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Karl Menger was a young geometry professor at the University of Vienna and Arthur Cayley was a British mathematician who specialized in algebraic geometry. Menger extended Cayley's algebraic results to propose a new axiom of metric spaces using the concepts of distance geometry up to congruence equivalence, known as the Cayley–Menger determinant.
All vertices of a Reeve tetrahedron are lattice points (points whose coordinates are all integers). No other lattice points lie on the surface or in the interior of the tetrahedron. The volume of the Reeve tetrahedron with vertex (1, 1, r) is r/6. In 1957 Reeve used this tetrahedron to show that there exist tetrahedra with four lattice points ...
The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume: =. where is the base' area and is the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apices to the opposite faces are inversely proportional to the areas of ...
The four altitudes of an orthogonal tetrahedron meet at its orthocenter. Edges AB, BC, CA are perpendicular to, respectively, edges CD, AD, BD. In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means
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°.
1 5-tetrahedron: 2 8-tetrahedron: 2 4-cube: 4 6-octahedron: 20 30-tetrahedron: 12 10-dodecahedron: Inscribed 120 in 120-cell 675 in 120-cell 2 16-cells 3 8-cells 25 24-cells 10 600-cells Great polygons: 2 squares x 3 4 rectangles x 4 4 hexagons x 4 12 decagons x 6 100 irregular hexagons x 4 Petrie polygons: 1 pentagon x 2 1 octagon x 3 2 ...
The tetrahedron itself may also be defined as the unit of volume (see below). The four quadrays may be linearly combined to provide integer coordinates for the inverse tetrahedron (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0), and for the cube, octahedron, rhombic dodecahedron and cuboctahedron of volumes 3, 4, 6 and 20 respectively, given the ...
A Heronian tetrahedron [1] (also called a Heron tetrahedron [2] or perfect pyramid [3]) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles (named for Hero of Alexandria). Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are ...