The Bridges Conference: Mathematical
Connections
|
The following papers are to be presented at the Bridges Conference in August 2006. They are in no particular order.
The draft programme schedule of talks and workshops is available for you to consult.
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 knotted 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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