3OSME Abstracts of Talks Plenary Talks Mathematics
of Origami Origami in Education
Science and Applications
Energetics of Crumpling
Abstract: When a thin sheet of elastic material is confined within a shrinking volume it does not deform uniformly, but instead forms a network of singular point and ridges. This crumpling process can be viewed as a condensation of elastic energy onto the area surrounding ridge lines, an area fraction of the sheet which becomes arbitrarily small as the thickness of the sheet goes to zero. The energy condensation is driven by a competition between bending and stretching energy costs, which are coupled by the geometric relation between curvature and strain. In this lecture I will review the energy scaling properties of ridges in thin elastic sheets. I will then present the results of our studies on crumpling of thin sheets in four spatial dimensions. Changing the spatial dimension changes the geometric coupling between energies. In simulations of very thin sheets in 3 or 4 spatial dimensions, small deviation of the sheet into the fourth dimension effectively negated the strain field around ridges  leading to greatly reduced total elastic energy and different energy scaling. I will discuss whether the two cases converge to the same configuration in the unstretchable limit. I will also comment on the insights this research gives into the nature of singularities in regular crumpling. For more information and illustrations, go to http://vonkarman.uchicago.edu/gallery/.
Fold Paper and Enjoy Mathematics: ORIGAMICS
Abstract: The subject of our meeting ORIGAMI is an international
word which is derived from the Japanese word origami. They have same
spelling, but they differ somewhat in sense as well as pronunciation.
The accent of the former falls on the third syllable (ga) while the latter
on second one (ri). In Japan, origami usually means a handicraft
hobby for children, so almost all books about origami are arranged in the
juvenile section of bookstore even if some of them are for enthusiasts.
Mathematical Origami and Alhazen's Problem
Abstract: I will give a survey of the axioms of mathematical
origami and their connections with the theory of fields and constructions
with conics. I will illustrate how to use the algebraic information from
fields and curves to construct a solution to the famous optical problem
of Alhazen and indicate how this leads to a solution by origami.
Origami and Origami Numbers
Abstract: The art of origami is from Japanese traditional culture
and produces many interesting geometrical shapes. In the present paper,
we study which shapes are possible to construct using origami. Then we
obtain the necessary and sufficient condition for which a real number is
an origami number constructed by origami. Also, a regular polygon of p
(p:odd prime) sides can be constructed using origami if and only if p is
an origami prime. Those were described by H.Huzita and B.Scimeni in the
Proceedings of the First Meeting of Origami Science and Technology, but
our proofs are different from theirs. In our proofs, the trisection of
an angle by H.Abe, the duplication of the cube volume and the Cardano's
formula of the cubic equation play essential parts. Moreover, we study
two classical construction problems of Pappus and Philo. It is impossible
to construct these problems using a compass and straight edge, but we can
solve these problems using origami.
Computational Tools for Origami Tessellations
Modelling the Folding of Paper into Three
Dimensions
Abstract: We model the folding of ordinary paper via piecewise
isometries from R^2 to R^3. The collection of crease lines and vertices
in the unfolded paper is called the crease pattern; we mainly consider
the case of crease patterns with a single vertex. Our results generalize
the previously known necessity conditions from the more restrictive case
of folding paper flat (into R^2); if the crease pattern is foldable, then
the product (in a nonintuitive order!) of the associated rotational matrices
is the identity matrix. This condition holds locally in a multiple
vertex crease pattern and can be adapted to a global condition. Sufficiency
conditions are significantly harder, and are not known except in the twodimensional
singlevertex case. We have achieved partial results in this arena.
The Complexity of Flat Origami
Abstract: We study a basic problem in mathematical origami: determine
if a given crease pattern can be folded to a flat origami. We show that
assigning mountain and valley folds is NPhard. We also show that determining
a suitable overlap order for flaps is NPhard, even assuming a valid mountain
and valley assignment.
A DiskPacking Algorithm for an Origami Magic
Trick
Abstract: We present an algorithm for a magic trick. Given a
polygon with holes P, our algorithm determines a folding of a rectangular
sheet of paper such that a single straight cut suffices to cut out P.
Recent Results in Computational Origami
Abstract: Computational origami is a recent branch of computer
science studying efficient algorithms for solving paperfolding problems.
This field essentially began with Robert Lang's work on algorithmic origami
design, starting around 1993. Since then, the field of computational origami
has grown significantly. The purpose of this paper is to survey the
work in the field, with a focus on recent results, and to present several
open problems that remain. Along the way, we present some new related results.
Just Like Young Gauss Playing with a
Square: Folding the Regular 17gon
Abstract: When Carl Friedrich Gauss wrote about constructing
a regular 17gon with straightedge and compass in his Disquisitiones Arithmeticae,
he gave the world of mathematics a playful little morsel embedded in a
work of seminal importance. Despite the fact that all straightedge and
compass constructions can readily be replaced by origami constructions,
folding a regular 17gon is not at all a simple matter if we try to follow
the usual constructions. In this talk, we present a precise method of
folding such an ngon as derived from CFG’s ideas, including all the math
needed to see that the method actually must work, even if sloppy folding
makes us doubt the validity of the method.
Counting MountainValley Assignments for Flat
Folds
Abstract: By a flat fold we mean a finite sheet of paper that
has been folded into a potentially complicated object which lies flat (can
be pressed in a book without crumpling). Flat folds can be described
by their crease patterns, C, a collection of lines in the paper which indicate
where the paper is folded. A mountainvalley (MV) assignment is a
function f : C to {M,V} which indicates which crease lines
are convex and which are concave, respectively. We examine the problem
of counting the number of valid MV assignments for a given crease pattern.
In particular we develop recursive functions that count the number of valid
MV assignments for flat vertex folds, crease patterns with only
one vertex. We also provide examples, especially those of Justin,
that illustrate how difficult the general, multivertex case is.
The Geometry of Orizuru
Abstract: It is possible to fold the classic orizuru (crane)
from paper other than a square. We can make an orizuru with long wings
using rhombic paper. However, we cannot make a beautiful orizuru from a
rectangle.
Trees and Circles: an Efficient Algorithm for
Origami Design
Abstract: A fundamental problem in origami of both mathematical
and practical interest is the problem: given a desired folded form, construct
a crease pattern in a square which, when folded, gives the initial target
shape. This is a problem that is often phrased, "How do you fold a (blank)?"
Surprisingly, a large class of target shapes may be folded by a deterministic
process of design. Specifically, the problem of folding any shape
topologically equivalent to a tree graph that satisfies certain simple
geometric conditions may be solved by a constrained nonlinear optimization
and simple geometric constructions. In this paper, I will describe
the theory and the constructions and will present several examples, including
a demonstration of the nonintuitive result that it is possible to fold
a shape with unbounded perimeter from a finite sheet of paper.
The Definition of Isoarea Folding
Abstract: Origami models whose obverse and reverse sides appear identically are called "isoarea folds". The concept of isoarea folding was presented by Kawasaki, and he made isoarea cubes, isoarea octahedron and others. It is a purely geometric concept, but the definition of it hasn't been public as far as I know.
The most general definition of isoarea folding is
to regard it as a kind of "rotatory inverse symmetry" which is a concept
of crystallography. When a figure (solid) isn't changed by n (n:natural
number) times rotatory inverse operation,
Perfect Bird Base From Nonsquare Quadrilaterals
Abstract: There are five steps needed to take a square sheet
and cut a nonsquare quadrilateral of one's choice and fold it into a perfect
bird base as defined by Jacques Justin (ISIS Symmetry V5, No. 2, "Mathematical
Remarks About Origami Bases," 153166).
Paperfolding Performs More Classical Constructions
in Elementary Geometry
Abstract: It is wellknown that some famous geometric constructions,
such as duplicating a cube or trisecting an angle, cannot be achieved by
ruler and compass but are successfully treated by paperfolding, provided
some proper basic procedures are permitted (see e.g. Huzita, H. and
Scimemi B., The algebra of paperfolding, Proceedings of the First International
Meeting of Origami Science and Technology, H. Huzita ed. (1989), 215222).
Here we shall apply p.f. to solve other classical, though less popular
geometrical problems. As an example, we shall treat minimum problems,
such as constructing a point on a given parabola whose distance from a
given point is minimal, or finding a line within a given pencil which cuts
a minimal length segment on two given lines (l'Huillet's problem). Other
problems will regard the construction of a triangle for which a set of
notable points is given or special relationships hold, regarding its perimeter,
inradius etc. While our constructions are purely syntetical and achieved
by a chain of permittable p.f. procedures, their proofs will require some
elementary analytical geometry.
Origami as a Model for Development in Organisms
Abstract: An important field in modern biology is the study
of the developmental processes in organisms. The Human Genome Project
is cataloguing the sequence of nucleotides in our DNA, but the research
on the conversion of this information (the genotype), through growth and
development, into an organism (the phenotype) is essential to our understanding
of life and life processes. To help students understand this morphogenesis
("form creation"), we use origami. An origami crease pattern is the
analog of the genome. The information is there, but is generally
not sufficient for producing the finished model. Information about
the folding sequence, however, makes this process much easier. Origami
models with similar, but not identical, crease patterns and folding sequences
can be compared to closely related species with similar, but not identical,
genomes and developmental sequences.
Origami in Mathematics Textbooks for Prospective
Elementary School Teachers
Abstract: This paper reports on a survey that was conducted on
the nature and extent of paperfolding applications in contemporary United
States textbooks intended for use in mathematics content courses for future
primary and intermediate school teachers.
Changing Origami Directions to Mathematical
Language
Abstract: "Fold the paper in half." Many origami
directions include similar statements, but they are not written using mathematical
language. "Mathematics generally has the reputation of having a precision
that no
Triangular Boxes From Arbitrary Sized Rectangular
Sheets of Paper: a Pedagogical Study that Incorporates Origami, Plane
Abstract: The problem of designing a specific class of isosceles
triangular boxes starting with single rectangular sheets of paper of arbitrary
dimensions provides an interesting pedagogical example of origami, plane
geometry, trigonometry, and calculus. As such, it may be a useful
source of enrichment in high school level mathematics classes. Some
aspects of the problem may also be effective for introducing informal geometric
ideas in the lower grades. The parameter space (length L and width
W of starting rectangle, height of box H, and angle q
made by the two equal length sides of the triangle) is explored.
Theoretically, boxes with any specified q
may be folded, but for some parameter values, the folding would be physiologically
impossible. H is determined as a function of L, W, and q
.
For given values of L and W, the values of theta that correspond to the
maximum area of the triangular face and to the maximum volume of the folded
box are determined. The maximum area of the face of the box corresponds
to q = 90*,
and the maximum box volume to theta greater than 90*.
Applications of Origami to the Teaching
of Sophisticated Communication Techniques
Abstract: Apart from its wellknown and successful uses in the
physical and mental development of children and in specialized therapies,
the learning and teaching of origami can be extremely practical for internalizing
strategies and techniques necessary for successful adult communication,
both interpersonal and public. It facilitates transferring the focus of
attention of the interlocutor from her or himself to the other(s) in a
communicative act; it develops consciousness of other viewpoints,
other interests, and the value and methods of adapting to those variables.
At the same time, it can be used to hone the development of selfpresentation
strategies, to enhance successful participation in communicative dyads.
Making More Use of Origami in the Elementary Classroom
Abstract: In this paper the author, in her retirement, has been
able to spend more time collaborating with other teachers who are also
interested in using origami in their classrooms. We look at
some specific mathematical applications particularly with fractions.
We have observations of groups of children in classrooms improving their
ability to get along with other classmates by teaming as they learn origami.
Another successful application has been teaching one group of children
to teach others. Selfesteem has been our goal as well as the improvement
of communication skills. Appreciating the fine art of origami is
a special application we also want to have happen with students.
Origami for Adult ESL Learners
Abstract: As an educational tool origami has become increasingly
prominent in various disciplines such as science, mathematics, and art.
It also carries numerous benefits in physical and psychological therapy.
However, its applications in language teaching, especially for adult ESL
(English as a Second Language) learners is quite nebulous. Therefore, the
presenters intend to define the role of paperfolding in the teaching of
ESL. They will show how origami appeals to the kinesthetic, tactile,
aesthetic, and visual styles of some learners. They will discuss
certain characteristics, unique to adult ESL teaching, that must be considered
in the choice of suitable materials or techniques especially with regard
to the subject of origami. The presenters will demonstrate and explain
how to create several ESL lessons using common origami models as devices
to enhance the learning process. Other sample lessons include using
paper as props, creating map matrices, and diagram reading. The presenters,
both skilled ESL teachers, will explain how to adapt storigami materials
to adult ESL teaching. They will show how paper can be used to elicit
new or newly learned vocabulary. There will also be a presentation
of ESL activities that require the students to fold an 8 1/2 x 11 paper
into parts.
Logic of Axiomatics in Origami Geometry
Abstract: It is too hard for everyone to understand deeply the
historical importance of the Euclidean Axioms, especially for school children
and even for school teachers. I think that nowadays all the instruments
necessary for the study of geometry should be easily fabricated and easy
to use. Anything less would be contrary to the intentions of Mr.
Euclid, but remember that the geometry of the ancient Greeks is called
called the Geometry of Ruler and Compass.
To Fold or Not to Fold
Abstract: Most books on origami contain information on symbols,
basic folds and elementary technique. This set of facts, in general, is
quite enough for working on origami material. But for creative purposes,
for satisfying natural inquisitiveness, especially with 3dimensional origami,
additional material is needed.
Reflections on a Broken Heart
Abstract: Folding, teaching and discussing origami models are
mathematical activities. In this presentation and paper, the authors focus
on the delightful action model "Broken Heart" by Edwin Young. A set of
questions and challenges are formulated and addressed for the heart model
and compared with the preliminary and waterbomb bases.
Computer Methods to Facilitate Production of
Origami Directions
Abstract: Directions for making origami models follow a moreorless
fixed methodology in most published books. Formal diagramming, however,
is not the only way to convey the procedure of making a model.
Using Origami to Promote Geometric Reasoning
and Collaboration in the Classroom
Abstract: This presentation focuses on the ways in which origami
can serve as a context for developing mathematical ideas and insights and
for posing mathematical questions. Students can explore such mathematical
topics as combinations, permutations, angle measures, numerical and geometric
patterns, perimeter, area, and volume relationships, the characteristics
of polyhedra, symmetry, and possible map colorings.
Using Planar Graphs to Devise Coloring Schemes
for Origami Polyhedra
Abstract: Graph theory has many interesting and important applications
in such fields as communication networks, work assignments, scheduling,
and coloring problems. This presentation will show some ways that
planar graphs can be used to explore coloring problems with origami polyhedral
structures.
Investigating the Role of Axioms with Paperfolding
Abstract: In a democratic society, it is critical to judge the
claims of scientists, journalists, politicians, and advertisers. Validity
and reliability of such information sources depend on the methods by which
natural scientists, social scientists, historians, and others come to "know"
things. Axioms play an important role in the ways that we "know"
mathematics, and the comparison to other fields of knowledge is illuminating.
In Praise of the Papercup: Mathematics and Origami
at the University
Abstract: The traditional papercup was the starting point
of a workshop held for students majoring in Primary Education at the Second
University of Milan, Italy. First, it is an easy model to teach people
with no previous knowledge of origami and origami symbols. Most importantly,
it lends itself to many mathematical observations: a worksheet was designed
to help students focus their attention on them (some formal proofs in Euclidean
geometry were also requested). Finally, some easy modules derived
from it can be used to introduce modular origami. A similar approach
was used elsewhere in the workshop, whose main goal was to make students
aware of the geometry underlying origami. Other topics included:
construction of modular regular polyhedra (and some stellations) fulfilling
some assigned colour requirements, rectangles and similarity, an origami
geometry theorem on triangles, approximate division of a sheet of paper
in any number of equal parts.
Abstract: In this talk we will explore a standard flat splint
square to round basket and explore some of its mathematical implications.
Among these are the transition from flat materials to threedimensional
containers and a concrete way to consider the ratio that we call pi.
Teaching is Impossible!
Abstract: Thoki will address the topic "Learning Origami" and
writes that he will touch upon 1) the difference between teaching and learning,
2) the difficulties in learning in mixed groups, 3) the advantages of person
to person instructions, 4) how to get a person in a mood of willingness
to learn, 5) adjusting the method according to the level of skill in the
pupil, and 6) the importance of terminology. However, Thoki has also
sent us the following message:
Science and Applications of Origami
Square Cycles: an Analysis of Flexagons
Abstract: In 1939 while a graduate student at Princeton, Arthur
Stone invented what is known as a hexaflexagon, a hexagonal flat figure
folded out of a strip of equilateral triangles. This object can be
flexed and rotated to bring different faces to view. At the time
of their invention, the flexagons were an interesting mathematical recreation.
They were also useful to topologists as an example of a Mobius strip, albeit
an unusual one. We intend to give a brief summary of hexaflexagons,
their properties, and some classification information. We will then
turn to tetraflexagons, littleknown relatives of the hexaflaxagon that
are folded from squares. The latter are barely discussed in the literature,
and in many ways seem to be more interesting than their betterknown cousins.
Exploring the Possibilities of a Module
Abstract: Rona Gurkewitz defined a modular system as a family
of polyhedra models made from the same module. Usually the system contains
only few models, because of constrains builtin into the module.
We concentrate on edge modules, i.e. modules which correspond to edges
of a polyhedra. Modules of this type join in polyhedra to form vertices.
There is no constraint on vertex degree, but the angle between two adjacent
modules is bounded by some maximal value. This constraint leads to
maximal number of sides of each face. An open frame 2 module by Tomoko
Fuse is an example of edge module. Here the angle between two adjacent
modules is limited by a right angle, so this module permits only triangle
and square faces of open frame system models. The construction of
a dome allows us to go beyond this limitation. Two types of domes: small
dome and large dome will be described. Domes give pentagon, hexagon, octagon
and decagon polygons, which are required to build all of the regular (Platonic
and Archimedian) polyhedra. The mixture of small and large domes
and different rotations of domes give additional effects. Examples
of the application of this technique to build polyhedra models are presented.
A Folded Deep Pot
Abstract: Folded deep pots can be made from a single sheet of
paper. Unlike other paper pots that consist of several pieces of
paper held together with adhesives, these pots are made from a single piece
of paper and no adhesive is necessary. They can serve as paper containers
for a variety of uses. The design of these pots is based upon mathematical
and engineering principles, and they can be made on a production line.
Multimodular Origami Polyhedra and their
Duals, Made From a Family of Vertex Modules
Abstract: Modular origami polyhedra are generally made from a
number of identical modules. In this paper we describe new polyhedra that
are made up of more than one type of module from a family or system of
modules.The family consists of the onepiece triangle, square, pentagon,
hexagon and octagon modules by Simon, Arnstein and Gurkewitz. The onepiece
triangle and square modules are onepiece analogues of the twopiece triangular
and square gyroscope modules by Lewis Simon.
Airbag Folding: Origami Design applied to an
Engineering Problem
Abstract: Today modern automobiles all have multiple airbags
installed. Airbags are complex engineering systems with a folded textile
structure as its core component. During the development of airbag systems,
numerical simulation models are used extensively. The mathematical methods
used, allow both the folding and forced unfolding of the airbag model.
The paper demonstrates how folded airbag models are created and how their
unfolding is simulated. It also shows how origami design techniques have
been used to create simulation models for complex 3Dairbags.
Use of Origami in Fielding Very Large Space Telescopes
Abstract: Very large, 2550 meter aperture, spacebased telescopes
are crucial for advanced tasks such as the direct detection and observation
of Earthlike planets around other stars. Fielding such large telescopes
in space is particularly challenging because the telescope must be extremely
lightweight, must be tightly stowed during launch, and yet deploy into
a highprecision optical instrument.
Polypolyhedra in Origami
On the Validity of the Orb and Other Origami
Models
Abstract: The folding and construction sequence of an origami
model, together with certain assumptions about the properties of mathematically
ideal paper, imply a system of constraints that defines the geometry of
the model. If the system of constraints has a solution, we call the
model "valid". We give examples of valid and invalid models, along
with their constraint systems. In particular, we analyze a questionable
model known as "The Orb" and find it to be valid.
Organizing Global Shape through Local Interactions
using Origami
Abstract: Amorphous computing is a project to build and understand
systems that organize themselves to achieve predetermined goals, using
only local computation and local communication. The dual goal is to learn
how to engineer selforganizing systems as well as understand such phenomena
in nature. In this talk I will present a language for describing the folding
of a sheet of identicallyprogrammed deformable cells to make predetermined
global shapes, and a means of compiling that language into instructions
for the cells. The global language is based on Huzita's axioms and the
local instructions are inspired by mechanisms from developmental biology.
I will show how the properties of Origami shed light on the creation of
pattern and form in biology.
Circular Origami: a Survey of Recent Results
The geometry of paper folding finds particularly successful application through the use of circles. Though hardly traditional, the use of circular paper for Origami has produced, directly or indirectly, some interesting results. This paper proposes a survey of these results which we have obtained in the fields of Fine Arts, Education and Mathematics. In particular, some of the spin offs of the research include "Project Geraldine", the construction of a 10 foot Endopentakisicosidodecahedron [1]. The author and her partner have also had the opportunity to use circular Origami to teach geometry to middle and high school children [2,3] as well as to explore classes of deltahedra with results carrying over into number theory [4]. These projects are the result of the first year and a half of systematic research into circular Origami, the chronology of which is the purpose of this paper. [1]Bridges 1999: Barnraising an Endopentakisicosidodecahedron
Bridges can be found at http://www.sckans.edu/~bridges/
and ISAMA at http://www.math.albany.edu/isama/2000/
Folded Tubes as Compared to "Kikko" ("Tortoiseshell")
Bamboo
Developable surfaces with three pairs of scored sinusoidal lines are
folded in extremely light and stiff tubular structures. This research started
from workshops with French students in industrial design and bionics. The
structures can be compared to abnormally grown "kikko" ("tortoiseshell"
pattern) bamboo of particular mophologies and mechanics. This bamboo is
highly appreciated for its strength and stiffness in Japan and shows an
interesting mechanical feedback between growth and form. The bamboo could
give some indications for design optimization of the folded tubes. Computer
simulations of both structures are currently under research.
Origami with Trigonometric Functions
Abstract: When we create some origami works, especially geometrical ones, trigonometric functions (sin,cos and tan) are very useful tool for us. These functions connect length with angle. Length is easy to handle via paperfolding because we know various origamic ways to divide an edge of paper. On the other hand, almost all angles are hard to handle except for specific ones, for example 90,60,30 and 45 degrees. At the 2nd. International Meeting of Origami Science and Scientific Origami, I presented a systematic way to construct arbitrary angles using trigonometric functions[1]. This method was as follows; first we provide a value of the trigonometric function for a certain angle. Next we approximate this value to an appropriate fraction. Finally, we can get this angle on a paper by folding. Moreover, we can use trigonometric functions to solve an equation or an inequality, which gives some conditions for forms of origami works (or usually modules/units for me). Here we make use of the property that trigonometric functions exchange an angle value to a ratio of two lengths. Trigonometric functions and origami are very good friends, I believe. These functions will make our origami life rich. I hope that trigonometric functions become a standard tool for origami people. [1] Miyuki Kawamura, "Polyhedron Origami:A Possible Formulation by 'Simple
Units'", Proceedings of The 2nd International Meeting of Origami
Science and Scientific Origami in Otsu, Japan, 1995.
The Application of Origami Science to Map and
Atlas Designs
Abstract: This paper reports applications of origami science
to map and atlas designs. In terms of technical words, this is the problem
of packaging 2dimensional information printed on a sheet of paper into
a smaller envelope.
A Study of Twist Boxes
Abstract: Two types of twist boxes are presented; each of which is folded from a square sheet of paper and is surrounded on every side with four equivalent parallelograms. Each parallelogram in the Type B5 is folded mountainlike along a shorter diagonal and that in the Type B6 is folded valleylike along a longer diagonal as shown in Figs.(a) and (b). The properties of the twist of these boxes are characterized by the shape of the parallelograms, which are determined by the box's height and the positions of four corners of the bottom of the box on the square sheet. Here the structures of the two types of twist boxes will be analyzed mathematically and the results will be reported.
Abstract: Sets of Laves tiles with all combinations of C and
S curves that allow spiral arrangements, studies of their properties and
expressions of them as crease patterns on the plane will be presented.
The Foldin'ator Modeler and Document Generator
Abstract: Sheet materials such as paper are ubiquitous in our
society, and form the basis of numerous manufactured products. However,
artisans working in the domain of folding and shaping sheet materials
have been hampered by lack of availability of visualization and modeling
tools common to other areas of design and production, working instead in
the actual materials with all their cumbersome, incumbent overhead, and
having no easy means of documenting their work.
"Is what you see what you think it is?" Unexpected
Properties of the Square Frame in Origami Puzzle Design
Abstract: An interactive presentation of two new and challenging
origami puzzles will be presented. This provides an exploration of
the way visual perception can both assist and inhibit thinking. The solutions
show relationships with general Cantor Sets and Chladni figures.
