Encyclopedia/All topics/Geometry

Geometry

Articles 1–60 of 264. Alphabetical by title.

• Geometry Affine Indep Interior

  For any nondegenerate tetrahedron, the angle between two adjacent faces is always strictly between 0 and 180 degrees, never touching the endpoints.

• Geometry Affine Indep Interior Adjacent Face Normals Independent Iff Cross Ne Ze

  A theorem in the framework's machine-checked library gives a simple cross product test for when two faces of a tetrahedron meet at a genuine angle.

• Geometry Affine Indep Interior Adjacent Face Normals Independent Of Affine Indep

  A tetrahedron's geometry guarantees that the angle between any two faces sharing an edge is never a flat 180 degrees, a fact a machine-checked proof now nails down.

• Geometry Affine Indep Interior Adjacent Face Normals Independent Of Triple Ne Ze

  In a tetrahedron, two faces meeting at an edge have normals that are never parallel, a fact the framework's machine-checked library proves from a single nonzero triple product

• Geometry Affine Indep Interior Dihedral Cos3 Sq Strict Interior Of Affine Indepe

  In a non-degenerate tetrahedron, the cosine of every dihedral angle lies strictly between -1 and 1, never reaching the endpoints.

• Geometry Affine Indep Interior Dihedral Cos3 Sq Strict Interior Of Face Normals

  A machine-checked theorem guarantees that in any non-degenerate tetrahedron, the cosine of every dihedral angle lies strictly between -1 and 1, never touching the endpoints that wo

• Geometry Affine Indep Interior Face Normal Ne Zero Of Edge Vectors Linear Indepe

  In a non-degenerate tetrahedron, two independent edge vectors guarantee that a face's normal vector is never zero, a fact that keeps dihedral angles well-defined.

• Geometry Affine Indep Interior Geometric Dihedral Cos Strict Interior Of Affine

  In a nondegenerate tetrahedron, the cosine of every dihedral angle lies strictly between -1 and 1, never touching the endpoints.

• Geometry Affine Indep Interior Geometric Dihedral Cos Strict Interior Of Face No

  A machine-checked proof shows that two independent face normals of a tetrahedron always define a dihedral angle that is neither flat nor fully open.

• Geometry Cayley Menger

  A formula from 1841 that computes a tetrahedron's volume from its edge lengths alone, and the machine-checked scaffold that Recognition Science builds on it.

• Geometry Cayley Menger Cayley Menger Cert

  A machine-checked certificate packages the classical formula that computes a tetrahedron's volume from its six edge lengths, for the regular case.

• Geometry Cayley Menger Derivatives

  A machine-checked library works out the full derivative structure of a classical geometry polynomial, giving an exact formula for how a tetrahedron's volume changes when you s

• Geometry Cayley Menger Derivatives Cm3 Cubic Single Perturb

  A single coordinate change in a tetrahedron's edge lengths produces a pure cubic term in the Cayley-Menger polynomial, a fact with a surprisingly simple proof.

• Geometry Cayley Menger Derivatives Cm3 Quadratic Single Perturb

  The Cayley-Menger determinant tells whether six lengths can form a tetrahedron; this theorem isolates how that determinant bends when just one edge changes.

• Geometry Cayley Menger Derivatives Has Deriv At Cm3 Partial0

  A machine-checked theorem pins down how the Cayley-Menger polynomial changes when exactly one edge of a tetrahedron is stretched.

• Geometry Cayley Menger Derivatives Has Deriv At Cm3 Partial1

  A machine-checked theorem proves that the Cayley-Menger polynomial, which decides whether six lengths fit a tetrahedron, has a well-defined slope when one edge length changes.

• Geometry Cayley Menger Derivatives Has Deriv At Cm3 Partial3

  A machine-checked theorem gives the exact slope of a geometric volume formula as one edge length changes, and it does not claim anything about which edge lengths form a real tetrah

• Geometry Cayley Menger Derivatives Has Deriv At Shifted Cubic

  A single-variable calculus fact about cubic polynomials, proved exactly, that is a stepping stone in a larger geometric argument.

• Geometry Cayley Menger Matrix

  A 5 by 5 table of squared edge lengths that encodes a tetrahedron's shape, and whose determinant vanishes exactly when the six lengths can form a tetrahedron.

• Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix

  A single determinant in a tetrahedron's distance matrix equals -1, a small but exact step in a machine-checked geometry library.

• Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix12

  A 4 by 4 matrix with a single changed entry has determinant negative one, a small but exact fact in the geometry of a tetrahedron.

• Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix13

  A 4 by 4 matrix cut from a tetrahedron's distance table has determinant 1, a fact that anchors a larger geometric computation.

• Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix14

  A 4 by 4 matrix from tetrahedron geometry has determinant -1, a fact the Recognition Science library proves by machine-checked calculation.

• Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix23

  A small matrix determinant inside the Cayley-Menger formula for a tetrahedron's volume turns out to be exactly -1 for a regular tetrahedron with unit edges.

• Geometry Cayley Menger Matrix Det Regular Unit Off Diag Minor Matrix24

  A 4 by 4 matrix built from a tetrahedron's edge lengths has determinant 1, a fact the framework's machine-checked library proves.

• Geometry Cayley Menger Matrix Regular Unit Vertex Diag Cofactor

  For a regular tetrahedron with unit edges, the diagonal cofactors of its Cayley-Menger matrix all equal -3, a fact that anchors the geometry of dihedral angles.

• Geometry Cayley Menger N

  A single determinant that gives the volume of any simplex, from a line segment to a tetrahedron and beyond, using only its edge lengths.

• Geometry Cayley Menger N Cm Det N

  A single determinant gives the volume of a triangle, tetrahedron, or any higher-dimensional simplex from its edge lengths alone.

• Geometry Cayley Menger N Cm Index Vertex

  A tiny function that labels the rows of a distance matrix, and the first step toward volumes in any dimension.

• Geometry Cayley Menger N Cm Matrix N

  A single matrix encodes all pairwise squared distances of an n-simplex, and its determinant yields the simplex volume in any dimension.

• Geometry Cayley Menger N Simplex Squared Distances

  A formula from 1919 that computes a simplex's volume from its edge lengths alone, now formalized for any number of dimensions.

• Geometry Cayley Menger N Simplex Volume Sq N

  A single formula gives the squared volume of a triangle, tetrahedron, or any higher-dimensional simplex from its edge lengths alone.

• Geometry Cayley Menger Polynomial

  A single polynomial in six edge lengths decides whether a tetrahedron is real and how big it is, and a machine-checked library now forces the result exactly.

• Geometry Cayley Menger Polynomial Cm3 Const Sq

  For a tetrahedron with all six edges equal, the Cayley-Menger polynomial simplifies to a single power law, a result the framework's machine-checked library proves.

• Geometry Cayley Menger Polynomial Cm3 Cont Diff

  The Cayley-Menger polynomial for a tetrahedron is infinitely differentiable; a machine-checked proof shows why later derivative computations are safe.

• Geometry Cayley Menger Polynomial Cm3 Regular Unit

  A machine-checked theorem confirms the classical formula for a tetrahedron's volume on the regular unit case, and nothing more.

• Geometry Cayley Menger Polynomial Cm3 Right Angle Unit

  A machine-checked theorem confirms the Cayley-Menger formula gives volume 1/6 for a unit right tetrahedron, a check that anchors later geometry work.

• Geometry Cayley Menger Polynomial Cm3 Scaling

  Stretch every edge of a tetrahedron by the same factor, and its volume-squared polynomial grows as the cube of that factor, a fact the framework's machine-checked library prov

• Geometry Cayley Menger Polynomial Cont Diff Eval

  A tetrahedron's volume, written as a polynomial in its edge lengths, is smooth enough for every derivative a geometer might need.

• Geometry Cayley Menger Polynomial Right Angle Unit Sq Edges

  The Cayley-Menger polynomial is a formula that decides whether six lengths can form a tetrahedron; a machine-checked proof verifies it on a right-angle unit tetrahedron.

• Geometry Cayley Menger Regular Cm Positive

  For a regular tetrahedron, the Cayley-Menger determinant is always positive, a fact that certifies the shape has a genuine volume.

• Geometry Cayley Menger Regular Cm Volume Identity

  A classic 19th-century formula, machine-checked for the regular tetrahedron, links edge lengths to volume and opens a bridge to general relativity.

• Geometry Cayley Menger Regular Not Flat

  A machine-checked proof confirms that a regular tetrahedron, with all edges equal and positive, cannot degenerate to a flat shape.

• Geometry Cayley Menger Tet Cmdata

  A tetrahedron's volume can be recovered from its six edge lengths alone; the framework's machine-checked library records that fact as a named object, not as a proved theo

• Geometry Cofactor Derivatives

  A machine-checked library now makes explicit the calculus of geometric cofactors, the building blocks that describe how a tetrahedron's shape responds to changes in its edge l

• Geometry Cofactor Derivatives Dihedral Cofactor Product Poly Ne Zero Of Non Dege

  A machine-checked theorem guarantees that a certain geometric expression, built from the cofactors of a tetrahedron's edge-length matrix, is never zero for any non-degenerate

• Geometry Cofactor Derivatives Dihedral Cofactor Product Poly Nonneg Of Non Degen

  A machine-checked theorem guarantees that a certain geometric product, built from the edges of a non-degenerate tetrahedron, is never negative.

• Geometry Cofactor Derivatives Dihedral Cofactor Product Poly Pos Of Non Degenera

  For any non-degenerate tetrahedron, a certain product of cofactor polynomials is always positive, a fact that keeps later derivative formulas well-defined.

• Geometry Cofactor Derivatives Dihedral Cos3 Sq Closed Form Deriv Eq Generic

  A machine-checked theorem in the Recognition Science library shows that a geometric derivative has a closed form matching the general quotient rule.

• Geometry Cofactor Derivatives Dihedral Denom3 Closed Deriv Value Eq Poly

  A machine-checked theorem shows that two different ways of writing the derivative of a tetrahedron's dihedral denominator are exactly the same expression.

• Geometry Cofactor Derivatives Dihedral Denom3 Poly Ne Zero Of Non Degenerate

  For any non-degenerate tetrahedron, a certain geometric denominator can never be zero, which keeps the calculus of its angles well-defined.

• Geometry Cofactor Derivatives Dihedral Denom3 Poly Pos Of Non Degenerate

  A machine-checked theorem guarantees that a certain geometric denominator, built from Cayley-Menger cofactors, is strictly positive for any non-degenerate tetrahedron.

• Geometry Cofactor Derivatives Has Deriv At Dihedral Cos3 Sq From Cofactors

  A machine-checked theorem gives the exact rate at which a squared cosine of a tetrahedron's dihedral angle changes when one edge length varies.

• Geometry Cofactor Polynomial

  A machine-checked library expands every tetrahedral Cayley-Menger cofactor into an explicit polynomial in the six squared edge lengths.

• Geometry Cofactor Polynomial Cm Cofactor3 Opposite Diag Eq Poly

  A Cayley-Menger cofactor is a determinant that encodes a tetrahedron's volume; this theorem rewrites one such cofactor as an explicit polynomial in the six squared edge length

• Geometry Cofactor Polynomial Cm Cofactor3 Poly 34 Update Polyform

  A machine-checked library rewrites every tetrahedral cofactor into a plain polynomial, so angle calculus can stop wrestling with opaque derivative terms.

• Geometry Cofactor Polynomial Cm Cofactor3 Poly Update Polyform

  A machine-checked library rewrites every tetrahedral geometry cofactor into an explicit polynomial in the six edge lengths, making derivative calculations concrete and checkable.

• Geometry Cofactor Polynomial Has Deriv At Cm Cofactor3 34 Along Coord

  A machine-checked theorem states that a certain geometric cofactor changes smoothly when one edge of a tetrahedron is stretched, and it names the exact rate of change.

• Geometry Cofactor Polynomial Has Deriv At Cm Cofactor3 Along Coord

  A machine-checked theorem turns a complicated geometric formula into a simple, named rate of change, making it safe to use in further calculations.

• Geometry Cofactor Polynomial Has Deriv At Cm Cofactor3 Poly 34 Along Coord

  This theorem gives a named, explicit formula for the rate of change of a specific cofactor of a tetrahedron's Cayley-Menger matrix as one edge length varies.