I. Relations Among the Three Symmetries

Translational symmetry corresponds to straight lines,
rotational symmetry corresponds to curves.

A straight line is a curve with zero curvature.
Therefore, translational symmetry is inherently a special case of rotational symmetry.

Reflection (folding) symmetry is the mirror inversion of curves and straight lines.

The three symmetries share the same origin, the same structure, and the same essence.

 

II. Strict Geometric Correspondence

1. Rotational Symmetry

Corresponds to: circle (circumference)
Curvature: \kappa = \dfrac{1}{R} \neq 0
Symmetry group: SO(2), the rotation group

2. Translational Symmetry

Corresponds to: straight line
Curvature: \kappa = 0
Symmetry group: \mathbb{R}, the translation group

3. Key Unification: Radius R\to\infty

As the radius of a circle R\to\infty, the circumference approaches a straight line, and the curvature:

\kappa = \frac{1}{R} \to 0

Meanwhile:

\text{rotation} \xrightarrow{R\to\infty} \text{translation}

Thus:
Translational symmetry = rotational symmetry in the limit R\to\infty (curvature \kappa\to 0).

 

III. The Unified Status of Reflection (Folding) Symmetry

Reflection (mirror symmetry) can be understood as:

- Mirror with respect to a straight line = ordinary axial symmetry
- Mirror with respect to a circle/curve = inversion, circular reflection, or symmetry in hyperbolic geometry

Unified description:

Reflection symmetry = the discrete dual of continuous symmetries (rotation/translation): inversion.

Therefore:

- Rotation: continuous, orientation-preserving, curved
- Translation: continuous, orientation-preserving, linear (special case of rotation)
- Reflection: discrete, orientation-reversing, mirror (dual of the above two)

Together they form the Euclidean plane isometry group E(2):

E(2) = SO(2) \ltimes \mathbb{R}^2

Rotation + translation + reflection = three manifestations of one and the same symmetric structure.