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 Mosc. Math. J., 2010, Volume 10, Number 2, Pages 337–342 (Mi mmj383)

Interlocking of convex polyhedra: towards a geometric theory of fragmented solids

A. J. Kanel-Belovabc, A. V. Dyskind, Y. Estrinef, E. Pasternakg, I. A. Ivanov-Pogodaevh

a Moscow Institute of Open Education, Moscow, Russia
b Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
c International University Bremen, Bremen, Germany
d School of Civil and Resource Engineering, The University of Western Australia, Crawley, WA, Australia
e ARC Centre of Excellence for Design in Light Metals, Department of Materials Engineering, Monash University, Clayton, Vic., Australia
f CSIRO Division of Manufacturing and Materials Technology, Clayton, Vic., Australia
g School of Mechanical Engineering, The University of Western Australia, Crawley, WA, Australia
h Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

Abstract: The article presents arrangements of identical regular polyhedra with very special and curious properties. Namely, the solids are situated in a sort of a layer and are interlocked in the sense that no one of them can be moved out without disturbing others. This situation cannot happen in the plane. First examples of this sort (composed of irregular convex polyhedra) were complicated and were constructed in a non regular way by G. Galperin. The examples presented here were constructed in framework of applied studies by the authors, C. Khor and M. Glickman and were not described in mathematical publications. The full version of this paper is presented here: http://arxiv.org/abs/0812.5089.

Key words and phrases: interlocking structures, combinatorial geometry, convex polyhedron, tilling.

DOI: https://doi.org/10.17323/1609-4514-2010-10-2-337-342

Full text: http://www.ams.org/.../abst10-2-2010.html
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Bibliographic databases:

MSC: 52B10, 74R
Received: November 7, 2006; in revised form January 7, 2007
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Citation: A. J. Kanel-Belov, A. V. Dyskin, Y. Estrin, E. Pasternak, I. A. Ivanov-Pogodaev, “Interlocking of convex polyhedra: towards a geometric theory of fragmented solids”, Mosc. Math. J., 10:2 (2010), 337–342

Citation in format AMSBIB
\Bibitem{KanDysEst10} \by A.~J.~Kanel-Belov, A.~V.~Dyskin, Y.~Estrin, E.~Pasternak, I.~A.~Ivanov-Pogodaev \paper Interlocking of convex polyhedra: towards a~geometric theory of fragmented solids \jour Mosc. Math.~J. \yr 2010 \vol 10 \issue 2 \pages 337--342 \mathnet{http://mi.mathnet.ru/mmj383} \crossref{https://doi.org/10.17323/1609-4514-2010-10-2-337-342} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2722801} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279342400004} 

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• http://mi.mathnet.ru/eng/mmj/v10/i2/p337

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Estrin Yu., Dyskin A., Pasternak E., Schaare S., “Topological interlocking in design of structures and materials”, Architecture multifunctional materials, Materials Research Society Symposium Proceedings, 1188, 2009, 117–129
2. Estrin Y., Dyskin A.V., Pasternak E., “Topological interlocking as a material design concept”, Materials Science & Engineering C-Materials for Biological Applications, 31:6 (2011), 1189–1194
3. Li J., Xue J., Xiao J., Wang Y., “Block theory on the complex combinations of free planes”, Computers and Geotechnics, 40 (2012), 127–134
4. Molotnikov A., Gerbrand R., Bouaziz O., Estrin Yu., “Sandwich Panels with a Core Segmented Into Topologically Interlocked Elements”, Adv. Eng. Mater., 15:8 (2013), 728–731
5. Dyskin A.V., Pasternak E., Shufrin I., “Structure of Resonances and Formation of Stationary Points in Symmetrical Chains of Bilinear Oscillators”, J. Sound Vibr., 333:24 (2014), 6590–6606
6. Li J., Yuan G., Zhang Yu., Zhao Ya., “Study on a General Cutting Algorithm of Complex Blocks”, Geotech. Geol. Eng., 33:5 (2015), 1193–1203
7. Zheng Yinhe, Xia Lu, Yu Qingchun, “a Method For Identifying Three-Dimensional Rock Blocks Formed By Curved Fractures”, Comput. Geotech., 65 (2015), 1–11
8. Molotnikov A., Gerbrand R., Qi Y., Simon G.P., Estrin Y., “Design of Responsive Materials Using Topologically Interlocked Elements”, Smart Mater. Struct., 24:2 (2015), 025034
9. Siegmund T., Barthelat F., Cipra R., Habtour E., Riddick J., “Manufacture and Mechanics of Topologically Interlocked Material Assemblies”, Appl. Mech. Rev., 68:4 (2016), 040803
10. Djumas L., Molotnikov A., Simon G.P., Estrin Yu., “Enhanced Mechanical Performance of Bio-Inspired Hybrid Structures Utilising Topological Interlocking Geometry”, Sci Rep, 6 (2016), 26706
11. Zareiyan B., Khoshnevis B., “Effects of Interlocking on Interlayer Adhesion and Strength of Structures in 3D Printing of Concrete”, Autom. Constr., 83 (2017), 212–221
12. Djumas L., Simon G.P., Estrin Yu., Molotnikov A., “Deformation Mechanics of Non-Planar Topologically Interlocked Assemblies With Structural Hierarchy and Varying Geometry”, Sci Rep, 7 (2017), 11844
13. Piirainen V.Y., Estrin Y.Z., “Topological Interlocking as a Principle of Engineering Design in Consruction of Marine and Coastal Structures”, J. Min. Inst., 226 (2017), 480–486