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Serbian Association for Geometry and Graphics th The 7 International Scientific Conference on Geometry and Graphics L th st September 18 - 21 , Belgrade, Serbia 2020 BOOK OF ABCTRACTS Serbian Association for Geometry and Graphics The 7th International Scientific Conference on Geometry and Graphics moNGeomatrija2020 BOOK OF ABSTRACTS September 18th – 21st 2020, Belgrade, Serbia ISBN 978-86-6060-047-1 SUGIG Beograd, 2020. The 7th International Scientific Conference on Geometry and Graphics moNGeomatrija2020 Publishers Serbian Society for Geometry and Graphics (SUGIG) Faculty of Mechanical Engineering, University of Belgrade Title of Publication BOOK OF ABSTRACTS Editor-in-Chief Zorana Jeli Co-Editors Branislav Popkonstantinović Slobodan Mišić Djordje Djordjević Maja Petrović Graphic design Djordje Djordjević Maja Petrović Formatters Djordje Djordjević Maja Petrović Ivana Cvetković Nikola Popović Number of copies: 80 ISBN 978-86-6060-047-1 The 7th International Scientific Conference on Geometry and Graphics moNGeomatrija2020 Conference Organizers Serbian Society of Geometry and Graphics (SUGIG) Faculty of Mechanical Engineering, University of Belgrade Co-organizers Faculty of Civil Engineering, University of Belgrade Faculty of Architecture, University of Belgrade Faculty of Forestry, University of Belgrade Faculty of Applied Arts, University of Arts in Belgrade Faculty of Transport and Traffic Engineering, University of Belgrade Under the auspicies of Ministry of Education, Science and Technological Development of Republic of Serbia Scientific Committee: Hellmuth Stachel – Austria Emil Molnar – Hungary Gunter Weiss – Germany Djuro Koruga– Serbia Naomi Ando – Japan Radovan Štulić – Serbia Branislav Popkonstantinović – Serbia Ratko Obradović – Serbia Branko Malešević – Serbia Lidija Matija – Serbia Grga Djurica – Serbia Daniel Lordick – Germany Albert Wiltsche – Austria Milena Stavric – Austria Ajla Aksamija– USA Olga Timcenko – Denmark Ema Jurkin – Croatia Alina Duta – Romania Dragos-Laurentiu Popa – Romania Carmen Marza – Romania Laszlo Voros – Hungary Zsuzsanna Balajti – Hungary Monika-Sroka Bizon – Poland Tashko Rizov – Macedonia Milan Petrović – Serbia Maja Čavić – Serbia Vesna Stojaković – Serbia Aleksandar Čučaković – Serbia Marija Obradović – Serbia Sonja Krasić – Serbia Milan Ristanović – Serbia Slobodan Mišić – Serbia GianLucca Stasi – Spain Maja Ilić – Republic of Srpska, Bosnia and Herzegovina Djordje Djordjević – Serbia Mirjana Devetaković – Serbia Marijana Paunović – Serbia Magdalena Dragović – Serbia Gordana Djukanović – Serbia Maja Petrović – Serbia Emil Veg – Serbia Miša Stojićević – Serbia Bojan Banjac – Serbia Organizing Committee: Zorana Jeli Aleksandar Čučalović Marija Obradović Slobodan Mišić Djordje Djordjević Mirjana Devetaković Marijana Paunović Magdalena Dragović Gordana Djukanović Maja Petrović Emil Veg Miša Stojićević Bojan Banjac Ana Petrović Boris Kosić Ivana Vasiljević Ivana Cvetković Aleksandra Dragićević Aleksandra Joksimović Reviewers: Prof. Hellmuth Stachel – Austria Prof. Gunter Weiss – Germany Prof. Naomi Ando – Japan Prof. Radovan Štulić – Serbia Prof. Branislav Popkonstantinović – Serbia Prof. Ratko Obradović – Serbia Prof. Branko Malešević – Serbia Prof. Lidija Matija – Serbia Associate Prof. Milena Stavrić – Austria Associate Prof. Ajla Aksamija – USA Associate Prof. Ema Jurkin – Croatia Associate Prof. Alina Duta – Romania Associate Prof. Dragos-Laurentiu Popa – Romania Associate Prof. Carmen Marza – Romania Associate Prof. Laszlo Voros – Hungary Associate Prof. Zsuzsanna Balajti – Hungary Associate Prof. Monika-Sroka Bizon – Poland Associate Prof. Tashko Rizov – Macedonia Associate Prof .Aleksandar Čučaković – Serbia Associate Prof. Marija Obradović – Serbia Associate Prof. Sonja Krasić – Serbia Associate Prof. Slobodan Mišić – Serbia Associate Prof. Vesna Stojaković – Serbia Associate Prof. GianLucca Stasi – Spain Associate Prof. Zorana Jeli – Serbia Ass. Prof. Maja Ilić – Republic of Srpska, Bosnia and Herzegovina Ass. Prof. Magdalena Dragović – Serbia Ass. Prof. Emil Veg – Serbia Ass. Prof. Miša Stojićević – Serbia Ass. Prof. Djordje Djordjević – Serbia Ass. Prof. Mirjana Devetaković – Serbia Ass. Prof. Marijana Paunović – Serbia Ass. Prof. Gordana Djukanović – Serbia Ass. Prof. Bojan Banjac – Serbia Ass. Prof. Maja Petrović – Serbia BOOK OF ABSTRACTS CONTENT Topic 1: Theoretical geometry and graphics……………………………………………………….11-28 Topic 2: Applied geometry and graphics…………………………………………………………..29-82 Topic 3: Engineering computer graphics and geometry…………………………………….....….83-110 Topic 4: Geometry, graphics education and teaching methodology…………………………….111-128 Student`s papers…………………………………………………………………….….129-155 9 TOPIC 1: THEORETICAL GEOMETRY AND GRAPHICS CONTENT Marija Obradović / Invited Lecturer: GEOMETRIC PROPERTIES OF THE “FLOWER” CONCAVE ANTIPRISMS OF THE SECOND SORT…………..………..….………………………………………..13-14 László Vörös: SETS OF PLANAR AND SPATIAL TESSELLATIONS BASED ON COMPOUND 3D MODELS OF THE 8D AND 9D CUBES AND ON RESTRUCTURED CONVEX UNIFORM HONEYCOMBS…..……………………………………………..15-16 Gordana Djukanović, Djordje Djordjević, Mirjana Devetaković: OBTAINING PENCILS OF CURVES OF HIGHER ORDER BY APPLYING A SUPERSYMMETRY TO PENCILS OF CONICS……………………………...…...….17-18 Hellmuth Stachel: PLÜCKER’S CONOID, HYPERBOLOIDS OF REVOLUTION, AND ORTHOGONAL HYPERBOLIC PARABOLOIDS……….……………….………...….19-20 Zsuzsa Balajti, József Ábel: THE BIJECTIVE PART OF THE MONGE CUBOID FOR THE MAPPING OF THE HELIX AND THE SPATIAL CURVE ARC…………..………..……………………….21-22 Slobodan Mišić, Marija Backović: CONCAVE PYRAMIDS OF FOURTH SORT....………..….….……………………….23-24 Maja Petrović, Branko Malešević, Radovan Štulić, Ema Jurkin, Radomir Mijailović: SPATIAL INTERPRETATION OF ERDÖS-MORDELL INEQUALITY FOR POLYGONS OVER WEBERIAN FOCAL-DIRECTORIAL SURFACES………….….25-26 Michal Zamboj: VISUALIZATION OF SINGULAR QUADRIC SECTIONS OF FOUR-DIMENSIONAL CONES…………..………..………………………………..….27-28 11 M. OBRADOVIĆ.: GEOMETRIC PROPERTIES OF “FLOWER” CONCAVE ANTIPRISMS OF THE SECOND SORT - INVITED LECTURE - GEOMETRIC PROPERTIES OF THE “FLOWER” CONCAVE ANTIPRISMS OF THE SECOND SORT Marija Obradović University of Belgrade, Faculty of Civil Engineering PhD., Associate Professor, [email protected] ABSTRACT This study presents a continuation of the research on the concave polyhedra of the second sort, adding to this family a new group of related polyhedra. They are formed over a specific type of isotoxal concave polygons that allow geometric arrangement of a double row of equilateral triangles into formations which enclose a deltahedral lateral surface without overlaps and gaps. As in all other representatives of the concave polyhedra of the second sort, we expect to find here the "major" and "minor" type, depending on the way we fold the net. This research has identified both these polyhedra types, which have the same planar net, but are formed over different basic concave polygons. The origination, constructive methods and properties of these solids are elaborated in the paper. Keywords: concave polyhedron; concave polygon; polygon elevation; antiprism; isotoxal; constructive geometry REFERENCES 1. Bowers J. (2012). Regular Polygons and Other Two Dimensional Shapes, Page created by, 2012. Polytope.net http://www.polytope.net/hedrondude/polygons.htm 2. Coxeter, Harold Scott Macdonald. (1969) Introduction to geometry, Second edition, 2.8 Star polygons p.36-38. 3. Grünbaum, B., & Shephard, G. C. (1986). Tilings and patterns. WH Freeman & Co. 4. Grünbaum, Branko, Geoffrey C. Shephard. (1977). Tilings by regular polygons, Mathematics Magazine Vol. 50. No5: 227-247. 5. Huybers, P. (2001) Prism based structural forms. Engineering structures, 23(1), 12-21. 6. Johnson, N. W. (1966). Convex polyhedra with regular faces. Canadian Journal of Mathematics, 18(1), 169-200. 7. Kepler, Johannes. 1619/ 1968. Harmonices mundi. Libri V. Culture et Civilisation. 8. Mišić S. 2013. Constructive – geometric generating of cupolae with concave polyhedral surfaces, Doctoral Dissertation, University of Belgrade - Faculty of Architecture, 2013. 9. Mišić S., Obradović M., Đukanović G., 2015. Composite Concave Cupolae as Geometric and Architectural Forms, Journal for Geometry and Graphics, Vol.19. No 1. pp 79-91. Heldermann Verlag 2015. 10. Obradović M., Mišić S. (2008). Concave Regular Faced Cupolae of Second Sort, In: Proceedings of 13th ICGG (ICGG 2008, Dresden, August 2008), ed. Gunter Weiss, Dresden: ISGG/ Technische Universität Dresden El. Book: 1-10. 11. Obradović M. 2012. A Group Of Polyhedra Arised As Variations Of Concave Bicupolae Of Second Sort, In: Proceedings of the 3rd International Scientific Conference MoNGeometrija 2012, ed. Ratko Obradović, FTN Novi Sad, June 21-24. 2012. pp. 95-132. THE 7TH ICGG CONFERENCE MONGEOMETRIJA 2020 BOOK OF ABSTRACTS Belgrade: 18-21 September 2020 1 3 M. OBRADOVIĆ.: GEOMETRIC PROPERTIES OF “FLOWER” CONCAVE ANTIPRISMS OF THE SECOND SORT 12. Obradović M., Mišić S., (2019). Concave deltahedral rings based on the geometry of the concave antiprisms of the second sort, GEOMETRIAS’19: Polyhedra and beyond, Porto, 05 - 07 September 2019. Book of abstracts, Aproged, pp. 85-89 . 13. Obradović M., Mišić S., Popkonstantinović B. (2014). Concave Pyramids of Second Sort - The Occurrence, Types, Variations, In: Proceedings of the 4th International Scientific Conference on Geometry and Graphics, 2. MoNGeometrija 2014, June 20-22.Vlasina, Serbia, ed. Sonja Krasić, Faculty of Civil Engineering and Architecture in Niš and Serbian Society for geometry and graphics