Glossary

Algebraic definitions and quadratic invariants referenced across the 28 principles of the GEOMETOR Wildberger codex.

centroid

The affine, metric-independent center \(G = (A_1 + A_2 + A_3)/3\) (\(X(2)\) when \(\operatorname{char} F \neq 3\)) where the three medians and all three Chromogeometry Euler lines concur.

Chromogeometry

Wildberger’s triality of planar geometries acting on the same coordinates: Blue Euclidean (\(\Delta x^2 + \Delta y^2\)), Red Minkowski (\(\Delta x^2 - \Delta y^2\)), and Green hyperbolic (\(2 \Delta x \Delta y\)).

circumcenter

circumquadrance

The unique intersection \(C\) (\(X(3)\)) of the side perpendicular bisectors of a non-null triangle, having equal quadrance \(K = Q_1 Q_2 Q_3 / \mathcal{A}\) (circumquadrance) to all three vertices.

cross

The complementary line invariant \(c(\ell_1, \ell_2) = 1 - s(\ell_1, \ell_2) = (a_1 a_2 + b_1 b_2)^2 / ((a_1^2 + b_1^2)(a_2^2 + b_2^2))\) (equal to \(\cos^2\theta\) over \(\mathbb{R}\)).

Cross Law

Law IV of Rational Trigonometry: \((Q_1 + Q_2 - Q_3)^2 = 4 Q_1 Q_2 (1 - s_3) = 4 Q_1 Q_2 c_3\), replacing the Law of Cosines without angle-sign ambiguity.

dihedral spread

The spread \(S(\Pi_1, \Pi_2) = 1 - (n_1 \cdot n_2)^2 / ((n_1 \cdot n_1)(n_2 \cdot n_2))\) between two planes in 3D space, equal to pure rationals \((8/9, 1, 8/9, 4/9, 4/5)\) across all five Platonic solids.

ellipson

The compact cubic algebraic surface inside \([0, 1]^3\) defined by the Triple Spread Formula \((s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4 s_1 s_2 s_3\).

golden ratio

divine proportion

The positive root \(\phi = (1 + \sqrt{5})/2\) of \(\phi^2 = \phi + 1\), governing Kepler’s Golden Right Triangle (W.11), pentagonal spreads (W.19), and icosahedral/dodecahedral coordinates (W.27).

grammola

The metrical conic locus \(Q(X, \ell_1) \pm Q(X, \ell_2) = R\) relative to two non-null lines.

Hansen’s problem

Peter Andreas Hansen’s 1841 geodetic problem of determining all quadrances among two known landmarks \(A, B\) and two unmeasured observer stations \(C, D\) from baseline \(Q(A, B)\) and four spreads at \(C, D\).

incenter

inquadrance

One of the four symmetric points \(I_0, I_1, I_2, I_3\) equidistant in quadrance (\(R_i\)) from all three sidelines of a triangle whose vertices are square over \(F\).

nine-point circle

The universal circle centered at \(N = (C + O)/2\) (\(X(5)\)) of quadrance \(K_9 = K / 4\) passing through the three side midpoints, three altitude feet, and three Euler midpoints.

null line

isotropic line

A line \(\langle a : b : c \rangle\) whose quadratic norm \(a^2 + b^2\) (or \(Q_M(u)\)) vanishes, occurring in Euclidean geometry over fields containing \(\sqrt{-1}\) (such as \(\mathbb{F}_{13}\)) and along light cones in Red/Green relativistic metrics.

orthocenter

The unique intersection \(O\) (\(X(4)\)) of the three altitudes of a non-null triangle.

parabola

The metrical conic locus \(Q(X, F_0) = Q(X, \ell_{\text{dir}})\) of points equidistant in quadrance from a focus \(F_0\) and a non-null directrix \(\ell_{\text{dir}}\).

Pentagonal spread

Pentagonal spreads

The two golden roots \(\alpha = (5 - \sqrt{5})/8 = (3 - \phi)/4\) and \(\beta = (5 + \sqrt{5})/8 = (2 + \phi)/4\) of \(S_5(s)/s = (16 s^2 - 20 s + 5)^2 = 0\), with exact ratio \(\beta / \alpha = \phi^2\).

quadrance

The exact quadratic separation \(Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2\) (or \((A_2 - A_1)^T M (A_2 - A_1)\) under a general symmetric bilinear form \(M\)), replacing irrational Euclidean distance over any field \(F\) (\(\operatorname{char} F \neq 2\)).

quadrea

Archimedes’ function

The symmetric quadratic polynomial \(\mathcal{A}(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2)\) of a triangle’s side quadrances, equal to \(16 \Delta^2\) over \(\mathbb{R}\).

quadrola

Wildberger’s irreducible bifocal conic \(\mathcal{A}(Q(X, F_1), Q(X, F_2), S) = 0\), unifying classical ellipses (\(S > Q(F_1, F_2)\)) and hyperbolas (\(0 < S < Q(F_1, F_2)\)) into one algebraic curve.

Quadruple Quad Formula

The symmetric degree-4 relation \(Q(a, b, c, d) = ((a+b+c+d)^2 - 2(a^2+b^2+c^2+d^2))^2 - 64 a b c d = 0\) satisfied by sequential quadrances of four collinear points.

Quadruple Spread Formula

The degree-8 self-dual polynomial relation \(R(s_{12}, s_{23}, s_{34}, s_{14}) = 0\) governing four coplanar lines, identically invariant under \(s_{ij} \leftrightarrow 1 - s_{ij}\).

ribbon

The degree-2 conic locus \(Q(X, \ell) = R\) of points having constant quadrance \(R\) to a fixed non-null line.

spread

The algebraic separation \(s(\ell_1, \ell_2) = (a_1 b_2 - a_2 b_1)^2 / ((a_1^2 + b_1^2)(a_2^2 + b_2^2))\) between two unoriented non-null lines (equal to \(\sin^2\theta\) over \(\mathbb{R}\)), vanishing when parallel and equaling \(1\) when perpendicular.

Spread Law

Law III of Rational Trigonometry: in any non-null triangle, \(s_1 / Q_1 = s_2 / Q_2 = s_3 / Q_3 = 1 / (4 K)\).

spread polynomial

spread polynomials

Wildberger’s universal integer-coefficient polynomial family \(S_n(s) \in \mathbb{Z}[s]\) satisfying \(S_0(s) = 0\), \(S_1(s) = s\), \(S_n(s) = 2(1 - 2s)S_{n-1}(s) - S_{n-2}(s) + 2s\), and semigroup composition \(S_n(S_m(s)) = S_{nm}(s)\).

Stewart’s theorem

The exact quadratic cevian identity relating \(Q(A_3, D)\) for any point \(D\) on \(A_1 A_2\) to the side quadrances \(Q_1, Q_2, Q_3\) and segment quadrances on \(A_1 A_2\).

Triple Quad Formula

Law II of Rational Trigonometry: three points with pairwise quadrances \(Q_1, Q_2, Q_3\) are collinear if and only if \(\mathcal{A}(Q_1, Q_2, Q_3) = 0\).

Triple Spread Formula

Law V of Rational Trigonometry: three coplanar non-null lines have pairwise spreads satisfying \((s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4 s_1 s_2 s_3\).

turn

The signed rational tangent \(u(\ell_1, \ell_2) = (a_1 b_2 - a_2 b_1)/(a_1 a_2 + b_1 b_2)\) between two ordered lines, whose square is the twist \(t = u^2\) and which satisfies \(u_1 + u_2 + u_3 = u_1 u_2 u_3\).

twist

The ratio \(t(\ell_1, \ell_2) = s(\ell_1, \ell_2) / c(\ell_1, \ell_2)\) for two non-perpendicular non-null lines (equal to \(\tan^2\theta\) over \(\mathbb{R}\)).

vertex bisector

square vertex

A line of equal quadrance \(Q(X, \ell_1) = Q(X, \ell_2)\) to two intersecting lines; exists over \(F\) iff the vertex is a square vertex where \((a_1^2 + b_1^2)(a_2^2 + b_2^2)\) is a square in \(F\).