13 Questions
What is the approximate percentage of the total area made up by the circles in the packing problem?
78.54%
Which geometric shape has its total area represented as approximately90% in the packing problem?
Hexagon
What does symmetry in nature indicate?
An object can be divided into mirror images along an imaginary line
Which type of symmetry is exhibited by Leonardo da Vinci’s Vitruvian Man?
Bilateral symmetry
What is the definition of a pattern as per the text provided?
Regular, repeated, recurring forms or designs
What does identifying patterns in nature help individuals do?
Find logical connections to form generalizations
Which of the following terms is used to describe when you can draw an imaginary line across an object and have mirror images?
Symmetry
In what way does rotational symmetry differ from bilateral symmetry?
(Rotational symmetry) ≠ (Bilateral symmetry)
What is the smallest measure of angle that a figure can be rotated while still preserving the original position called?
Angle of rotation
If a figure has a rotational symmetry of order 6, what is the angle of rotation?
60°
Why do bees use hexagons in making honeycombs?
Hexagons are the most efficient way to pack the honeycombs
What is the packing problem referring to?
Finding the most efficient way to pack any shape into a given space
What is the order of rotational symmetry for a snowflake pattern that repeats six times?
6-fold
Study Notes
Patterns in Nature
- A pattern is a regular, repeated, or recurring form or design.
- Examples of patterns in nature include:
- Butterfly's bilateral symmetry
- Leonardo da Vinci's Vitruvian Man's proportion and symmetry
- Starfish's rotational symmetry
Symmetry
- Symmetry indicates that an object can be divided into two identical parts, like a mirror image.
- Types of symmetry include:
- Bilateral symmetry (left and right sides are identical)
- Rotational symmetry (object remains unchanged after rotation)
- Examples of symmetry include:
- Butterfly's bilateral symmetry
- Starfish's rotational symmetry
Packing Problem
- The packing problem involves finding the optimal method of filling a given space with shapes.
- Calculations for packing problem:
- Area of a circle: 𝜋𝑐𝑚2
- Area of a square: 𝑠𝑖𝑑𝑒2 ∙ 3/4
- Area of a hexagon: 6 × 3 𝑐𝑚2
- Efficiency of packing shapes:
- Square packing: ≈ 78.54% efficient
- Hexagonal packing: ≈ 90.69% efficient
Rotational Symmetry
- Rotational symmetry occurs when an object remains unchanged after rotation by a certain angle.
- Order of rotation (n) is the number of times an object can be rotated by a certain angle and still appear unchanged.
- Formula for angle of rotation: 𝐴𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 = 360° / 𝑛
- Example: The snowflake has a 6-fold rotational symmetry, with an angle of rotation of 60°.
Honeycomb
- Honeycomb's hexagonal structure is an efficient way to pack shapes.
- Bees use hexagons to make honeycomb because it allows for maximum storage and minimum use of wax.
Enhance your ability to identify patterns in nature and regularities in the world with Mr. Augustus Ceazar L. Lodronio, LPT. Delve into the significance of Mathematics in everyday life and appreciate the concept of patterns, regularity, and relationships in nature.
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