{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:37:46Z","timestamp":1760243866364,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2011,12,28]],"date-time":"2011-12-28T00:00:00Z","timestamp":1325030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g \u2265 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 \u2264 g \u2264 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 \u2264 g \u2264 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan\u2019s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.<\/jats:p>","DOI":"10.3390\/sym4010001","type":"journal-article","created":{"date-parts":[[2011,12,28]],"date-time":"2011-12-28T12:01:29Z","timestamp":1325073689000},"page":"1-14","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Convex-Faced Combinatorially Regular Polyhedra of Small Genus"],"prefix":"10.3390","volume":"4","author":[{"given":"Egon","family":"Schulte","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northeastern University, Boston, MA 02115, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00f6rg M.","family":"Wills","sequence":"additional","affiliation":[{"name":"Department Mathematik, University of Siegen, Emmy-Noether-Campus, D-57068 Siegen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2011,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/BF02760539","article-title":"Equivelar polyhedral manifolds in E3","volume":"41","author":"McMullen","year":"1982","journal-title":"Isr. 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