Encyclopedia/All topics/Geometry

Geometry

Articles 241–264 of 264. Alphabetical by title.

• Geometry Schlaefli Tetrahedron Tetra Schlaefli Derivative Data Of Equation

  A single tetrahedron obeys a fixed relation among its edge lengths and dihedral angles, and a machine-checked library now records it as a reusable package.

• Geometry Schlaefli Total Deficit Flat

  When every hinge in a piecewise-flat complex is locally flat, the total deficit vanishes: a theorem that anchors Regge calculus.

• Geometry Schlaefli Triangulation3 D

  A three-dimensional shape built from tetrahedra obeys a hidden bookkeeping rule: the total change in its edge lengths and angles always cancels to zero.

• Geometry Schlaefli Triangulation3 D Global Schlaefli Lhs

  A sum over every tetrahedron in a 3D triangulation cancels exactly to zero, a machine-checked identity with a precise scope.

• Geometry Schlaefli Triangulation3 D Global Schlaefli Of Local

  A theorem about triangulated 3D space shows that a certain sum of edge-length changes over all tetrahedra always cancels to zero, a fact that links local geometry to a global invar

• Geometry Schlaefli Triangulation3 D Global Schlaefli Rhs

  In a 3D triangulation, a sum over all tetrahedra of a certain angle-derivative product always equals zero.

• Geometry Schlaefli Triangulation3 D Triangulation Schlaefli Data

  A machine-checked identity shows that in any finite 3D triangulation, the sum of local tetrahedral angle variations cancels exactly, leaving a global invariant.

• Geometry Tetrahedron Realization

  A tetrahedron is a pyramid with four triangular faces; Recognition Science builds one from six squared edge lengths and proves its volume formula.

• Geometry Tetrahedron Realization Basis Edge Vector

  A tetrahedron in space is fixed by three vectors from one vertex; the framework's basisEdgeVector names exactly those three.

• Geometry Tetrahedron Realization Det Gram3 Eq 36 Volume Sq

  For any nondegenerate tetrahedron in Euclidean 3-space, the determinant of its Gram matrix equals 36 times the square of its volume.

• Geometry Tetrahedron Realization Gram Cayley Menger Volume Theorem

  For any tetrahedron built from four points in ordinary space, two different formulas for its volume are forced to agree.

• Geometry Tetrahedron Realization Gram3 Symm

  For any tetrahedron placed in ordinary three-dimensional space, the matrix of edge dot products is symmetric, a simple fact with a long reach.

• Geometry Tetrahedron Realization Realized Tet

  A tetrahedron is classically a solid with four triangular faces; the framework's RealizedTet pins down exactly what it means for six edge lengths to come from actual points in

• Geometry Tetrahedron Realization Sq Edge Of Points

  A tetrahedron's six edge lengths, squared, are the bridge between abstract geometry and actual points in space.

• Geometry Tetrahedron Realization Sq Edge Of Points Nonneg

  In Euclidean geometry, the squared length of any edge of a tetrahedron is never negative; a machine-checked proof makes this trivial fact explicit.

• Geometry Tetrahedron Realization Volume Sq From Gram

  A tetrahedron's volume can be computed from its six edge lengths alone; this page explains the squared-volume formula and its exact scope.

• Geometry Triangulation3 Dconsistency

  A triangulation of space is consistent when every tetrahedron's edges agree with the global shape, a condition that lets a key geometric identity be proven.

• Geometry Triangulation3 Dconsistency Global Schlaefli From Geometry

  A machine-checked theorem shows that a 3D triangulation's local geometry alone guarantees a global identity, without storing extra data on each tetrahedron.

• Geometry Triangulation3 Dconsistency Global Schlaefli From Incidence

  A theorem in the framework's machine-checked library shows that matching edge data across tetrahedra is enough to prove a global geometric identity, without storing extra loca

• Geometry Triangulation3 Dconsistency Incidence Consistent

  A triangulation of space is consistent when every tetrahedron agrees with its neighbors about shared edge lengths, a condition that lets a global geometric structure be built from

• Geometry Triangulation3 Dconsistency Incidence Geometry

  How a 3D mesh of tetrahedra keeps its edges consistent, and what that consistency alone can and cannot prove.

• Geometry Triangulation3 Dconsistency Local Edge Length Eq Global Edge Length

  In a 3D triangulation, the length of an edge as seen from inside any tetrahedron equals its length as seen from the whole structure.

• Geometry Triangulation3 Dconsistency Local Sq Edge Eq Global Sq Edge

  A machine-checked theorem in the Recognition Science library proves that a tetrahedron's local edge lengths always match the global triangulation's edge lengths, a consis

• Geometry Triangulation3 Dconsistency Nonempty Triangulation Schlaefli Data Of In

  A machine-checked proof that a consistent 3D mesh always carries the local angle data needed for a global geometric identity, with no extra assumptions.