Encyclopedia/All topics/Geometry

Geometry

Articles 121–180 of 264. Alphabetical by title.

• Geometry Freudenthal Two Cube Strip

  A Freudenthal triangulation splits a cube into six tetrahedra; joining two cubes tests whether the pieces fit together cleanly at every shared edge.

• Geometry Freudenthal Two Cube Strip Global Sq Edge

  A machine-checked library proves that two cubes glued face to face can be split into tetrahedra with a single, consistent numbering of every shared edge.

• Geometry Freudenthal Two Cube Strip Local Edge Complete

  A machine-checked proof that in a strip of two cubes, every one of the 72 local edge positions in the 12 tetrahedra is occupied by a real, global edge.

• Geometry Freudenthal Two Cube Strip Two Cube Strip Edge Slot Bookkeeping

  Two cubes sharing a face, each cut into six tetrahedra, produce a test case for how a discrete geometry keeps track of its edges.

• Geometry Freudenthal Two Cube Strip Two Cube Strip Edge Slot Partition

  A pair of cubes sharing a face, each sliced into six tetrahedra, gives the smallest test of whether a mesh can keep consistent track of its own edges.

• Geometry Freudenthal Two Cube Strip Two Cube Strip Incidence Consistent

  Two cubes glued face to face, each split into six tetrahedra, form a test object that proves how shared edges stay consistent across the seam.

• Geometry Gram Cayley Menger

  Two classical formulas, one from dot products and one from edge lengths, both compute the same tetrahedron volume; a machine-checked proof shows they agree.

• Geometry Gram Cayley Menger Cm3 Sq Edge Of Points Eq 8 Det Gram

  A machine-checked theorem ties a tetrahedron's squared edge lengths to the determinant of its Gram matrix, linking two classical ways to compute its volume.

• Geometry Gram Cayley Menger Cm3 Sq Edges From Gram Eq 8 Det

  A machine-checked identity links the squared edge lengths of a tetrahedron to a single determinant, a bridge between two classical formulas.

• Geometry Gram Cayley Menger Gram Cayley Menger Det Target Equiv

  A machine-checked theorem shows that two classical formulas for a tetrahedron's volume, one from edge lengths and one from a Gram matrix, always agree.

• Geometry Gram Cayley Menger Gram Cayley Menger Realized

  For any tetrahedron built from actual points in ordinary space, two classical volume formulas, one based on edge lengths and one on a Gram matrix, always agree.

• Geometry Gram Cayley Menger Sq Dist Eq Base Gram

  A single theorem in the machine-checked library restates the law of cosines in a form that anchors tetrahedron geometry to a Gram matrix.

• Geometry Gram Cayley Menger Sq Edge Of Points Eq Sq Edges From Gram

  A theorem shows that the six squared edge lengths of a tetrahedron are fully determined by a 3 by 3 matrix of inner products, and that the two classic volume formulas agree.

• Geometry Periodic Freudenthal Torus

  A torus is a shape like the surface of a donut, and a periodic Freudenthal torus is a way of filling that shape with a repeating pattern of tetrahedra, the three-dimensional analog

• Geometry Periodic Freudenthal Torus Canonical Edge In Tet Eq Some Implies

  In a periodic tetrahedral grid, a single theorem guarantees that when a global edge is found inside a tetrahedron, the identification is exact and unambiguous.

• Geometry Periodic Freudenthal Torus Canonical Edge Slot Eq Some Of No Dup

  In a periodic tetrahedral mesh, each edge belongs to exactly one slot in each tetrahedron, and a machine-checked proof guarantees the bookkeeping never double-assigns.

• Geometry Periodic Freudenthal Torus Canonical Encoded Periodic K Tet Verts Eq

  A machine-checked library proves any finite encoding of a periodic Freudenthal torus has the edge structure needed for a key physics theorem.

• Geometry Periodic Freudenthal Torus Canonical Encoded Periodic Tet Equiv Eq

  A machine-checked library proves that every finite periodic tetrahedral mesh can be indexed by a simple typed description, a bridge between abstract geometry and concrete computati

• Geometry Periodic Freudenthal Torus Canonical Encoded Periodic Tet Verts Add Ver

  A periodic tetrahedral mesh can be encoded by a few bits per vertex, and the Recognition Science library proves that encoding is unique.

• Geometry Periodic Freudenthal Torus Freudenthal Tet Sq Edge Eq Periodic Disp Sq

  A machine-checked proof shows that a periodic tetrahedral mesh has only seven possible edge lengths, a fact that anchors a larger geometric framework.

• Geometry Periodic Freudenthal Torus Local Edge Of Endpoints Match Tet Verts

  In a periodic tetrahedral mesh, the theorem guarantees that every edge of every tetrahedron is recorded with its two true endpoints, so the mesh's geometry and its bookkeeping

• Geometry Realisability Cone

  A tetrahedron exists only when its six squared edge lengths satisfy two inequalities; this cone collects exactly those length combinations.

• Geometry Realisability Cone Realisable Tet Cone

  A tetrahedron's six edge lengths must satisfy a precise inequality to fit in ordinary space; this declaration names the open region where they do.

• Geometry Realisability Cone Regular Unit Mem Realisable Tet Cone

  The regular unit tetrahedron, with all six edges of length 1, passes the first geometric test for being a real tetrahedron in Euclidean space.

• Geometry Realisability Cone Right Angle Unit Mem Realisable Tet Cone

  A tetrahedron with three mutually perpendicular edges of length 1 is a real geometric object, and the framework's machine-checked library proves it.

• Geometry Regge Action Concrete

  A machine-checked proof shows that a discrete model of curved space has a well-defined second-order approximation, a key step toward a theory of quantum geometry.

• Geometry Regge Action Concrete Canonical Dirichlet Equals Edge Stencil Of Sum Co

  In a triangulated space, the energy of a field can be written as a sum over vertices or a sum over edges; a machine-checked proof shows when these two descriptions coincide.

• Geometry Regge Action Concrete Canonical Edge Pair Weight Reindex Of No Self Loo

  A machine-checked lemma shows that a certain sum over vertex pairs collapses to a single edge term, but only when the triangulation has no self-loops.

• Geometry Regge Action Concrete Canonical Edge Stencil Dirichlet Energy Nonneg

  A discrete geometry construction used in numerical relativity shows why a certain measure of deformation energy can never be negative, and what that does not imply.

• Geometry Regge Action Concrete Canonical Regge Hessian Off Diag Eq Neg Weight

  In a discrete model of spacetime, the second derivative of the action between two different points is always the negative of a certain geometric weight.

• Geometry Regge Action Concrete Canonical Regge Hessian Quadratic Eq Dirichlet

  In a triangulated space, the second-order change in a geometric action equals a simple sum of squared edge differences, a fact the framework's machine-checked library proves.

• Geometry Regge Action Concrete Canonical Regge Hessian Quadratic Expanded

  A machine-checked identity rewrites the second variation of a discrete gravity action as a familiar quadratic form, opening the way to stability analysis.

• Geometry Regge Action Concrete Canonical Regge Hessian Quadratic Nonneg

  A machine-checked proof shows that a standard discrete model of spacetime geometry is stable against small perturbations, a key step toward a concrete theory of quantum gravity.

• Geometry Regge Action Concrete Regge Action Second Order Second Variation

  For a triangulated 3D space, the second variation of the Regge action takes a simple quadratic form, and the framework proves it exactly.

• Geometry Regge Action Cubic Taylor Bound

  A machine-checked proof that the error in a discrete approximation to gravity shrinks at least as fast as the cube of the perturbation, a key step toward showing the approximation

• Geometry Regge Action Cubic Taylor Bound Canonical Remainder Iterated Fderiv3 Lo

  A machine-checked theorem shows that near a flat configuration, the error term of the nonlinear Regge action is locally controlled by the cube of the perturbation size.

• Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Cont Diff At Z

  A machine-checked theorem shows that the error left over when a curved space is approximated by flat pieces behaves smoothly near the flat configuration, a technical step toward a

• Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Has Deriv At Z

  A machine-checked theorem shows that the leftover error in a discrete gravity action vanishes at least as fast as the cube of a small perturbation.

• Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Quadratic Tayl

  A machine-checked theorem shows that for a flat geometry, the leftover error in a quadratic approximation to the Regge action grows no faster than the cube of the perturbation.

• Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Taylor Data Of

  How a machine-checked proof shows the leftover error in a discrete gravity action shrinks at least as fast as the cube of a small perturbation.

• Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Third Deriv Bo

  A machine-checked theorem in the Recognition Science framework shows that a cubic error bound for a geometric action follows from two simpler analytic conditions.

• Geometry Regge Action Cubic Taylor Bound Iterated Deriv Within One Canonical Rem

  A machine-checked theorem bounds the error when a curved space is approximated by flat pieces, and it does so with a third-power estimate that makes the approximation's accura

• Geometry Regge Action Cubic Taylor Bound Regge Action Remainder Second Variation

  A Taylor bound that controls how much a curved space's action can deviate from its quadratic approximation, and the precise conditions under which that control holds.

• Geometry Regge Action First Variation

  The Regge action, a discrete stand-in for Einstein's equations, has a flat-space critical point that the framework's machine-checked library proves by cancellation.

• Geometry Regge Action First Variation Conformal Schlaefli Cancellation Of Length

  A machine-checked proof shows that a certain geometric action has no first-order change at flat space, a key consistency test for a theory of discrete geometry.

• Geometry Regge Action First Variation Conformal Schlaefli Incidence Bookkeeping

  A machine-checked theorem shows that a certain bookkeeping structure for tracking edges in a triangulation is enough to make a local geometric cancellation identity hold.

• Geometry Regge Action First Variation Directional Critical Of First Variation Fo

  In Regge calculus, a discrete model of spacetime, the flat geometry is a stationary point of the action: the first tiny change in any direction leaves the total action unchanged.

• Geometry Regge Action First Variation Directional First Variation Formula Of Def

  In Regge calculus, the first variation of the action vanishes at a flat geometry; a machine-checked theorem records the precise analytic condition.

• Geometry Regge Action First Variation Local Angle Length Chain Deriv Eq Sq Edge

  In a curved three-dimensional space built from flat tetrahedra, a machine-checked theorem proves that two different ways of measuring how angles respond to a deformation always agr

• Geometry Regge Action First Variation Local Deficit Angle Contribution Has Deriv

  In a triangulated space, the rate of change of the angle deficit around an edge is exactly the sum of the rates of change of the dihedral angles of the tetrahedra that meet there.

• Geometry Regge Action First Variation Regge Action Critical At Zero Of First Var

  In discrete geometry, the Regge action measures curvature concentrated along edges; this theorem states that at a perfectly flat configuration, that action is stationary.

• Geometry Regge Action Nonlinear Correspondence

  In a triangulated space, the full Regge action matches a simple quadratic energy near flatness, with an error that shrinks like the cube of the disturbance.

• Geometry Regge Action Nonlinear Correspondence Canonical Jquadratic Term Eq Diri

  A machine-checked theorem shows that, near flat space, the full Regge action of discrete gravity matches a simple quadratic energy, with the error controlled by a cubic remainder.

• Geometry Regge Action Nonlinear Correspondence Nonlinear Regge Local Corresponde

  A machine-checked theorem shows that a discrete gravity action and a cost-based action agree near flat space, up to a controlled error.

• Geometry Regge Action Nonlinear Correspondence Strongest True Regge Jcost Replac

  A machine-checked theorem shows that near flat space, the full nonlinear Regge action matches a simple quadratic energy up to a controlled cubic error, without claiming the two are

• Geometry Regge Action Nonlinear Hessian Proof

  A machine-checked proof shows that the full nonlinear Regge action has the same second variation at flat space as its standard quadratic approximation.

• Geometry Regge Action Nonlinear Hessian Proof Action Derivative Tangency To Quad

  A machine-checked proof establishes that a complex geometric action behaves like a simple quadratic form near flat space, a key step in a larger calculation.

• Geometry Regge Action Nonlinear Hessian Proof Canonical Remainder Derivative Ide

  A machine-checked theorem shows that a certain error term in a discrete gravity action has a derivative that vanishes at the flat configuration, a step toward reducing the nonlinea

• Geometry Regge Action Nonlinear Hessian Proof Canonical Remainder Line Different

  A machine-checked theorem shows that if the nonlinear Regge action is smooth along every line through the flat configuration, then the remainder term that isolates its curvature is

• Geometry Regge Action Nonlinear Hessian Proof Canonical Remainder Second Variati

  A machine-checked proof shows that near a flat configuration, the nonlinear Regge action and its quadratic approximation agree to second order, with the remainder vanishing.