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  • 1
    Electronic Resource
    Electronic Resource
    Polyeder mit zehn Ecken vom Geschlecht drei (1981)
    Springer
    Geometriae dedicata 11 (1981), S. 119-124 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00183194
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  • 2
    Electronic Resource
    Electronic Resource
    Smooth approximation of polyhedral surfaces regarding curvatures (1982)
    Springer
    Geometriae dedicata 12 (1982), S. 435-461 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we prove that every closed polyhedral surface in Euclidean three-space can be approximated (uniformly with respect to the Hausdorff metric) by smooth surfaces of the same topological type such that not only the (Gaussian) curvature but also the absolute curvature and the absolute mean curvature converge in the measure sense. This gives a direct connection between the concepts of total absolute curvature for both smooth and polyhedral surfaces which have been worked out by several authors, particularly N. H. Kuiper and T. F. Banchoff.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00147585
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  • 3
    Electronic Resource
    Electronic Resource
    The weakly neighborly polyhedral maps on the torus (1985)
    Springer
    Geometriae dedicata 18 (1985), S. 227-238 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without a boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just five distinct w.n.p. maps on the torus, and that only three of them are geometrically realizable as polyhedra with convex faces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00181222
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  • 4
    Electronic Resource
    Electronic Resource
    A polyhedron of genus 4 with minimal number of vertices and maximal symmetry (1989)
    Springer
    Geometriae dedicata 29 (1989), S. 53-64 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We construct a symmetric polyhedron of genus 4 in R 3 with 11 vertices. This shows that for given genus g the minimal numbers of vertices of combinatorial manifolds and of polyhedra coincide in the first previously unknown case g=4 also. We show that our polyhedron has the maximal symmetry for the given genus and minimal number of vertices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00147470
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  • 5
    Electronic Resource
    Electronic Resource
    The shape invariant of triangles and trigonometry in two-point homogeneous spaces (1990)
    Springer
    Geometriae dedicata 33 (1990), S. 59-76 
    Details
    ISSN: 1572-9168
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces ℂP n and ℍP n (n≥2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces ℂH n and ℍH n (n≥2).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00147601
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  • 6
    Electronic Resource
    Electronic Resource
    15-vertex triangulations of an 8-manifold (1992)
    Springer
    Mathematische Annalen 294 (1992), S. 167-193 
    Details
    ISSN: 1432-1807
    Keywords: 57Q15 ; 52B70 ; 53C42
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934320
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  • 7
    Electronic Resource
    Electronic Resource
    The weakly neighborly polyhedral maps on the 2-manifold with euler characteristic −1 (1986)
    Springer
    Discrete & computational geometry 1 (1986), S. 355-369 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A weakly neighborly polyhedral map (w.n.p. map) is a two-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just four w.n.p. maps with Euler characteristic −1 and we describe them.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02187707
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  • 8
    Electronic Resource
    Electronic Resource
    Nonuniqueness results for X-ray problems with point sources (1992)
    Springer
    Discrete & computational geometry 8 (1992), S. 153-169 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Let $$X \subseteq \mathbb{R}^d (d \geqslant 2)$$ be any finite set with no three points on a line. Then we construct explicitly families of bounded Borel sets which are not uniquely determined by their X-ray images from the points inX. This result is in sharp contrast to the known uniqueness results for convex sets (Hammer's X-ray problem).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02293041
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  • 9
    Electronic Resource
    Electronic Resource
    Neighborly maps with few vertices (1992)
    Springer
    Discrete & computational geometry 8 (1992), S. 93-104 
    Details
    ISSN: 1432-0444
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX〉−10 (i.e., genusg〈6, if orientable) or, equivalently, all the neighborly maps withV〈12 vertices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02293037
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  • 10
    Electronic Resource
    Electronic Resource
    Manifold structures on abstract regular polytopes (1995)
    Springer
    Aequationes mathematicae 49 (1995), S. 12-35 
    Details
    ISSN: 1420-8903
    Keywords: Primary 51M20 ; Secondary 57M50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS 3, connected sums of handlesS 1 × S 2 , euclidean and spherical space forms, and other examples with non-trivial fundamental group.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01827926
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