Any changes to a mesh done in Grasshopper translates to TouchDesigner in real time. This is a proof of concept and unoptomized for large meshes.
site-specific projection sculpture
concrete, plaster, multiuple projectors, computer, custom software
12' x 3.5' x 3.5'
2 channel projection
Site Specific, New Dehli
In collaboration with Vishal K. Dar
Release - New Delhi, 2012
When artists, Vishal K Dar and Gabriel L Dunne, presented their work in a small space in Mehrauli, on the fringe of the community’s development, the children of this neighborhood had immediately made up their minds about the creature's true nature and the reasons for its appearance. To them it was a wish fulfilling sea-serpent, silent and evocative, which had found abode in this unused space to hibernate during the cold winter months of Delhi.
Interestingly, we found how myths originate in such communities that are untouched by contemporary art and technology. Their chancing upon an object that is beyond comprehension, allows them to put their faith in notions of folk-lore and myth making.
Vishal K Dar / Gabriel L Dunne wanted to challenge the notion of sculpture as a static object. Their organic sculptural form has its roots in CAD software and is further skinned with a generative algorithm. Different parts of the sculpture move differently, as if a form had coiled onto itself. The viewer is liberated since the sculpture does not rely on prescribed grammar and the experience helps resolve a paradigm shift in our viewing registers.
The sculptural form wraps itself around an existing architectural column. The surface texture is similar in material and color to that of the existing walls. During the day it appears to be sedentary, but after sun down, the surface starts to glimmer and move. The projected animations are mapped onto the surface from two positions, covering a 360 degree viewing angle. The projections move in tandem with the surface segments, creating a mesmerizing rhythm.
Live-coding GLSL editor with audio spectrum data for creating audio-reactive shaders in WebGL. Coded numbers in the shader become converted into performance sliders as they are typed, allowing any variable to be performed. Live audio data is made avaiable as a variable to affect visuals.
The form is composed of 7 panels trimmed and scaled from a full icosidodecahedron. The piece represents my practice and experiences in regards to my own growth and experience. Geometrically, the form is inspired by sacred polyhedra that represent forms of transition. The choice of an icosidodecahedron represents transition of two forms and frequencies, represented by an icosahedron and its dual polyhedron, a dodecahedron.
Software and audio toolsets for ICOSI are procedurally generated from the form. A factor I didn't anticipate was how this project dramatically split my focus into two realms (sound and visuals), which was a juggle that I didn't anticipate as I was developing the visual software on one split-screen before the performance. My goal was to create a unified performance tool, yet still resulted in a focus split. I am inspired to continue pursuing interfaces and controllers to overcome the separation of creating audio and visuals simultaneously.
The multiple mediums (sculpture, sound, visual media) of this particular project was fascinating. Each iteration represents a process of my curiosities in technology, consciousness, spirituality, design, and fabrication.
7' x 7' x 8"
Wood, Fabric, Java/Processing, Ableton Live, Max/MSP
A one dimensional visualization of the universe represented as varying
length of vibration and time cycles. concentric rings representing
lengths of time oscillate between the age of the universe, planck
time, and the current moment.
the central point of a circle is the beginning of a one-dimensional
universal construct. the steps involved in creating any circle is a
metaphor for the ongoing creating process itself. anywhere you place a
center you can radiate a circle and symbolically create the space of
the universe. a true point is impossible to create, having no
dimension. consider the center point of monad as the current consciousness
of the present moment, and the edge of the circle as an infinite expansion of
natures forms are represented to us by invisible forces that make them
visible. these forces are vibratory cycles that oscillate through all
matter. as we explore frequencies of vibration on a universal scale,
we are metaphorically repeating the principal of the Monad: the
opening of light, space, and time in all directions.
cycles are the core principal of the universe, and are all pervasive.
we are thoroughly enmeshed in cycles and the periodic rhythms, which
span the frequencies of space and time, are of only the most obvious
on a daily basis: our sleep patterns, the weather, the seasons,
sunrise, sunset, the rising and falling of emotion. On a larger
magnitude: the age of our bodies, the rise and decline of species, the
life cycle of stars. On a smaller, our breathing rate, our heart rate,
the audio spectrum of music, speaking, the visible light spectrum,
radio waves, cosmic rays, and ultimately the smallest measurable
amount of time we can comprehend, described as Planck time. The
appearance of the entire world and all existence with its natural and
technological cycles are rooted in the archetypal, cyclical sinusoidal
principals of the monad.
each ring that moves across the surface represents a length of a time
cycle. as they bass by, audible ticks are heard. higher pitched,
rapid clicks count the current order of magnitude, ranging from 10^-18
to 10^-43. When passing the visual spectrum, it is show as color,
while passing the audible spectrum, frequencies of the chakras are heard.
Miles Stemper, Stephanie Sherriff for fabrication assistance.
Jeff Lubow for audio engineering assistance.
Ryan Alexander for projection-alignment coding assistance.