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The Bridges Conference: Mathematical Connections
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The Bridges Conference: Mathematical Connections
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Please note this is an uncorrected pre-conference draft of the paper abstracts. They are listed in no particular order.
Collaboration on the
Integration of Sculpture and Architecture in The Eden Project
This paper and talk document
my collaboration with Jolyon Brewis of Grimshaw Architects on the design of a
new education building for the Eden Project, Cornwall. The roof structure of
the building is based on plant geometry in the form of spiral phyllotaxis and
incorporates a granite sculpture which will be sited in it's own specially
designed chamber at the centre of the building. This very large sculpture is
based on the same growth pattern as the roof and has involved collaboration
with professionals from many disciplines including quarrymen, stone masons,
engineers and computer experts.
The Work of Foster
and Partners Specialist Modelling Group
The following paper is a
brief introduction to Foster and Partners and the work of its Specialist
Modelling Group (SMG). The SMG was formed in 1997 and has been involved in over
100 projects. The SMG expertise encompasses architecture, art, math and
geometry, environmental analysis, geography, programming and computation, urban
planning, and rapid prototyping. The SMG brief is to carry out project-driven
research and development. The group consults in the
area of project workflow,
advanced three-dimensional modelling techniques, and the creation of custom
digital tools. The specialists in the team are a new breed of architectural
designer, requiring an education based in design, math, geometry, computing,
and analysis.
The Borromean Rings
- A Tripartite Topological Relationship
On Mathematics,
Music and Autism
A discussion of research
into the psychology of mathematicians, especially in relation to autism and the
possible links to the psychology of musicians.
Cultural Insights
from Symmetry Studies
Washburn and Crowe have
published texts and studies documenting the procedure for and application of
the use of plane pattern symmetries to classify cultural patterns [8, 9]. This
paper contrasts the difference in cultural insights gained between pattern studies
that simply describe patterns by motif type and shape and those that describe
the way motifs are repeated by plane pattern symmetries.
Non-Euclidean
Symmetry and Indra's Pearls
Escher's well known picture
of devils and angels is an example of a symmetrical tiling of two dimensional
hyperbolic space. We discuss similar symmetries of three dimensional hyperbolic
space, modelled as the inside of a solid ball. The `shadows' of the solid tiles
on the boundary of the ball themselves form patterns governed by a new kind of
symmetry, that of Möbius maps on the complex plane. All aspects of such
pictures, together with instructions for making them, are explored in the
authors' book Indra's Pearls. We give examples of beautiful fractal patterns
created in this way.
Bridging the gap - a
search for a braid language
As a braidmaker, my work
encompasses both maths and art. However, language can be a bridge, or a
barrier, between different disciplines and without a 'mathematical language' it
has been difficult for me to access work done in this field. This paper
describes my search for a visual language thatprovides me with a practical and
theoretical way of comparing and analysing braid structure. From this comes the
means of discovering all possible braid structures for a set of given
constraints. Although braids have been made for millennia, they tend to be
limited to certain types of structure. These have usually evolved from the
characteristics found within the methods of production. Approaching the subject
from a mathematical viewpoint, enables me to find new structures from the
wealth of possibilities that have yet to be explored.
Love, Understanding,
and Soap Bubbles
As an artist my interest in
mathematics has evolved through a love of nature and a desire to better understand the 'nature of
things'. An evolving interest in natural efficiencies has recently led to a
thorough investigation of soap bubble foam, where I have found the relationship
between pressure differentials and geometric organisation of particular
interest. Through this study I have developed a physical modelling system,
which is the foundation of my latest Artwork(s).
Creating
Penrose-type Islamic Interlacing Patterns
Some of the most interesting
Islamic interlacing patterns involve ten-pointed stars or ten-petalled
rosettes. These motifs have local ten-fold symmetry, yet they are often
included as part of a plane periodic pattern, which can have no overall five-
or ten-fold symmetries. Instead of using these motifs in periodic patterns, can
we incorporate them in patterns based in some way on Penrose tilings (which
have many local five-fold symmetries)?
Steve Reich's
Clapping Music and the Yoruba Bell Timeline
Steve Reich’s Clapping Music
consists of a rhythmic pattern played by two performers each clapping the
rhythm with their hands. One performer repeats the pattern unchangingly
throughout the piece, while the other shifts the pattern by one unit of time
after a certain fixed number of repetitions. This shifting continues until the
performers are once again playing in unison, which signals the end of the
piece. Two intriguing questions in the past have been: how did Steve Reich
select his pattern in the first place, and what kinds of explanations can be
given for its success in what it does. Here we compare the Clapping Music
rhythmic pattern to an almost identical Yoruba bell timeline of West Africa,
which strongly influenced Reich. Reich added only one note to the Yoruba
pattern. The two patterns are compared using two mathematical measures as a
function of time as the piece is performed. One measure is a dissimilarity
measure between the two patterns as they are being played, and the other is a
measure of syncopation computed on both patterns, also as they are played. The
analysis reveals that the pattern selected by Reich has greater rhythmic
changes and a larger variety of changes as the piece progresses. Furthermore, a
phylogenetic graph computed with the dissimilarity matrix yields additional
insights into the salience of the pattern selected by Reich.
Illuminating Chaos ‑
Art on Average
At first sight, chaos and
structure seem antithetical. Yet there is an intimate connection between
randomness and structure. In this talk we explain some of the ideas we have
used for creative artistic design that depend on results from the study of
chaotic dynamics. Our intention is to avoid the Platonistic perspective that
the role of the mathematician is to dig out and discover the beauty hidden
within the mathematics. Our view will be more that of an engineer. How can we
use mathematics in a creative way to produce aesthetically pleasing art? (as
opposed to ‘pretty patterns’.) How can we achieve the effects we want to
emphasize in a particular design? We illustrate the talk with examples of
(symmetric) designs, many of which have appeared in art exhibitions in the
Americas and Europe. As well we give some visual demonstrations and
explanations of chaos and, if there is time, indicate some practical
applications of these ideas to teaching art students (some mathematics) and
mathematics teachers (some art).
Magic Stars and
Their Components
Magic Stars is the title of
a musical work based on mathematical objects of the same name. Six six-pointed
magic stars provide six two-dimensional 12-tone structures, which constitute
the building blocks of the work. These structures are subjected to analysis,
transformations, disintegration and recombination of their components. The
parts of the score, which is richly visual, look like tables rather than
traditional musical pieces. While pitches (‘space’) are fixed, time is not,
giving ultimate freedom to the performer, who may find out his own time and
thereby meet quite mathematical and objective things in a very personal and
intimate way.
Introducing the
Precious Tangram Family
The Author of this paper has
developed a family of Precious Tangrams based upon dissections of the first six
regular polygons. Each set of tiles has similar properties to that of the
regular tangram. In particular the property called Preciousness. It includes a
discussion of some of the mathematical aspects of the dissections with examples
of non periodic tessellating patterns. It continues with examples of the unique
way in which they can produce an infinite number of designs. It explains the
iterative nature of the process as applied to designs for mosaics, quilts and animation.
Sand Drawings and
Gaussian Graphs
Sand drawings form a part of
many cultural artistic traditions. Depending on the part of the world in which
they occur, such drawings have different names such as sona, kolam, and
Malekula drawings. Gaussian graphs are mathematical objects studied in the
disciplines of graph theory and topology. We uncover a bridge between sand
drawings and Gaussian graphs, leading to a variety of new mathematical problems
related to sand drawings. In particular, we analyze sand drawings from
combinatorial, graphtheoretical, and geometric points of view. Many new
mathematical open problems are illuminated and listed.
Symmetric
Characteristics of Traditional Hawaiian Patterns: a Computer Model
Most of Hawaiian quilts,
fabrics and traditional handicrafts are lavishly decorated with patterns.
Reflecting the culture of Hawaii, Hawaiian flora and fauna find their creation
in a fabric of symmetrical patterns. Although this exotic and highly balanced
symmetry is an essential component of many traditional handicrafts in Hawaiian
patterns, the symmetric principles of Hawaiian patterns have rarely been
discussed. To provide insight into the creation of Hawaiian patterns, this
article analyzes the symmetric characteristics of the traditional Hawaiian
patterns. In addition, the article presents a computer model using a java
applet that has been developed to generate an exponential number of different
Hawaiian patterns.
Circle Folded
helices
Helices are explored as
functions of circle reformation using observations that the circle functions as
both Whole and parts in ways no other shape or form demonstrates. The
generalization of tubes and cones, parallel surface and non-parallel surface,
is fundamental to reforming the circle revealing countless variations in the
helix and conical helices. The circle can generate forms that in multiples will
model natural growth systems revealing a dynamic process reflecting the
interrelated nature of universe order. The helix and conical helix are uniquely
demonstrated in the first right angle movement of the circle to itself and
fundamental to all subsequent folding of the circle.
The Taming of
Roelofs Polyhedra
Roelofs polyhedra form a
vast collection of polyhedra containing many interesting solids and including
very irregular ones. The purpose of this paper is to consider two special
subsets: polyhedra with the symmetry of the prism and polyhedra with just two
different types of vertices. Beside the
figures in the paper PowerPoint pictures, all made by Rinus Roelofs, will be
presented.
A Program to
Interpolate (and Extrapolate) Between Turtle Programs
People have been creating
geometric figures with computer programs consisting of turtle commands such as
forward and right since the late 1960s [1]. Here I describe a program that
takes in two such programs and produces a new program capable of producing both
figures and all the intermediate figures. It can produce a figure that is one
third circle and two thirds triangle or one that is half star and half
pentagon. The program produced by interpolating, say, a square and a circle
program takes in a number between zero and one and produces a figure between a
square and a circle. If, however, it is given a number greater than one, or a
negative number, it will produce an extrapolation between a square and
circle. Interpolated programs can be the
basis of playful aesthetic explorations. The intermediate forms can be drawn on
the same image. Or animations can be generated where the figures morph into
(and beyond) each other. Colours and other attributes of the turtle pen can
also be interpolated. Unlike conventional morphing programs, we are
interpolating between computational processes rather than static images.
The Programmer as
Poet
In Tennyson's Now Sleeps the
Crimson Petal, the poet requests from his lover, "...slip into my bosom
and be lost in me." This theme is poetically developed by seeking an
oneness with nature: The poet reviews many natural events which have a cycle of
energetic wakefulness followed by a state of relative rest. A similar method of
poetic development, by analogy with several other domains, occurs in other
poetic passages: for example, Job wishes death by metaphorically seeking that
his day of birth be lost, stained, unlit, not allowed to come to the calendar,... Computer
scientists will immediately recognize this technique of poetic development as
resembling polymorphism, which allows the naming of an abstract concept by its
instantiation in one particular domain. This paper explores use of computer
concepts to classify poetic technique; it also advocates enriching computer science
curriculum with the teaching of poetic technique.
Minkowski Sums and
Spherical Duals
At Bridges 2001, Zongker and
Hart gave a construction for ‘blending’ two polyhedra using an overlay of dual
spherical nets. The resulting blend, they noted, is the Minkowski sum of the
original polyhedra. They considered only a restricted class of polyhedra, with
all edges tangent to some common sphere. This note defines spherical duals of
general convex polyhedra and proves that the Zongker/Hart construction is
always valid. It can be used visually, for instance, to ‘morph’ from any
polyhedron to any other.
Polygon Foldups in
3D
The software Cabri 3D allows
the nets of polyhedra to be constructed using one or more sets of connected
polygons where the angle between all connected polygons is the same. These
collections can be folded into the polyhedron by dragging a point controlling
the angle between the polygons. Viewed from above, the polygons act as a
kaleidoscope as the angle changes, and when the angle is decreased so that
polygons intersect, surprisingly beautiful symmetric figures emerge, which can
be constructed as physical artefacts or experienced as dynamic computer
animations.
Portraits of Groups
This paper represents some
small finite groups as groups of transformations of a compact surface of small
genus. In particular, we start with a designated pair of regions of this
surface and each region is labeled with the group element, which transforms the
designated region into it. This gives a
portrait of that finite group. These surfaces and the regions corresponding to
the group elements are shown in this paper. William Burnside first gave a
simple example of such a portrait in his 1911 book, 'Theory of Groups of Finite
Order'.
A New Use of the
Basic Mathematical Idea of Twelve-Tone Music
We here briefly describe a
collection of pieces which we have written, and which have been performed for
large audiences, in which mathematics is used. Specifically, every one of the
12 major chords, and every one of the 12 minor chords, appears in each of these
pieces. We argue that this is a more pleasing use of the number 12 in music
than the twelve-tone system of Schönberg.
A Braided Effort: A
Mathematical Analysis of Compositional Options
Artist James Mai created a
system of forms in the developmental stages of his work Epicycles. This system
offered mathematician Daylene Zielinski opportunities to provide mathematical
analysis and to contribute to the final compositional organization of
Epicycles. A set of eight new permutational forms are developed from a revised
interrogation of a previously developed system of eighteen forms. The new set
of forms lends itself to a variety of compositional arrangements including,
with contributions from Zielinski, a 'braided' ordering that creates a coherent
sequence of the forms in the final work. This paper not only explicates the
system of forms used in the resulting work, but it also illustrates the
benefits and insights gained from interdisciplinary interactions between an
artist and a mathematician during the development of a mathematically based
work of art.
Constellations of
Form: New Compositional Elements Related to Polyominoes
A predominant theme of
artist James Mai's compositions is the development of finite sets of related
objects derived from permutational processes. Each element is distinct, yet all
of them share particular features. Thus, he develops families of objects that
are at once diverse since each object is visually distinct and integral since
the set of objects is exhaustive. These objects provide the elements for
combination and composition in paintings and digital prints. Recent permutational
investigations by Mai have yielded objects we call point arrays and strutforms,
which are related to polyominoes via dual graphs. These new objects, however,
have greater variety than polyominoes and offer some new opportunities for a
different interpretation of tilings. The results of these investigations are
visible in the digital print, Heart of Sky, which includes the complete sets of
3- and 4-strutforms in a 'close-packed' or minimal area arrangement. Mai is
currently working on compositions with the set of 5-strutforms.
Slide-Together
Structures
About ten years ago I
discovered an interesting way to construct a tetrahedral shape by sliding
together four rectangular planes in a certain way. By using halfway cuts in the
planes it was possible to slide them together, all at once, to become the
enclosed tetrahedron. This way of constructing objects and structures, finite
and infinite, has been one of my interests from then on. In this paper I will
give an insight into some of the results of my research in this field. Besides
halfway cuts I examined some other ways of slide-together structures.
Repeated Figures
This paper illustrates the
development of two types of design from the beginning concept through execution
onto enhancement for final presentation. Emphasis is on a structured, modular
process suitable for instruction in either an art or beginning programming
curriculum.
Seville’s Real
Alcázar: Are All 17 Planar Crystallographic Groups Represented Here?
Contemporary with the
Alhambra, the Real Alcázar of Seville, Spain was rebuilt in 1364 as a palace
for Dom Pedro, Christian king of Castile (1334 - 1369) in the Mudejar style.
(Muslims who chose to live under Christian rule were known as mudéjares).
Although there have been alterations and additions over the centuries, this
remarkably well-preserved palace was originally built by a Christian ruler in
the Islamic style of Andalusia and retains its Islamic character, containing
some of the most beautiful examples of Mudejar alicatado (Spanish, for cut
tiles, derived from the Arab verb qata’a,
“to cutâ€)
from this time period. Since all 17 planar crystallographic groups are now
believed to be represented n the tilings
of the Alhambra, one wonders if the same may be said of the ornament found in
the Alcázar. This paper will briefly discuss the history of the Alcázar,
illustrate and classify some of the planar designs as to the isometries they
permit and then attempt to answer the salient question broached in the title of
this paper.
Math must be
Beautiful
I present here a video
installation inspired by the famous performance of Marina Abramovic 'Art must
be Beautiful., Artist must be Beautiful' It addresses the theme of teaching as
a performance art.
The Integrated Scale
Desirability Function: A Musical Scale Consonance Measure Based on Perception
Data
Tiled Artworks Based
on the Goldbach Conjecture
A simply, stated though
still unproved, mathematical conjecture by Christian Goldbach is utilized to
make two-dimensional artworks. Tile patterns with even numbers of tiles are
divided into two sets. Each set consists of a prime number of tiles that
reflects Goldbach's conjecture that any even number greater than two has at
least one pair of primes that sum to that number.
Sculpture Puzzles
A series of novel
sculpture-puzzles is illustrated, with mathematical explanation. Each consists
of a set of identical parts that snap together into a symmetric form. The parts
are flat, so they can be cut out or stamped from sheet materials such as wood,
metal, plastic, or cardboard. High accuracy is required for the parts to mate
properly, so computer-controlled fabrication technologies are useful. The
examples shown were made by laser-cutting, by solid freeform fabrication
techniques, or by scissors and paper. Their intricate geometric forms make for
challenging assembly puzzles and attractive artworks. A template and
instructions show how to make one from paper.
The Mechanical
Drawing of Cycloids, The Geometric Chuck
This paper discusses
cycloids and their construction using the 19th century mechanical drawing
instrument known as the Geometric Chuck. The first part of the paper is a brief
history and description of the Geometric Chuck. The last part of the paper is
devoted to a discussion the definition of cycloids and examples showing the
results that various settings of the Geometric Chuck have on the cycloid
patterns produced. This paper is an attempt, in part, to respond to the comment
in the Savory book "As this book does not aim at giving a scientific
account of the principles on which it works. It might be an exceedingly
interesting subject for the scientific person, the scientific knowledge
required to understand a three-part chuck would be so great that I doubt if
there is the person existing who could describe the course of a line that would
be produced."
Sashiko: the
Stitched Geometry of Rural Japan
Shashiko comes, not from the
imperial courts, but from the humble origins of rural Japan. This textile
tradition requires only needle, thread and countless hours of patient
stitching. No fancy machinery or clever devices are used. It is just cloth,
single or layered, held together by running stitches. The results are beautiful:
geometric patterns interlock with precision and grace, stunning tessellations
emerge. Some of the traditional patterns are easy to decipher but others are
less obvious. This paper will examine how these patterns are drawn on the cloth
and what design principles the stitcher uses to guide the needle.
Literatronic: Use of
Hamiltonian Cycles to Produce Adaptivity in Literary Hypertext
Literatronic is an adaptive
hypermedia system for hypertext fiction. Its adaptive features are based on an
algorithm that simulates a Hamiltonian cycle on a weighted graph. The algorithm
maximizes narrative continuity and minimizes the probability of loosing a
reader's attention. The metric for this optimization is defined as the
minimization of hypertextual friction and hypertextual attraction. We consider
the challenges involved with modeling such hypertext, and we offer specific
examples of this type of adaptivity.
Responsive
Visualization for Musical Performance
We present a framework that
facilitates the visualization of live musical performance using virtual and
augmented reality technologies. In order to create a framework suitable for
developing technologically augmented artistic applications, we have defined our
system in a way that is modular and incorporates intuitive development
processes when possible. In this paper we present a method of musical feature
extraction and provide three examples of music visualization applications that
we have developed using our system. Our visualizations illustrate features in
live singing and keyboard playing using responsive virtual characters,
responsive video imagery, and responsive virtual spaces.
The Necessity of
Time in the Perception of Three Dimensions: A Preliminary Inquiry
In working with 3-D computer
models I came to realize that there would not be much advantage to presenting
them as a three dimensional representation rather than on a flat screen. In
either case, they would have to be manipulated, over time, in some way to offer
much information. This paper is a non-rigorous exploration of why that is true.
It begins by presenting some of the mechanisms by which we orient ourselves in
space and how we perceive it. The most important of these are visual, but they
do not yield much information in a static situation, since they are vulnerable
to misinterpretation and illusion. The paper then goes on to examine the
importance of a changing point of view in the perception of space, how points
of view have been depicted in art, and how time affects point of view. The
example of motion pictures provides foundation for the idea that certain
perceptions are essentially free of time, while others occur over time. It goes
on to discuss time and how it becomes essential to the perception of space.
Finally, it offers some insight into the perception of time.
An
Interactive/Collaborative Su Doku Quilt
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.
Patterns on the
Genus-3 Klein Quartic
Projections of Klein's
quartic surface of genus 3 into 3D space are used as canvases on which we
present regular tessellations, Escher tilings, knot- and graph-embedding
problems, Hamiltonian cycles, Petrie polygons and equatorial weaves derived
from them. Many of the solutions found have also been realized as small
physical models made on rapid-prototyping machines.
The Lorenz Manifold:
Crochet and Curvature
We present a crocheted model
of an intriguing two-dimensional surface known as the Lorenz manifold which
illustrates chaotic dynamics in the well-known Lorenz system. The crochet
instructions are the result of specialized computer software developed by us to
compute so called stable and unstable manifolds. The implicitly defined Lorenz
manifold is not only key to understanding chaotic dynamics, but also emerges as
an inherently artistic object.
Playing Musical
Tiles
In this survey paper, I
describe three applications of tilings to music theory: the representation of
tuning systems and chord relationships by lattices, modeling voice leading by
tilings of n-dimensional space, and the classification of rhythmic tiling
canons, which are essentially one-dimensional tilings.
Mathematics and the
Architecture: The Problem and the Theory in Pre-Modern Cultures
There is always a mystery on
pre-modern architecture practice on the relation between dimensions and ratios.
The reasons of using certain proportions used on the design of religious
buildings/ spaces are the result of the application of numerical symbolism and
Pythagorean triangle. Thus, the paper will be focused on the unity of theory in
premodern architecture practice via giving some special examples of pre-modern
architecture through the human history, such as Antique Egyptian and Antique
Greek temples, Roman churches, Gothic cathedrals, and so on.
Towards
Pedagogability of Mathematical Music Theory: Algebraic Models and Tiling
Problems in computer-aided composition
The paper aims at clarifying
the pedagogical relevance of an algebraic-oriented perspective in the
foundation of a structural and formalized approach in contemporary
computational musicology. After briefly discussing the historical emergence of
the concept of algebraic structure in systematic musicology, we present some
pedagogical aspects of our MathTools environment within OpenMusic graphical
programming language. This environment makes use of some standard elementary
algebraic structures and it enables the music theorist to visualize musical
properties in a geometric way by also expressing their underlying combinatorial
character. This could have a strong implication in the way at teaching
mathematical music theory as we will suggest by discussing some tiling problems
in computer-aided composition.
Streptohedrons
(Twisted polygons)
Imagine a simple form, a
cone with a symmetrical cross-section. Now split that cone from apex to base,
twist the two halves and re-join. Before your eyes a new, complex form is
produced. Imagine more intricate geometric solids which are split, twisted and
re-joined, magically producing shapes which coil and twirl - shapes not seen
before, unexplored shapes. Remove the inner form of some of these twisted
shapes and a path or ribbon remains. These shapes, these ribbons, this idea,
will excite the Mathematician, the Sculptor and artist alike.
Fractal Tilings
Based on Dissections of Polyominoes
Polyominoes, shapes made up
of squares connected edge-to-edge, provide a rich source of prototiles for
edge-to-edge fractal tilings. We give examples of fractal tilings with 2-fold
and 4-fold rotational symmetry based on prototiles derived by dissecting
polyominoes with 2-fold and 4-fold rotational symmetry, respectively. A
systematic analysis is made of candidate prototiles based on lower-order
polyominoes. In some of these fractal tilings, polyomino-shaped holes occur
repeatedly with each new generation. We also give an example of a fractal knot
created by marking such tiles with Celtic-knot-like graphics.
Vortex Maze
Construction
Labyrinths and mazes have
existed in our world for thousands of years. Spirals and vortices are important
elements in maze generation. In this paper, we describe an algorithm for
constructing spiral and vortex mazes using concentric offset curves. We join vortices
into networks, leading to mazes that are difficult to solve. We also show some
results generated with our techniques.
Models of cubic
surfaces in polyester
Historically, there are many
examples of model building of mathematical surfaces. In particular, models of a
very special cubic surface called the Clebsch diagonal have been built in
plaster and clay since the 19th century. The sculptor Cayetano Ramírez has
succeeded in building this surface using polyester. With this material, the
resulting sculpture shows all the mathematical properties of the surface. We
first give a short mathematical introduction and an overview of the models that
have been built in the past to represent it. Next, we proceed to describe the
work of Cayetano, explaining the techniques used by him in the whole procedure.
“Geometry” in Early
Geometrical Disciplines: Representations and Demonstrations
This paper discusses various
manifestations of geometry in early geometrical disciplines with reference to
specific cases from the Islamic 'Middle Ages', a period of intense scientific
activity falling intermediately between the initial reception of Greek
scientific material in the early Islamic period (8th-9th centuries AD), and
their subsequent diffusion within both Islamic and to European lands (12-13th
centuries AD). The paper begins with the classification of mathematical
sciences in ancient Greek and early Arabic sources, and proceeds with the
identification and distinction of aspects of geometry such as geometrical
'representation' and 'demonstration' through a case study of specific
geometrical disciplines. The case study covers sample problems from four early
geometrical disciplines: optics, mechanics, surveying and algebra: optics and
mechanics are subdivisions of plane and solid geometry in Aristotelian
classifications, surveying and algebra are the respective subdivisions of each
in early Arabic Classifications. The samples include geometrical
representations (definitions, figures, models) and geometrical demonstrations
(illustrations, constructions, proof), as representatives of a range of Arabic
and Persian scientific sources from the Islamic Middle Ages.
Ant Paintings using a
Multiple Pheromone Model
Ant paintings are
visualizations of the paths made by a simulated group of ants on a toroidal
grid. Ant movements and interactions are determined by a simple but formal
mathematical model that often includes some stochastic features. Previous ant
paintings used the color trails deposited by the ants to represent the
pheromone, but more recently color trails and pheromones have been considered
separately so that pheromone evaporation can be modelled. Here, furthering an
idea of Urbano, we consider simulated groups of ants whose movements and
behaviors are influenced by both an external environmentally generated
pheromone and an internal ant generated pheromone. Our computational art works
are of interest because they use a formal model of a biological system with
simple rules to generate abstract images with a high level of visual
complexity.
Verbogeometry: The
Confluence Of Words And Analytic Geometry
Verbogeometry is a form of
art which is interested in creating an aesthetic experience with poetic
structures of mathematical / verbal metaphors. I am introducing Verbogeometry
as a subset of a small movement of mathematical poetry occurring globally but
mostly in America and Finland. This particular mathematical poetry movement has
some connections to the visual poetry movement in the English speaking world.
This paper on Verbogeometry is a primer and also an ongoing investigation.
Zome-inspired
Sculpture
"There's something
irritating about doing something right by accident"-- S. Rogers
An invitation to build 1)
permanent, Zome-inspired sculptures 2) designed and built as a collaborative
effort under the name of fictitious artist(s), 3) as much about art as
mathematics, 4) which could serve as the basis for large-scale architectural
projects for the 21st century 5) to be installed at Bridges venues, as
possible, on an ongoing basis. I'll give a little background about Zome, survey
some sculptures and artists, and discuss the guidelines above in more detail.
There are no designs yet. This is an invitation to get started!
Developable
Sculptural Forms of Ilhan Koman
Ilhan Koman is one of the
innovative sculptors of the 20th century [9, 10]. He frequently used mathematical concepts in
creating his sculptures and discovered a wide variety of sculptural forms that
can be of interest for the art+math community. In this paper, we focus on
developable sculptural forms he invented approximately 25 years ago, during a
period that covers the late 1970's and early 1980's.
On a Family of
Symmetric, Connected and High Genus Sculptures
This paper introduces a
design guideline to construct a family of symmetric, connected sculptures with high number of
holes and handles. Our guideline provides users a creative flexibility. Using
this design guideline, sculptors can easily create a wide variety of sculptures
with a similar conceptual form.
Transformations of
Vertices, Edges and Faces to Derive Polyhedra
Three geometric
transformations produced a large number of polyhedra, each originating from an initial polyhedron. In the
first transformation, vertices were slid along edges and across faces producing
nested polyhedra. A second transformation produced dual polyhedra, whereby
edges of the initial polyhedron were rotated and scaled and the end points of
these edges derived the dual polyhedra. In a third transformation, faces of an
initial polyhedron were rotated and scaled producing snub polyhedra. The
vertices of these rotated and scaled faces were used to derive other polyhedra.
This geometric approach which derives new vertices from previous vertices,
edges and faces, produced precise results. A CD-ROM accompanying this paper
contains three animations and data for all the derived polyhedra. This CD-ROM
can be obtained by sending me email.
Chromatic
Fantasy: Music-inspired Weavings Lead to
a Multitude of Mathematical Possibilities
As part of my thesis work
for my MFA in Fibers at the University of Oregon, I wove five panels that were
inspired by Johann Sebastian Bach’s ˜Chromatic Fantasy”. The many possible combinations
of these weavings led me to create a flipbook of their images, as well as a
computer-animated video of the weavings dancing to the music from which they
were inspired.
Asymmetry vs.
Symmetry in a New Class of Space-Filling Curves
A novel Peano curve
construction technique shows how the self-referential interplay between
symmetry and asymmetry based on the translation, rotation, scaling, and
mirroring of a single angled line segment that traverses a square evinces rich
visual beauty and optical intrigue.
Modular Perspective
and Vermeer's Room
The room's dimensions of the
Music Lesson (ML), as deduced in my first perspective analysis, corroborate
that the projected image on its back wall approximates the real size of the
painting, as Steadman first pointed out. It seems unlikely that the tiled
floors in Vermeer's paintings were done at random. Instead, some of them seem
to have a consistent image formation of about 90º of aperture of visual field,
which speaks on behalf of the use of the camera obscura. Steadman based his
consistency analysis of the underlying tiled floor grids of Vermeer's paintings
in the inverse perspective method, finding that about six of them seem to
depict the very same room. Following this idea, but instead of deducing the
room's plan and elevation as he did, I will proceed directly in perspective
with the aid of my Modular Perspective method. Thus overlaying the floor grid
of the ML to another painting's floor grid, I will prove if they are consistent
or not. In addition, if they are so, the real size of the second floor grid
will be deduced. As far as I know, such a perspective proof has never been
attempted before.
On the Bridging
Powers of Geometry In the Study of Ancient Theatre Architecture
The on-going popularity of
the Vitruvian layout for the Latin theatre is largely due to its capacity to
bridge across several disciplines, which seems to appeal to a certain
conception of material culture that assumes the existence of a plurality of
formally similar structures of culture beyond surface phenomena. These tend to
be not merely potent in their explanatory force but also gratifying
aesthetically and ethically. Modern scholarship has forcefully promoted such a
conjunction of truth, beauty, and goodness in the link between the Theatre in
the Asklepieion at Epidauros and Pythagorean speculation. However, similar
cognitively-significant structural or formal bridges would seem difficult to
establish in all examples. In their absence, the search for a perfect geometry
of perfect shapes beyond the extant remains may turn into a purely formalist
exercise made possible by the capability of geometry to serve as an analytical
tool through a reduction of the architectural code to a geometric code. This is
a dilemma intrinsic in the difficult relation between architecture and
geometry. In fact, Vitruvius seems to have noticed the problem long ago and
tried to build a material bridge between his geometric assembly and the
architectural project by recognizing the necessity to give up symmetry in the
latter, wherever required by the nature of the site or the size of the project.
The Gemini Family of
Triangles
There are a series of
triangles in the pentagon/pentagram figure that can be used advantageously in
quilting. We are going to investigate these triangles both mathematically and
artistically.
Taitographs:
Drawings made by machines
If a machine is instructed
to make drawings and the results are viewed in the same way that a person's
drawings are read, then speculation about the nature of creativity and art is
not only possible but desirable. The decision making process becomes transparent
because the maths, mechanics and after treatment are available for scrutiny,
unlike the partially subconscious aspects of a person's drawing activity. It is
proposed that the ideal way to meet the "Bridges" aspirations is to
follow Harold Cohen's exhortation that the most important task at the end of
the 20th C (and beginning of the 21st) is to study how art works. My machines
are electro-mechanical devices; from simple instructions they produce rich and
complex images. Questions raised by machine drawings will be examined below.
Photography and the
Understanding of Mathematics
This paper considers ways in
which photographs help our understanding and teaching of mathematics. Some
historical landmarks are considered from Muybridge's galloping horses to
mathematics trails snapped with mobile phones. The possibilities have always
been limited by the available technology and have been shaped by changing
attitudes to mathematics teaching. It is argued that in mathematics teaching,
photographs are not just for illustration. They provoke discussion, pose
problems and provide data. We can measure them and model them with graphs. The
approach adopted for developing the Problem Pictures calendars and CD-ROMs is
described together with some of the ways these resources are used.
Inference and Design
in Kuba and Zillij Art with Shape Grammars
We present a simple method for
structural inference in African Kuba cloth, and Moorish zillij mosaics. Our
work is based on Stiny and Gips' formulation of 'Shape Grammars'. It provides
art scholars and geometers with a simple yet powerful medium to perform
analysis of existing art and draw inspiration for new designs. The analysis
involves studying an artwork and capturing its structure as simple shape
grammar rules. We then show how interesting families of artworks could be
generated using simple variations in their corresponding grammars.
Green Quaternions,
Tenacious Symmetry, and Octahedral Zome
We describe a new Zome-like
system that exhibits octahedral rather than icosahedral symmetry, and
illustrate its application to 3-dimensional projections of 4-dimensional
regular polychora. Furthermore, we explain the existence of that system, as
well as an infinite family of related systems, in terms of Hamilton's
quaternions and the binary icosahedral group. Finally, we describe a remarkably
tenacious aspect of H4 symmetry that 'survives' projection down to three
dimensions, reappearing only in 2-dimensional projections.
Mathematics and
Music: Models and Morals
The intimate association
between mathematics and music can be traced to the Greek culture. It is
well-represented in the prevailing Western musical culture of the 18th and 19th
centuries, where the traditional cycle of fifths provides a mathematical model
for classical harmony that originated with the well-tempered, and later the equal-tempered,
keyboard. Equal-temperament gives equivalent status to all twelve tonal centres
in the chromatic scale, leading to a high degree of symmetry and an underlying
group structure. This connection seems to endorse the Pythagorean concept of
music as exemplifying an ideal mathematical harmony. This paper examines the
relationship between abstract mathematics and music more critically,
challenging the idealized view of music as rooted in pure mathematical
relations and instead highlighting the significance of music as an association
between form and meaning that is negotiated and pragmatic in nature. In
passing, it illustrates how the complex and subtle relationship between
mathematics and music can be investigated effectively using principles and techniques
for interactive computer-based modelling [17] that in themselves may be seen as
relating mathematics to the art of computing, a theme that is developed in a
companion paper.
Teaching Design
Science: An Exploration of Geometric Structures
The late Dr. Arthur Loeb,
professor in the Department of Visual and Environmental Studies at Harvard
University, developed and taught Design Science/Synergetics, an exploration of
three-dimensional space, and Visual Mathematics, which explored the parameters
of structure in two and three dimensions for more than two decades. The main
foci of design science were geometry, mathematics, design and the beauty that
resulted from this marriage. Dr. Loeb’s
widow, Charlotte Loeb, donated the Design Science Teaching Collection to the
Edna Lawrence Nature Lab at Rhode Island School of Design in 2003. In its new
environment, the Teaching Collection is inspiring both faculty and students.
This paper includes examples of models made by RISD students in response to questions
arising from the study of geometry and design science.
More “Circle Limit
III” Patterns
M.C. Escher used the
Poincaré model of hyperbolic geometry when he created his four 'Circle Limit'
patterns. The third one of this series, Circle Limit III, is usually considered
to be the most attractive of the four. In Circle Limit III, four fish meet at
right fin tips, three fish meet at left fin tips, and three fish meet at their
noses. In this paper, we show patterns with other numbers of fish that meet at
those points, and describe some of the theory of such patterns.
J-F. Niceron's La
Perspective Curieuse Revisited
J-F Niceron's well known
work on the mathematics of anamorphism La Perspective Curieuse is a much quoted
but perhaps less read classic. In particular the templates he provides for
various transformations are commonly used as a starting point by those artists
who occasionally practise the anamorphic art. Some of these templates are known
to be approximations and some are exact. In the process of casting the
mathematical descriptions of these templates into modern notation suitable for
computation, a peculiar error has been found in Niceron's analysis of
transformations onto the surface of a cone or pyramid. The correct relationships
are presented and possible reasons for the error are discussed.
A meditation on
Kepler's Aa
Kepler's Harmonice Mundi
includes a mysterious arrangement of polygons labeled Aa, in which many of the
polygons have fivefold symmetry. In the twentieth century, solutions were
proposed for how Aa might be continued in a natural way to tile the whole
plane. I present a collection of variations on Aa, and show how it forms one
step in a sequence of derivations starting from a simpler tiling. I present
alternate arrangements of the tilings based on spirals and substitution
systems. Finally, I show some Islamic star patterns that can be derived from
Kepler-like tilings.
Approximating
Mathematical Surfaces with Spline Modelers
Computer modeling permits
the creation and editing of mathematical surfaces with only an intuitive
understanding of such forms. B-splines used in most commercial modeling
packages permit the approximation of a wide variety of mathematical surfaces.
Such programs may contain tools for aiding in the production of these surfaces
as physical sculptures. We outline some techniques for non-mathematical
designers and sculptors to produce these objects with conventional modeling.
The Lost Harmonic
Law of the Bible
The ethnomusicologist Ernest
McClain has shown that metaphors based on the musical scale appear throughout
the great sacred and philosophical works of the ancient world. This paper will
present an introduction to McClain's harmonic system and how it sheds light on
the Old Testament.
New ways in symmetry
This proposal presents the
continuation of the task assumed some years ago by this interdisciplinary
research team about the relations between Mathematics and Design. The basic
objectives in this proposal are:
1. To research about the
syntactic, generative and methodological possibilities of mathematical models
and fundamentally, geometric structures, as a base for the morphologycal
definition of the objects, in their widest significance.
2. To study the transference
of these knowledges to the educational level, through the implementation of
learning situations that imply not only to offer the model, but also the ways
of manipulation, extracting from it all its compositive possibilities. The idea
is to establish a work methodology that can be applied to different situations,
moving the students to be involved in each possible stage of the search.
3. To develop a systemic
approach that allows the use of different informatical programs to promote
creative development of students in the teaching- learning tasks.
Linkages to Op-Art
Many artists using
mathematical curves to generate lines in their work use Lissajous figures or
cycloids. There are many other curves which can be drawn 'mechanically' and
linkages do not appear to have been used in an obvious way. In my op-art period
many years ago, I used a simple linkage and I have resurrected this to create
some new ideas following a particular interest in the lemniscate.
D-Forms: 3D forms
from two 2D sheets
Is there a significant
branch of geometry that has been overlooked? Unlikely as it may seem, D-Form
geometry provides designers, architects, sculptors and artists with a vast, new
vocabulary of three-dimensional forms that are easy to play with and make. Easy
as they are to fabricate, D-Forms are proving equally hard to predict with
computing. This geometry exploits some interesting properties of developable
surfaces that, among other things, will enable you to 'square the circle'.
Visualizing Escape
Paths in the Mandelbrot Set
This paper describes a
method for producing a striking animation of the explosions that take place as
the parameter c that defines the Mandelbrot Set is allowed to traverse a path
from inside the large cardioid component of the Mandelbrot Set into one of the
attached ‘bulbs’ or other regions just outside the set. The presentation will
include the animation itself, as well as some of the colorful images obtained
by stopping the animation at various points.
The Math of Art:
Exploring connections between math and color theory
Simultaneous contrast and
extension are fundamental principles in color theory, which directly relate to mathematics. Color
study includes study of the proportions of colors and their effects. Using
these concepts of the interrelationship of proportions and color can broaden
expression much like adding extra colors to a painter's palette.
Islamic Art: An
Exploration of Pattern
As an historian of Islamic
art in the Department of Art History at the Maryland Institute College of Art,
I am continually learning as I endeavor to teach my students about pattern.
Teaching about pattern in Islamic art has facilitated my own exploration of
geometry in ways that also benefits my students. This visual presentation
explores the results of a single assignment that pertains to coloring a linear
plate reproduced in Bourgoin's classic work, Arabic Geometrical Pattern and
Design.
In Search of
Demiregular Tilings
Many books on mathematics
and art discuss a topic called demiregular tilings and claim that there are 14
such tilings. However, many of them give different lists of 14 tilings! In this
paper we will compare the lists from some standard references that give a total
of 18 such tilings. We will also show that unless we add further restrictions,
there will in fact be infinitely many such tilings. The "fact" that
there are 14 demiregular tilings has been repeated by many authors. The goal of
this paper is to put an end to the concept of demiregular tilings.
Tribute to the
Atomium
This paper describes the
project of a sculpture stemming from the pattern of the outside skin layout of
the Brussels' Atomium spheres. Two dual polyhedra are considered, the Catalan
disdyakis dodecahedron and the Archimedean truncated cuboctahedron. The special
projected location and setup of this sculpture makes it a good candidate for
celebrating the upcoming 50th anniversary of the Atomium in Brussels built for
the World Fair 1958.
RHYTHMOS: An
Interactive System for Exploring Rhythm from the Mathematical and Musical
Points of View
This paper introduces
RHYTHMOS: an interactive software system designed as a tool-kit for the
visualization, exploration, understanding, analysis, praxis, and composition of
musical notated (symbolic) rhythms. As such it provides user-friendly bridges
between art (music composition), performance (praxis), mathematics (cyclic
polygons and the distance geometry of point sets on a circle), and science (the
psychology of music perception). A description is provided of the system’s
capability and interactive graphical user interface. Applications to teaching,
learning, and practicing rhythms are discussed. Examples are given of the kinds
of research that RHYTHMOS facilitates. These include the testing of rhythmic features
for the classification, clustering, and phylogenetic analyses of families of
rhythms.
Spidron Domain: The
Expanding Spidron Universe
A number of new discoveries
have been made since the last Bridges conference in the area of Spidron
research. Shown here are samples of what will be presented in London.
An Introduction to
Medieval Spherical Geometry for Artists and Artisans
The main goal of this
article is to present some geometric constructions that have been performed on
the sphere by a medieval Persian mathematician, Abul Wafa al-Buzjani, which is
documented in his treatise On Those Parts of Geometry Needed by Craftsmen.
These constructions, which have been illustrated as flat images, could be considered
the bases of the arts and designs that artists and artisans have created on
both the exterior and interior surfaces of a dome. Therefore, such a dome art
design is a result of cooperation between mathematicians and artists. This
article also shows that the construction of the icosahedron on a sphere
presented in that treatise is not mathematically correct. However, the
construction of the spherical dodecahedron is exact. The article also presents
flat images of constructions of some Archimedean solids according to the
treatise.
Fabric Sculpture -
Jacob's Ladder
This paper develops ideas
from a paper folding idea known as Jacob's Ladder into a fabric sculpture. It
shows how, as an artist, I became aware of mathematics in my work. Translating
origami concepts into fabric constructions, the nature of fabric affects the
form. The opportunities fabric creates suggest possible developments.
Eva Hild:
Topological Sculpture from Life Experience
This is an introduction to
the ceramic sculpture of Eva Hild.
Interdisciplinary
Bridges: A Novel Approach for Teaching Mathematics
This paper describes
examples of interdisciplinary exploration opportunities that encourage students
to use critical and creative thinking skills as they gain understanding and
ownership of mathematical ideas. Students enjoy a stimulating journey on a road
of discovery.
Concerning the
Geometrical in Art
It is the expanse of thought
from earlier twentieth-century Modern art that has been in part an inspiration
to my recent painting entitled, The Blue Rider. There is History serving as a
discipline and the elements that shape the boundaries of style, vision,
repetition, method, and constraint in art. This is the role of suggestibility
for our perceptions. Expression is not isolated, limited, or confined to a
single notion, or arbitrary method. To escape from the perpetual forces of
society, tradition, and attitude would be to escape History itself. A process
that is introspectively palpable and its individuated, continual, motivated,
thematic, imaginative integration of geometrical configuration with color
serves as a vehicle to this discipline.
Knot Designs from
Snowflake Curves
The Koch snowflake curve is
one of the best-known self-similar fractals. Natural modifications of the
polygons that represent the early stages of its generation provide templates
for knotwork designs, some of which have been used in bookbindings. The
boundary of another well-known self-similar fractal, the Sierpinski gasket, is
closely related, and suggests a way to construct fractal knots.
Asymmetry in Persian
Symmetrical Art and Architecture
Since ancient times, the
integration of asymmetry in the design of composition has been a common
practice in Iranian art and architecture in order to avoid problems such as
topography and winds, and/or to comply with cultural and religious believes.
This is manifested in mosques where the Mehrabs are1 turned to the Qebla2 to face
in the direction of Mecca; in some entrances of mosques, public bath houses, or
houses, in order to provide more privacy for the users; in town planning of
large cities, in order to emphasize the old existing Friday mosques, or to
avoid the direct access to a castle or governmental building; in the design of
staircases, wind catchers, or in water distribution system; and in decorations
such as tiling and miniatures.
Cultural Statistics
and Instructional Designs
In online education, a
student’s first point of contact is the Web interface, a GUI that must induce
good feelings and trust. To achieve this, designer needs to be aware of
cultural trends shared between members of target group. This requires
mathematical formulas and statistical feedbacks so results can be stored,
categorized, processed, and retrieved. For example a resulting bar chart can
give a designer a vital clue as to what extend a target group tolerates
teacher’s interference. Efforts towards statistical representation of culture
started since late twentieth century. They mostly concentrated on multinational
organizations, but now with the table turned and employer being the end-user
(students), more sensitivity to the cultural issues must be paid. This paper is
a Call to the statisticians inviting them to explore this much-needed young
science, with applications that go beyond just commerce and education.
Musical Scales,
Integer Partitions, Necklaces, and Polygons
A musical scale can be
viewed as a subsequence of notes taken from a chromatic sequence. Given
integers (N,K) N > K we use particular integer partitions of N into K parts
to construct distinguished scales. We show that a natural geometric realization
of these scales results in maximal polygons.
Affine Regular
Pentagon Sculptures
In this paper we shall
describe how to apply symmetric linear constructions to a random non-planar
pentagon to construct mathematically and artistically interesting sculptures, such
as in Figure A. This process will always produce a nested set of affine regular
stellar pentagons. This generalizes a procedure created by Jesse Douglas.
1927 Two processes of creating form in music
It is intended to examine
form in two pieces of music written/realised in 1927: Webern's Opus 20 (string
trio) and Louis Armstrong's Wild Man Blues (Hot Seven) as a method of
evaluating their significance and diametric social relationship. Reference is
made to visual art movements and ideas from this period as well as a glance at
scientific and mathematical theory which may be seen to have a coincidental
relationship with some ideas in art in 1927.
The Effect of
Music-Enriched Instruction on the Mathematics Scores of Pre-School Children
While a growing body of
research reveals the beneficial effects of music on education performance the
value of music in educating the young child is not being recognized,
particularly in the area of Montessori education. This study was an experimental
design using a two-group post-test comparison. A sample of 200 Montessori
students aged 3 to 5-years-old were selected and randomly placed in one of two
groups. The experimental treatment was an ‘in-house’ music-enriched Montessori
program and children participated in 3 half-hour sessions weekly, for 6 months.
This program was designed from appropriate early childhood educational
pedagogies and was sequenced in order to teach concepts of pitch, dynamics,
duration, timbre, and form. The instrument used to measure mathematical
achievement was the Test of Early Mathematics Ability-3 to determine if the
independent variable, music instruction had any effect on Students’ mathematics
test scores, the dependent variable. The results showed that subjects who
received music-enriched Montessori instruction had significantly higher
mathematics scores. When compared by age group, 3 year-old students had higher
scores than either the 4 or 5 year-old children.
Celtic knotwork and
knot theory
Celtic knotwork is a form of
decoration in use for over a thousand years. The designs fill spaces or borders
with a pattern derived from plaiting. The designs have no loose ends and may
contain more than one closed loop. As in a plait (or braid) of hair, each strand
bounces back and forth like a billiard ball to form a pattern of diagonal lines
between the edges of the rectangle while crossing over and under others
alternately. The dimensions of a rectangular plaited panel can be expressed as
the number of bounces there are along the long and short edges. The number of
closed loops, referred to as knots by knot theorists, is the greatest common
divisor of these two numbers. This paper shows how one can predict the number
of loops there will be as a piece of knotwork is created from the panel by
removing some of the crossing places and rejoining the loose ends, without
crossing to make a gap either looking like ) ( to make what will be called a
horizontal gap, or like this shape turned through 90° to make a vertical gap.
As each of the chosen crossing places is removed and the loose ends rejoined in
this way a more intricate interlaced design is formed. It is easy enough to
trace round the resulting design with coloured pens to find out how many closed
loops there are, but the results proved and demonstrated in the paper enable
one to predict how the number of loops will change at each stage of the
creation of the interlaced design. Such a prediction is not addressed by
current knot theory. The first thing to notice is whether a loop crosses itself
at the crossing to be removed or whether two different loops cross there. In
the first case one or two questions must be answered before the number of loops
can be predicted. In the second case the two different loops get combined into
one single loop by the rejoining of the loose ends. The difficult part of the
research was to devise questions which could be proved to make reliable
predictions possible. One’s common understanding of how one might take a
shortcut back to the start while following a nature trail provides the last
link in the chain leading to the prediction. The method will be applied to
successive designs produced as each crossing is removed.
Mathematics
Investigations in Art-Based Environments
This paper presents two
sources of information about mathematics and art integration. The first source
is a brief outline of concepts that will be introduced through the workshop
series at this conference. The second source is a collection of insights and
resources about mathematics and art integration provided by a group of
elementary education teacher candidates.
A Geometric
Inspection of Pennsylvanian Dutch Hex Signs
This paper discusses the mathematics that is
involved in the construction of “Hex Signs” and describes the construction of
such signs. Hex Signs are circular discs with intricate geometric designs with
specific meanings that were hung on barns in the “Pennsylvania Dutch” region of
the United States. Common designs include: Rosettes, Birds, and Star Polygons.
Creating Sliceforms
with 3D Modelers
3D or CAD modeling programs can provide tools for
the beginner to quickly create mathematical models known as Sliceforms, or, in
the terminology of computer graphics, raster surfaces. This tutorial and
workshop provides the novice with the tools and procedures for modeling and physically constructing these models using their PC, a printer, craft
knife, glue and paperboard.
Paper Sculptures
with Vertex Deflection
This workshop presents mathematical concepts
vertex deflection and Gauss-Bonnet Theorem with hands on experiences using
paper, plastic, stapler and glue. We show how to create sculptor Ilhan Koman’s
mathematically motivated developable surfaces [1, 3, 4]. We also present how
one can construct a variety of shapes creating saddle, maxima and minima using
nip and tuck.
Understanding the
Mathematics Based Formulation on Dome Tessellation in Architect Sinan's Mosques
Design
K-12 education is
a comprehensive learning program that includes curriculum, tools, materials and
an innovative lesson delivering system. In college education it is not always
easy for teachers to keep students’ attention in history lessons. Following the
example of K-12 education, some new teaching methods should be suggested to
students on their history learning program to help them to understand the
history.
Moving Beyond Geometric Shapes: Other Connections Between Mathematics and the Arts for Elementary-grade Teachers
When classroom teachers are asked to identify connections between mathematics and art, they typically refer to geometric concepts. In an attempt to broaden their understanding of potential connections, this paper presents activities that involve common vocabulary, probability, and imagery.
Mandala and 5, 6 and
7 fold Division of the Circle
The Compass is perhaps
oldest of all math and drawing tools. It
is commonly known that with only compass, ruler and pencil, a six-fold division
of the circle can be made. An amazing array of 2 and 3 dimensional
possibilities then follow, to form bridges between Math, Art, History, Culture
and Science and even Mythology and Magic! Mathematics is learned through the
hands, creativity and social interaction. Further, the compass, when coupled
with the phi proportion, can be used to obtain 5 and 7 fold division of the
circle. The Initiate, interested in mastering the compass, must begin this
journey of exploration by ensuring precision. Often, the compass user grips the
device too firmly, pressing harder in an effort to ensure quality. The result
of this 'muscling' is often that the point makes an overly large hole in the
paper, the compass opens from the pressure, making a spiral, and the paper
slips. The proper way to grasp the compass is to twirl the upper post between
thumb and index finger, so that it pirouettes.
In this way it makes a crisp circle. The image may be faint but we can
twirl the compass more times for better definition, rather than pressing
harder. With brief explanations, we will now proceed rapidly through a
multitude of forms.
Mathematical Book
Forms for Teachers
The sequential properties of basic mathematics
facilitate the creation of math art book forms. This workshop presents three
artists book forms with mathematical significance for school teachers.
Scissors, glue stick, protractor, straight edge and pre-cut paper are the only
required equipment to make all three forms.
The Aréte of Line
Designs
This workshop will explore the historical,
philosophical, and pedagogical nature of line designs, with a focus on good
designs and what constitutes the proper context and good environment ensuring
"joy in work" is realized, now and in the future.
The Plato Bead A
Bead Dodecahedron
Creating polyhedra with
beads is another way to learn the properties of regular and semi-regular
solids. The instructions given below are for the dodecahedron (The Plato Bead).
In a bead polyhedron each face becomes open space; each edge becomes one bead;
each vertex becomes a thread void. The structure is light and open. The size of
the overall bead changes with the size of beads used.
Building Simple and
Not So Simple Stick Models
Physical models are
invaluable for conveying concepts in geometry. In this paper, I explain how to
build stick models based on the Platonic polyhedra. Supplies for these models
were thin bamboo shish kebab sticks from a grocery store, and vinyl tubing from
a hardware store; both supplies are inexpensive and readily available. The
tools used were a ruler for measuring the length of sticks, a clipper to cut
the sticks, a scissor to cut the tubing, and a punch to make holes in the
tubing. These tools are also reasonably inexpensive and readily available.
Grade school, high school, undergraduate and graduate level students have made
models with these supplies and tools and all of them have taken away something
meaningful related to their existing level of knowledge.
Topological Mesh
Modeling
This workshop presents Topological Mesh Modeling
with hand-on experiments using our topological modeler, TopMod. Our modeler
provides a wide variety of interactive techniques that allow to create unusual
and interesting shapes by changing the topology of 2-manifold meshes.
Vermeer's the Music
Lesson in Modular Perspective
This BTTB Workshop has the aim to recreate the
perspective outline of the Music Lesson. The reader may notice in Figure 1, how
the painting’s image formation clearly fits in my RMS90
Modular Scale. We will learn the basic use of this scale to
directly deduce all the elements of the scene in perspective. Therefore,
neither a plan nor an elevation is required for the practice, just a good
photograph copy of the painting is needed. I will provide these copies and the
scales as well, while the participants should bring some A4 sheets, a
portable drawing board, squares, eraser, and pencils (gray and yellow). It would
take about 75 minutes to perform it. You will remember those high school days.
Zellij Multipuzzle
Using the technique of laser cutting I recently
made a game I named “zellij Multipuzzle”. It is a set of 669 zellij-style
tiles. One side colored in white, the other in a different color, so each tile
can be used in a “positive” or “negative” configuration, according to the
necessary alternation of colors. There is also a set of units with which you
can make frames of different sizes. This make the game easier for beginners.
This is the most efficient way I have experimented for an introduction to the
art of geometrical arabesque : direct immersion in the galaxy of zellij
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