Lunar and Planetary Science XXXII (2001)
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HOW INTERACTIVE GRAPHICAL MODELING HELPS SPACE SCIENCE AND GEOMETRY EDUCATION IN HUNGARY. Szilassi L.1 Karsai J.2, Pataki T.3, Kabai S.4, Bérczi Sz.5, 1Szeged University, Juhász Gyula College, H-6720 Szeged, Boldogasszony sugárút 6. Hungary. 2Szeged University, Med. Fac. Institute of Medical Informatics, H-6720 Szeged, Korányi fasor 9. Hungary, 3 MIMIKRISTUDIO, H-1111 Budapest, Bertalan L. u. 26. Hungary, 4UNICONSTANT, H-4150 Püspökladány, Honvéd u. 3. Hungary, 5Eötvös University, Dept. G. Physics, H-1117 Budapest, Pázmány Péter sét. 1/a. Hungary, (
[email protected]) ABSTRACT We (lecturers on geometry and engineering) formed a course on geometry and interactive graphical design for students who are interested in space probe construction and planetary science modeling. Geometry of space structures is more attractive if graphical projections and movements can be visualized. We show some results of this curriculum. Several computer programs were selected to realize design products. Spatial relations were constructed, visualized, transformed, and by vrlm extension the observer could fly over the objects as if he/she traveled in spacecraft. INTRODUCTION In history of science geometry was always strongly connected with space research studies [1], [2]. There are several computer graphic programs (Mathematica [3], Graf, Maple, vrlm, etc.) which help learning not only how to design geometric projections of a space structure, but how to move (around, outside, inside) to see objects. We selected those type of programs which specially help spatial view and spatial arrangement transformations during construction.
design of great structures with modular elements can be learned quickly, and spatial movements around
Fig. 2. L. Szilassi: Mutually invading spatial knots in a pentagondodecahedral form.
Fig. 3. One thoroidal helix along the equator of a (3,5,3,5) skeletal structure.
Fig. 1. Kabai: Icosahedral station with fullerene outer skeleton (5,6,6).
Complexity of these programs allows handling parametric equations of constructing functions. Computer
a space object (sp. station, asteroid) if visualized, serves as feedback for the designer. SPATIAL (SPACE STATION?) STRUCTURES Just in these weeks astronauts assembled the great rod system of the International Space Station. Important principles in construction of stable space skeletons: rigidity, modularity, [4], folding/opening abilities [5], optimization of surfaces depending on heat absorption or emission, etc. Specially principles come from the enfolding-launch-folding out operations, which open the use of both western [6] and tra-
Lunar and Planetary Science XXXII (2001)
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INTERACTIVE GRAPHICAL MODELING IN SPACE SCIENCE: L. Szilassi et al.
ditional Japanese origami like [5] folding technologies [7-9]. Together with classical geometry constructions (which forms the concept of locus) and spherical astronomy (coordinate systems) lessons were included. First example shows how we can arrange a great skeletal space construction, by using Mathematica [3], and (Fig. 1.) Fig. 2. exhibits a modular structure for teaching spatial relations with symmetry, a knot defined along the rod system of an icosidodecahedral frame. First step is construction of the module, the helical thoroid unit with 5 helical screw (together with the skeletal gray network it is shown on Fig. 3.) Using symmetry of the icosahedral group it can be easily proved that if two such helical thoroids are not intersecting each other, then any of the six thoroids will do the same. {The knot structure consists of 6 such helical thoroids, screwing along icosi-dodecahedral truncated (3,5,3,5) (Steiner symbol) [10-11].} Another basic principle is coming from rod systems, centuries earlier studied in Japan [12]. Made from bamboo the structure of self supporting rod system does not contain gravity dependence [4], [13].
Fig. 4. This frame of bamboo system holds together without any force over friction (Ogawa, Teshima, Watanabe, 1996 [4], [13].)
Fig. 5. shows a space station skeleton with oriented conical grid system of bipyramid with icosagon basic equatorial plate (almost circular at its equator). Not only constructional principles but navigation and celestial coordinate systems can be linked to this topics. This connection of local rigid space geometry with astronomical coordinate systems in teaching is fruitful to show the interrelations of disciplines used in space science. Over the geometry as common background, the living and motion in space, the using of space is the basic common source in this visual modeling curriculum.
Fig. 5. Icosagon-bipyramidoidal space station structure. (Kabai, [14],)
SUMMARY Our space geometry and structures graphic study curriculum wanted to show examples how interactive computer programs help spatial geometry studies, design and motions in and around such structures. During the use of these constructing methods various functions were used (for example the outer mantle's curvature of Fig. 5. is astroid). Many new works on symmetry/space geometry papers show that unfolding structures, many modern crystallography principles has increasing role in space science/geometry studies. ACKNOWLEDGMENTS: One of authors (B.Sz.) thanks to Koryo Miura the worthy discussions on origami organized space structures, and we also thank the help of the Bolyai János Mathematical Institute Library, of Szeged University. REFERENCES: [1] Kepler, J. (1611): The hexagonal snowflake. Oxford Univ. Press. (1966); [2] Thompson, D'A. W. (1917): On Growth and Form. 2nd ed. Cambridge Univ. Pr.; [3] Wolfram, S. (1988): Mathematica. Addison-Wesley; [4] Ogawa T., Teshima Y., Watanabe Y. (1996): Geometry and crystallography of self supporting rod structures. (In: Katachi U Symmetry, Ogawa T., Miura K., Masunari T., Nagy D. Eds.) 239-246. Springer, Tokyo; [5] Miura, K., Sakamaki M. (1994): Mathematics of Form and its Relation to Design: A Look at Space Structures. FORMA, 9. 239-251. Tokyo; [6] Schattschneider, D., Walker. W. (1977) M.C. Escher Kaleidocycles, Ballantine, New York; [7] Wester, T. (1995): (In: Katachi U Symmetry) 223. Springer; [8] Ichikawa S., Kawamura H., Tani A. (1995): (In: Katachi U Symmetry) 165-172. Springer; [9] Miyazaki K., Takada I., Nakata H. (1995): (In: Katachi U Symmetry) 215. Springer; [10] Bérczi Sz., Kabai S., et al. (1999): LPSC XXX, #1037; [11] MacKay A. (1986): Generalized Crystallography. CAMWA, 12B, No. 1-2. 21-37.; [12] Bérczi Sz., Cech V., Hegyi S., Sz. Fabriczy A., Lukács B. (1998): LPSC XXIX, #1371; [13] Teshima Y., Watanabe Y., Ogawa T. (1994): The periodic six-axes-structures in rod systems. Katachi U Symmetry Conf. Abstract Vol. p. 154-157. Tsukuba; [14] Kabai S. (2000): Les. on Space Structure Constructions (1-75). Püspökladány;
