Doument Mth. 419 Voronoi Digrms n Deluny Tringultions: Uiquitous Simese Twins Thoms M. Lieling n Lionel Pournin 2010 Mthemtis Sujet Clssifition: 01A65, 49-03, 52C99, 68R99, 90C99, 70-08, 82-08, 92-08 Keywors n Phrses: Voronoi, Deluny, tesseltions, tringultions, flip-grphs 1 Introution Coneling their rih struture ehin pprent simpliity, Voronoi igrms n their ul Simese twins, the Deluny tringultions onstitute remrkly powerful n uiquitous onepts well eyon the relm of mthemtis. This my e why they hve een isovere n reisovere time n gin. They were lrey present in fiels s iverse s stronomy n rystllogrphy enturies efore the irth of the two Russin mthemtiins whose nmes they rry. In more reent times, they hve eome ornerstones of moern isiplines suh s isrete n omputtionl geometry, lgorithm esign, sientifi omputing, n optimiztion. To fix ies, let us efine their most fmilir mnifesttions (in the Eulien plne) efore proeeing to sketh of their history, min properties, n pplitions, inluing glimpse t some of the tors involve. A Voronoi igrm inue y finite set A of sites is eomposition of the plne into possily unoune (onvex) polygons lle Voronoi regions, eh onsisting of those points t lest s lose to some prtiulr site s to the others. The ul Deluny tringultion ssoite to the sme set A of sites is otine y rwing tringle ege etween every pir of sites whose orresponing Voronoi regions re themselves jent long n ege. Boris Deluny hs equivlently hrterize these tringultions vi the empty irle property, wherey tringultion of set of sites is Deluny iff the irumirle of none of its tringles ontins sites in its interior. These efinitions re strightforwrly generlizle to three n higher imensions.
420 Thoms M. Lieling n Lionel Pournin Figure 1: From left to right: Johnnes Kepler, René Desrtes, Crl Frierih Guss, Johnn Peter Gustv Lejeune Dirihlet, John Snow, Emon Lguerre, Georgy Feoosevih Voronoi, n Boris Nikolevih Delone. The first seven pitures hve fllen in the puli omin, n the lst one ws kinly provie y Nikoli Dolilin. One my woner wht Voronoi n Deluny tesselltions hve to o in this optimiztion histories ook. For one they re themselves solutions of optimiztion prolems. More speifilly, for some set of sites A, the ssoite Deluny tringultions re me up of the losest to equilterl tringles; they re lsotherounestinthtthttheymximizethesumofriiofinsrieirles to their tringles. Moreover, they provie the mens to esrie fsinting energy optimiztion prolems tht nture itself solves [37, 18]. Furthermore Voronoi igrms re tools for solving optiml fility lotion prolems or fining the k-nerest n frthest neighors. Deluny tringultions re use to fin the minimum Eulien spnning tree of A, the smllest irle enlosing the set, n the two losest points in it. Algorithms to onstrut Voronoi igrms n Deluny tringultions re intimtely linke to optimiztion methos, like the greey lgorithm, flipping n pivoting, ivie n onquer [31]. Furthermore the min t strutures to implement geometri lgorithms were rete in onjuntion with those for Voronoi n Deluny tesselltions. Exellent soures on the notions of Voronoi igrms n Deluny tringultions, their history, pplitions, n generliztions re [12, 2, 3, 28]. 2 A glne t the pst The olest oumente tre of Voronoi igrms goes k to two gints of the Renissne: Johnnes Kepler (1571 Weil er Stt 1630 Regensurg) n René Desrtes (1596 L Hye en Tourine, now Desrtes 1650 Stokholm). The ltter use them to verify tht the istriution of mtter in the universe forms vorties entere t fixe strs (his Voronoi igrm s sites), see figure 2 [9]. Severl ees erlier, Kepler h lso introue Voronoi n Deluny tesselltions generte y integer ltties while stuying the shpes of snowflkes n the ensest sphere pking prolem (tht lso le to his fmous onjeture). Two enturies lter, the British physiin John Snow (1813 York 1858 Lonon) one more me up with Voronoi igrms in yet totlly ifferent ontext. During the 1854 Lonon holer outrek, he superpose the mp of holer ses n the Voronoi igrm inue y the sites of the wter
Voronoi Digrms n Deluny Tringultions 421 Figure 2: Left: Voronoi igrm rwn y René Desrtes[9], n its relultion isplying yellow Voronoi regions, with the ul Deluny tringultion in lue. Right: The Voronoi region entere on Bro Street pump, skethe y John Snow [33] using otte line. pumps, see figure 2 [33], therey ientifying the infete pump, thus proving tht Voronoi igrms n even sve lives. His igrm is referre to in [26] s the most fmous 19th entury isese mp n Snow s the fther of moern epiemiology. Aroun the time when John Snow ws helping to fight the Lonon holer epiemi, the eminent mthemtiin Johnn Peter Gustv Lejeune Dirihlet (1805 Düren 1859 Göttingen) ws in Berlin, prouing some of his seminl work on qurti forms. Following erlier ies y Kepler(see ove) n Crl Frierih Guss (1777 Brunshweig -1855 Göttingen), he onsiere Voronoi prtitions of spe inue y integer lttie points s sites [10]. Therefore, to this y, Voronoi igrms re lso lle Dirihlet tesseltions. Thirty yers lter, Georges Voronoi (1868 Zhurvky 1908 Zhurvky) extene Dirihlet s stuy of qurti forms n the orresponing tesselltions to higher imensions [34]. In the sme pper, he lso stuie the ssoite ul tesselltions tht were to e lle Deluny tringultions. Voronoi s results ppere in Crelle s journl in 1908, the yer of his untimely eth t the ge of 40. He h een stuent of Mrkov in Sint Petersurg, n spent most of his reer t the University of Wrsw where he h eome professor even efore ompleting his PhD thesis. It ws there tht young Boris Delone Russin spelling of the originl n usul Frenh Deluny (1890 Sint Petersurg 1980 Mosow) got introue to his fther s ollegue Voronoi. The ltter me lsting impression on the teenger, profounly influening his susequent work [11]. This my hve prompte the Mthemtil Genelogy Projet [25] to inorretly list Voronoi s Delone s PhD thesis visor just s they i with Euler n his stuent Lgrnge. Atully, Lgrnge never otine PhD, wheres Delone proly strte to work on his thesis, ut efinitely efene it well fter Voronoi s eth. Delone generlize Voronoi igrms n their uls to the se of irregulrly ple sites in -imensionl spe.
422 Thoms M. Lieling n Lionel Pournin He pulishe these results in pper written in Frenh [7], whih he signe Deluny. During his long life spnning nerly whole entury, he ws not only elerte s rillint mthemtiin, ut lso s one of Russi s foremost mountin limers. Inee, sie from his tringultions, one of the highest peks (4300m) in the Sierin Alti ws nme fter him too. For etile ount of Boris Deluny s life, reers re referre to the eutiful iogrphy written y Nikoli Dolilin [11]. Deluny s hrteriztion of his tringultions vi empty irles, respetively empty spheres in higher imensions lter turne out to e n essentil ingreient of the effiient onstrution of these strutures (see in setion 4 elow). At lest hlf ozen further isoveries of Voronoi igrms in suh misellneous fiels s gol mining, rystllogrphy, metllurgy, or meteorology re reore in [28]. Oly, some of these seemingly inepenent reisoveries tully took ple within the sme fiels of pplition. In 1933, Eugene Wigner (1902 Bupest 1995 Prineton) n Freerik Seitz (1911 Sn Frniso 2008 New York City) introue Voronoi igrms inue y the toms of metlli rystl [36]. Previously Pul Niggli (1888 Zofingen - 1953 Zürih) [27] n Deluny [6] h stuie similr rrngements n lssifie the ssoite polyher. To this y, physiists inifferently ll the ells of suh Voronoi igrms Wigner-Seitz zones, Dirihlet zones, or omins of tion. It shoul e unerline tht, over the lst ees, Voronoi igrms n Deluny tringultions hve lso me their pperne in the fiels of sientifi omputing n omputtionl geometry where they ply entrl role. In prtiulr, they re inresingly pplie for geometri moeling [4, 24, 1, 32] n s importnt ingreients of numeril methos for solving prtil ifferentil equtions. 3 Generliztions n pplitions As esrie y Aurenhmmer [3], orinry Voronoi igrms n e interprete s resulting from rystl growth proess s follows: From severl sites fixe in spe, rystls strt growing t the sme rte in ll iretions n without pushing prt ut stopping growth s they ome into ontt. The rystl emerging from eh site in this proess is the region of spe loser to tht site thn to ll others. A generliztion in whih rystls o not ll strt their growth simultneously ws propose inepenently y Kolmogorov in 1937 n Johnson n Mehl in 1939 [20]. In the plnr se, this gives rise to hyperoli region ounries. On the other hn, if the growth proesses strt simultneously ut progress t ifferent rtes, they yiel the so-lle Apollonius tesselltions, with spheril region ounries, resp. irulr in the plne. These ptterns n tully e oserve in sop foms [35]. Apollonius tesseltions re in ft multiplitively weighte Voronoi igrms in whih weights ssoite to eh site multiply the orresponing istnes. These types of Voronoi igrm ptterns re lso forme y myeli s they
Voronoi Digrms n Deluny Tringultions 423 Figure 3: Simulte hyphl growth. Left: Initilly ten numeril spores using self-voine grow n oupy the surrouning two-imensionl meium, efining Voronoi igrm. Right: Hyphl wll growth moel using pieewise flt surfes n Voronoi igrms thereon. evolve from single spores n ompete for territory(see figure 3). The myelium is the prt of the fungus tht evelops unergroun s n roresene whose suessive rnhes re lle hyphe [18]. Certin mols tully exhiit n essentilly plnr growth. Hyphl growth in its intertion with the surrouning meium n e moele using the ssumption tht s they grow, hyphe serete sustne tht iffuses into the meium, whose onentrtion they n etet n try to voi, therey oth voiing eh other n lso elerting their own irulriztion. Thus the reltionship to Voronoi igrms eomes pprent. At more mirosopi level, growth of hyphl wlls n e simulte y moeling them s pieewise flt surfes tht evolve oring to iologilly n mehnilly motivte ssumptions [18]. Therein, Deluny tringultions n Voronoi igrms on pieewise liner surfes re useful tools. Lguerre igrms (or tesseltions) re itively weighte Voronoi igrms lrey propose y Dirihlet [10] ees efore Emon Niols Lguerre (1834 Br-le-Du 1886 Br-le-Du) stuie the unerlying geometry. In the erly nineteen eighties, Frnz Aurenhmmer, who lls Lguerre igrms power igrms, wrote his PhD thesis out them, resulting in the pper [2], whih to this te remins n uthorittive soure on the sujet. They h previously lso een stuie y Lszlò Fejes Toth (1915 Szege 2005 Bupest) in the ontext of pking, overing, n illumintion prolems with spheres [14, 15]. Power igrms yiel muh riher lss of prtitions of spe into onvex ells thn orinry Voronoi igrms. They re inue y set of positively weighte sites, the weights eing interprete s the squre rii of spheres entere t the sites. The region inue y some weighte site i.e. sphere onsists of those points whose power with respet to tht sphere is smller or equl to tht with respet to ll others [15, 12, 3]. Note tht some spheres my generte n empty region of the power igrm, whih hs to o with
424 Thoms M. Lieling n Lionel Pournin Figure 4: The growth of polyrystl moele using ynmi power igrms. From left to right, lrger monorystlline regions grow, eting up the smller ones the ft tht the power with respet to sphere is not metri sine it n e negtive. The ul tringultions of power igrms re lle weighte Deluny tringultions, or regulr tringultions. These ojets n e efine in Eulien spes of ritrry imension. Lguerre tesselltions turn out to e very powerful moeling tools for some physil proesses, s for instne metl soliifition or ermis sintering. During the proution of ermi mterils, polyrystlline struture forms strting from, sy lumin power (Al2 SO3 ). With the help of time, het n pressure, the polyristl, whih is onglomerte of unligne rystlline ells unergoes proess in whih lrger ells grow t the expense of the smller ones (see figure 4). It hs een shown tht t ny point in time, three-imensionl Lguerre tesselltions re equte representtions of suh self-similr evolving polyrystlline strutures [37]. Their growth is riven y surfe energy minimiztion, the surfe eing the totl surfe etween jent rystlline regions. Not only is it esy to ompute this surfe in the se of Lguerre tesselltions, ut lso its grient when the prmeters efining the generting spheres evolve. With the use of the hin rule, it is thus possile to set up motion equtions for the generting spheres of the Lguerre tesselltion, tht reflet the energy minimiztion. They remin vli s long s there is no topologil trnsformtion of this tesseltion (suh trnsformtion onsisting either in neighor exhnge or ell vnishing). Whenever suh trnsformtion tkes ple, the tesselltion n motion equtions hve to e upte n integrte until etetion of the following topologil trnsformtion, n so on. This proess n go on until the polyrystlline struture eomes mono-rystl. The growth of foms n e moele in similr fshion. All this hs een implemente in two n three imensions for very lrge ell popultions, n perioi ounry onitions. The ltter imply generliztion of Lguerre tesselltions to flt tori. Suh simultions remin the only wy to follow the ynmi phenomen tking ple in the interior of three-imensionl polyrystls. Doument Mthemti Extr Volume ISMP (2012) 419 431
Voronoi Digrms n Deluny Tringultions 425 Another pplition, lose to tht in [15] omes up in the numeril simultion of grnulr mei where the ehvior of ssemlies of mrosopi grins like sn, orn, rie, oke is stuie y repliting trjetories of iniviul grins. Inrese omputing power in onjuntion with the power supplie y mthemtis now llows simultion of proesses involving hunres of thousns of grins. The min hllenge involve is threefol: relisti moeling of iniviul grin shpes eyon simple spheres; relisti physil moeling of the intertion etween ontting oies; effiient ontt etetion metho. The ltter is where Deluny tringultions re use. Inee, they yiel methos tht permit to effiiently test ontts within very lrge popultions of spheril grins. The unerlying property eing tht whenever two spheril grins re in ontt, their enters re linke y n ege of the ssoite regulr tringultion. Using this metho requires n effiient n numerilly stle upting proeure of regulr tringultions ssoite to ynmilly evolving sites. Using sphero-polyherl grins ( sphero-polyheron is the Minkowski sum of sphere with onvex polyheron), this proeure n e strightforwrly generlize to suh quite ritrrily shpe non-spheril grins. With this pproh, lrge-sle simultions of grin rystlliztion, mixing n unmixing, n omption proesses in nture n tehnology hve een performe (see figure 5). In priniple, Voronoi igrms n e efine for sets of sites on ritrry metri spes, suh s girffe n rooile skins, turtle shells, or isrete ones suh s grphs with positive ege weights stisfying the tringle inequlity, giving rise to lssil grph optimiztion prolems. 4 Geometry n lgorithms The previously introue -imensionl power igrms n the ssoite regulr tringultion n lso e viewe s the projetions to R of the lower ounries of two onvex (+1)-imensionl polyher. In ft, this projetive property n e use s efinition. In other wors, suivision of R into onvex ells is power igrm if n only if one n efine pieewise-liner onvex funtion from R to R whose regions of linerity re the ells of the igrm (see [3], n the referenes therein). The sme equivlene is lso true for regulr tringultions, where the given funtion is efine only on the onvex hull of the sites n hs simpliil regions of linerity. In this light, regulr tringultions n e interprete s proper sulss of the power igrms. In other wors, they re the power igrms whose fes re simplies. Note tht y fr, not every prtition of spe into onvex polyherl ells n e interprete s n orinry Voronoi igrm. As shown y Chnler Dvis [5], power igrms onstitute muh riher lss of suh
426 Thoms M. Lieling n Lionel Pournin Figure 5: Grnulr mei simultion using regulr tringultions. Left: All the ontts ourring in set of two-imensionl iss re etete y testing the eges of regulr tringultion. This tringultion is epite in lk n its ul power igrm in light gry. Right: Simultion of the output of funnel with very low frition, involving out 100 000 spheril prtiles. Contts re teste using regulr tringultions. prtitions. In ft, in imension higher thn 2, every simple onvex prtition is power igrm. In nlogy to simple polytopes, simple prtitions onsist of regions suh tht no more thn of them re jent t ny vertex. In this ontext it is interesting to note tht Kli hs shown tht the Hsse igrm of simple polytope n tully e reonstrute from its 1-skeleton [22]. Rell tht the 1-skeleton of polytope is the grph forme y its verties n eges. Hene the sme lso hols for simple power igrms. An importnt implition of the projetion property is tht softwre for onvex hull omputtion n e iretly use to ompute power igrms [16]. Sine the nineteen-seventies, mny other speilize lgorithms hve een evelope tht ompute these igrms. Toy, onstruting 2-imensionl Voronoi igrm hs eome stnt homework exerise of every si ourse in lgorithms n t strutures. In ft, the optiml ivie n onquer lgorithm y Shmos n e onsiere s one of the ornerstones of moern omputtionl geometry (see [31]). In this reursive lgorithm of omplexity O(n log(n)), the set of n sites is suessively prtitione into two smller ones, whereupon their orresponing Voronoi igrms re onstrute n sewn together. Unfortuntely, no generliztion of this lgorithm to higher imensions or to power igrms is known. Severl lgorithms tht ompute regulr tringultions re known, though, n y ulity, one n esily eue the power igrm generte y set of weighte sites from its ssoite regulr tringultion. Note in prtiulr
Voronoi Digrms n Deluny Tringultions 427 e e e Figure 6: Four types of flips in 2-imensions (left) n 3-imensions (right). The flips t the top insert or remove ege {,} n the flips t the ottom insert or remove vertex. tht one otins the Hsse igrm of power igrm y turning upsie own tht of the orresponing regulr tringultion. Plne Deluny tringultions n e onstrute using flip lgorithms suh s first propose y Lwson [23]. While their worst-se omplexity is O(n 2 ), in prtil ses they re not only lot fster thn tht, ut lso hve other esirle numeril properties. Consier tringultion of set of n points in the plne. Whenever two jent tringulr ells form onvex qurilterl, one n fin new tringultion y exhnging the igonls of this qurilterl. Suh n opertion is lle n ege flip n the flippe eges re lle flipple (see figure 6). A qurilterl with flipple ege is lle illegl if the irumirle of one of its tringles lso ontins the thir vertex of the other in its interior. Otherwise, it is legl. It is esy to see tht flip opertion on n illegl qurilterl mkes it legl n vie-vers. The simple lgorithm tht onsists in flipping ll illegl qurilterls to leglity, one fter the other in ny orer, lwys onverges to Deluny tringultion. Testing the leglity of qurilterl mounts to heking the sign of ertin eterminnt. Along with the flip opertion, this eterminnt-test generlizes to higher imensions [8]. Moreover, the forementione flip-lgorithm n e generlize to regulr tringultions with weighte sites y simply introuing n itionl type of flip to insert or elete (flip in/flip out) verties (see figure 6) n testing slightly moifie eterminnt. Unfortuntely, in this se, this lgorithm n stll without rehing the esire solution. For rigorous tretment of flips using Ron s theorem on minimlly ffinely epenent point sets, see [8]. The inrementl flip lgorithm [19] for the onstrution of regulr tringultions is metho tht lwys works. Therein, sequene of regulr tringultions is onstrute y suessively ing the sites in n ritrry orer. An initil tringultion onsists of properly hosen suffiiently lrge rtifiil tringle tht will ontin ll given sites in its interior n will e remove one
428 Thoms M. Lieling n Lionel Pournin the onstrution is finishe. At ny step new site is flippe in (see figure 6), suiviing its ontining tringle into three smller ones, the new tringultion possily not eing Deluny tringultion yet. However, s shown in [19], it is lwys possile to mke it eome one y sequene of flips. This inrementl flip lgorithm hs een generlize in [13] to the onstrution of regulr tringultions in ritrry imension. Any pir of regulr tringultions of given set of sites is onnete y sequene of flips [8]. If t lest one of the tringultions is not regulr, this nee not e the se. This issue gives rise to interesting questions tht will e the mentione in this lst prgrph. Consier the grph whose verties re the tringultions of finite -imensionl set of sites A, with n ege etween every pir of tringultions tht n e otine from one nother y flip. Wht Lwson prove [23] is tht this grph, lle the flip-grph of A, is onnete when A is 2-imensionl. The sugrph inue y regulr tringultions in the flip-grph of A is lso onnete (it is tully isomorphi to the 1-skeleton of the so-lle seonry polytope [17]). Furthermore, so is the lrger sugrph inue in the flip-grph of A y tringultions projete from the ounry omplex of ( + 2)-imensionl polytopes [29]. To this te, it is not known whether the flip grphs of 3- or 4-imensionl point sets re onnete, n point sets of imension 5 n 6 were foun whose flipgrph is not onnete [8] (the ltter hving omponent onsisting of single tringultion!). Finlly, it hs een shown only reently tht the flip-grph of the 4-imensionl ue is onnete [30]. 5 Conlusion This hpter hs esrie few milestones on journey tht strte when Kepler n Desrtes use wht were to eome Voronoi igrms to stuy the universe from snowflkes to glxies. These igrms n their ul Deluny tringultions hve menwhile eome powerful engineering esign, moeling, n nlysis tools, hve given rise to mny interesting questions in mthemtis n omputer siene, n hve helpe solving others (in prtiulr, Kepler s onjeture! See for instne [21])). The journey is y fr not ene n will ertinly le to still other fsinting isoveries. Referenes [1] N. Ament, S. Choi, R. K. Kolluri, The power rust, unions of lls, n the meil xis trnsform, Comput. Geom. 19, 127 153 (2001) [2] F. Aurenhmmer, Power igrms: properties, lgorithms n pplitions, SIAM J. Comput. 16, 1, 78 96 (1987) [3] F. Aurenhmmer, Voronoi igrms survey of funmentl geometri t struture, ACM Computing Surveys 23, 3, 345 405 (1991)
Voronoi Digrms n Deluny Tringultions 429 [4] CGAL, Computtionl Geometry Algorithms Lirry, http://www.gl. org [5] C. Dvis, The set of non-linerity of onvex pieewise-liner funtion, Sript Mth. 24, 219 228 (1959) [6] B.N. Deluny, Neue Drstellung er geometrishen Kristllogrphie, Z. Kristllogrph. 84, 109 149 (1932) [7] B. N. Deluny, Sur l sphère vie, Bull. A. Siene USSR VII: Clss. Si. Mth., 193 800 (1934) [8] J. A. e Loer, J. Rmu, F. Sntos, Tringultions: strutures for lgorithms n pplitions, Algorithms n Computtion in Mthemtis 25, Springer (2010) [9] R. Desrtes, Prinipi philosophie (1644) [10] G. L. Dirihlet, Üer ie Reuktion er positiven qurtishen Formen mit rei unestimmten gnzen Zhlen, J. Reine Angew. Mth. 40, 209 227 (1850) [11] N. P. Dolilin, Boris Nikolevih Delone (Deluny): Life n Work, Proeeings of the Steklov Institute of Mthemtis 275, 1 14 (2011) [12] H. Eelsrunner, Algorithms in Comintoril Geometry, Springer, Heielerg (1987) [13] H. Eelsrunner, N. R. Shh, Inrementl topologil flipping works for regulr tringultions, Algorithmi 15, 223 241 (1996) [14] L. Fejes Tóth, Regulr figures, Pergmon Press (1964) [15] L. Fejes Tóth, Illumintion of onvex iss, At Mth. Ae. Sient. Hung. 29, 355 360 (1977) [16] K. Fuku, Polyherl Computtions, MOS-SIAM Series in Optimiztion, 2012 (to pper) [17] I. M. Gel fn, M. M. Kprnov n A. V. Zelevinsky, Disriminnts of polynomils of severl vriles n tringultions of Newton polyher, Leningr Mth. J. 2, 449 505 (1990) [18] C. Inermitte, Th. M. Lieling, M. Troynov, H. Clémençon, Voronoi igrms on pieewise flt surfes n n pplition to iologil growth, Theoretil Computer Siene 263, 263 274 (2001) [19] B. Joe, Constrution of three-imensionl Deluny tringultions using lol trnformtions, Comput. Aie Geom. Design 8, 123 142 (1991)
430 Thoms M. Lieling n Lionel Pournin [20] W. A. Johnson, R.F. Mehl, Retion kinetis in proesses of nuletion n growth, Trns. Am. Instit. Mining Metll. A. I. M. M. E. 135, 416 458 (1939) [21] M. Joswig, From Kepler to Hles, n k to Hilert, this volume. [22] G. Kli, A simple wy to tell simple polytope from its grph, J. Com. Theor. Ser. A 49, 381 383 (1988) [23] C. L. Lwson, Trnsforming tringultions, Disrete Mth. 3, 365 372 (1972) [24] LEDA, Lirry of Effiient Dt Types n Algorithms, http://www. lgorithmi-solutions.om [25] The Mthemtis Genelogy Projet: http://www.genelogy.ms.org [26] M. S. Mee, Coneptul n Methoologil Issues in Meil Geogrphy, Chpel Hill (1980) [27] R. Niggli, Die topologishe Strukturnlyse, Z. Kristllogrph. 65 391 415 (1927) [28] A. Oke, B. Boots, K. Sugihr,S. N. Chiu, Sptil Tesselltions, Wiley (2000) [29] L. Pournin, A result on flip-grph onnetivity, Av. Geom. 12, 63 82 (2012) [30] L. Pournin, The flip-grph of the 4-imensionl ue is onnete, rxiv:1201.6543v1 [mth.mg] (2012) [31] M. I. Shmos, D. Hoey, Closest-point prolems. In Proeeings ot the 16th Annul IEEE Symposium on FOCS, 151 162 (1975) [32] J. R. Shewhuk, Generl-Dimensionl Constrine Deluny n Constrine Regulr Tringultions, I: Comintoril Properties, Disrete Comput. Geom. 39, 580 637 (2008) [33] J. Snow, Report on the Choler Outrek in the Prish of St. Jmes, Westminster, uring the Autumn of 1854 (1855) [34] G. Voronoi, Nouvelles pplitions es prmètres ontinus à théorie es formes qurtiques, J. Reine Angew. Mth. 134, 198 287 (1908) [35] D. Weire, N. Rivier, Sop, ells, n sttistis-rnom ptterns in two imensions, Contemp. Phys. 25, 59 99 (1984) [36] E. Wigner, F. Seitz, On the onstitution of metlli soium, Phys. Rev. 43, 804 810 (1933)
Voronoi Digrms n Deluny Tringultions 431 [37] X. J. Xue, F. Righetti, H. Telley, Th. M. Lieling, A. Moellin, The Lguerre moel for grin growth in three imensions, Phil. Mg. B 75 (1997) 567 585. Thoms M. Lieling EPFL Bsi Sienes Mthemtis MA A1 417 Sttion 8 1015 Lusnne Switzerln thoms.lieling@epfl.h Lionel Pournin EFREI 30 32 venue e l Répulique 94800 Villejuif Frne lionel.pournin@efrei.fr
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