Self-similar figures (such as fractals) primarily exhibit scale symmetry
Self-similar figures (such as fractals) primarily exhibit scale symmetry ((also called dilation symmetry or scaling symmetry), which is a type of similarity symmetry.
Here’s a detailed elaboration:
---
1. What scale symmetry means
In scale symmetry, an object looks the same (or statistically the same) when magnified or shrunk by certain factors.
Mathematically, if you scale the figure by a factor \( r \), the result is congruent or similar to the original — but unlike translational or rotational symmetry, the “copy” is a different size.
---
2. Discrete vs. continuous scale symmetry
- **Discrete scale symmetry**: Invariant under scaling by specific factors only (e.g., the Sierpiński triangle is invariant under scaling by factor \( \frac{1}{2} \), meaning if you zoom in by factor 2 on certain points, you see the same pattern).
- Continuous scale symmetry: Invariant under scaling by any factor (e.g., a perfect straight line or a logarithmic spiral’s shape is preserved under any scaling factor about its center, though position may shift).
---
1. Relationship to other symmetries
Scale symmetry is separate from the familiar Euclidean symmetries (rotation, reflection, translation) because it involves changing scale.
However, many fractals combine scale symmetry with other symmetries. For example:
- The Sierpiński triangle has rotational symmetry (order 3) **and** scale symmetry.
- The Koch snowflake has both 6-fold rotational symmetry and discrete scale symmetry.
---
4. Statistical self-similarity
Many natural fractals (coastlines, clouds) have statistical scale symmetry: when scaled, their statistical properties (e.g., roughness, density fluctuations) remain the same, even if the exact pattern is not identical.
---
5. Invariance under dilation
In group-theoretic terms, scale symmetry corresponds to invariance under the dilation transformation \( x \mapsto \lambda x \) for certain \( \lambda \).
For fractals, this is usually about a specific point (the *fixed point* of the dilation).
---
Conclusion
Self-similar figures exhibit **scale symmetry** (dilation symmetry), which can be discrete or continuous, exact or statistical. This distinguishes them from objects that are only symmetric under rigid motions (translations, rotations, reflections).
