Morphological & Geometric Glossary

Technical Terminology of Growth, Capillarity, Phyllotaxis & Dynamic Symmetry

Affine Shear (Oblique Coordinates) (English / Mathematical · x’ = x + y cot ω, y’ = y sin ω)

A linear coordinate transformation tilting rectangular axes through an oblique angle ω while preserving parallelism and collinearity ratios—exemplified by Thompson’s 70° deformation of Argyropelecus Olfersi into Sternoptyx diaphana (Ch. XVII, Figs. 373–374).

Catenoid (English / French (Caténoïde) · Minimal Surface of Revolution)

The surface generated by revolving a catenary around its directrix, shown by Plateau to have equal and opposite principal curvatures everywhere (zero mean curvature, 1/R₁ + 1/R₂ = 0), realized by an open soap film stretched between two coaxial wire rings (Ch. V, pp. 214–222).

Coaxial / Curvilinear Transformation (English / Projective-Conformal · Orthogonal Circular-Hyperbolic Coordinate Net)

Thompson’s non-linear coordinate deformation mapping one family of Cartesian lines into nested coaxial circles and the perpendicular family into orthogonal hyperbolae or diverging rays, transforming the porcupine-fish Diodon into the oceanic sunfish Orthagoriscus mola (Ch. XVII, Figs. 381–382).

δυνάμει σύμμετροι (dynamei symmetroi) (Greek (Euclid Book X, Def. 2) · Commensurable in Square / Commensurable in Power)

Magnitudes whose linear lengths are irrational with respect to one another (such as 1 and √2, √3, √5) yet whose squares (quadrances) have exact integer ratios (1 : 2 : 3 : 4 : 5). Identified by Jay Hambidge (1919–1920) as the exact classical origin of ‘Dynamic Symmetry’.

Divergence Angle (Phyllotactic) (English / Botanical · Δθ = 360° / φ² ≈ 137° 30′ 28″)

The azimuthal angular step around the stem axis between consecutively initiated leaves (n and n+1) along the genetic spiral. Convergents of the Schimper–Braun fractions approach the golden angle 2π(2 - φ) = 2π/φ².

Dynamic Symmetry (English (Jay Hambidge, 1919) · Active Proportional Symmetry Commensurable in Square)

Proportional architecture derived from organic growth (phyllotaxis and equiangular shells) and Euclidean commensurability in square, built upon root-rectangles (√2, √3, √5) and the whirling-square golden rectangle (φ = 1.6180…) whose diagonals and reciprocals subdivide areas into self-similar gnomonic parts.

Equiangular / Logarithmic Spiral (English / Latin (Spira mirabilis) · r = a e^(θ cot α))

The planar growth curve studied by Descartes, Bernoulli, Moseley, and Thompson (Ch. XI) whose tangent at every point makes a constant angle α with the radius vector OP, so that continuous terminal accretion increases size without altering geometric form.

Fermat–Torricelli–Steiner Point X(13) (English / Triangle Geometry · 120° Minimal-Network Center)

For a triangle with all angles less than 120°, the unique interior point P = X(13) from which all three sides subtend exact 120° angles (rational spread s = 3/4), minimizing the sum PA + PB + PC—physically realized instantaneously by Plateau soap films (Ch. VII).

Genetic Spiral (Grundspirale) (English / German · Ontogenetic Leaf-Initiation Helix)

The single theoretical helix connecting all leaves on a stem in strict chronological order of emergence (0, 1, 2, 3, …), contrasted by Bravais, Tait, Church, and Thompson with the visually dominant secondary parastichies (Ch. XIV).

γνώμων (gnōmōn) (Greek · Gnomon / Self-Similar Growth Increment)

Originally the carpenter’s L-square surrounding a square to make a larger square (Aristotle, Categories 15a30; Euclid II Def. 2), generalized by Hero of Alexandria (Def. 58) and D’Arcy Thompson (Ch. XI, p. 509) to any figure whose addition to a given figure leaves the composite figure geometrically similar to the original.

Golden / Sublime Triangle (Euclid IV.10) (English / Euclidean · 36°–72°–72° Isosceles Triangle)

The isosceles triangle constructed in Euclid Elements IV.10 whose base angles are double the vertical angle (72°, 72°, 36°) and whose leg-to-base ratio is φ. Bisecting a base angle partitions it into a similar 36°–72°–72° triangle and an obtuse 108°–36°–36° gnomon (Ch. XI, Fig. 254).

Maraldi / Tetrahedral Angle (English · arccos(-1/3) = 109° 28′ 16.39″ (spread s = 8/9))

The exact central angle of a regular tetrahedron, equal both to the angle at which four Plateau soap-film edges meet in 3D foam equilibrium and to the obtuse planar angle of the three rhombi capping the honeybee’s hexagonal cell (measured by Maraldi in 1712 and proved synthetic-geometrically by Maclaurin in 1743; Ch. VII).

Mean Curvature (Laplace–Young Law) (English / Differential Geometry · H = ½(1/R₁ + 1/R₂), ΔP = T(1/R₁ + 1/R₂))

Half the sum of the two orthogonal principal curvatures of a surface. Under uniform surface tension T and constant transmembrane pressure ΔP, fluid cells and droplets assume Delaunay surfaces of constant mean curvature (Ch. V, pp. 209–226).

Method of Coordinates (Theory of Transformations) (English · x’ = f(x, y), y’ = g(x, y))

Thompson’s crowning morphological principle in Chapter XVII: inscribing an organism’s outline in a Cartesian coordinate system and subjecting the entire coordinate plane to a smooth global deformation so that related genera and species emerge as holistic transformations of a single law of growth.

Nodoid (English / French (Nodoïde) · Self-Intersecting Constant-Mean-Curvature Roulette)

The surface of revolution generated by the focus of a hyperbola rolling along an axis, completing Delaunay’s and Plateau’s classification of axisymmetric constant-mean-curvature soap-film surfaces alongside the sphere, cylinder, catenoid, and unduloid (Ch. V, pp. 216–224).

Orthogonal Principal Stress Trajectories (English / Graphic Statics · Lines of Principal Compression & Tension)

Two mutually perpendicular families of curves inside an elastic loaded body along which shear stress vanishes completely, leaving pure compression along one family and pure tension along the other—realized in cancellous bone by Meyer’s trabeculae and Culmann’s crane (Ch. XVI, Fig. 335).

Parastichy (Contact Spiral) (English / Greek (παραστιχίς) · Secondary Intersecting Phyllotactic Helix (m, n))

One of the conspicuous counter-winding families of steep helices or logarithmic spirals connecting physically adjacent scales, florets, or seeds (typically consecutive Fibonacci pairs such as (5, 8), (8, 13), (21, 34), or (34, 55); Ch. XIV).

Phyllotaxis (English / Neo-Latin · Leaf-Arrangement)

The geometric arrangement of leaves on a plant stem, scales on a pine cone, or florets in a composite capitulum, governed by cylindrical or apical packing lattices and Fibonacci parastichy numbers (Ch. XIV).

Plateau’s Laws of Foam Equilibrium (English · 120° Films & 109° 28′ 16″ Tetrahedral Vertices)

The two universal local rules governing minimal-area liquid films and biological cell-aggregates (Ch. VII): (1) three smooth film surfaces meet along a liquid edge at exact 120° dihedral angles; (2) four such edges meet at a point at the tetrahedral Maraldi angle arccos(-1/3) ≈ 109° 28′ 16″.

Quadrance (English (Rational Trigonometry) · Q(A, B) = (x₂ - x₁)² + (y₂ - y₁)²)

Squared Euclidean distance—corresponding directly to Greek dynamis (‘square/power’, Euclid Book X). In Hambidge’s root rectangles (1 × √n), the linear sides are irrational for n = 2, 3, 5, whereas their side quadrances are the consecutive integers Q = 1, 2, 3, 4, 5.

Reciprocal Rectangle (English (Dynamic Symmetry) · Similar Sub-Rectangle Cutting the Whole Diagonal at 90°)

In Hambidge’s Dynamic Symmetry, a rectangle of aspect ratio 1/r cut off across the end of a parent rectangle of aspect ratio r; its diagonal intersects the diagonal of the whole rectangle at an exact right angle (90°), generating recursive harmonic subdivisions. For a √n rectangle, the reciprocal equals 1/n of the parent area.

Rhombic Dodecahedron (English / Polyhedral Geometry · 12-Rhombus Space-Filling Dual of the Cuboctahedron)

The convex zonohedron discovered by Kepler (1611) bounded by twelve congruent rhombi with diagonals in ratio 1 : √2 (obtuse angle 109° 28′ 16″, acute angle 70° 31′ 44″). Three of its faces form the pyramidal base of the honeybee’s cell (Ch. VII).

Root Rectangle (√2, √3, √4, √5) (English (Dynamic Symmetry) · Rectangle of End 1 and Side √n)

A family of rectangles unfolded constructively from the unit square by swinging each rectangle’s diagonal down to the baseline with a compass: the diagonal of the unit square yields √2, the diagonal of √2 yields √3, √3 yields √4 = 2 (two squares), and 2 yields √5 (parent of the whirling-square φ rectangle).

Principle of Similitude (English · Dimensional Scaling of Similar Figures)

The fundamental law of Chapter II (after Euclid VI, Galileo, and Newton): in geometrically similar bodies scaled by linear factor L, areas (surface tension, muscle cross-section, wing area, intestinal absorption) vary as L², whereas volumes and weights (gravity, inertia, metabolic mass) vary as L³.

Spicule (Sponge & Radiolarian) (English / Latin (Spiculum) · Calcareous or Siliceous Skeletal Needle / Triaxon / Tetraxon)

Mineralized skeletal elements secreted along intercellular boundaries in sponges, holothurians, and Radiolaria; shown by Thompson (Ch. IX) to adopt triradiate (120°) and tetraradiate (109° 28′) forms molded by Plateau’s intercellular capillary Triple/Quadruple junctions rather than crystalline habit alone.

Spread (English (Rational Trigonometry) · s = sin² θ)

Exact algebraic measure of line separation in Wildberger’s Rational Trigonometry. Both universal Plateau angles highlighted by Thompson possess pure rational spreads over Q: the 120° planar soap-film angle has spread s = 3/4, and the 109° 28′ 16″ 3D tetrahedral Maraldi angle has spread s = 8/9.

Square-Cube Law (English · S ∝ L², V ∝ L³, S/V ∝ L⁻¹)

Galileo’s mechanical consequence of geometric similitude (1638): enlarging an organism isometrically causes weight to outstrip bone and muscle strength as L, while shrinking to microscopic cellular dimensions causes surface forces (capillarity, viscosity, Brownian motion) to overwhelm gravity (Ch. II).

Static Symmetry (English (Jay Hambidge, 1919) · Radial / Rational-Grid Symmetry)

In Hambidge’s classification, symmetry based on fixed linear commensurability, radial repetition, crystals, regular polygons (equilateral triangle, square, regular hexagon), and rational grid multiples—contrasted with the √5 and φ Dynamic Symmetry of growing shells, leaves, and classical Greek art.

Surface Tension (Capillarity) (English · Interfacial Free Energy per Unit Area)

The molecular contractile tension along fluid interfaces which acts to minimize surface area for given enclosed volumes, governing the forms of solitary cells, ciliates, foraminiferal chambers, soap froths, and cellular tissues throughout Chapters V–VIII.

Tetrakaidecahedron (Kelvin Cell) (English / Greek (14-hedron) · Space-Filling Truncated Octahedron (8 Hexagons + 6 Squares))

Lord Kelvin’s minimal-partitional space-filling polyhedron (1887) having 14 faces, 24 vertices, and 36 edges; identified by Thompson (Ch. VII, pp. 338–345) as the idealized three-dimensional form of unspecialized parenchymatous plant cells and soap froth.

Trabeculae (Substantia Spongiosa) (Latin / Anatomical · Cancellous Bone Struts & Lamellae)

The delicate intersecting bony plates inside the epiphyses of vertebrate bones (such as the head of the human femur), which continually remodel under mechanical load (Wolff’s / Thompson’s functional adaptation) to align along Culmann’s orthogonal compression and tension trajectories (Ch. XVI).

Unduloid (English / French (Onduloïde) · Wave-Profiled Constant-Mean-Curvature Roulette)

Delaunay’s axisymmetric surface of constant mean curvature generated by the focus of a rolling ellipse, invoked by Thompson in Chapter V (pp. 216–248) to explain beaded protoplasmic threads, Lagena shells, and peristaltic ciliate forms.

Whirling-Square Rectangle (Golden Rectangle) (English (Hambidge & Thompson) · 1 : φ = 1 : (1 + √5)/2 ≈ 1 : 1.618034)

The rectangle whose reciprocal is obtained by subtracting a unit square (gnomon) from its end, leaving a smaller golden rectangle turned by 90°; repeating the square gnomon removal indefinitely spirals (‘whirls’) around the pole O of an inscribed equiangular spiral (Ch. XI, Fig. 252; Hambidge 1920).