Encyclopedia/All topics/Geometry

Geometry

Articles 61–120 of 264. Alphabetical by title.

• Geometry Cofactor Polynomial Has Deriv At Cm Cofactor3 Poly Along Coord

  A theorem in a machine-checked geometry library states that each Cayley-Menger cofactor polynomial has a named derivative along any edge coordinate, a fact that makes dihedral-angl

• Geometry Deficit Linearization

  A method from 1980s Regge calculus that lets physicists treat slightly curved space as flat space plus small corrections, and what a machine-checked library proves about it.

• Geometry Deficit Linearization Deficit Linearization Cert

  A machine-checked certificate that small geometric wobbles produce no first-order energy change, a fact classical Regge calculus already knew.

• Geometry Deficit Linearization Edge Perturbation

  In Regge calculus, a small change to an edge length shifts the angles of the surrounding simplices; this page records how that shift is packaged and what it proves.

• Geometry Deficit Linearization Linear Regge Vanishes

  In a flat simplicial complex, the first-order change in the Regge action under edge-length perturbations is exactly zero, a result that makes the action quadratic at leading order.

• Geometry Deficit Linearization Linearization Coefficients

  Around a flat grid of triangles, a small nudge in edge lengths changes the angles; the linearization coefficients record exactly how much each nudge bends each corner.

• Geometry Deficit Linearization Well Shaped Data

  A machine-checked package certifies when a curved space can be treated as a flat one with small wobbles, and proves the wobbles cost energy only at second order.

• Geometry Dihedral Angle

  A dihedral angle is the angle between two planes, like the opening of a book, and it is the key to measuring curvature in a folded space.

• Geometry Dihedral Angle Cubic Lattice Deficit Zero

  A machine-checked proof that four right angles around a cube's edge sum to a full turn, the geometric fact behind flat three-dimensional space.

• Geometry Dihedral Angle Cubic Lattice Flat Sum

  Four right angles meeting at a cube's edge sum to a full turn, a fact the framework's machine-checked library records as a formal theorem.

• Geometry Dihedral Angle Deficit Eq Zero Of Flat

  In a flat space, the angles around any hinge must add up to a full circle; this is the formal proof that the leftover angle is exactly zero.

• Geometry Dihedral Angle Dihedral Angle Cert

  A dihedral angle is the angle between two faces of a solid, and a machine-checked certificate now bundles the key facts about them.

• Geometry Dihedral Angle Regular Tet Dihedral In Open Interval

  A machine-checked proof pins the dihedral angle of a regular tetrahedron to arccos(1/3), about 70.53 degrees, and confirms it lies strictly between 0 and 180 degrees.

• Geometry Dihedral Angle Regular Tet Dihedral Theta

  A regular tetrahedron, the simplest of the Platonic solids, has a dihedral angle of about 70.53 degrees, a value with a long history in geometry.

• Geometry Dihedral Cayley Menger

  A classical formula lets you compute the angle between two faces of a tetrahedron from only its six edge lengths, and a machine-checked library now proves it for the regular case.

• Geometry Dihedral Cayley Menger Dihedral Angle3 Regular Unit

  For a regular tetrahedron with unit edges, a classical formula for dihedral angles reduces to the familiar value whose cosine is one third.

• Geometry Dihedral Cayley Menger Dihedral Angle3 Regular Unit Of Cofactor Check

  For a regular tetrahedron with unit edges, a cofactor formula for dihedral angles provably yields the familiar angle with cosine 1/3.

• Geometry Dihedral Cayley Menger Dihedral Cos3 Regular Unit

  A machine-checked proof shows that a standard formula for tetrahedral angles gives the familiar value 1/3 for a regular tetrahedron.

• Geometry Dihedral Cayley Menger Dihedral Cos3 Regular Unit Of Cofactor Check

  A machine-checked proof that the Cayley-Menger cofactor formula yields the familiar 1/3 cosine for a regular tetrahedron, with no hidden assumptions.

• Geometry Dihedral Cayley Menger Opposite Cmvertices

  A small lookup table that names the two tetrahedron vertices opposite each edge, the first step in a machine-checked formula for dihedral angles.

• Geometry Dihedral Cofactor Formula

  A tetrahedron's dihedral angles are determined by its six edge lengths, and a machine-checked proof now shows the formula.

• Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge0 Diag Product Eq Sixteen De

  A machine-checked theorem ties a tetrahedron's dihedral angle to a ratio of determinants, with the number 16 as the bridge.

• Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge0 Right Diag Eq Neg Four Nor

  For a tetrahedron, a certain algebraic expression involving squared edge lengths turns out to be exactly negative four times the squared area of one face, a fact a machine-checked

• Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge1 Diag Product Eq Sixteen De

  A machine-checked proof shows that, for any tetrahedron, a certain product of cofactors equals sixteen times a squared geometric quantity; here is what that means and what it leave

• Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge2 Diag Product Eq Sixteen De

  In a tetrahedron, the product of two specific cofactor entries equals sixteen times the squared denominator of the dihedral cosine, a bridge between algebraic and geometric descrip

• Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge2 Right Diag Eq Neg Four Nor

  A machine-checked theorem links the geometric angle of a tetrahedron to an algebraic formula, with a surprising factor of four.

• Geometry Dihedral Cofactor Formula Dihedral Cos3 Sq Sq Edge Of Points Interior O

  For any non-flat tetrahedron, the cosine of a dihedral angle is never exactly 1 or -1 unless the tetrahedron is degenerate.

• Geometry Dihedral Cofactor Formula Geometric Dihedral Cos Edge0 Eq Cofactor Rati

  The cosine of the angle between two faces of a tetrahedron can be written in two very different-looking ways; this page explains the equivalence and its limits.

• Geometry Dihedral Derivatives

  The angle between two faces of a tetrahedron, and the exact rule for how that angle changes when the tetrahedron is deformed.

• Geometry Dihedral Derivatives Arccos Endpoint Hypotheses Of Interior

  A small lemma about the arccos function guarantees that a dihedral angle derivative is well-defined, provided the angle is not exactly 0 or 180 degrees.

• Geometry Dihedral Derivatives Arccos Endpoint Hypotheses Of Realized Ne Endpoint

  A machine-checked theorem states the exact conditions under which a tetrahedron's dihedral angle has a well-defined derivative, and what it leaves open.

• Geometry Dihedral Derivatives Dihedral Angle Derivative Along

  A dihedral angle is the corner angle between two faces of a tetrahedron; this page explains how that angle changes as the shape deforms, and what the framework's formal librar

• Geometry Dihedral Derivatives Has Deriv At Arccos Comp

  A theorem in the framework's machine-checked library states the ordinary calculus rule for differentiating an angle expressed as arccos of a cosine, under one condition.

• Geometry Dihedral Derivatives Has Deriv At Dihedral Angle3 Sq Along

  A machine-checked theorem spells out how a tetrahedron's dihedral angle changes as its edges stretch, and the exact conditions under which that rate exists.

• Geometry Dihedral Derivatives Has Deriv At Dihedral Angle3 Sq Explicit

  In a tetrahedron, a dihedral angle is the angle between two faces; a machine-checked theorem now gives its exact rate of change as an edge length varies.

• Geometry Dihedral Derivatives Has Deriv At Dihedral Angle3 Sq From Cofactors

  A machine-checked theorem states exactly how a tetrahedron's dihedral angle changes when its edge lengths shift, under specific non-degeneracy conditions.

• Geometry Discrete Bianchi

  A machine-checked proof shows that the discrete Bianchi identity, a key constraint in Regge calculus, is exactly the Schläfli identity from simplicial geometry.

• Geometry Discrete Bianchi Discrete Bianchi Contracted Cert

  In simplicial geometry, a local identity links the change in area of a triangle's faces to its angles, and a machine-checked library proves the discrete version of Einstein&#x

• Geometry Discrete Bianchi Discrete Bianchi Contracted Cert Inhabited

  A machine-checked proof shows that a discrete version of Einstein's equations has at least one solution: empty, flat space.

• Geometry Discrete Bianchi Discrete Bianchi Contracted From Schlafli

  In Regge calculus, the contracted Bianchi identity is a geometric fact about how the curvature of a triangulated space responds to moving a vertex.

• Geometry Discrete Bianchi Discrete Bianchi Contracted One Statement

  The contracted Bianchi identity, which makes Einstein's equations consistent, has a discrete counterpart in Regge calculus, and a machine-checked library now proves the struct

• Geometry Discrete Bianchi Discrete Bianchi Eq Schlafli

  In Regge calculus, a discrete version of Einstein's constraint is identical to a classical identity about simplicial geometry, and a machine-checked proof now records that equ

• Geometry Discrete Bianchi Flat Regge Data Schlafli

  A machine-checked proof shows that a completely flat, zero-curvature simplicial space satisfies a key identity of Regge calculus, providing a non-vacuous starting point for discret

• Geometry Discrete Bianchi Schlafli Regge Data Inhabited

  A machine-checked proof shows that at least one geometry satisfies a key identity of discrete gravity, but it does not prove the identity for all geometries.

• Geometry Freudenthal Cube Triangulation

  A cube can be cut into six identical tetrahedra; a machine-checked library verifies the bookkeeping of that cut.

• Geometry Freudenthal Cube Triangulation Cm3 Freudenthal Tet Sq Edges

  A unit cube can be cut into six identical tetrahedra; a machine-checked proof confirms each one is a genuine, non-flat solid.

• Geometry Freudenthal Cube Triangulation Edge In Tet Iff Local Edge Of

  A cube can be sliced into six tetrahedra; a machine-checked theorem guarantees that every edge in that slice has exactly one home.

• Geometry Freudenthal Cube Triangulation Edge In Tet Vertices

  A machine-checked proof that in the standard division of a cube into six tetrahedra, each edge of each tetrahedron is exactly one of the cube's 19 edges.

• Geometry Freudenthal Cube Triangulation Freudenthal Cube Edge Slot Partition

  A cube can be cut into six tetrahedra; the Freudenthal triangulation is the standard way, and its edge bookkeeping is now machine-checked.

• Geometry Freudenthal Cube Triangulation Freudenthal Cube Incidence Consistent

  A cube can be cut into six tetrahedra along one diagonal; the framework's machine-checked library proves the bookkeeping of that cut is consistent.

• Geometry Freudenthal Cube Triangulation Local Sq Edge Eq Global

  A machine-checked theorem proves that the six tetrahedra inside a unit cube agree on the lengths of the edges they share.

• Geometry Freudenthal Regge Component

  A machine-checked module shows that, for one specific eight-vertex local geometry, the second-order Regge action exactly equals a Dirichlet form, a key step toward linking geometry

• Geometry Freudenthal Regge Component Concrete M Off Diag Eq Neg Area Weight

  In a finite model of spacetime geometry, the framework proves that the second-order variation of the Regge action is exactly the negative of the geometric area weight off the diago

• Geometry Freudenthal Regge Component Concrete Regge Second Variation Eq Jcost Di

  A machine-checked proof shows that, in a specific finite model, the second-order Regge action equals a Dirichlet form, linking discrete geometry to a cost function.

• Geometry Freudenthal Regge Component Freudenthal Regge Component Cert

  A machine-checked certificate that, for one specific eight-vertex model, the second-order Regge action equals a geometric Dirichlet form; it is not a proof for all triangulations.

• Geometry Freudenthal Regge Component Has Deriv At Regular Dihedral Uniform Scale

  A machine-checked theorem confirms that uniformly scaling a regular tetrahedron leaves its dihedral angle unchanged, a small but concrete step in a larger physical framework.

• Geometry Freudenthal Regge Component Has Deriv At Regular Triangle Area

  A machine-checked theorem confirms that the standard formula for a regular triangle's area changes with side length exactly as calculus says, and it does so without any new ge

• Geometry Freudenthal Regge Component Regular Tetrahedral Dihedral Angle Eq

  A regular tetrahedron's dihedral angle is the angle whose cosine is 1/3, about 70.53 degrees, and a machine-checked library proves it.

• Geometry Freudenthal Regge Component Regular Triangle Area Nonneg

  A machine-checked proof that the standard formula for the area of an equilateral triangle can never give a negative number, and why that small fact matters for a larger geometric p

• Geometry Freudenthal Regge Component Regular Triangle Area Pos

  The area of an equilateral triangle is positive whenever its side length is positive, a fact the framework's machine-checked library proves from the standard formula.