Morley’s trisector Theorem for isosceles tetrahedron

We extend Morley’s trisector theorem in the plane to an isosceles tetrahedron in three-dimensional space. We will show that the Morley tetrahedron of an isosceles tetrahedron is also isosceles tetrahedron. Furthermore, by the formula for distance in barycentric coordinate, we introduce and prove a general theorem on an isosceles tetrahedron.

     

Regular simple geodesic loops on a tetrahedron

A regular simple geodesic loop on a tetrahedron is a simple geodesic loop which does not pass through any vertex of the tetrahedron. It is evident that such loops meet each face of the tetrahedron. Among these loops, the minimal loops are those which meet each face exactly once. Necessary and sufficient conditions for the existence of minimal loops are obtained. These conditions fall naturally into two categories, conditions in the first category being called coherence conditions and conditions in the second category being called separation conditions. It is shown that for the existence of three distinct minimal loops through any point on the face of a tetrahedron it is necessary and sufficient that the tetrahedron be isosceles, which, in turn, amounts to the tetrahedron satisfying three coherence conditions. All other regular simple geodesic loops on an isosceles tetrahedron are then classified. Finally, coherence conditions for the existence of similar loops on an arbitrary tetrahedron are found.     

Irregularly shaped space-filling truncated octahedra

For any parent tetrahedron ABCD, centroids of selected sub-tetrahedra form the vertices of an irregularly shaped space-filling truncated octahedron. To reflect these properties, such a figure will be called an ISTO. Each edge of the ISTO is parallel to and one-eighth the length of one of the edges of tetrahedron ABCD and the volume of the ISTO is 3/16-th the volume of the tetrahedron. The ISTO is symmetric about the centroid of tetrahedron ABCD and each face is symmetric about a centre and has an opposite face that is parallel and congruent. The area of the faces of the ISTO is not proportional to that of the generating tetrahedron.     

四面体的十二点二次曲面

垂心四面体中四条高的垂足,四个面的重心及从各顶点与四面体的垂心连线的三等分点,共十二个点共球.试图把垂心改为四面体内的任意点,相应地把四条高线改换为过该点与每个顶点连线的共点直线组时,则将把垂心四面体的十二点球有趣地推广为四面体的十二点二次曲面.     

三维Poisson方程特征值的四种有限元解及比较

应用三维EQ1rot元、三维Crouzeix-Raviart元、八节点等参数元、四面体线性元计算三维Poisson方程的近似特征值.计算结果表明:三维EQ1rot元和三维Crouzeix-Raviart元特征值下逼近准确特征值,八节点等参数元、四面体线性元特征值上逼近准确特征值,三维EQr1ot元和三维Crouzeix-Raviart元外推特征值下逼近准确特征值.计算结果还表明三维Crouzeix-Raviart元是一种计算效率较高的非协调元.     

A volume formula for ℤ<Subscript>2</Subscript>-symmetric spherical tetrahedra

We obtain formulas for the volume of a spherical tetrahedron with ℤ₂-symmetry realized as rotation about the axis passing through the midpoints of a pair of skew edges. We show the dependence of the volume formula on the edge lengths and dihedral angles of the tetrahedron. Several different formulas result whose scopes are determined by the geometric characteristics of the tetrahedron.     

Euler Characteristics for Generalized Tetrahedron Groups

This paper is concerned with a class of groups which generalize the ordinary tetrahedron groups. This class of groups was introduced by Fine and Rosenberger and independently by Vinberg. Some conditions for the generalized tetrahedron groups to have a rational Euler characteristic are shown here and the values of the Euler characteristics in these cases are calculated. In addition here are some geometrical applications of the Euler characteristic of generalized tetrahedron groups.^{1 2}     

Volume of a tetrahedron in terms of dihedral angles and circumradius

The volume of a tetrahedron is represented in terms of the six dihedral angles between the faces and the radius of the sphere circumscribing the tetrahedron.     

16个和20个自由度的四面体有限元的场函数表达式的显式

本文写出了16个和20个自由度的四面体有限元的场函数表达式的显式.它们是用体积座标L₁,L₂,L₃,L₄来表达的.     

On the volume and surface area of hyperbolic polyhedra

A truncated tetrahedron is a building block of hyperbolic 3-manifolds with totally geodesic boundary. We study the relation between the volume of a truncated tetrahedron and the area of its faces which form the boundary of manifolds.     

The combinatorics of some tetrahedron manifolds

We study a family of closed connected orientable 3-manifolds (which are examples of tetrahedron manifolds) obtained by pairwise identifications of the boundary faces of a standard tetrahedron. These manifolds generalize those considered in previous papers due to Grasselli, Piccarreta, Molnár and Sieradski. Then we completely describe our tetrahedron manifolds in terms of Seifert fibered spaces, and determine their Seifert invariants. Moreover, we obtain different representations of our manifolds as 2-fold coverings, and give examples of non-equivalent knots with the same tetrahedron manifold as 2-fold branched covering space.     

On the Schlafli differential formula based on edge lengths of tetrahedron in <Emphasis Type="Italic">H</Emphasis><Superscript>3</Superscript> and <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

We obtain a new version of Schlafli differential formula based on edge lengths for the volume of a tetrahedron in hyperbolic and spherical 3-spaces, by using the edge matrix of a hyperbolic(or spherical) tetrahedron and its submatrix.      

Volume formula for a ℤ2-symmetric spherical tetrahedron through its edge lengths

The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle π in the middle points of a certain pair of its skew edges.     

Regge?s Einstein-Hilbert functional on the double tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge?s Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics.     

Geodesics on the regular tetrahedron and the cube

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.     

The Word Problem for Pride Groups

Pride groups are defined by means of finite (simplicial) graphs, and examples include Artin groups, Coxeter groups, and generalized tetrahedron groups. Under suitable conditions, we calculate an upper bound of the first order Dehn function for a finitely presented Pride group. We thus obtain sufficient conditions for when finitely presented Pride groups have solvable word problems. As a corollary to our main result, we show that the first order Dehn function of a generalized tetrahedron group, containing finite generalized triangle groups, is at most cubic.     

Continuous flattening of truncated tetrahedra

Each Platonic polyhedron P can be folded using a continuous folding process into a face of P so that the resulting shape is flat and multilayered, while two of the faces are rigid during the motion. In previous works, explicit formulas of continuous functions for such motions were given and the same result as above was shown to hold for any tetrahedron. In this paper, we show that a truncated regular tetrahedron can be folded continuously via explicit continuous folding mappings into a flat (folded) state, such that two of the hexagonal faces are rigid. Furthermore, given any general tetrahedron P and any truncated tetrahedron Q of P, we show that if Q contains the largest inscribed sphere of P and satisfies some condition, then Q can be folded continuously into a flat folded state such that two of the hexagonal faces of Q are rigid during the motion.     

Exact bound on the number of limit cycles arising from a periodic annulus bounded by a symmetric heteroclinic loop

In this paper, the bound on the number of limit cycles by Poincare bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between $3$ and $4$. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.     

三维空间中的有理样条插值

对三维空间某个多面体区域的四面体剖分，通过在每个四面体胞腔的棱和顶点设置适当的插值结点．本文给出了（１，１）型Ｃ⁰及Ｃ¹光滑的非奇异有理样条存在的充分必要条件．     

Construction of a 3/4-Ideal Hyperbolic Tetrahedron out of Ideal Tetrahedra

We show how to construct an arbitrary 3/4-ideal hyperbolic tetrahedron out of 10 ideal tetrahedra.