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The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...
Net. In four-dimensional geometry, the 24-cell is the convex regular 4-polytope [1] (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C 24, or the icositetrachoron, [2] octaplex (short for "octahedral complex"), icosatetrahedroid, [3] octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.
A vertex moving on the isocline visits three other vertices that are √ 2 apart before reaching the fourth vertex that is √ 4 away. [o] The eight-cell ring is chiral: there is a right-handed form which spirals clockwise, and a left-handed form which spirals counterclockwise. The 16-cell contains one of each, so it also contains a left and a ...
The circumcenter of a tetrahedron can be found as intersection of three bisector planes. A bisector plane is defined as the plane centered on, and orthogonal to an edge of the tetrahedron. With this definition, the circumcenter C of a tetrahedron with vertices x 0, x 1, x 2, x 3 can be formulated as matrix-vector product: [35]
One more interesting line (in some sense dual to the Newton's one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.
Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is the 16-cell with Schläfli symbol {3,3,4}, with which it can be combined to form the compound of tesseract and 16-cell. Each edge of a regular tesseract is of the same length.
The 8-vertex 16-cell has 4 long diameters inclined at 90° = 𝜋 / 2 to each other, often taken as the 4 orthogonal axes or basis of the coordinate system. The 24-vertex 24-cell has 12 long diameters inclined at 60° = 𝜋 / 3 to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 ...
If the parabola intersects the x-axis in two points, there are two real roots, which are the x-coordinates of these two points (also called x-intercept). If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the contact point between the graph and parabola.