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On Stability of Polytopes ON STABILITY OF POLYTOPES BY Wendy Finbow SUBSIITTED IX P-iFtïL-U. FCZFILLMEST OF THE REQLXRE-UIEXTS FOR THE DEGREE OF @ Copyright by Wendy Finbon*. 1998 The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la Natio~liflLiirary of Canada to Bibliothèque nationale du Canada de repfoduce, 10- distriMe or seil reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous papex or electronic formats. la forme de microfiche/£ïh, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur consewe la propriété du copyright in this thesis. Neidier the droit d'auteur qui protège cette thèse. thesis nor substantial extracts from it Ni la thèse ni des extraits substantiels may be printed or othefwise de celle-ci ne doivent être imprimés repfoduced doutthe auihor's ou autrement reprodrrits sans son permission. airtorisation. Canada Contents List of Tables List of Figures vii Acknowledgements X Abstract xi Abbreviations and Symbols Used xii 1 Introduction 1 2 Preliminary Results 2.1 Minkowski's Theorem . 2.2 Further Results . 3 Monostatic Simplexes in 8, 9, and 10 Dimensions 24 3.1 The Search Method . 25 3.2 Cornputer Algorithm . 26 3.3 Error .4nalysis. 33 3.4 The Main Results . 37 4 Loaded Polytopes 44 4.1 Preliminary Results . 44 4.2 Nonloadable Polytopes .......................... 47 4.3 Loadable Polytopes ............................ 52 4.4 Related Results .............................. 89 Bibliograp hy 91 List of Tables 3.1 Maximum contribution of each simplex with five facets ........ 30 3.2 Maximumcontributionofeachsimplexwithsixfacets ......... 31 3.3 Maximum contribution of each simplex with seven facets ........ 32 3.4 Mawimum contribution of each sirnplex with eight facets ....... 33 3.4 Maximum contribution of each simplex with eight facets, continued . 34 3.5 Maximum contribution of trees with 5, 6. 7. 8. and 9 nodes ...... 38 3.6 Contributions of vectors ......................... 39 4.1 Loadable/Nonloadable Polytopes .................... 48 4.1 Loadable/rYonloadable Polytopes. continued .............. 49 List of Figures 2.1 Illustration of the Projection Criterion (tetrahedron) ......... 3.1 Example of a straight tree ......................... 3.2 Example of a tree with six nodes ..................... 4.1 Overview of a truncated tetrahedron .................. 4.2 Overlap of opposite triangular facets (icosahedron) ........... 4.3 Overlapping facets of the icosahedron .................. 4.4 Overlap of opposite pentagonal facets (dodecahedron)......... 4.5 Overlapping facets of the dodecahedron ................. 4.6 Overlapping facets of the tmncated icosahedron ............. 4.7 Overlapping facets of the truncated dodecahedron ............ 4.8 Overlapping facets of the cuboctahedron ................ 4.9 Overlap of a pentagonal facet onto a triangular facet of the icosidodec- ahedron .................................. 4.10 Overlap of a triangular facet onto a pentagonal facet of the icosidodec- ahedron .................................. 1.11 Overlap of square facets of the rhornbicoçidodecahedron ........ 4.12 Overlap of a square facet onto a pentagonal facet of the rhombicosido- decahedron ................................ 4.13 Overlap of square facets onto triangular facets of a rhombicosidodeca- hedron ................................... vii 4.14 Overlap of triangular facets onto square facets of the rhornbicosidodec- ahedron .................................. 4.13 Overlap of pentagonal facets onto square facets of the rhombicosido- decahedron ................................ 4.16 Overlapping facets of the rhombicosidodecahedron ........... 4-17 Overlapping square facets of a snub cube ............ 4-18 Overlap of square facets onto triangular facets of the snub cube ... 4.19 Overlap of triangular facets of the snub cube and snub dodecahedron 4.20 Overlap of triangular facets onto square facets of the snub cube ... 4.21 Overlapping facets of the snub cube ................... 4.22 Overlapping pentagonal facets of the snub dodecahedron ....... 4.23 Overlap of the pentagonal facets of the snub dodecahedron ...... 4.24 Overlap of triangular facets onto pentagonal facets of the snub dodec- ahedron .................................. 4.25 Overlapping facets of the snub dodecahedron .............. I dedzcate my thesis to my Dad. Acknowledgements 1 should like to thank my supervisor, Dr. Dawson, for his support and encourage- ment over the last two years, and Dr. Thornpson, for his endless supply of corrections. 1 should also like to thank Dr. M. Heukaeufer' a professor in the department of Modem Languages at Saint Mary's University, and Connie hk. for their help in translating the paper by Minkowski, (111, from Gerrnan to English. Lastly, 1should like to thank my fnends and famil. for ail their love and support. Abstract A polytope is said to be monostatic if its center of gravity is above only one facet. We show that no monostatic simplex exists in eight or fewer dimensions?and there is no monostatic simplex in ten dimensions that fdls sequentially onto dl its facets. We cda polytope loadable if there is a point in the interior of the polytope that is above only one facet. We classify some convex polytopes as either loadable or nonloadable. Abbreviat ions and Symbols Used e,= (0,. 0,1,0,. ,O)=, where the 1 is in the ith position of the vector. 0 u 5 V. where u = (ul,.. , and v = (vlo... v,JT implies that ui 5 viVi = 1, ...,72 xii Chapter 1 Introduction -4closed and bounded region in space is said to be convez if given aqv tn-O points p and q in the region, then the line joining them is aiso completely contained in the region. (In other w-or&. tp + (1- t )q is inside the region for every t E [O. 11). A conves set wïth nonempty interior is called a convez body. Let a E Rn. and let b E R. Then a hyperplane is the set of d points x = (xl. .x,) satisfiing the equation a - x = b. The vector a = (alt... .a,) points in the direction normal to the hyperplane. and if llall = 1. b is the distance fiom the origin 0. to the point in the hyperplane ciosesr to it. -4 polytope is definecl. see. for example. [4]to be a conves region in Rn bounded by a finite number of hwyperplanes. -1 h-yperplane that bounds a polytope is cailed a supportzng hyperplane, or a support plane of the polytope. If HI.. Hm is a finite set of hwyperplanesthat bound a polytope P. Hi n P. where i = 1.. m is said to be a facet of P provided that it is maximal in the sense that no Hj n P. j = 1. m. j # i properly contains it. If H, f~ P is a facet? then H* is called a facet hypeîplane. For any facet F* of a polytope, the facet vector vi is the vector with length equal to the (n - 1)-dimensional volume of the facet, directeci outward and normal to the facet. We may describe a polytope P mith m facets in the following -: The facet Fi lies in the hyperplane ai-x = bit V2 = 1,. .,m, with 11aJ1 = 1. The facet vector vi that describes Fi is a multiple of the vector Vi = 1, . , m. Then if R-edefine Yi = ~~/ll~~ll,and let A = (Yl,.. ,y,)=. b = (bl,.- . .bJT, P is descrîbed as the region defined by -4.x 5 b. ( For v = (vil .. .,un) E Rn.w = (wl,.. ., w,) E Rn-we Say v 5 w ifui 5 wi, Vz = 1,. ,n.) Let a polytope P E Rn be given. Let density of P. p(x), be a non-oegative, real- valued function defined on the points x E P. If the volume of P is V. then define the mas of P to be m = J Jv p(x)dCi. The center of grauity of P is If for ail x,y E P, p(x) = p(y), then we say the polytope has unzfom density. We Say that a point is above a facet of a polytope if the line parallel to the facet vector v and through the point meets the interior of the facet. .A polytope is said to be stable on a facet if and only if its center of gravity is above that facet. Therefore the position of the center of gravity of a polytope determines whether or not the polytope is stable on a particular facet. A regular polytope (constructed) of uniform density will be stable on dl of its facets, since its center of gravity will be located at the center of the polytope, above every facet. Imagine a pardelogram with two 45' angles, such that two sides are tnice as long as the other ones. Such a pardelogram will be stable on any of the long sides, but if set on a short side, it will fa11 over, since the center of gravity of the object is not above that face. A convex polytope of uniform density will fall from its ith facet to another facet if the center of gravity of the polytope is not above the ith facet. The dihedral angle between the ith facet and the facet onto which the polytope falls must be obtuse, and if the polytope is stable on its jth facet, the center of gravity di be above that facet. Because the angle between any two facets in a polytope such that it falls from one facet to the other must be obtuse, the angle between the facet vectors must be acute. Note that when a polytope falls boom one facet to another. the position of the center of gavity of the polytope is lowered. so the polytope cannot continue to fa11 kom one facet to another facet forever. In fact, a polytope will never fall onto a facet from which it has already fallen. This means a polytope must be stable on at least one of its facets, a resting jacet; a polytope which is stable on only one of its facets is said to be monostatic.
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