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# Art and Mathematics

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Publications authored by Eva Knoll pertaining to Art and Mathematics.

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- ItemBarn-Raising an Endo-Pentakis-Icosi-Dodecahedron(Bridges: Mathematical Connections in Art, Music, and Science, 1999) Knoll, Eva; Morgan, Simon
Show more The workshop is planned as the raising of an endo-pentakis-icosi-dodecahedron with a 1 meter edge length. This collective experience will give the participants new insights about polyhedra in general, and deltahedra in particular. The specific method of construction applied here, using kite technology and the snowflake layout allows for a perspective entirely different from that found in the construction of hand-held models or the observation of computer animations. In the present case, the participants will be able to pace the area of the flat shape and physically enter the space defined by the polyhedron.Show more - ItemBuilding a Möbius Bracelet Using Safety Pins: A Problem of Modular Arithmetic and Staggered Positions(2008) Knoll, Eva
Show more This article reports on the resolution of a mathematical problem that emerged when two ideas were brought together. The first idea consists of a method for constructing a decorated bracelet made with safety pins that are strung together at both ends, creating a band. The other is suggested by the word band: why not introduce a twist and make the bracelet a Möbius band? As Isaksen and Petrofsky demonstrated in their paper [1] discussing the knitting of a Möbius band, the endless nature of the band’s single face and edge introduces an additional design constraint, particularly if the connection is to appear seamless. To make the creation appear seamless, the decoration applied to the design must itself be regular, as this helps the eye travel along the endless length. The paper discusses the mathematical and practical constraints of this result for a design that uses a repeating pattern throughout the band, first in the standard design, then in the Möbius bracelet. This resolution involves some simple modular arithmetic and an unusual way to lay out the pins in preparation for their being strung together.Show more - ItemCircular Origami: a Survey of Recent Results(A.K Peters, 2001) Knoll, Eva
Show more - ItemDecomposing Deltahedra(International Society of the Arts, Mathematics, and Architecture, 2000) Knoll, Eva
Show more Deltahedra are polyhedra with all equilateral triangular faces of the same size. We consider a class of we will call ‘regular’ deltahedra which possess the icosahedral rotational symmetry group and have either six or five triangles meeting at each vertex. Some, but not all of this class can be generated using operations of subdivision, stellation and truncation on the platonic solids. We develop a method of generating and classifying all deltahedra in this class using the idea of a generating vector on a triangular grid that is made into the net of the deltahedron. We observed and proved a geometric property of the length of these generating vectors and the surface area of the corresponding deltahedra. A consequence of this is that all deltahedra in our class have an integer multiple of 20 faces, starting with the icosahedron which has the minimum of 20 faces.Show more - ItemDeveloping a Procedure to Transfer Geometrical Constraints from the Plane into Space(International Society for Geometry and Graphics, 1998) Knoll, Eva
Show more Topology teaches us that the two dimensional plane and three dimensional space have a comparable structure. In fact, this apparent parallel is deeply rooted in our consciousness and is applied in many domains, including various fields in the design industry, through the use of such tools as descriptive geometry and perspective drawing. From the particular point of view of the designer, however, this parallel in structure has often been simplified to plans, sections and elevations i.e. 2-D slices through a 3-D object. It has therefore not been an integral part of the design process, but rather a tool of representation of the design process. In the following paper, the relationship between plane and space will be explored as a design element. The question will be answered whether it is possible, starting with a 2-dimensional system of design parameters, to construct a 3-dimensional object based on the spatial equivalents of the initial parameters. To illustrate this process, the painting Opus 84 of Hans Hinterreiter (1902-1992), a Swiss Concrete painter, will be re-interpreted in space.Show more - ItemDiscussing Beauty in Mathematics and in Art(Taylor & Francis Ltd., 2009-09-04) Knoll, Eva
Show more - ItemAn Exploration of Froebel’s Gift Number 14 leads to Monolinear, Re-entrant, Dichromic Mono-Polyomino Weavings(Bridges Coimbra Conference Proceedings, 2011) Knoll, Eva; Landry, Wendy
Show more When Froebel, the inventor of the Kindergarten [1] designed the “Gifts” and “Occupations” given to the children, he deliberately selected materials that provided a haptic dimension to their explorations. This physicality in the interaction with the gifts can create a significant potential learning focus making full use of concepts of spatial reasoning (front-back, over-under, etc.). For adults playing with the materials for the first time, and incorporating a reflective component in their doing and their thinking, the Gifts can provide a novel perspective on other, deeper mathematical concepts. The following paper and its accompanying workshop present some activities possible with Gift # 14, which involves the Occupation of paper weaving, and explore ideas in modular arithmetic, combinatorial geometry, ethnomathematics and more.Show more - ItemExploring Some of the Mathematical Properties of Chains(2009) Knoll, Eva; Taylor, Tara
Show more This workshop aims to explore various mathematical topics that emerge from examining classes of chains and their properties. Basic concepts are taken from topology, an area of mathematics that is concerned with notions like connectedness, how many holes there are, and orientability; geometry, including symmetries; and collapsibility and degrees of freedom. These topics are explored through an examination of a small number of chain designs including examples that are not topologically linked at all, examples in which the relative position of the links determine the symmetries, degrees of freedom, and the way in which their structure is analogous to that of a Moebius band, and finally a model of a chain design with a fractal structure. The workshop will include building human models to explore various properties and other activities where the participants will be able to play with necklace models to better understand the theory and to come up with their own questions to investigate.Show more - ItemFinding the Dual of the Tetrahedral-Octahedral Space Filler(Bridges: Mathematical Connections in Art, Music, and Science, 2003) Knoll, Eva
Show more The goal of this paper is to illustrate how octahedra and tetrahedra pack together to fill space, and to identify and visualize the dual to this packing. First, we examine a progression of 2-D and 3-D space-filling packings that relate the tetrahedral-octahedral space-filling packing to the packing of 2-D space by squares. The process will use a combination of stretching, truncation and 2-D to 3-D correspondence. Through slicing, we will also relate certain stages of the process back to simple 2-D packings such as the triangular grid and the 3.6.3.6 Archimedean tiling of the plane. Second, we will illustrate the meaning of duality as it relates to polygons, polyhedra and 2-D and 3-D packings. At a later stage, we will reason out the dual packing of the tetrahedral-octahedral packing. Finally, we will demonstrate that it is indeed a 3-D space filler in its own right by showing different construction methods.Show more - ItemFrom a Subdivided Tetrahedron to the Dodecahedron: Exploring Regular Colorings(Bridges: Mathematical Connections in Art, Music, and Science, 2002) Knoll, Eva
Show more The following paper recounts the stages of a stroll through symmetry relationships between the regular tetrahedron whose faces were subdivided into symmetrical kites and the regular dodecahedron. I will use transformations such as stretching edges and faces and splitting vertices. The simplest non-adjacent regular coloring, which illustrates inherent symmetry properties of regular solids, will help to keep track of the transformations and reveal underlying relationships between the polyhedra. In the conclusion, we will make observations about the handedness of the various stages, and discuss the possibility of applying the process to other regular polyhedra.Show more