{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T15:18:06Z","timestamp":1759331886728,"version":"build-2065373602"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,2,22]],"date-time":"2024-02-22T00:00:00Z","timestamp":1708560000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,2,22]],"date-time":"2024-02-22T00:00:00Z","timestamp":1708560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001343","name":"University of Pretoria","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001343","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2025,10]]},"abstract":"Abstract<\/jats:title>\n For \n \n $$d\\in {\\mathbb {N}}$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, a compact sphere packing of Euclidean space \n \n $${\\mathbb {R}}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a set of spheres in \n \n $${\\mathbb {R}}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial d<\/jats:italic>-complex that covers all of \n \n $${\\mathbb {R}}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We are motivated by the question: For \n \n $$d,n\\in {\\mathbb {N}}$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with \n \n $$d,n\\ge 2$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, how many configurations of numbers \n \n $$0<r_{0}<r_{1}<\\cdots <r_{n-1}=1$$<\/jats:tex-math>\n \n \n 0<\/mml:mn>\n <<\/mml:mo>\n \n r<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n <<\/mml:mo>\n \n r<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <<\/mml:mo>\n \u22ef<\/mml:mo>\n <<\/mml:mo>\n \n r<\/mml:mi>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can occur as the radii of spheres in a compact sphere packing of \n \n $${\\mathbb {R}}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> wherein there occur exactly n<\/jats:italic> sizes of sphere? We introduce what we call \u2018heteroperturbative sets\u2019 of labeled triangulations of unit spheres and we discuss the existence of non-trivial examples of heteroperturbative sets. For a fixed heteroperturbative set, we discuss how a compact sphere packing may be associated to the heteroperturbative set or not. We proceed to show, for \n \n $$d,n\\in {\\mathbb {N}}$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with \n \n $$d,n\\ge 2$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and for a fixed heteroperturbative set, that the collection of all configurations of n<\/jats:italic> distinct positive numbers that can occur as the radii of spheres in a compact packing is finite, when taken over all compact sphere packings of \n \n $${\\mathbb {R}}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> which have exactly n<\/jats:italic> sizes of sphere and which are associated to the fixed heteroperturbative set.<\/jats:p>","DOI":"10.1007\/s00454-024-00628-y","type":"journal-article","created":{"date-parts":[[2024,2,22]],"date-time":"2024-02-22T10:02:58Z","timestamp":1708596178000},"page":"719-742","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Compact Packings of Euclidean Space with Spheres of Finitely Many Sizes"],"prefix":"10.1007","volume":"74","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0162-3229","authenticated-orcid":false,"given":"Miek","family":"Messerschmidt","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eder","family":"Kikianty","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,2,22]]},"reference":[{"issue":"7860","key":"628_CR1","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1038\/s41586-021-03492-5","volume":"593","author":"I Cherniukh","year":"2021","unstructured":"Cherniukh, I., Rain\u00f2, G., St\u00f6ferle, T., Burian, M., Travesset, A., Naumenko, D., Amenitsch, H., Erni, R., Mahrt, R.F., Bodnarchuk, M.I., Kovalenko, M.V.: Perovskite-type superlattices from lead halide perovskite nanocubes. 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