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With edge length , the inscribed sphere of a cube is the sphere tangent to the faces of a cube at their centroids, with radius . The midsphere of a cube is the sphere tangent to the edges of a cube, with radius 2 2 a {\textstyle {\frac {\sqrt {2}}{2}}a} .
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
A midsphere of a three-dimensional convex polyhedron is defined to be a sphere that is tangent to every edge of the polyhedron. That is to say, each edge must touch it, at an interior point of the edge, without crossing it. Equivalently, it is a sphere that contains the inscribed circle of every face of the polyhedron. [2]
For example, one sphere that is described in Cartesian coordinates with the equation x 2 + y 2 + z 2 = c 2 can be described in spherical coordinates by the simple equation r = c. (In this system—shown here in the mathematics convention—the sphere is adapted as a unit sphere, where the radius is set to unity and then can generally be ignored ...
Three mutually perpendicular golden ratio rectangles, with edges connecting their corners, form a regular icosahedron. Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint.
If the sphere is isometrically embedded in Euclidean space, the sphere's intersection with a plane is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant Euclidean distance (the extrinsic radius) from a point in the plane (the extrinsic center). A great circle lies ...
In Euclidean space, the dual of a polyhedron is often defined in terms of polar reciprocation about a sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius.
The first sphere of this row only touches one sphere in the original row, but its location follows suit with the rest of the row. The next row follows this pattern of shifting the x-coordinate by r and the y-coordinate by √ 3. Add rows until reaching the x and y maximum borders of the box.