LC+WS 1 What is Symmetry ? The term "Symmetry" is a composite of "sym" (common) and "metry" (measure), hence it has a meaning of standard of beauty. From a general idea that symmetrical shapes are beautiful, the people might have begun to call this property "Symmetry". In a narrow sense "Symmetry" means that a shape does not change by mirror reflection. It is sometimes called a mirror symmetry or reflection symmetry. In a broader sense "Symmetry" is defined as a property of a shape, which does not change by a certain type of operation on the shape. According to the type of operation, the property is called rotational symmetry, translational symmetry, etc. Mirror symmetry (reflection symmetry) Rotational symmetry Translational symmetry 2 How images are born ? Let us obtain images by repeating reflections in opposite sense of orders for the intersection angle of 120 deg. Then, these two images do not in general coincide in positions. If they coinside in positions, their shapes do not coinside (one is a reflection of the other). How to obtain images of an object in intersecting mirrors? 1. Obtain an image by a miror 2. Obtain another image of the image by the second mirror. 3. Repeat this process. Let the intersecting angle be 360 / n ( n is integer ) n = even: coincidence n = odd: no coincidence EX1: Draw images of an object in intersecting mirrors with angles 60 deg and 72 deg. EX2: Cut craft paper with mirror surface (Off-metal, silver) along the solid line (see the right figure), and fold along the dashed line. Fix the angle between the parts A and B to 60, 72 or 120 deg, and observe images of an oblect which is put between A and B. Note: Make a thin ditch along the dashed line by a knife. Then, you can fold the paper sharply. 3 Cylindrical Kaleidoscope Kaleidoscope is a toy to apply multi-reflections by intersectiong mirrors. The kaleidoscope with mirrors arranged to form a prism is called a prism kaleidoscope (this term is proposed by the lecturer of this course). In conventional kaleidoscopes any angle between mirrors is 360deg / n ( n is integer). A kaleidoscope whose angles are expressed by even n is called an ieal kaleidoscope. A kaleidoscope including angles with odd n is called a semi-ideal kaleidoscope. We have only four types of ideal kaleidoscopes, as shown below. EX3: Draw images in the ideal kaleidoscopes below other than the rectangular one. EX4: Add one more folding line (as shown in the right) to the mirror craft paper, which was used in EX2, and a make prizm kaleidoscope. Choose angles to combinations (45, 45, 90) and (72, 54, 54) also. The latter leads to a semi-ideal kaleidoscope. Try to see how images are formed. 4 Polyhedral kaleidoscope Combination of mirrors forming a conical shape with special combination of angles is called a polyhedral kaleidoscope. Good combination of angles is obtained by extracting a cone from one or two faces of geodesic polyherdron. A geodesic polyheron is defined as a group of geodesic circles which are drawn through the following processes. 1. Inscribe one of Platonic regular polyhedra in a sphere. 2. Project all edges of the polyhedron to the sphere with a light source at the spherical center. 3. Extend these projected lines to form full geodesic circles. We have four types of geodesic polyhedra as shown below. Note that these geodesic polyhedra are sometimes called complete geodesic polyhedra, while groups of geodesic circles with any orientations are called geodesic polyhedra. From one of spherical tringles on a geodesic polyhedron we can make a cone by collecting planes, which are determined by connecting edges of a spherical triangle and the spherical center. The simpelst one is a cone made of three perpendicular planes, which is produced from the geodesic octahedron (the left-most one in the above table). Let us construct a polyhedral kaleidoscope by replacing planes of a cone with mirrors. Since angles between planes are devisions of 360 deg into even numbers (confirm this fact for the above figures), the resulting kaleidoscope should be an ideal kaleidoscope. On the other hand, if you try to make a kaleidoscope from a cone, which is a combination of two cones (see hatched section in the rightmost figure above), one angle is 1/5 of 360 deg, and the resulting kaleidoscope is not an ideal one. Each plane of a cone has a shape of triangle (whose angle at the spherical center is called here a vertex angle). Angels between planes of a cone and vertex angles of triangles of a cone are given in the table below. The vertex angles are important when you try to make a cone. Angles between planes Vertex angles Geodesic octahedron 90 90 90 90 90 90 Geodesic 24-hedron 90 60 60 70.5 54.75 54.75 Geodesic 48-hedron 90 60 45 54.75 45 35.25 Geodesic 120-hedron 90 60 36 37.4 31.7 20.9 The polyhedral kaleidoscope below is constructed from a combined cone of geodesic 120-hedron (the hatched section of the above figure). It is made by Prof. Caspar Schwabe at Kurashiki University of Science and Art, who named it "Pentakis kaleidoscope". He put no obejct inside but made rows of holes on the mirrors and cut the vertex of the cone. The, wonderful image appears inside. Pentakis kaleidoscope by Caspar Schwabe CR1: Make a polyhedral kaleidoscope from the geodesic octahedron (see figures below) with mirror craft paper. A better one may be made by the use of alminum plate with mirror surface. Make an object to be placed inside as you like. Tooles and Materials for WS: Craft paper with mirror surface (A4 type), glue, scale, graduator, knife, cellophane tape, compass Results of CR1 (click each photo for larger size)

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<Updated: 9/15/04>