Knots & Polygons

Research expedition via Wikipedia, cut-and-paste.

Tessellation (computer graphics)

In computer graphics, tessellation is used to manage datasets of polygons (sometimes called vertex sets) presenting objects in a scene and divide them into suitable structures for rendering. Especially for real-time rendering, data is tessellated into triangles, for example in OpenGL and Direct3D 11.

In computer-aided design

In computer-aided design the constructed design is represented by a boundary representation topological model, where analytical 3D surfaces and curves, limited to faces, edges, and vertices, constitute a continuous boundary of a 3D body. Arbitrary 3D bodies are often too complicated to analyze directly. So they are approximated (tessellated) with a mesh of small, easy-to-analyze pieces of 3D volume—usually either irregular tetrahedra, or irregular hexahedra. The mesh is used for finite element analysis.

Polygonal chain

In geometry, a polygonal chain is a connected series of line segments.

See also

 

 

 

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician’s knot differs in that the ends are joined together so that it cannot be undone.
A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot

A knot diagram of the trefoil knot

 

History

Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC (see Chinese knotting). The endless knot appears in Tibetan Buddhism, while the Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.

Intricate Celtic knotwork in the 1200-year-old Book of Kells

A mathematical theory of knots was first developed in 1771 by Alexandre-Théophile Vandermonde who explicitly noted the importance of topological features when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Gauss, who defined the linking integral (Silver 2006). In the 1860s, Lord Kelvin‘s theory that atoms were knots in the aether led to Peter Guthrie Tait‘s creation of the first knot tables for complete classification. Tait, in 1885, published a table of knots with up to ten crossings, and what came to be known as the Tait conjectures. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology.

In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. Many knots were shown to be hyperbolic knots, enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in 1984 (Sossinsky 2002, pp. 71–89), and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology.

Chinese knotting

Eight examples of one traditional Chinese knot.

Chinese knotting (Chinese: 中國結; pinyin: Zhōngguó jié) is a decorative handicraft art that began as a form of Chinese folk art in the Tang and Song Dynasty (960-1279 AD) in China. It was later popularized in the Ming). The art is also referred to as “Chinese traditional decorative knots”.[1] In other cultures, it is known as “decorative knots”.

Chinese knots are usually lanyard type arrangements where two cords enter from the top of the knot and two cords leave from the bottom. The knots are usually double-layered and symmetrical.

File:Thistlethwaite unknot.svg
This is a unknot described by Morwen Thistlethwaite.

 

…I find all this interesting because as with leads or yarn, the default state of loose strands seems to be tangled.

 

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Web v1.0

One of my current works-in-progress is a composition of white phyllotaxis designs of varying sizes, drawing on concepts of weaving, mythology, space and science-fiction. The piece will be constructed in phases. I like to think of each individual design as representing an interval of time, emotions and thoughts. More will be added to the overall composition as stockpiles form of these translated time-periods.

The eventual ideal setting for its installation was initially envisioned as a big public space,  showcased at night accompanied by a projector with colourful, changing visuals.

After a brief chat with a colleague it became apparent that different fibers reflect light differently, some better than others, and I realized that before doing an initial test installation at uni, I should first do one in my studio.

So here’s a summary of the results:

Pre-production:
Initial construction composition (around 20 individual designs at this stage)
Sourcing of public domain projection footage; ‘The Thread of Life‘, ‘Sperm‘ and ‘Electric Sheep

White fluorescent light –

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‘The Thread of Life’ –

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Electric Sheep –

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(one of the more interesting results IMO)

electricsheepgif

Sperm –
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[Back wall with shadows]
[[I found the footage itself quite beautiful, however underutilised in this application.]]

 

 

Wearable Art Update

Brief: 3x wearable head-pieces for female dance performers.

Result: New half-mask design incorporating a symmetrical-spiral forehead and loose threads which can be woven into the performer’s hair and fix the mask into place. These pieces are lightweight compared to full head-covering designs and potentially more practical in physical performance applications.

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