Special Issue "Topological Modeling"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (20 December 2018).

Special Issue Editor

Prof. Dr. Ergun Akleman
Website
Guest Editor
Visualization Department, Texas A&M University, College Station, TX 77843-3137, USA
Interests: shape modeling; image synthesis; artistic depiction; image based lighting; computer aided caricature; electrical engineering and computer aided architecture

Special Issue Information

Dear Colleagues,

Topological Modeling is an umbrella term that covers all shape modeling approaches that includes topological modifications. Topological modeling researchers usually borrow some relatively obscure mathematical ideas and turn them into applications to design interesting shapes. Applications include but not limited to modelling orientable 2-manifold surfaces, modeling knots and links, modelling non-orientable 2-manifold surfaces, modeling Seifert Surfaces, designing regular maps, branched covering surfaces, immersions of 3-manifolds, woven and knitted objects, and origami. The subjects also include areas related to shape construction, such as paper unfolding, and physical shape constructions with developable surfaces.

Prof. Dr. Ergun Akleman
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Keywords

  • Modelling orientable 2-manifold surfaces,
  • Modelling non-orientable 2-manifold surfaces,
  • Modelling knots and links,
  • Modeling and visualization of Seifert surfaces,
  • Designing regular maps,
  • Branched covering surfaces,
  • Immersions of 3-manifolds,
  • Woven and knitted objects,
  • Origami and curved origami,
  • Geometric unfolding algorithms,
  • Developable surfaces
  • Piecewise planar surfaces
  • Modeling with simplicial complexes,
  • Modeling with cellular complexes,
  • Topological graph theory applications.
  • Minimal surfaces
  • Applications of Morse theory,
  • Morse-Smale complexes,
  • Applications of Gauss-Bonnet theorem
  • D-forms and pita forms
  • Hyperbolic crochet
  • Discrete differential geometry

Published Papers (4 papers)

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Research

Open AccessArticle
Sculpture from Patchwise Modules
Mathematics 2019, 7(2), 197; https://doi.org/10.3390/math7020197 - 19 Feb 2019
Abstract
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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Open AccessArticle
A 6-Letter ‘DNA’ for Baskets with Handles
Mathematics 2019, 7(2), 165; https://doi.org/10.3390/math7020165 - 13 Feb 2019
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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Open AccessArticle
Turning Hild’s Sculptures into Single-Sided Surfaces
Mathematics 2019, 7(2), 125; https://doi.org/10.3390/math7020125 - 25 Jan 2019
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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Open AccessArticle
Secondary, Near Chaotic Patterns from Analogue Drawing Machines
Mathematics 2019, 7(1), 86; https://doi.org/10.3390/math7010086 - 15 Jan 2019
Cited by 1
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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