Content of type 'topic'

W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads

The integer-coefficient spread polynomials S_0(s) = 0, S_1(s) = s, S_n(s) = 2(1 - 2s)S_{n-1}(s) - S_{n-2}(s) + 2s satisfy the semigroup composition law S_n(S_m(s)) = S_{nm}(s) over every commutative ring, and the fivefold polynomial S_5(s) = s(16s^2 - 20s + 5)^2 vanishes precisely at the Pentagonal Golden Spreads s = (5 -/+ sqrt(5))/8 = (3 - phi)/4 and (2 + phi)/4.

W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea

Three planar points A_1, A_2, A_3 with pairwise quadrances Q_1, Q_2, Q_3 are collinear if and only if Archimedes’ symmetric quadratic form A(Q_1, Q_2, Q_3) = (Q_1 + Q_2 + Q_3)^2 - 2(Q_1^2 + Q_2^2 + Q_3^2) vanishes; for any triangle, A(Q_1, Q_2, Q_3) equals 4 times the squared coordinate determinant (16 * Delta^2), defining the triangle’s quadrea.

W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas

For any three coplanar non-null lines l_1, l_2, l_3 with pairwise spreads s_1, s_2, s_3 and crosses c_i = 1 - s_i, the spreads satisfy the cubic 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 and the crosses satisfy (c_1 + c_2 + c_3 - 1)^2 = 4*c_1*c_2*c_3; for four lines, the Quadruple Spread polynomial R(s_12, s_23, s_34, s_14) = 0 is identically invariant under s_ij <-> 1 - s_ij.

W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle

In a non-null triangle, Q_1 = Q_2 iff s_1 = s_2 (with apex spread s_3 = (Q_3/Q_1)(1 - Q_3/(4*Q_1))), and Q_1 = Q_2 = Q_3 iff s_1 = s_2 = s_3 = 3/4; in Kepler’s Golden Right Triangle with side lengths (1, sqrt(phi), phi), the nested radical vanishes in quadrance to give Q = (1, phi, phi^2) and exact golden spreads (s_1, s_2, s_3) = (2 - phi, phi - 1, 1) = (1/phi^2, 1/phi, 1).

W.12 Vertex Bisectors, Square Vertices & The Four Incenters

The locus of points X having equal quadrance Q(X, l_1) = Q(X, l_2) to two intersecting non-null lines l_1, l_2 splits into two orthogonal vertex bisectors over F if and only if (a_1^2 + b_1^2)(a_2^2 + b_2^2) is a square in F; when two vertices of a triangle have bisectors in F, so does the third, producing four symmetric incenters I_0, I_1, I_2, I_3.

W.15 Circumcenter, Orthocenter & Universal Affine Barycentrics

Every non-null triangle A_1 A_2 A_3 over a field F (char F != 2) has a unique circumcenter C at the intersection of its side perpendicular bisectors and a unique orthocenter O at the intersection of its altitudes, with affine barycentric weights beta_i = Q_i*(Q_j + Q_k - Q_i)/(2*A) for C and gamma_i = (Q_i + Q_j - Q_k)*(Q_k + Q_i - Q_j)/(2*A) for O.

W.16 The Euler Line & Nine-Point Circle Over Arbitrary Fields

In any non-null triangle A_1 A_2 A_3 over a field F (char F != 2), the nine-point center N = (C + O)/2 satisfies 2*N - C = O (and 3*G = 2*C + O when char F != 3), placing C, G, N, O on the Euler line with quadrance proportions Q(C, G) : Q(C, N) : Q(C, O) = 4 : 9 : 36; the three side midpoints M_i, three altitude feet F_i, and three Euler midpoints E_i = (A_i + O)/2 all have exact quadrance K / 4 from N.

W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points

On a circle Q(X, O_c) = K over F, a diameter subtends spread 1 at every other circle point (Thales), any chord P_1 P_2 subtends constant spread s_3 at the circumference with Q(P_1, P_2) = 4*K*s_3 and central spread s(O_c P_1, O_c P_2) = S_2(s_3) = 4*s_3*(1 - s_3), and any external or internal point A_0 has invariant secant quadrance product Q(A_0, P_1)*Q(A_0, P_2) = (Q(A_0, O_c) - K)^2.

W.18 Cyclic Quadrilaterals, Ptolemy & Euler’s Four-Point Relation

Four non-collinear points A_1, A_2, A_3, A_4 on a circle c satisfy equal opposite spreads s(A_1 A_2, A_1 A_4) = s(A_3 A_2, A_3 A_4), Ptolemy’s Triple-Quad Law A(Q_12*Q_34, Q_23*Q_14, Q_13*Q_24) = 0 among product quadrances, and the cyclic quadrea / circumquadrance formulas, while four collinear points satisfy Q(Q_12, Q_23, Q_34, Q_14) = 0.

W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons

A regular n-star of lines has consecutive spread s satisfying S_n(s) = 0; for n = 5, a regular pentagon P_0..P_4 has side-to-diagonal spread alpha = (5 - sqrt(5))/8 = (3 - phi)/4, exterior/star spread beta = (5 + sqrt(5))/8 = (2 + phi)/4, and exact golden quadrance ratio Q_diag / Q_side = beta / alpha = phi^2 = phi + 1, existing rationally in F_p^2 whenever 5 and -1/5-dependent discriminants split (e.g. regular 5-stars in F_19).

W.20 Metrical Conics: Circles, Ribbons, Parabolas, Quadrolas & Grammolas

A conic is the zero locus of a quadratic polynomial proportion C = <a:b:c:d:e:f>; metrically, Q(X, l) = R defines a ribbon, Q(X, F_0) = Q(X, l_dir) defines a parabola, the squared-sum/difference bifocal locus A(Q(X, F_1), Q(X, F_2), S) = 0 defines a quadrola (unifying ellipses and hyperbolas into one irreducible algebraic curve), and Q(X, l_1) +/- Q(X, l_2) = R defines a grammola.

W.21 Taylor Conics, Algebraic Tangents & Optical Reflection Laws

Translating any conic f(x, y) = 0 to a point A = [x_0, y_0] on the conic isolates a unique linear term l_1(x - x_0, y - y_0) = 0 defining its algebraic tangent t_tan without limits; for a parabola, t_tan is the perpendicular bisector of focus F_0 and directrix foot D_0 (giving equal spreads with F_0 A and the axis), and for a quadrola, t_tan makes equal spreads with both focal lines F_1 P_q and F_2 P_q.

W.22 Projective Cross-Ratio, Harmonic Pencils & The Pappus Bridge

If four concurrent lines r_1, r_2, r_3, r_4 through O_p meet a transversal l_a at collinear points A_1, A_2, A_3, A_4 and another transversal l_b at B_1, B_2, B_3, B_4, then the squared cross-ratio in quadrances (Q(A_1, A_3)*Q(A_2, A_4)) / (Q(A_1, A_4)*Q(A_2, A_3)) equals the pencil spread cross-ratio (s_13 * s_24) / (s_14 * s_23) and is invariant across all transversals.

W.23 Chromogeometry & Universal Symmetric Bilinear Forms

Replacing the Euclidean quadratic form with any non-degenerate symmetric bilinear form M on F^2 preserves all Five Laws of Rational Trigonometry; in Chromogeometry—pairing Blue (dx^2 + dy^2), Red (dx^2 - dy^2), and Green (2*dx*dy) metrics on the same triangle—the three Euler lines concur at the centroid G, C_B C_R O_B C_G is a parallelogram, and N_B = (C_R + C_G)/2.

W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem

In Snellius-Pothenot resection, an observer P measuring spreads to three known beacons A, B, C lies on the algebraic intersection of two subtended-spread conics; in Hansen’s problem, knowing one baseline quadrance Q(A, B) and four sighting spreads at unobserved stations C, D determines every side and diagonal quadrance exactly via Triple Spread Systems and Alternate Spreads.

W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity

A Galilean projectile launched from O with initial speed-squared v^2 and elevation spread s traces a parabola whose apex height h = v^2*s/(2*g), horizontal range quadrance Q_range = 16*(v^4/(4*g^2))*s*(1 - s) = (v^4/g^2)*S_2(s), and focus-directrix geometry depend rationally on s; Snell’s refraction law becomes n_1^2 * s_1 = n_2^2 * s_2 in spreads.

W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads

In 3D space over F, quadrance is Q(A_1, A_2) = dx^2 + dy^2 + dz^2, a right-corner tetrahedron satisfies De Gua’s Quadrea Theorem A_base = A_xy + A_yz + A_zx, and the dihedral (face) spreads of ALL FIVE regular Platonic solids—Tetrahedron (8/9), Cube (1), Octahedron (8/9), Icosahedron (4/9), and Dodecahedron (4/5)—are exact rational numbers in Q!

W.28 Rational Polar Coordinates, Classical Curves & The Ellipson

Replacing polar coordinates (r, theta) with rational polar coordinates [s, Q] where s = x^2 / (x^2 + y^2) (spread to the y-axis) and Q = x^2 + y^2 converts classical polar curves into exact polynomial relations—Bernoulli’s lemniscate s = (Q + 2)/4 (or Q = a*(2*s - 1)), Descartes’ folium (2*s - 1)^2*(s^2 - s + 1)^2*Q^2 - 18*s*(1 - s)*(1 - 3*s + 3*s^2)*Q + 81*s^2*(1 - s)^2 = 0, and the deltoid/ellipson S_3(s) law—while (s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4*s_1*s_2*s_3 cuts out the tetrahedral Ellipson.