Mathematics in the Modern World Chapter 1

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Questions and Answers

What is defined as patterns in nature that can be mathematically modeled?

Textures

Colors

Shapes

Symmetries (correct)

Which type of symmetry is characterized by reflectional properties?

Rotational symmetry

Bilateral symmetry (correct)

Six-fold symmetry

Fivefold symmetry

Which type of symmetry is not found in true crystals?

Cubic symmetry

Hexagonal symmetry

Fivefold symmetry (correct)

Octahedral symmetry

What is the primary reason for studying patterns in living organisms?

Biological processes Signup and view all the answers

What type of symmetry can be observed in the crown-shaped splash pattern formed by a drop of water?

Rotational symmetry Signup and view all the answers

Which of the following is NOT a type of pattern mentioned in nature?

Randomness Signup and view all the answers

In which groups is fivefold symmetry predominantly found?

Echinoderms Signup and view all the answers

What mathematical concept can explain the patterns formed during crystallization?

Symmetry Signup and view all the answers

What is the characteristic feature of fractals?

They involve infinite iteration. Signup and view all the answers

Which of the following best describes spirals?

Curves moving away from a point while revolving around it. Signup and view all the answers

What is the primary function of stripes in animals?

To increase survival chances for offspring. Signup and view all the answers

How many wallpaper groups of tessellations exist?

17 Signup and view all the answers

What is the smallest possible surface area for a given volume in bubble formation?

Sphere Signup and view all the answers

What does the pattern of cracks in materials indicate?

The level of stress the material can withstand. Signup and view all the answers

What is the rule identified in the Fibonacci sequence example provided?

Adding the last two terms to get the next term. Signup and view all the answers

In the sequence 1, 10, 100, 1000,... what will be the 6th term?

$10^6$ Signup and view all the answers

What is the 7th term in the Fibonacci sequence?

34 Signup and view all the answers

Which of the following correctly represents the rule for the Fibonacci sequence?

X n = X n −1 + X n −2 Signup and view all the answers

What is the value of the 8th term in the Fibonacci sequence?

34 Signup and view all the answers

Which of the following statements accurately describes the pattern observed in multiples according to the provided information?

Every 3rd number is a multiple of 2. Signup and view all the answers

In the Fibonacci sequence, what is the sum of the 6th and 7th terms?

55 Signup and view all the answers

How does the Fibonacci sequence relate to patterns in nature?

It represents mathematical relationships that can be observed everywhere. Signup and view all the answers

What is the first number that is a multiple of 5 in the described sequence of every 5th number?

5 Signup and view all the answers

If you continue the described spiral to find the Fibonacci sequence, what is the 30th Fibonacci number?

832040 Signup and view all the answers

If a Fibonacci sequence starts with 0 and 1, what will be the next four terms?

1, 2, 3, 5 Signup and view all the answers

What pattern is shown by the Fibonacci sequence when represented graphically?

A spiral Signup and view all the answers

Based on the information given about n, which description is correct regarding multiples?

Every nth number is a multiple of n. Signup and view all the answers

What is the correct continuation of the pattern observed with X=3, X=4 and X=5?

Every 6th number is a multiple of 6. Signup and view all the answers

What is the term number for the value 144 in the Fibonacci sequence?

12 Signup and view all the answers

Study Notes

Chapter 1: Nature of Mathematics

Objectives

Identify and differentiate patterns in nature.
Understand the Fibonacci sequence.
Appreciate the beauty of mathematics through patterns and numbers in nature and the world.

Patterns in Nature

Visible regular forms found throughout the natural world.
Can be modeled mathematically, including types like symmetries, spirals, etc.
Represent underlying biological processes in living organisms.
Computer models are utilized to simulate a variety of patterns.

Types of Patterns

Symmetry: Balanced proportions where different sides are alike.
- Bilateral Symmetry: Reflectional symmetry; figures remain unchanged upon reflection.
- Rotational Symmetry: Pattern remains unchanged when rotated (e.g., splash patterns, planet rings).
- Fivefold Symmetry: Found in echinoderms like starfish; reason behind this symmetry is unclear.
- Sixfold Symmetry: Common in snowflakes, mirroring growth patterns on each arm.
Fractals: Infinite self-similar patterns; nature only approximates these due to limitations in iteration.
Spirals: Curves increasing in distance from a central point as they revolve around it.
Tessellations: Repeating tiled patterns covering flat surfaces; 17 wallpaper groups exist.
Bubbles/Foams: Soap bubbles form spheres with minimal surface area, optimizing volume.
Stripes: Patterns often explained by evolutionary functions helping offspring survival.
Cracks: Linear patterns forming in materials to relieve stress, indicating elasticity.

Fibonacci Sequence

An ordered list of numbers where each term is the sum of the two preceding terms.
Starts with 0, 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Key Features of Fibonacci Sequence

Terms are indexed starting from 0.
Rule: X_n = X_(n-1) + X_(n-2)
Patterns observed in nature with Fibonacci numbers, such as the arrangement of leaves, flowers, and fruits.

Notable Patterns within Fibonacci Sequence

Every third number (2, 8, 34, ...) is a multiple of 2.
Every fourth number (3, 21, 144, ...) is a multiple of 3.
Every fifth number (5, 55, 610, ...) is a multiple of 5.

Activities

Capture the Beauty of Mathematics: Identify and document at least 10 patterns found in the environment.
Fill the Spiral: Extend the Fibonacci sequence to term number 30 by finding and writing the subsequent numbers.

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Description

Explore the fascinating nature of mathematics in this first chapter of the module. Gain insight into patterns found in nature and the significance of the Fibonacci sequence. Appreciate the beauty of mathematics as it connects to the world around us.