Bilingual Technical Glossary

Algebraic Constructibility, Cyclotomy, Transformation Groups, and Transcendence

This lexicon defines 25 foundational bilingual concepts across Klein’s Famous Problems (1895/1897) and Erlangen Program (1872/1893). Every inline :term: role links directly to its entry below.

analysis situs (topology)

  • Tradition / Domain: Latin / German (Erlangen § 8.2)

  • Concise Formulation: Geometry of properties invariant under arbitrary continuous bijective deformations

Following Riemann and Leibniz, Klein characterizes topology in § 8.2 as the invariant theory of the group of all one-to-one bicontinuous distortions (Verzerrungen), preserving genus, connectivity, and sheet-branching.

cissoid of Diocles (Cissoide des Diokles)

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. V, § 3)

  • Concise Formulation: Rational nodal/cuspidal cubic curve (x^2 + y^2)x - y^2 = 0 solving cube roots

Circular cubic curve with a cusp at the origin and asymptote x = 1 passing through the circular points at infinity, invented by Diocles (c. 180 BC) to extract cube roots.

conchoid of Nicomedes (Konchoide des Nikomedes)

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. V, § 4)

  • Concise Formulation: Quartic curve (x^2 + y^2)(x - a)^2 - b^2 x^2 = 0 accomplishing neusis angle trisection

Circular quartic locus traced by laying off constant length +-b along rays from the origin beyond directrix x = a; its eight Bézout intersections with an auxiliary circle isolate the three angle-trisection roots.

conjugate values (konjugierte Werte)

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. I, § 7)

  • Concise Formulation: The 2^m algebraic values obtained by independently toggling signs of m independent square roots

In Klein Part I, Ch. I (§§ 7–14), flipping the signs of the m independent square roots entering a constructible expression produces 2^m conjugates whose symmetric product F(x) = [phi(x)]^nu is a rational power of its irreducible minimal polynomial, forcing deg(phi) = 2^h.

contact-transformation (Berührungstransformation)

  • Tradition / Domain: English / German (Erlangen § 9)

  • Concise Formulation: Transformation of surface-elements (x, y, z, p, q) preserving first-order tangency

Transformations studied by Jacobi and Sophus Lie that map points into curves or surfaces while preserving the incidence condition dz - p dx - q dy = 0 of infinitely near elements (such as projective duality and pedal/parallel-surface maps).

Delian problem (Delisches Problem)

  • Tradition / Domain: English / German (Famous Problems Intro & Part I, Ch. II, § 1)

  • Concise Formulation: Duplication of the cube: constructing x such that x^3 = 2

Ancient problem attributed to the oracle at Delos of doubling a cubical altar; algebraically equivalent to solving the irreducible cubic x^3 - 2 = 0 of degree 3 != 2^h.

domain of rationality (Rationalitätsbereich)

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. II, § 2)

  • Concise Formulation: Ground field generated by given geometric magnitudes or parameters under +, -, *, /

Kronecker’s term adopted by Klein for the field closed under rational operations generated by the given data (either Q for numerical segments or Q(lambda) / Q(cos phi, sin phi) for variable geometric constructions).

Fermat prime (Gauss’sche Primzahl 2^(2^μ) + 1)

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. III, §§ 1–4)

  • Concise Formulation: Prime number of the form p = 2^(2^μ) + 1 governing odd-prime cyclotomic constructibility

An odd prime p for which p - 1 is a power of 2, forcing p = 2^(2^mu) + 1; the five known Fermat primes correspond to mu = 0 (3), mu = 1 (5), mu = 2 (17), mu = 3 (257), and mu = 4 (65537).

Gaussian periods (Gauss’sche Perioden η, η’, η’’)

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. IV, §§ 2–6)

  • Concise Formulation: Galois-orbit sums of primitive p-th roots of unity ordered by powers of a primitive root g

For p = 17 and primitive root g = 3, partitioning the 16 roots epsilon^(3^k) into alternating subsets of 8, 4, and 2 terms produces real sums x_i (8-periods), y_i (4-periods), and z_i = 2*cos(2*pi*m/17) (2-periods) chained by four quadratic equations.

geometry of reciprocal radii (Geometrie der reciproken Radien)

  • Tradition / Domain: English / German (Erlangen § 6)

  • Concise Formulation: Conformal Möbius geometry generated by similarities and circle/sphere inversions

The geometry whose group is generated by adjoining circle or sphere inversions $P \mapsto O + R^2(P - O)/|P - O|^2$ to the Principal Group; straight lines and circles merge into a single invariant family of generalized circles while angles and 4-point circular cross-ratios remain invariant.

group-body (Körper im Sinne Dedekinds)

  • Tradition / Domain: English / German (Erlangen § 5)

  • Concise Formulation: Submanifold invariant as a whole under a prescribed transformation group

Klein’s geometric adaptation (borrowed by analogy from Dedekind’s algebraic number fields, Körper) for an aggregate of elements closed as a totality under every operation of a transformation group, such as the invariant quadric in line or sphere geometry.

height of an algebraic equation (Höhe N)

  • Tradition / Domain: English / German (Famous Problems Part II, Ch. I, § 2)

  • Concise Formulation: Positive integer $N = n - 1 + |a_0| + |a_1| + \dots + |a_n|$ well-ordering algebraic numbers

Dedekind and Cantor’s finite-bucket index grouping every irreducible integer equation of degree $n$ and coefficient magnitudes $|a_i|$ into finite sets of size $\varphi(N)$, proving that all algebraic numbers form a countable sequence.

integraph (Integraph von Abdank-Abakanowicz)

  • Tradition / Domain: English / German (Famous Problems Part II, Ch. V / Anhang)

  • Concise Formulation: Continuous-motion mechanical linkage tracing the integral curve Y(x) = ∫ y(x) dx

Precision mechanical instrument built by G. Coradi of Zürich whose sharp-edged steering wheel maintains slope dY/dx = y(x), integrating the circle x^2 + y^2 = r^2 into a transcendental curve cutting off ordinates +-pi*r^2/2.

invariant theory of a group (Invariantentheorie)

  • Tradition / Domain: English / German (Erlangen §§ 1–2, § 10)

  • Concise Formulation: Study of configurations and relations unaltered by every transformation of G

Klein’s generalization of classical algebraic invariant theory (Cayley, Sylvester, Clebsch, Gordan) from the full linear/projective group to any chosen transformation group G acting on a manifold.

Lie’s sphere geometry (Lie’sche Kugelgeometrie)

  • Tradition / Domain: English / German (Erlangen § 7)

  • Concise Formulation: Contact-transformation geometry of oriented spheres and planes (including points as zero-radius spheres)

Sophus Lie’s higher extension of reciprocal-radii geometry on a 4D quadric in P^5, preserving oriented tangential contact of spheres and corresponding via Klein’s line-to-sphere map to Plücker’s projective line geometry.

manifoldness (Mannigfaltigkeit)

  • Tradition / Domain: English / German (Erlangen § 10, Notes III–IV)

  • Concise Formulation: n-tuply extended space parameterized by coordinates (x_1, …, x_n)

Riemann’s and Grassmann’s concept of an n-dimensional space of determinations, liberated by Klein from visual spatial intuition to encompass high-dimensional configuration spaces of geometric elements.

order of a square-root expression (Ordnung eines Wurzelausdrucks)

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. I, § 2)

  • Concise Formulation: Depth mu of nested square-root extractions over rational quantities

Klein’s inductive measure of radical nesting: rational expressions have order 0, sqrt(R) with R rational has order 1, and expressions nesting square roots above order mu - 1 have order mu.

Poncelet–Steiner theorem

  • Tradition / Domain: English / German (Famous Problems Part I, Ch. IV, §§ 7–9)

  • Concise Formulation: Equivalence of ruler-and-compass constructibility with straightedge + one fixed circle with center

The theorem of Poncelet (1822) and Steiner (1833) that once a single fixed circle and its center are drawn on the plane, every quadratic (compass-and-straightedge) construction—including the regular 17-gon—can be executed with the straightedge alone.

Principal Group (Hauptgruppe)

  • Tradition / Domain: English / German (Erlangen § 1)

  • Concise Formulation: Group of spatial similarities: rigid motions, reflections, and isotropic scalings

Klein’s foundational concept in § 1 of the Erlangen Program: the totality of motions, symmetric reflections, and similarity transformations in space, characterised by leaving all ‘essential’ elementary geometric properties (angles, parallelism, and ratios of distances) unchanged.

principle of transference (Übertragungsprincip)

  • Tradition / Domain: English / German (Erlangen §§ 4–5)

  • Concise Formulation: Transfer of geometric theorems between equivalent group actions via bijective representation (Abbildung)

Generalized from Otto Hesse (1866): whenever a manifold A under group G is mapped bijectively onto A’ so that G induces group G’, the geometries (A, G) and (A’, G’) are structurally identical and every theorem transfers automatically.

quadratrix of Hippias and Dinostratus (Quadratrix)

  • Tradition / Domain: English / German (Famous Problems Part II, Ch. II, § 2)

  • Concise Formulation: Transcendental locus y/x = tan(πy/2) rectifying the circle via lim x = 2/π

Ancient kinematic curve generated by uniform rotation of a radius and uniform translation of a horizontal line; Dinostratus showed its x-intercept equals 2/pi, though reaching that limit point physically fails at y = 0.

space-element (Raumelement)

  • Tradition / Domain: English / German (Erlangen § 5)

  • Concise Formulation: Generating geometric entity (point, line, plane, sphere, or higher locus) parameterizing space

In § 5 of the Erlangen Program, Klein shows that ordinary space need not be viewed solely as a 3-manifold of points: taking straight lines (Plücker, 4D), spheres (Lie, 4D), or conics as primary elements yields higher-dimensional manifolds governed by the same transformation group.

the Absolute (das Fundamentalgebilde / Unendlichferne)

  • Tradition / Domain: English / German (Erlangen §§ 1–2, § 10.2, Note V)

  • Concise Formulation: Distinguished invariant conic or quadric defining a Cayley–Klein metric subgroup

The fixed quadric locus whose stabilizer inside the projective group yields a constant-curvature metric geometry: the imaginary spherical circle at infinity yields Euclidean similarity geometry, an imaginary non-degenerate quadric yields elliptic geometry, and a real non-degenerate quadric yields Lobachevskian hyperbolic geometry.

transcendental number (transcendente Zahl)

  • Tradition / Domain: English / German (Famous Problems Part II, Chs. I–IV)

  • Concise Formulation: Real or complex number satisfying no non-zero polynomial equation with integer coefficients

Any number outside the countable field of algebraic numbers—demonstrated by Liouville (1844), Cantor (1874/1891), Hermite for e (1873), and Lindemann for pi (1882).

transformation group (Transformationsgruppe)

  • Tradition / Domain: English / German (Erlangen § 1)

  • Concise Formulation: Closed system of space-transformations containing compositions and inverses

A family G of transformations of a manifold M such that the composition of any two transformations of G (and the inverse of each) again belongs to G; every geometry on M is defined by Klein as the theory of properties invariant under G.