Overview
This lecture introduces patterns and numbers in nature, focusing on different types of symmetry and the efficiency of hexagonal packing in natural structures.
Patterns and Symmetry in Nature
- Patterns in nature show structure, organization, and sometimes suggest intelligent design.
- Symmetry means an object can be divided so parts are mirror images; common in natural and human-made forms.
- Bilateral symmetry occurs when an object (like a butterfly) has identical left and right sides.
- The Vitruvian Man by Leonardo da Vinci illustrates human body symmetry and proportion.
- Starfish exhibit rotational symmetry, specifically fivefold symmetry.
- Rotational symmetry means an object looks the same after a certain degree of rotation.
- The angle of rotation is the smallest angle a figure can turn and appear unchanged.
- Order of rotation is the number of times a figure matches itself in a full 360° rotation.
- Formula for angle of rotation: 360° divided by the order of symmetry (n).
- Snowflakes have sixfold symmetry; their angle of rotation is 60° (360°/6).
Hexagonal Packing and Natural Optimization
- Honeybees use hexagons in honeycombs for efficient use of space.
- The packing problem seeks the most efficient way to fill space with objects (e.g., circles in a plane).
- Square packing of circles covers 78.54% of the area.
- Hexagonal packing of circles covers 90.69% of the area.
- Hexagonal formation is more optimal than square formation for space usage.
- This efficiency explains why hexagonal shapes are prevalent in nature (e.g., honeycombs).
Key Terms & Definitions
- Symmetry — A balanced arrangement where parts are mirror images across a dividing line.
- Bilateral Symmetry — Symmetry with identical left and right halves.
- Rotational Symmetry — When an object looks the same after a partial rotation.
- Angle of Rotation — The smallest angle that turns a figure onto itself.
- Order of Rotation — The number of times an object matches itself in one full turn.
- Packing Problem — Finding the most efficient way to fill a space with given shapes.
Action Items / Next Steps
- Review symmetry types and be able to identify them in natural and human-made objects.
- Practice calculating angle of rotation using 360°/n.
- Compare percentage areas of different packings as shown in the lesson.