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The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".
Flipping is used to improve quality measures of a triangle such as skewness. Mesh smoothing enhances element shapes and overall mesh quality by adjusting the location of mesh vertices. In mesh smoothing, core features such as non-zero pattern of the linear system are preserved as the topology of the mesh remains invariant.
Each layer represents one of the first five triangular numbers. A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The n th tetrahedral number, Te n, is the sum of the first n triangular numbers, that is,
In a triangle, any arbitrary side can be considered the base. The two endpoints of the base are called base vertices and the corresponding angles are called base angles. The third vertex opposite the base is called the apex. The extended base of a triangle (a particular case of an extended side) is the line that contains the base.
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number , other examples being square numbers and cube numbers . The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural ...
This definition rules out, for example, the square pyramid (since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces of that triangular bipyramid would be equilateral triangles, that is, congruent and regular, some ...
Given the edge length .The surface area of a truncated tetrahedron is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume is: [2] =, =.. The dihedral angle of a truncated tetrahedron between triangle-to-hexagon is approximately 109.47°, and that between adjacent hexagonal faces is approximately 70.53°.
If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume [2] [3] =. The altitude h satisfies [4] = + +. The area of the base is given by [5] =. The solid angle at the right-angled vertex, from which the opposite face (the base) subtends an octant, has measure π /2 steradians, one eighth of the surface area of a unit sphere.