Mathematics in the Modern World: Patterns, Symmetry, and Modeling

Introduction to Patterns and Numbers in Nature

• Definition of Patterns: Patterns are defined as regular, repeated, or recurring forms or designs. These regularities can be observed throughout the world, both in the natural environment and in human-made structures.

• Examples of Patterns:

  • Human Endeavor: The layout of floor tiles, the architectural designs of buildings, and the utilitarian method of how we tie our shoelaces.

  • Natural World: Patterns exist in biological structures and geographical formations.

• Importance of Studying Patterns: Studying patterns is fundamental for identifying relationships and finding local connections. This process allows for the formation of generalizations and the ability to make accurate predictions based on observed data.

Pattern Recognition and Logical Series Examples

• Example 1: Geometric Rotation: A series of figures involves a base figure rotating at an angle of $45^\circ$ in the counterclockwise direction. For this specific pattern, identifying the next step requires continuing this rotation.

• Example 2: Clockwise Rotation: A series where the base figure rotates at an angle of $90^\circ$ in the clockwise direction.

• Example 3: Vertical Inversion: In this series, a figure is followed by a combination of itself and its vertical inversion.

• Example 4: Quadratic Numerical Series: Given the series $10, 17, 26, 37, ?$

  • Logic: Beginning with the number $3$, each number in the series follows the formula $n^2 + 1$, where $n$ is the succeeding integer.

  • Calculation:

    • $3^2 + 1 = 10$

    • $4^2 + 1 = 17$

    • $5^2 + 1 = 26$

    • $6^2 + 1 = 37$

    • $7^2 + 1 = 50$

  • Result: The next number should be $50$.

• Example 5: Grid Relationship Matrix: Given the matrix of numbers:

  • Row 1: $17, 8, 5, 5$

  • Row 2: $13, 7, 5, 4$

  • Row 3: $6, 12, 6, 3$

  • Row 4: $10, 6, 4, ?$

  • Logic: For each row, the sum of the first two columns is equal to the product (multiple) of the last two columns.

  • Verification:

    • Row 1: $17 + 8 = 25$; $5 \times 5 = 25$

    • Row 2: $13 + 7 = 20$; $5 \times 4 = 20$

    • Row 3: $6 + 12 = 18$; $6 \times 3 = 18$

    • Row 4: $10 + 6 = 16$; $4 \times ? = 16$

  • Result: The value replacing the question mark is $4$.

Concepts of Symmetry

• Definition: Symmetry indicates that an imaginary line can be drawn across an object such that the resulting parts are mirror images of each other.

• Examples of Symmetry:

  • Natural Organisms: The wings of a butterfly and the structure of a starfish.

  • Art and Anatomy: Leonardo da Vinci's Vitruvian Man, which demonstrates the mathematical proportion and symmetry of the human body.

• Bilateral Symmetry: This occurs when an object is symmetric about a single axis, and the left and right portions are exactly the same (e.g., the butterfly).

• Rotational Symmetry: A figure has rotational symmetry if it can be rotated by a specific angle and still achieve the same appearance as the original position.

  • Example: A starfish can be rotated by $72^\circ$ to match its original orientation.

• Angle of Rotation: The smallest measure of the angle that a figure can be rotated while still preserving its original appearance.

• Order of Rotation (n-fold rotational symmetry): A figure has rotational symmetry of order $n$ if $\frac{1}{n}$ of a complete turn ($360^\circ$) leaves the figure unchanged.

  • Formula for Angle of Rotation:

    • $\text{Angle of rotation} = \frac{360^\circ}{n}$

The Packing Problem and Hexagonal Structures

• The Honeycomb Conjecture: Bees utilize hexagonal structures in making honeycombs rather than other polygons because hexagons allow for the maximum area to be covered using the least amount of material (wax).

• Packing Problem Definition: This involves finding the optimum method of filling a given space, such as a cubic or spherical container.

• Mathematical Proof (Hexagonal vs. Square Packing):

  • Assumptions: Suppose circles have a radius of $1\,\text{cm}$, resulting in an area of $\pi\,\text{cm}^2$ for each circle.

  • Case 1: Square Packing:

    • Each square has a side of $2\,\text{cm}$ (diameter of the circle) and an area of $4\,\text{cm}^2$.

    • Each square fits exactly one circle.

    • Percentage of area covered: $\frac{\pi\,\text{cm}^2}{4\,\text{cm}^2} \times 100\% \approx 78.54\%$

  • Case 2: Hexagonal Packing:

    • A hexagon can be viewed as being composed of six equilateral triangles with side lengths of $2\,\text{cm}$.

    • Area of one equilateral triangle: $A = \frac{\sqrt{3}}{4} \times \text{side}^2 = \frac{\sqrt{3}}{4} \times (2\,\text{cm})^2 = \sqrt{3}\,\text{cm}^2$

    • Total area of the hexagon: $6 \times \sqrt{3}\,\text{cm}^2 = 6\sqrt{3}\,\text{cm}^2$

    • Capacity: A single hexagon fits 3 full circles (one whole circle in the middle and 6 fragments that are each one-third of a circle).

    • Total area of circles inside: $3\pi\,\text{cm}^2$

    • Percentage of area covered: $\frac{3\pi\,\text{cm}^2}{6\sqrt{3}\,\text{cm}^2} \times 100\% \approx 90.69\%$

  • Conclusion: Hexagonal packing is significantly more efficient than square packing, covering approximately $12.15\%$ more area.

Mathematical Modeling in Biology and Populations

• Animal Patterns (Turing Patterns): British Mathematician Alan Turing proposed that the formation of hyena spots and tiger stripes is governed by a set of "reaction-diffusion equations."

  • Morphogens: Turing suggested that two unidentified chemicals (morphogens) interact inside an embryo, reacting with each other and diffusing through the embryo's surface to create patterns.

• Natural Sequences: Fibonacci numbers are frequently found in the petal counts of flowers and the spiral growth of nautilus shells.

• Population Growth Modeling: Mathematics is used to model the growth of populations using the exponential growth formula:

  • $A = Pe^{rt}$

  • Where:

    • $A$ is the size of the population after growth.

    • $P$ is the initial population size.

    • $e$ is Euler's constant (approximately $2.718$).

    • $r$ is the rate of growth.

    • $t$ is the time elapsed.

• Application Example:

  • Model: $A = 30e^{0.02t}$, where population is in thousands and $t$ is years after 1995.

  • Question 1: What was the population in 1995?

    • Set $t = 0$. $A = 30e^{0.02(0)} = 30$. Population was $30,000$.

  • Question 2: What will the population be in 2017?

    • Calculate $t = 2017 - 1995 = 22$.

    • Substitute into model: $A = 30e^{0.02(22)} = 30e^{0.44} \approx 46.5813$.

    • Result: Approximately $46,581$.

References

• Mathematics in the World, published by Rex Book Store, Inc. (RBSI).

• Pattern recognition resources from iqtestexperts.com.