{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,25]],"date-time":"2026-01-25T01:13:24Z","timestamp":1769303604787,"version":"3.49.0"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,19]],"date-time":"2022-02-19T00:00:00Z","timestamp":1645228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal\u2019s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal\u2019s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries.<\/jats:p>","DOI":"10.3390\/sym14020411","type":"journal-article","created":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T08:34:47Z","timestamp":1645432487000},"page":"411","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On a Surface Associated with Pascal\u2019s Triangle"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8185-956X","authenticated-orcid":false,"given":"Valeriu","family":"Beiu","sequence":"first","affiliation":[{"name":"Faculty of Exact Sciences, \u201cAurel Vlaicu\u201d University of Arad, 310032 Arad, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2770-023X","authenticated-orcid":false,"given":"Leonard","family":"D\u0103u\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2718-8133","authenticated-orcid":false,"given":"Marilena","family":"Jianu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2033-8394","authenticated-orcid":false,"given":"Adela","family":"Mihai","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020396 Bucharest, Romania"},{"name":"Interdisciplinary Doctoral School, Transilvania University of Bra\u015fov, 500036 Bra\u015fov, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3782-2983","authenticated-orcid":false,"given":"Ion","family":"Mihai","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bucharest, 010014 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,19]]},"reference":[{"key":"ref_1","unstructured":"Beiu, V. 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