sinussinus.uni-bayreuth.de/math/tnt_math_15.pdf · develop a braided model corresponding to a cube...

Peter Baptist Carsten Miller ISSN 2192-7596Dagmar Raab University of Bayreuth (Eds.) www.sinus-international.net

SINUSinternational

Towards New Teaching in Mathematics

15 / 2013

Hans Walser

Hands-On Geometry 15


Issue 15Hans WalserHands-On GeometryBayreuth, Germany 2013

EditorsPeter Baptist, Carsten Miller, Dagmar RaabUniversity of Bayreuth, Germany

AuthorHans Walser Universität Basel – Mathematisches InstitutRheinsprung 21CH–4051 Basel

PublisherCentre for Math e matics and Science Educa tion (Z-MNU) Chair of Mathematics and Mathematics EducationUniversity of Bayreuth95440 BayreuthGermany

TranslatorReineke-Team, Heidelbergwww.reineke-heidelberg.de

The English translation and layout were financially supported by the Federal Ministry of Education and Research and the European Union’s Seventh Framework Programme within the European Fibonacci Project.

LayoutCarsten MillerUniversity of Bayreuth

German originalGeometrie zum Anfassen, BLK-Tagung Leipzig, 18. bis 21. September

SINUSinternational

DISSEMINATING INQUIRY-BASED SCIENCE

AND MATHEMATICS EDUCATION IN EUROPE

www.sinus-international.net

ISSN 2192-7596

Hans Walser

Hands-On Geometry

4

Towards New Teaching in Mathematics 15 / 2013

Hands-on geometry

Introduction

School geometry is usually two-dimensional, presumably in keeping with the two-dimensional nature of notebooks and blackboards. However, the world we live in is three-dimensional, and so are most practical geometrical problems.Learning geometry – especially in three dimensions – means constructing models. Of the vari-ous methods used in this context, the braiding technique stands out by virtue of its simplicity. All you need is strips of paper, nothing else. Braided models help to make three-dimensional geometry comprehensible on a hands-on basis. The symmetry of these models leads to com-binatorial insights, on the one hand, and to spherical geometry, on the other. This results in an understanding of particular pavement patterns and dense spherical packs and their signifi-cance in crystallography and chemistry. The braiding structures pave the way to topological considerations and invariants.Students discover amazing technical and methodological simplifications. In addition, the models are very appealing in aesthetic terms.

5

Hands-On Geometry

1 Introduction

Making geometry a hands-on experience means working with models. They are usually con-structed using thick paper and glue – an irreversible work process that is repeatedly inter-rupted due to the necessary drying time for the bonding agent. Braided models, by contrast, require no glue and, in principle, can be broken down into strips again. The braiding struc-tures of these models have both a topological and a combinatorial aspect. The braiding struc-tures can be visualized using great circle models and thus provide a connection to spherical geometry. Braided models of different kinds are described in [Cundy/Rollet 1961], [Hilton/Pedersen 1994], [Pedersen 1981] and [Hilton/Pedersen/Walser 2003]; Pargeter [Pargeter 1959] even showed that every polyhedron can be constructed as a braided model.

2 Braiding technique

The braiding or weaving technique is one of the oldest techniques of human civilization [Gerdes 1990], [Pedersen 1983]. Two sets of parallel horizontal and vertical strips suffice for making a flat braiding structure.

Three types of strips are required, by contrast, to form the three-dimensional corners of a cuboid basket, for example.

Braiding: flat and with corner

6


3 Braided model of a cube

The simplest braided model of a cube consists of three strips of paper. Each strip is composed of six fields that are nearly square. For braiding-related reasons (leeway) the strip width has to be somewhat smaller than the edge length of the cube. In practice a difference of ε ≈ 1 mm is sufficient. The first two fields of the strips have to overlap the last two in each case. They serve to stabilize the braided model.

Cube with three strips

If we imagine the strip width is reduced, we get an insight into the braiding structure.

This braiding structure can be even further abstracted by means of a spherical model, which you can construct using the three great circles consisting of the equator, the 0° / 180° meridian and +90° / –90° meridian such that they run inside or outside each other as ap-propriate at the intersection points. The braiding structure is topologically composed of three entwined rings with the following characteristic: if one of the three rings is removed, the other

Braiding structure: square and round

7

Hands-On Geometry

The Borromean rings

two fall apart. This figure was the em-blem of the Borromeo family. One of the members of this aristocratic Italian fam-ily dating back to the 13th century was Carlo Borromeo (1538-1584), a prelate who played a major role in the Counter-Reformation.

4 Cuboid and parallelepiped

When treating the parallelepiped product (volume of the parallelepiped) in vector geometry, I noticed that students had great trouble recognizing a parallelepiped in a two-dimensional illustration of one. Since students get used to “orthogonalizing” the distortedly displayed angles between the edges in oblique projections of cubes and cuboids, i.e. interpreting them as right angles, they also see a parallelepiped merely as a cuboid. It is essential to familiar-ize students with a “genuine”, i.e. three-dimensional, parallelepiped. That is why I tried to develop a braided model corresponding to a cube for both a cuboid and a parallelepiped.

4.1 Braided model of a cuboid

Students easily find the three strips required for the braided model of a cuboid with the edge lengths a, b and c.

Strips for a cuboid

8


4.2 Braided model of a parallelepiped

Working out the three strips needed for a parallelepiped offers the opportunity to learn from error.

First of all, the students make an attempt with strips that are affine distortions of the cor-responding cuboid strips. When such a strip is folded, it turns out that the strip does not close, but rises like a staircase. The key insight is that it “has to go back down again”. After several attempts – including use of an intermediate step with a special parallelepiped that is bordered by four squares and only two “genuine” parallelograms and can be depicted well, for instance by shearing a stack of books – the pattern for the three strips is found in the end.

4.3 Glide reflection symmetry

The strips for the parallelepiped do not display any symmetry at first glance. That is not true; in fact, when viewed as a bracelet ornament, i.e. continued infinitely at both ends, the strips have a so-called glide reflection symmetry.

This strip doesn’t do the job

The three strips for the parallelepiped

Glide reflection symmetry

9

Hands-On Geometry

This type of symmetry can be found in many decorative creeper designs. It is also the sym-metry of a footprint in the sand.

5 Rotational symmetries

5.1 In cubes

Various rotations are possible with a cube where the cube after the rotation appears identical to the cube before the rotation: a quarter rotation, a one-third rotation and a half rotation.

Footprints in the sand

Creeper design with glide reflection symmetry

Rotations in a cube

10


5.2 In the braided model

In our braided model not all of these rotations are possible (even leaving aside the different colors of the strips). Only the following rotations are permissible: half rotation and one-third rotation. The half rotation leads to no change in braiding and color structure while the one-third rotation results in a cyclic switching of colors.

Braiding structure and rotations

Question: Is there a braided model of a cube having the same possible rotations as the cube itself?

Another question: How can a cube-shaped package be tied?

The three strips of our braided models of cubes and cuboids run around these figures in the same way as a corresponding package would usually be tied, though a third cord has to be additionally imagined as an “equa-torial cord”.The calorie bomb

5.3 Diagonal strip cube

As with gift packages, there is also a diagonal tying method. In the simplest case of a cube, such a diagonally tied cord forms a regular hexagon. Since each of these hexagons has a di-agonal on the cube as the axis, there are four such hexagons altogether.

11

Hands-On Geometry

A braided model with four diagonal strips can be derived from this.

Topologically the braiding structure consists of four rings.

Diagonal tying

Diagonal strip braided model

Braiding structure

In this braided model the same rotations (leaving aside the colors of the strips) are possible as with a cube.

12


5.4 Spherical model

A spherical model of the braiding structure can be constructed using four strips. The strips are made of plastic material from the packaging industry and have seven holes with a diam-eter of about 3 mm at equal intervals.

The first and the last hole have to be identified. This results in a circle divided into six equal sections. The sphere can now be assem-bled using brass fasteners.

6 Combinatorics

If we use four different colors for the four strips, this results in a combinatorial play of colors. For four different elements there are 4! = 24 possible linear arrangements and 4! __

4 = 6 pos-

sible cyclic arrangements because with the cyclic arrangement the four possibilities resulting from rotations of multiples of 90° have to be identified.

It turns out that each of these six cyclic arrangements occurs exactly once on the six sides of the cube in the diagonal strip model.

Strips

Sphere composed of four strips

The six cyclic arrangements

13

Hands-On Geometry

7 Topology

A braiding structure with only horizontal and vertical strips is “flat”. We therefore assign an index j = 0 to each of the squares in the abstract braiding structure.

When a three-dimensional corner is formed, a triangle appears in the braiding structure with three strip intersections; we assign the following index to this triangle: j = 1 __

4

Index zero when braiding is flat

Index one fourth at one corner

Indices

Generally we assign the following index to a k corner (with k strip intersections) in the braid-ing structure: index j = 1 − k __

4

14


An index sum of 2 results for each of the two braiding structures of a cube we have come across thus far.

Index sum 2

The index sum is a topological invariant. Proof of this invariance can be provided either induc-tively according to the number of strips or through application of Euler’s theorem for polyhe-dra. The following figure shows the braiding structure of a three-dimensional cross consisting of seven cubes with the corresponding indices.

Three-dimensional cross and braiding structure

15

Hands-On Geometry

8 Other figures with the same braiding structure

8.1 The octahedron

An octahedron can also be braided using four strips. The strips are zigzag-shaped and based on a triangular grid.

Braided model of an octahedron

The braiding structure in this model is the same as in the diagonal strip model of a cube.

8.2 Rhombic dodecahedron

If we set up a pyramid with a height of half the cube edge on each side of the cube, this results in a figure that is bor-dered by twelve rhombi and has a diagonal ratio of

√__

2 , the so-called rhombic dodecahedron.

Its braided model requires four zigzag strips. The braiding structure is again the same as with the diagonal strip cube. The acute angle α of a rhombus with a diagonal ratio of

√__

2 measures α = 2arctan( 1 ____ √__

2 ) ≈ 70.53°.

This angle α also results as the angle of intersection of the diagonal on a DIN A sheet of

Rhombic dodecahedron

16


paper.The four zigzag strips are con-structed as follows: a rectangu-lar sheet of paper is folded three times. Rhombi with the required acute angle α ≈ 70.53° are now cut out of the now eight-layer sheet. When the rhombi are un-

folded, this results in zigzag strips.We obtain nice crystalline models by using transparent polyester slides (slides used for an overhead projector). We can also work with color slides. If we use three slides with the pri-mary colors yellow, red and blue as well as a colorless slide for the four strips, this results in a remarkable play of colors. The rhombic dodecahedron permits six views through two lateral rhombi opposite each other respectively. In three of these six cases the colorless strip inter-sects one of the strips with the three primary colors, i.e. we see rhombi with the three primary colors. In the other three cases two primary colors intersect, leading to the secondary colors orange, green and violet.

8.3 Space filler

The rhombic dodecahedron is a so-called “space filler”. A space can be completely filled with rhombic dodecahedrons of equal size [Coxeter 1973].To understand this, we first imagine a space with cubes of equal size whose corners form a cubic lattice. We further imagine these cubes as a three-dimensional checkered pattern al-ternately colored black and white. Starting from the center of the cube, we then break down each black cube into six pyramids whose base in each case is a lateral area square of the black cubes. If we now attach these black pyramids to the neighboring white cubes, we obtain a breakdown of the space into rhombic dodecahedrons.Without much effort we can now construct a large number of braided models of the rhombic dodecahedron out of paper and thus illustrate the space filler characteristic of the rhombic dodecahedron. To demonstrate this characteristic, it is advantageous to use a base surface

Rhombic dodecahedron and strips

Silhouette technique

17

Hands-On Geometry

made of egg boxes. Such surfaces can also be braided using zigzag strips. The length of the strips depends on the desired size of the base surface. There are two ways of braiding a base surface using such strips, one results in a “pointed” egg box, the other in a “rounded” egg box. However, space fillers formed by these different base surfaces are congruent and arise from one another by turning them over appropriately.

Pointed egg box

Rounded egg box

The inner spheres of the stacked rhombic dodecahedrons form an extremely dense spherical pack.

18


9 Mistakes can lead to new insights. Star-shaped figures

The models presented in the following resulted from a mistake made by a student. While making the zigzag strips for a rhombic dodecahedron, he worked with an angle that was too acute – the result was a star-shaped model with projecting corners. An infinite number of star-shaped bodies are possible in this way, all of which have the same braiding structure as a diagonal strip cube. The rhombi of the zigzag strips are provided with a “counter-fold” along a diagonal, thus forming two isosceles triangles out of the rhombi in each case. The follow-ing figure shows a model and strips of the so-called Kepler’s Star as an example. Here the isosceles triangles are even equilateral.

Kepler’s Star

19

Hands-On Geometry

Literature

> [Adam/Wyss 1994] Paul Adam / Arnold Wyss: Platonische und Archimedische Körper, ihre Sternformen und polaren Gebilde. 2. Auflage. Bern: Verlag Paul Haupt 1994. ISBN 3-258-04943-2

> [Chatani 1989] Chatani, Masahiro: Papierkunst. Dreidimensionales Falten. Stuttgart 1989.

> [Coxeter 1973] Coxeter, H.S.M.: Regular Polytopes. Third Edition. New York: Dover 1973. ISBN 0-486-61480-8

> [Cundy/Rollet 1961] Cundy, H.M. / Rollet, A.P.: Mathematical Models. Oxford: Claren-don Press 1961.

> [Fusè 1993] Fusè, Tomoko: Unit Origami. Multidimensional Transformations. Tokyo: Japan Publications 1993. ISBN 0-87040-852-6

> [Gerdes 1990] Gerdes, Paulus: Ethnogeometrie. Kulturanthropologische Beiträge zur Genese und Didaktik der Geometrie. Bad Salzdetfurth: Franzbecker 1990. ISBN 3-88120-189-0

> [Hilton/Pedersen 1994] Hilton, Peter / Pedersen, Jean: Build Your Own Polyhedra. Menlo Park: Addison-Wesley 1994. ISBN 0-201-49096-X

> [Hilton/Pedersen/Walser 2003] Hilton, Peter / Jean Pedersen / Hans Walser: Die Kunst der Mathematik. Von der handgreiflichen Geometrie zur Zahlentheorie. Dillingen: Akademie für Lehrerfortbildung und Personalführung. 2003

> [Kneißler 1999] Kneißler, Irmgard: Einfaches Origami. Berlin: Urania-Ravensburger 1999. ISBN 3-332-00731-9

> [Lörcher/Rümmele] Lörcher, G. A. / H. Rümmele: Körper falten. Hauptstelle RAA, Heßlerstr. 208-210, D-45329 Essen, 1995

> [Mitchell 1999] Mitchell, David: Mathematical Origami. Norfolk: Tarquin Publica-tions 1999. ISBN 1-899618-18-X

> [Pargeter 1959] Pargeter, A.R.: Plaited Polyhedra. The mathematical gazette 43, 1959, p. 88-101.

> [Pedersen 1981] Pedersen, J.J.: Some Isonemal Fabrics on Polyhedral Surfaces.The Geometric Vein (The Coxeter Festschrift), ed. by C. Davis, B. Grünbaum and F.A. Sherk. New York, Heidelberg, Berlin: Springer 1981, p. 99-122. ISBN 0-387-90587-1

> [Pedersen 1983] Pedersen, Jean: Geometry: The Unity of Theory and Practice. The Mathematical Intelligencer, Vol. 5 (1983), No. 4, p. 37-49

> [Steibl 1996] Steibl, Horst: Geometrie aus dem Zettelkasten. Hildesheim: Franz-becker 1996. ISBN 3-88120-269-2

> [Walser 1987] Walser, Hans: Flechtmodelle. Didaktik der Mathematik (15), 1-17 > [Walser 1994] Walser, Hans: Geometrie zum Anfassen. Mathematik Lehren, Heft 65, August 1994, p. 56-59.

> [Zeier 1983] Zeier, Franz: Papier. Versuche zwischen Geometrie und Spiel. Bern: Haupt 1983. ISBN 3-258-03309-9

15 / 2013ISSN 2192-7596

University of Bayreuthwww.sinus-international.net


SINUSinternational

DISSEMINATING INQUIRY-BASED SCIENCE

AND MATHEMATICS EDUCATION IN EUROPE

sinussinus.uni-bayreuth.de/math/tnt_math_15.pdf · develop a braided model corresponding to a cube...

Documents