"cross-polytope" meaning in All languages combined

{
  "forms": [
    {
      "form": "cross-polytopes",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "cross-polytope (plural cross-polytopes)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Geometry",
          "orig": "en:Geometry",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "2006, Ingemar Bengtsson, Karol Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, page 25",
          "text": "The case p = 1, which is actually important to us, is illustrated in Figure 1.14; we are looking at the intersection of a cross-polytope with the probability simplex.",
          "type": "quotation"
        },
        {
          "text": "2008, Robert Erdahl, Andrei Ordine, Konstantin Rybnikov, Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices, Matthias Beck, et al. (editors), Contemporary Mathematics 452: Integer Points in Polyhedra — Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, AMS-IMS-SIAM Joint Summer Conference, June 11-15, 2006, page 104,\nThe convex hulls of such crosses often appear as cells in Delaunay tilings – cross polytopes are examples, as are the more spectacular symmetric perfect Delaunay polytopes."
        },
        {
          "ref": "2009, Herbert W. Hamber, Quantum Gravitation: The Feynman Path Integral Approach, Springer, page 263",
          "text": "The cross polytope βₙ is the regular polytope in n dimensions corresponding to the convex hull of the points formed by permuting the coordinates (±1,0,0,...,0), and has therefore 2n vertices. It is named so because its vertices are located equidistant from the origin, along the Cartesian axes in n-space. The cross polytope in n dimensions is bounded by 2ⁿ (n - 1)-simplices, has 2n vertices and 2ⁿ(n - 1) edges.",
          "type": "quotation"
        },
        {
          "ref": "2010, Rainer Typke, Agatha Walczak-Typke, “Indexing Techniques for Non-metric Music Dissimilarity Measures”, in Zbigniew W. Raś, Alicja A. Wieczorkowska, editors, Advances in Music Information Retrieval, Springer, page 10",
          "text": "If one works with the l₁ norm, a ball (the set of all points whose distance lies within a certain radius around a point of interest) has the shape of a cross-polytope. A one-dimensional cross-polytope is a line segment, a two-dimensional cross-polytope is a square, for three dimensions, an octahedron, and so forth.",
          "type": "quotation"
        },
        {
          "ref": "2003, Branko Grünbaum, Convex Polytopes, 2nd edition, Springer, page 121b",
          "text": "While centrally symmetric d-polytopes with 2d vertices are affinely equivalent to the d-dimensional cross-polytope, a complete classification of the centrally symmetric d-polytopes with 2d + 2 vertices is out of reach.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A polytope that is the convex hull of points, two on each Cartesian axis of a Euclidean space, that are equidistant from the origin; an orthoplex."
      ],
      "id": "en-cross-polytope-en-noun-ubWqX70v",
      "links": [
        [
          "geometry",
          "geometry"
        ],
        [
          "polytope",
          "polytope"
        ],
        [
          "convex hull",
          "convex hull"
        ],
        [
          "Cartesian",
          "Cartesian"
        ],
        [
          "Euclidean",
          "Euclidean"
        ],
        [
          "orthoplex",
          "orthoplex"
        ]
      ],
      "raw_glosses": [
        "(geometry) A polytope that is the convex hull of points, two on each Cartesian axis of a Euclidean space, that are equidistant from the origin; an orthoplex."
      ],
      "related": [
        {
          "word": "Turán graph"
        }
      ],
      "synonyms": [
        {
          "sense": "polytope",
          "word": "cocube"
        },
        {
          "sense": "polytope",
          "word": "hyperoctahedron"
        },
        {
          "sense": "polytope",
          "word": "orthoplex"
        },
        {
          "word": "cross polytope"
        }
      ],
      "topics": [
        "geometry",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "cross-polytope"
      ]
    }
  ],
  "word": "cross-polytope"
}

{
  "forms": [
    {
      "form": "cross-polytopes",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "cross-polytope (plural cross-polytopes)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Turán graph"
    }
  ],
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Geometry"
      ],
      "examples": [
        {
          "ref": "2006, Ingemar Bengtsson, Karol Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement, page 25",
          "text": "The case p = 1, which is actually important to us, is illustrated in Figure 1.14; we are looking at the intersection of a cross-polytope with the probability simplex.",
          "type": "quotation"
        },
        {
          "text": "2008, Robert Erdahl, Andrei Ordine, Konstantin Rybnikov, Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices, Matthias Beck, et al. (editors), Contemporary Mathematics 452: Integer Points in Polyhedra — Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, AMS-IMS-SIAM Joint Summer Conference, June 11-15, 2006, page 104,\nThe convex hulls of such crosses often appear as cells in Delaunay tilings – cross polytopes are examples, as are the more spectacular symmetric perfect Delaunay polytopes."
        },
        {
          "ref": "2009, Herbert W. Hamber, Quantum Gravitation: The Feynman Path Integral Approach, Springer, page 263",
          "text": "The cross polytope βₙ is the regular polytope in n dimensions corresponding to the convex hull of the points formed by permuting the coordinates (±1,0,0,...,0), and has therefore 2n vertices. It is named so because its vertices are located equidistant from the origin, along the Cartesian axes in n-space. The cross polytope in n dimensions is bounded by 2ⁿ (n - 1)-simplices, has 2n vertices and 2ⁿ(n - 1) edges.",
          "type": "quotation"
        },
        {
          "ref": "2010, Rainer Typke, Agatha Walczak-Typke, “Indexing Techniques for Non-metric Music Dissimilarity Measures”, in Zbigniew W. Raś, Alicja A. Wieczorkowska, editors, Advances in Music Information Retrieval, Springer, page 10",
          "text": "If one works with the l₁ norm, a ball (the set of all points whose distance lies within a certain radius around a point of interest) has the shape of a cross-polytope. A one-dimensional cross-polytope is a line segment, a two-dimensional cross-polytope is a square, for three dimensions, an octahedron, and so forth.",
          "type": "quotation"
        },
        {
          "ref": "2003, Branko Grünbaum, Convex Polytopes, 2nd edition, Springer, page 121b",
          "text": "While centrally symmetric d-polytopes with 2d vertices are affinely equivalent to the d-dimensional cross-polytope, a complete classification of the centrally symmetric d-polytopes with 2d + 2 vertices is out of reach.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A polytope that is the convex hull of points, two on each Cartesian axis of a Euclidean space, that are equidistant from the origin; an orthoplex."
      ],
      "links": [
        [
          "geometry",
          "geometry"
        ],
        [
          "polytope",
          "polytope"
        ],
        [
          "convex hull",
          "convex hull"
        ],
        [
          "Cartesian",
          "Cartesian"
        ],
        [
          "Euclidean",
          "Euclidean"
        ],
        [
          "orthoplex",
          "orthoplex"
        ]
      ],
      "raw_glosses": [
        "(geometry) A polytope that is the convex hull of points, two on each Cartesian axis of a Euclidean space, that are equidistant from the origin; an orthoplex."
      ],
      "topics": [
        "geometry",
        "mathematics",
        "sciences"
      ],
      "wikipedia": [
        "cross-polytope"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "polytope",
      "word": "cocube"
    },
    {
      "sense": "polytope",
      "word": "hyperoctahedron"
    },
    {
      "sense": "polytope",
      "word": "orthoplex"
    },
    {
      "word": "cross polytope"
    }
  ],
  "word": "cross-polytope"
}