Research

13 pages, 10565 KiB

Open AccessArticle

by Stephen Luecking

Mathematics 2019, 7(2), 197; https://doi.org/10.3390/math7020197 - 19 Feb 2019

Viewed by 3244

The sculptor adapts the geometry of spline surfaces commonly used in 3D modeling programs in order to translate some of the topological nature of these virtual surfaces into his sculpture. He realizes the patchwise geometry of such surfaces by gluing square modules of [...] Read more.

The sculptor adapts the geometry of spline surfaces commonly used in 3D modeling programs in order to translate some of the topological nature of these virtual surfaces into his sculpture. He realizes the patchwise geometry of such surfaces by gluing square modules of neoprene rubber edge to edge to define the surface which he then torques and bends into sculptures. While limited by the nature of actual materials, the finished sculptures successfully incorporate the expressive tension and flow of forms sought by the sculptor. He presents images of finished works and provides an analysis of the emotive values of a select sculpture. Full article

(This article belongs to the Special Issue Topological Modeling)

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14 pages, 20746 KiB

Open AccessArticle

A 6-Letter ‘DNA’ for Baskets with Handles

by James Mallos

Mathematics 2019, 7(2), 165; https://doi.org/10.3390/math7020165 - 13 Feb 2019

Viewed by 2341

Abstract

Fabric surfaces, made using techniques such as crochet and net-making, are typically worked in a linear order that meanders, without crossing itself, to ultimately visit and build the entire surface. For a closed basket, whose surface is a topological sphere, it is known [...] Read more.

Fabric surfaces, made using techniques such as crochet and net-making, are typically worked in a linear order that meanders, without crossing itself, to ultimately visit and build the entire surface. For a closed basket, whose surface is a topological sphere, it is known that the construction can be described by a codeword on a 4-letter alphabet via Mullin’s encoding of plane graphs. Mullin’s code exemplifies the formal language known as the Shuffled Dyck Language with 2 Types of Parenthesis (

S D L_{2}

). Besides its 4-letter alphabet,

S D L_{2}

has some other similarities to DNA: Any word can be ‘evolved’ via a sequence of local mutations (rewriting rules), and ‘gene-splicing’ two

S D L_{2}

words, by an insertion or concatenation, produces another

S D L_{2}

word. However,

S D L_{2}

comes up short when we attempt to make a basket with handles. I show that extending the language to

S D L_{3}

, by addition of a third type of parenthesis, succeeds for orientable surfaces with handles—provided an appropriate choice of cut graph is made. Full article

(This article belongs to the Special Issue Topological Modeling)

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17 pages, 7462 KiB

Open AccessArticle

Turning Hild’s Sculptures into Single-Sided Surfaces

by Carlo H. Séquin

Mathematics 2019, 7(2), 125; https://doi.org/10.3390/math7020125 - 25 Jan 2019

Cited by 2 | Viewed by 4877

Abstract

Eva Hild uses an intuitive, incremental approach to create fascinating ceramic sculptures representing 2-manifolds with interesting topologies. Typically, these free-form shapes are two-sided and thus orientable. Here I am exploring ways in which similar-looking shapes may be created that are single-sided. Some differences [...] Read more.

Eva Hild uses an intuitive, incremental approach to create fascinating ceramic sculptures representing 2-manifolds with interesting topologies. Typically, these free-form shapes are two-sided and thus orientable. Here I am exploring ways in which similar-looking shapes may be created that are single-sided. Some differences in our two approaches are highlighted and then used to create some complex 2-manifolds that are clearly different from Hild’s repertoire. Full article

(This article belongs to the Special Issue Topological Modeling)

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15 pages, 10797 KiB

Open AccessArticle

Secondary, Near Chaotic Patterns from Analogue Drawing Machines

by Jack Tait

Mathematics 2019, 7(1), 86; https://doi.org/10.3390/math7010086 - 15 Jan 2019

Cited by 2 | Viewed by 7687

Abstract

Chaos is now recognized as one of three emergent topics of study in the 21c. It is seen as appropriate to examine this in art practice. Accordingly, this paper is written from an art perspective. It does not mimic a traditional mathematical or [...] Read more.

Chaos is now recognized as one of three emergent topics of study in the 21c. It is seen as appropriate to examine this in art practice. Accordingly, this paper is written from an art perspective. It does not mimic a traditional mathematical or science format, presenting hypothesis, repeat testing, and a conclusion. The art process operates differently, and chaos is seen in graphic terms, veers more to philosophy, and is obviously subjective. The intent in researching secondary patterns, near the edge of chaos, is to make expressive graphic art images as art works, testing how close they might come to a chaotic state whilst retaining visual coherence. This underpins the author’s current research, but it is recognised as being a very narrow and specialized subset of analogue art activity. The way in which analogue generative art differs from the more common use of digital computers is addressed. Unlike the latter, the work involves designing and making the machines, making the programmers, and writing the algorithms; this is implicit in the text. A brief look at drawing machine history is presented, demonstrating how the author’s machines differ from others. A contextual cross refence is also made, where appropriate, to artists using digital means. The author’s research has documented practitioners who choose an analogue route to make art. However, hardly any of them create programmes to generate coherent images. This shortage creates problems when attempting to cite similar work. Whilst the general principle underlying the work presented is algorithmic, a significant element of quasi-random input is incorporated, consistent with a study of chaos. Emergent facets are implicit, such as the art process, design problem solving, the relationship between quasi-random and determinism, the psychology of evaluation, and the philosophy of how art works. From the author’s Programmable Analogue Drawing Machines, two are selected for this paper which draw Lissajous figures, use X:Y axes, turntables, Direct Current motors, and an asynchronous pen-lift mechanism. Simple instructions generate complex patterns in a similar vein to Alan Turings topics of phyllotaxis and morphogenesis. These aspects will be discussed, presenting two machines that demonstrate these properties. Full article

(This article belongs to the Special Issue Topological Modeling)

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