LESSON 1.2 Patterns and Numbers in Nature

Question 1

What is the nature of mathematics according to the National Council of Teachers of Mathematics (1991)?

Answer 1

Mathematics is a study of patterns and relationship, a way of thinking, an art, a language, and a tool. It is about patterns and relationships. Numbers are just a way to express those patterns and relationships.

National Council of Teachers of Mathematics (1991)

Question 2

Where can patterns be observed?

Answer 2

Patterns are everywhere, deeply embedded all around us. They can be observed in colors, shapes, actions, lines or curves of buildings, pathways, grocery stores, etc.

Patterns observation

Question 3

What are some examples of number patterns?

Answer 3

Examples of number patterns include 2, 4, 6, 8 and 5, 10, 15, 20. These are among the first patterns encountered in younger years.

Number patterns examples

Question 4

What are some types of patterns learned in school?

Answer 4

In school, we learn number patterns, logic patterns, geometric patterns, and word patterns. These are examples of various patterns encountered in education.

School patterns

Question 5

What are some examples of natural patterns?

Answer 5

Natural patterns include symmetries, fractals, spirals, meanders, waves, foams, tessellations, cracks, and stripes. These patterns can be observed and sometimes modeled mathematically.

Nature patterns

Question 6

What is symmetry?

Answer 6

Symmetry is the fundamental ‘language’ of patterns. Symmetry occurs when there is congruence in dimensions, due proportions and arrangement. It provides a sense of harmony and balance.

Question 7

Where can symmetry be found?

Answer 7

Symmetry can be found everywhere, including nature, the arts and architecture, mathematics, and science.

Question 8

What is reflection or bilateral symmetry?

Answer 8

Bilateral or reflection symmetry is the simplest kind of symmetry. It can also be called mirror symmetry because an object with this symmetry looks unchanged if a mirror passes through its middle.
It only has one line of symmetry.

Question 9

What is radial symmetry?

Answer 9

Rotational symmetry around a fixed point known as the center. Images with more than one lines of symmetry meeting at a common point exhibits a radial symmetry.

Question 10

What are examples of shapes with radial symmetry?

Answer 10

An equilateral triangle and circles.

Question 11

How many different axes can you cut along on an equilateral triangle?

Answer 11

Three different axes.

Question 12

How many axes can a circle be cut along?

Answer 12

An infinite number of axes.

Question 13

Where can radial symmetry be found?

Answer 13

Both in natural and human made objects.

Question 14

What are the three classifications of symmetric patterns in the plane?

Answer 14

Rosette patterns, frieze patterns, and wallpaper patterns.

These are the three groups into which patterns in the plane are usually divided.

Question 15

What are rosette patterns?

Answer 15

Patterns that consist of taking a motif or an element and rotating and/or reflecting that element.

There are two types of rosette patterns: cyclic and dihedral.

Question 16

What is a cyclic rosette pattern?

Answer 16

A rosette pattern that only admits rotational symmetries.

This type of rosette pattern only has rotational symmetries.

Question 17

What is a dihedral rosette pattern?

Answer 17

A rosette pattern that admits both rotational symmetries and bilateral or reflectional symmetries.

This type of rosette pattern has both rotational and bilateral symmetries.

Question 18

What is a frieze pattern?

Answer 18

A pattern in which a basic motif repeats itself over and over in one direction. It extends to the left and right in a way that the pattern can be mapped onto itself by a horizontal translation

Question 19

Where can we usually find frieze patterns?

Answer 19

On the walls of buildings, fabrics, borders of rugs, and tiled floors.

Question 20

How many types of frieze patterns are there?

Answer 20

There are only seven types.

Question 21

Who invented the names of the seven types of frieze patterns?

Answer 21

Mathematician John B. Conway

Question 22

What is the defining characteristic of a wallpaper pattern?

Answer 22

A wallpaper pattern has translation symmetry in two directions.

Question 23

What are the three types of symmetries involved in wallpaper patterns?

Answer 23

The three symmetries are reflection, rotation, and glide reflection.

Question 24

According to Nocon (2016), what are the necessary conditions for a plane figure to be considered a wallpaper pattern?

Answer 24

It must have at least one basic unit, one copy by translation, and a copy of these two by translation in the second direction, with at least two rows, each at least two units long.

Question 25

How many distinct types of wallpaper patterns have mathematicians classified?

Answer 25

There are 17 distinct types of wallpaper patterns.

Question 26

What is a tessellation?

Answer 26

A tessellation is a repeating pattern of figures that covers a plane with no gaps or overlaps.

Question 27

What types of transformations can create tessellations?

Answer 27

Tessellations can be created with translations, rotations, and reflections.

Question 28

What are some examples of tessellations in nature?

Answer 28

Examples include pavements, snake skin, turtle shells, and honeycombs.

Question 29

What is a pattern?

Answer 29

A pattern is
an arrangement which helps observers anticipate what they might see or what happens next. A pattern
also shows what may have come before. A pattern organizes information so that it becomes more useful.

Question 30

What is the primary function of the human mind concerning data?

Answer 30

The human mind is programmed to make sense of data or bring order where there is disorder by discovering relationships and connections between seemingly unrelated bits of information.

Question 31

What types of number patterns are introduced as one advances in mathematics?

Answer 31

As one advances, they encounter more complex patterns that can be ascending, descending, or multiples of a certain number, often explored through functions and sequences like arithmetic and geometric sequences.