Tetrahedron with insphere in red (also midsphere in green, circumsphere in blue)In his 1597 book Mysterium Cosmographicum, Kepler modelled of the Solar System with its then known six planets' orbits by nested platonic solids, each circumscribed and inscribed by a sphere.
In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron's circumsphere.
The radius of the sphere inscribed in a polyhedron P is called the inradius of P.
In geometry, the inscribedsphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's...
polygon (or a sphere or ellipsoid inscribed in a convex polyhedron) is tangent to every side or face of the outer figure (but see Inscribedsphere for semantic...
instead. The receiver position can be interpreted as the center of an inscribedsphere (insphere) of radius bc, given by the receiver clock bias b (scaled...
inscribedsphere, a sphere tangent to all faces of a polyhedron. In the regular polyhedra, the inscribedsphere, midsphere, and circumscribed sphere all...
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical...
were set inside one another and separated by a series of inscribed and circumscribed spheres. Kepler proposed that the distance relationships between...
Toronto Studies. p. 4. Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to...
= r, assuming the sphere of radius r is centered at the origin. For most practical purposes, the volume inside a sphereinscribed in a cube can be approximated...
conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. For a cube centered at the...
the vertices to the points of contact of the opposite faces with the inscribedsphere of the tetrahedron. In a trirectangular tetrahedron the three face...
regular octahedron is the square root of 6 times the radius of an inscribedsphere (that is, the distance from the center of the solid to the center of...
258\,538\cdot a} (sequence A179296 in the OEIS) and the radius of an inscribedsphere (tangent to each of the regular dodecahedron's faces) is r i = a 1...
through the vertices of a triangle or is tangent to its sides Inscribedsphere – Sphere tangent to every face of a polyhedron Power of a point – Relative...
must have an inscribed circle (that is, it must be a tangential polygon), and all of these inscribed circles must belong to a single sphere. For example...
circumscribed sphere has radius (the circumradius): R = l 2 + m 2 + n 2 8 {\displaystyle R={\sqrt {\frac {l^{2}+m^{2}+n^{2}}{8}}}} and the inscribedsphere has...
by a the edge length of a rhombic dodecahedron, the radius of its inscribedsphere (tangent to each of the rhombic dodecahedron's faces) is r i = 6 3...
armillary sphere (variations are known as spherical astrolabe, armilla, or armil) is a model of objects in the sky (on the celestial sphere), consisting...
r_{u}={\frac {\sqrt {2}}{2}}a\approx 0.707\cdot a} and the radius of an inscribedsphere (tangent to each of the octahedron's faces) is r i = 6 6 a ≈ 0.408...
and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle...
belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron. The rhombic triacontahedron...
below). Failing that, for a polyhedron with a circumscribed sphere, inscribedsphere, or midsphere (one with all edges as tangents), this can be used. However...
allegedly inscribed on his tombstone discovered by Cicero), and so he asked for a sketch of a sphereinscribed in a cylinder to be inscribed on his grave...
the vertices to the points of contact of the opposite faces with the inscribedsphere of the tetrahedron. In an orthocentric tetrahedron the four altitudes...
circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term...
area of a square to the area of its inscribed circle and the volume of a cube and volume of the inscribedsphere should also be 42:32. In formula, with...
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