Publications

An integral part of my practice is to disseminate documents and articles that describe the process, and the reasoning, behind my findings, as well as some of their potential for teaching. These publications, many of whom were written in collaboration with mathematicians, artists and educators, find their way into academic and professional journals, conference proceedings, and newsletters.

See also Talks, Workshops and Exhibitions. For publications on other topics, see my university page.


2018
  • Experiencing Group Structure: Observing, Creating and Performing the Plain Hunt on 4 via Music, Poetry, Visual and Culinary Arts
    with Susan Gerofsky, Tara Taylor and Avalon Campbell-Cousins
  • This workshop introduces participants to a group structure, derived from change-ringing, and its interpretation using a variety of artistic modalities. The purpose of the proposed activities is to have participants experience diverse multisensory embodiments of equivalent structures, in a process that may contribute to more robust mathematical understanding. The workshop will span multiple artistic modalities including, music, poetry, visual, and culinary arts.

2017
  • Colourwave: Some Variations on a Mathematical Schema

    Using colour to track paths of traveling elements in permutation algorithms can not only help visualise what is happening from a mathematical perspective, but it also produces aesthetically pleasing configurations that can be the starting point for compelling artwork. In this paper, I walk the reader through the process that produced a series of art works and point out some of their mathematical properties.

  • The Aesthetics of Colour in Mathematical Diagrams
    with Wendy Landry, Tara Taylor, Paul Carreiro, Karyn Harrison and Katie Puxley
  • Mathematical diagramming serves many purposes in mathematics research and education. It may be conceptual or representational in form, but in all cases, producing graphic presentations benefits from carefully considered use of the various elements of graphical composition. The challenge of mathematical diagramming – indeed, of all diagramming – is how to do it well, so that the diagram fulfills its specific conceptual and communicative purposes. This paper and workshop will examine how the considered use of colour in diagramming and in corresponding artefacts can identify and clarify mathematically pertinent features with respect to various mathematical insights. The idea and role of aesthetics in diagramming will be considered with respect to various representations and artefacts of interlacing and some of the related sensitivities, especially colour sensitivity, will be exercised. The reader should note that since the proceedings are printed in black and white, it is worth consulting the electronic version in the Bridges Archive to see the colour images.

  • Dancing Rope and Braid Into Being: Whole-body Learning in Creating Mathematical/ Architectural Structures
    with Susan Gerofsky and James Forren
  • In this interactive, hands-on workshop, the three co-authors' workshop presenters (with experience in mathematics, math education, weaving and architecture) share insights into the mathematics of braided and twined structures and interests in large scale embodied mathematics learning and community construction techniques. Participants will explore the structures of braid- and rope-making, first through small-scale hand building, and then through collaborative ‘danced’ construction of twines, braids and Maypole-inspired helical lattices as architectural structures, with the aim of developing deeper understandings of patterns inherent in these patterns. The techniques used in this workshop would provide powerful techniques in classroom explorations of the geometries of ropes and braids for courses in mathematics, teacher education and architecture.

2015
  • The Aesthetics of Scale: Weaving Mathematical Understandings
    with Tara Taylor, Wendy Landry, Paul Carreiro and Susan Gerofsky
  • The use of visual arts applications to illustrate mathematical concepts is an old idea. Most instances, however, involve the observation and analysis of finished works and artefacts, rather than focusing on the making of them. We propose the idea of making with rigour, which incorporates the deliberate attending to mathematical structures into the process of making artefacts using specifically selected techniques. In this workshop, we suggest that there are additional insights to be gained by learners through the making process. Further, working in the same medium and technique at multiple scales can develop mathematical sensitivities. This enhances understanding by exposing mathematically essential properties that remain constant across multiple scales, yet are observed through the diverse perspectives afforded by the differing scales. The workshop will bring these ideas to light through participant experience and subsequent discussion.

  • Exploring Concepts from Abstract Algebra Using Variations of Generalized Woven Figure Eights
    with Tara Taylor and Wendy Landry
  • Students often struggle with concepts from abstract algebra. Typical classes incorporate few ways to make the concepts concrete. Using a set of woven paper artifacts, this paper proposes a way to visualize and explore concepts (symmetries, groups, permutations, subgroups, etc.). The set of artifacts used to illustrate these concepts is derived from our investigation of open-work woven mats produced in several cultures in the South Pacific. The exemplars that will be shown present variations of the figure eight, and can be created using readily available materials and straightforward instructions.

2014
  • Art and Mathematics: The Hilbert Curve and Other Stories
2013
  • Mat Weaving: Towards the Möbius Band
    with Wendy Landry and Tara Taylor
  • Traditional mat weaving, as practiced in several regions of Southeast Asia, presents the potential for mathematical exploration that can lead to the creation of plaited forms with interesting mathematical properties. This paper and its accompanying workshop introduce some of the possibilities that the writers have discovered and related mathematical properties and constraints.

2011
  • An Exploration of Froebel’s Gift Number 14 leads to Monolinear, Re-entrant, Dichromic Mono-Polyomino Weaving
    with Wendy Landry
  • When Froebel, the inventor of the Kindergarten 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.

  • Documentary Film Review "Max Bill: a Master's Vision"
  • Making Art, Doing Math
    with Tara Taylor
  • The connections between art and mathematics have a long tradition. This dates back to the time when knowledge disciplines were not as clearly segregated (as for example the development of the laws of perspective during the Renaissance). In more recent times, the connections have been maintained both from the perspective of mathematicians who create aesthetically pleasing representations of their ideas, and from the perspective of artists making explicit use of mathematical concepts in their work. In this working group, we expect most participants to come with a primarily mathematics perspective and background. Thus we choose to take the antithetical position, and approach the connection from the point of view of artists. This connection can take a variety of forms. For example, and as members of the Concrete Movement believed, art should:
    “emerge from its own means and rules, without having to call upon external natural phenomena… By the act of modeling, art works take on a concrete form, they are translated from their mental form into reality; they become objects, with a visual and spiritual use.”(Albrecht and Koella, 1982).
    In consequence, “released from its attachments to natural phenomena and bound to natural laws, this art gives the feeling and shaping mind, the creative imagination, the greatest possible freedom” (Rotzler, 1989, page 142). In the working group, we will explore and experience this freedom, focusing particularly on the ways in which mathematics can be integrated into the process of creating art. The three main (nonexclusive) ways are: the mathematics can simply be a tool for the creation of art, it can be the subject of the art piece, or it can be the source of inspiration. The focus of the working group is on mathematics as subject or inspiration for the creation of art.

2010
  • Creative Learning with Giant Triangles
    with Simon Morgan and Jacki Sack
  • Participants will review, explore and develop geometric concepts and vocabulary, through guided discovery, building polyhedra with giant brightly-colored connectable equilateral triangles. Activities will be geared for teachers of all levels. Learning issues and instructional strategies are related to teacher-student ‘hidden contracts’ and the Van Hiele Model of Geometric Thought.

  • Pattern Transference: Making a ‘Nova Scotia Tartan’ Bracelet Using the Peyote Stitch
  • The look and style of a hand-crafted object is in many cases closely connected to the specific techniques used in its creation. When designs and patterns are transferred from their traditional medium to a different one, these technical parameters can modify and sometimes even limit the results, as well as pose mathematical challenges. In this article, I examine the parameters under which the Nova Scotia tartan can be transferred into an off-loom beading technique, known as peyote stitch, gourd stitch or twill stitch, by using the concepts from tiling theory, in order to produce a piece of wearable art.

  • The 2009 Mathematical Art Exhibition at the Bridges Renaissance Banff II Conference

2009
  • Exploring Some of the Mathematical Properties of Chains
    with Tara Taylor
  • 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.

  • Transferring Patterns: From Twill to Peyote Stitch
  • Crafts are generally known for pieces whose structure and geometry are derived from the constraints of the techniques used. In particular, the look of specific patterns and textures are the natural product of the structure of the specific medium and technique applied to their production. The transfer of a pattern from its natural medium to another whose constraints may differ can sometimes present interesting mathematical challenges. In this workshop, this is exemplified through the transfer of a classic pattern resulting from Twill weaving, the Hound’s-Tooth Check as it is transferred to a different medium, known as Peyote, Gourd or Twill Stitch, whereby beads are strung in a traditional bricklaying pattern using an off-loom beading technique. This transfer presents the challenge of adapting a structure so that the transferred pattern still resembles the original, in as simple a way as possible. In the workshop, several possible result of this transfer are compared and materials are made available to both design and create Peyote-stitched hound’s tooth surfaces, thereby introducing the participants to some of the mathematical constraints of this type of transfer.

2008
  • Building a Möbius Bracelet Using Safety Pins: A Problem of Modular Arithmetic and Staggered Positions
  • 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.

  • Using D-Forms to Create a Calder Type Mobile
    with John Sharp and Roger Tobie
  • D-Forms have been the subject of papers at previous Bridges Conferences, but not as a workshop. They are created by joining the edges of two flat surfaces that have the same length of perimeter. A related problem is to create a baseball or tennis ball by joining the edges of two pieces of leather. This workshop will consider some relationships between these concepts and display the variations in a mobile of the type invented by the artist Alexander Calder.

2007
  • Discussing Beauty in Mathematics and in Art
     with David Reid
2006
  • An Interactive/Collaborative Su Doku Quilt
    with Mary Crowley
  • After introducing Su Doku, a popular number place puzzle, the authors describe a transformation of the puzzle where each number is replaced with a distinct colour. The authors investigate the nature of the experience of solving this transposed version. This, in turn, inspires a design process leading to the creation of an interactive quilt. This process, involving issues of choice of medium, level of interactivity, colour theory and aesthetics, is described. The resulting artefact is a textile diptych accompanied by a collection of coloured buttons, constituting a solvable puzzle and its solution.

2003
  • Finding the Dual of the Tetrahedral-Octahedral Space Filler
  • 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.

2002
  • Circular Origami: a Survey of Recent Results
  • Life after Escher: a (Young) Artist’s Journey
  • Learning about Perception through the Design Process
2000
  • From the Circle to the Icosahedron
  • The following exercise is based on experiments conducted in circular Origami. This type of paper folding allows for a completely different geometry than the square type since it lends itself very easily to the creation of shapes based on 30-60-90 degree angles. This allows for experimentation with shapes made up of equilateral triangles such as deltahedra. The results of this research were used in Annenberg sponsored activities conducted in a progressive middle school in Houston TX, as well as a workshop presented at the 1999 Bridges Conference in Winfield, KS. Not including preparatory and follow up work by the teacher, the activities in Houston were composed of two main parts, the collaborative construction of a three-yard-across, eighty-faced regular deltahedron (the Endo- Pentakis Icosi-dodecahedron) and the following exercise. The barn-raising was presented last year inWinfield, and the paper folding is the topic of this paper.

  • Decomposing Deltahedra
  • 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.


1999
  • Barn-Raising an Endo-Pentakis-Icosi-Dodecahedron
    with Simon Morgan
  • 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.

  • Digital Image Resolution: What it Means and How it Can Work for You
    with Anne Lemieux
  • Image resolution can be a real headache. The image looks great on your screen but the minute it comes out of your office printer, it ends up in the garbage, or the image slows down your whole online project. What went wrong? We propose to unravel the resolution mystery so you can publish your images online, in print or on the Web painlessly. The quality of the end product depends on such things as understanding resolution and using the proper color depth. We will sort it all out for you so you can feel confident you are making the most of your images. Scanners, laser copiers, computer monitors and professional printers measure the image resolution in different ways. What are the standards for professional-quality images? What are the properties of color and how do they affect your image? We will show you how to use the proper color model for each type of image printed, online or for the web. How many colors do you need to use to keep quality up and file size down? An image file needs to be optimized for each application. Color depth is the key. All the steps needed to get a good image into your document are waiting to be revealed to you!

1998
  • Developing a Procedure to Transfer Geometrical Constraints from the Plane into Space
  • 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.


© 2019 Eva Knoll. All rights reserved.
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