Core Euclidean Results

With the parallel postulate adjoined to the neutral trunk, we are in full Euclidean geometry — the flat plane $K=0$. This page collects the results that depend on the fifth postulate (and so fail in the hyperbolic and elliptic planes) and then classifies the rigid motions of the Euclidean plane, the concrete objects that the isometry-group page later abstracts into the group $E(2)$. Its purpose in the spine is twofold: to make vivid what the parallel postulate buys you, and to introduce isometries as the bridge to the group-theoretic viewpoint.

References: Euclid I.29–48; Coxeter, Introduction to Geometry; Hartshorne, Geometry: Euclid and Beyond, ch. 4; Martin, Transformation Geometry.

1. Consequences of the Fifth Postulate

Everything here is false in the non-Euclidean planes and is therefore, by the web of equivalents, just the parallel postulate wearing different clothes.

1.1 Angle sum and parallels

• Angle sum. Every triangle has angle sum exactly $\pi$; every convex $n$-gon has angle sum $(n-2)\pi$. Defect is identically zero.
• Alternate angles (Euclid I.29). A transversal cutting two parallels makes equal alternate interior angles and equal corresponding angles. This is the first proposition in the Elements to use Postulate 5.
• Transitivity of parallelism. If $\ell \parallel m$ and $m\parallel n$ then $\ell \parallel n$ — a clean statement that quietly needs uniqueness.

1.2 Similarity

AAA Similarity. If two triangles have equal corresponding angles, their corresponding sides are proportional: $\frac{a^{\prime }}{a}=\frac{b^{\prime }}{b}=>\frac{c^{\prime }}{c}$.

Similarity is the hallmark of a scale-invariant geometry: Euclidean figures can be magnified arbitrarily with no distortion, which is exactly what the hyperbolic plane forbids (there AAA implies congruence). The whole apparatus of trigonometry — sine, cosine, tangent as ratios attached to an angle rather than to a particular triangle — rests on AAA similarity and is thus a Euclidean privilege.

1.3 The Pythagorean theorem

Pythagoras (Euclid I.47). In a right triangle with legs $a,b$ and hypotenuse $c$, $a^2+b^2=c^2$.

There are hundreds of proofs; the similarity proof (drop the altitude to the hypotenuse, obtaining two triangles each similar to the whole) exhibits the dependence on the parallel postulate most transparently, since it runs through §1.2. Its non-Euclidean deformations are instructive:

$$\text{hyperbolic:}\cosh \frac{c}{R}=\cosh \frac{a}{R}\cosh \frac{b}{R},\text{spherical:}\cos \frac{c}{R}=\cos \frac{a}{R}\cos \frac{b}{R}.$$

Expanding either to second order in the small quantity (side)$/R$ recovers $a^2+b^2=c^2$ as the flat limit $R\to \infty$, i.e. $K\to 0$. Pythagoras is the infinitesimal shadow of the law of cosines on a curved space.

1.4 Coordinates and area

The parallel postulate is what lets one lay down a global Cartesian coordinate system: choose two perpendicular axes, and the existence of rectangles (a parallel-postulate equivalent) guarantees the coordinate grid closes up consistently, giving $\mathrm{dist}(P,Q{)}^2=(x_1-x_2{)}^2+(y_1-y_2{)}^2$. This is the analytic bridge to metric geometry: the Euclidean plane is $({\mathbb{R}}^2,{\delta }_{ij})$, the constant identity metric. Areas follow — triangle $\frac{1}{2}bh$, circle $\pi r^2$, and the additivity that makes area the model for the defect functional in the curved cases.

2. Isometries of the Euclidean Plane

An isometry (rigid motion) of the plane is a bijection $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ preserving distance: $|f(P)-f(Q)|=|P-Q|$ for all $P,Q$. Isometries are exactly Euclid's "superposition" (the move Hilbert made rigorous) made into first-class objects: two figures are congruent iff some isometry carries one onto the other.

2.1 The four types

Every plane isometry is one of exactly four kinds, distinguished by whether it preserves orientation and whether it has fixed points:

2.2 The classification theorem

Classification of plane isometries. Every isometry of ${\mathbb{R}}^2$ is a translation, a rotation, a reflection, or a glide reflection. Equivalently, every isometry is a composition of at most three reflections — one or two reflections give the orientation-preserving cases (rotation/translation), and two or three give the orientation-reversing cases (reflection/glide).

Sketch. An isometry fixing three non-collinear points is the identity (a point is pinned down by its distances to three such points). Given any isometry $f$, compose it with reflections to fix three points one at a time; at most three are needed, so $f$ is a product of at most three reflections. Reading off orientation and fixed points sorts the product into the four types. $■$

Reflections generate everything. This is the structural punchline: the mirror reflection is the atom of Euclidean congruence. It generalizes verbatim to the sphere and the hyperbolic plane, where "reflect across a geodesic" again generates the full isometry group — the Cartan–Dieudonné theorem.

2.3 Analytic form and the group $E(2)$

In coordinates, every Euclidean isometry has the form

$$f(\mathbf{x})=A\mathbf{x}+\mathbf{b},A\in O(2),\mathbf{b}\in {\mathbb{R}}^2,$$

with $A$ an orthogonal matrix ($\det A=+1$ for rotations/translations, $\det A=-1$ for reflections/glides) and $\mathbf{b}$ a translation vector. These compose by

$$(A_2,{\mathbf{b}}_2)\cdot (A_1,{\mathbf{b}}_1)=(A_2{A}_1,A_2{\mathbf{b}}_1+{\mathbf{b}}_2),$$

which is the multiplication of the semidirect product

$$E(2)={\mathbb{R}}^2⋊O(2),$$

the Euclidean group of the plane. This is precisely the $K=0$ member of the family of isometry groups classified in isometry-groups.md, and its structure — a normal subgroup of translations ${\mathbb{R}}^2$ acted on by the rotations/reflections $O(2)$ — is the group-theoretic fingerprint of a flat, homogeneous, isotropic space. For the general theory of semidirect products and $O(n)$ see group-theory/README.md.

3. Looking Ahead

These results mark the boundary of the Euclidean world. Every theorem in §1 is an avatar of $K=0$; loosening the parallel postulate deforms each of them in a controlled way, indexed by the curvature. The next page tells how mathematicians came to accept that loosening as legitimate, and the hyperbolic and elliptic pages carry out the deformation explicitly. The isometry group $E(2)$ reappears, alongside its curved siblings $O(3)$ and $O(2,1)$, in isometry-groups.md.