Because the gist of my work is about the process of exploring, problem finding/posing then solving, and designing, I often prefer to give workshops, taking the audience through one or more aspects of the thinking and playing involved.
See also Talks, Exhibitions and Publications.
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.
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 work well in classroom explorations of the geometries of ropes and braids for courses in mathematics, teacher education and architecture.
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.
We will present stations of art-making activities that explore mathematical concepts while building artistic skill and competency. Each activity will be explicitly related to outcomes and to the overarching [Essential Graduating Competencies]. Participants will be expected to engage in hands-on mathematical and aesthetic exploration. Stations:
MathWeave is a group of artists, mathematicians and educators studying the relationship between art and mathematics, with an emphasis on implication for education.
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.
Patterns surround us. Children delight in recognizing and making them. The popularity of rainbow looms, for example, show the hunger children have to make things with their own hands, and to play with the aesthetic variations possible in simple crafts. But have you considered that patternmaking crafts also link to both the social studies and the mathematics curricula? Weaving, for example, is a complex human skill that enables a generation raised on electronics to develop sophisticated understandings that cannot be made passively.
This presentation arises from the work of a local research group based at Mount Saint Vincent University pursuing the Study of Mathematics in Textiles.
The group examines principally the mathematical ideas manifested in textiles objects and processes, and the use of textiles crafts to acquire and explore mathematical understanding. The group also considers the manner in which aesthetics and mathematics intersect in patternmaking activities such as weaving.
This presentation will show how making simple paper woven creations, such as bookmarks, badges, and coasters, that are exciting to primary school children, can be used to understand and explore concepts which are key to both art and mathematics. Participants will learn simple weaving projects, hands-on, and be encouraged to consider how else they might engage with patternmaking activities and discussion in their classes. There will be discussion of the use of mathematical ideas to inspire creation. Educational resources underlying this presentation will also be made available.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 coloring1, 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.
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.
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.
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.