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23040 symmetries of hyperbolic tetrahedra  [PDF]
Peter Doyle,Gregory Leibon
Mathematics , 2003,
Abstract: We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices `beyond infinity'; and we uncover a group, which we call 22.5K, of 23040 scissors-class-preserving symmetries of the space of (suitably decorated) generalized hyperbolic tetrahedra. The group 22.5K contains the Regge symmetries as a subgroup of order 144. From a generic tetrahedron, 22.5K produces 30 distinct generalized tetrahedra in the same scissors class, including the 12 honest-to-goodness tetrahedra produced by the Regge subgroup. The action of 22.5K leads us to the Murakami-Yano formula, and to 9 others, which are similar but less symmetrical. From here, we can derive yet other volume formulas with pleasant algebraic and analytical properties. The key to understanding all this is a natural relationship between a hyperbolic tetrahedron and a pair of ideal hyperbolic octahedra.
Nonexistence for extremal Type II $\ZZ_{2k}$-Codes  [PDF]
Tsuyoshi Miezaki
Mathematics , 2009,
Abstract: In this paper, we show that an extremal Type II $\ZZ_{2k}$-code of length $n$ dose not exist for all sufficiently large $n$ when $k=2,3,4,5,6$.
Routh's Theorem for Tetrahedra  [PDF]
Semyon Litvinov,Franti?ek Marko
Mathematics , 2014,
Abstract: We give a geometric proof of the Routh's theorem for tetrahedra.
A smooth space of tetrahedra  [PDF]
Eric Babson,Paul E. Gunnells,Richard Scott
Mathematics , 1999,
Abstract: We construct a smooth symmetric compactification of the space of all labeled tetrahedra in P^3.
Orthologic Tetrahedra with Intersecting Edges  [PDF]
Hans-Peter Schr?cker
Mathematics , 2009,
Abstract: Two tetrahedra are called orthologic if the lines through vertices of one and perpendicular to corresponding faces of the other are intersecting. This is equivalent to the orthogonality of non-corresponding edges. We prove that the additional assumption of intersecting non-corresponding edges ("orthosecting tetrahedra") implies that the six intersection points lie on a sphere. To a given tetrahedron there exists generally a one-parametric family of orthosecting tetrahedra. The orthographic projection of the locus of one vertex onto the corresponding face plane of the given tetrahedron is a curve which remains fixed under isogonal conjugation. This allows the construction of pairs of conjugate orthosecting tetrahedra to a given tetrahedron.
Coxeter Decompositions of Hyperbolic Tetrahedra  [PDF]
A. Felikson
Mathematics , 2002, DOI: 10.1070/SM2002v193n12ABEH000702
Abstract: We classify Coxeter decompositions of hyperbolic tetrahedra, i.e. simplices in the hyperbolic space H^3. The paper completes the classification of Coxeter decompositions of hyperbolic simplices.
Enumeration of integral tetrahedra  [PDF]
Sascha Kurz
Mathematics , 2008,
Abstract: We determine the numbers of integral tetrahedra with diameter $d$ up to isomorphism for all $d\le 1000$ via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most $d$ in $O(d^5)$ time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral $4\times 4$ matrices with diameter $d$ fulfilling the triangle inequalities we derive an exact formula.
Sets of tetrahedra, defined by maxima of distance functions  [PDF]
Jo?l Rouyer,Costin V?lcu
Mathematics , 2012,
Abstract: We study tetrahedra and the space of tetrahedra from the viewpoint of local and global maxima for intrinsic distance functions.
A Dense Packing of Regular Tetrahedra  [PDF]
Elizabeth R. Chen
Mathematics , 2009, DOI: 10.1007/s00454-008-9101-y
Abstract: We construct a dense packing of regular tetrahedra, with packing density $D > >.7786157$.
No acute tetrahedron is an 8-reptile  [PDF]
Herman Haverkort
Computer Science , 2015,
Abstract: An $r$-gentiling is a dissection of a shape into $r \geq 2$ parts which are all similar to the original shape. An $r$-reptiling is an $r$-gentiling of which all parts are mutually congruent. This article shows that no acute tetrahedron is an $r$-gentile or $r$-reptile for any $r < 9$, by showing that no acute spherical diangle can be dissected into less than nine acute spherical triangles.
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