Our mathematical journey begins with the first appearance of mankind, which is believed to have occurred about 600,000 or 700,000 years ago. Mankind, though physically ill prepared to compete in a hostile world in terms of strength and natural armament, used ingenuity and a little luck to begin the long journey toward civilization. 

(Information from The Grolier Multimedia Encyclopedia)

The human hand is quite different from most other animals, in that we have what is called an "opposable thumb". Our thumb "opposes", or faces the opposite direction as our fingers, and this allows us to grasp a stick, a knife, a pencil, a paint brush. Human beings differ from other creatures in their ability to make and use tools, unlike any other creature on Earth. Perhaps our greatest achievement is our ability to create visual and written, as well as verbal means to communicate, to create, and to record our own history. The first time a man or woman picked up a stick and drew a line or a circle in the dirt, visual communication began, and with it, the beginnings of art, science, and mathematics.

The history of mathematics begins, then, with that first line drawn in the sand or carved into the wall of a cave. Archaeologists have discovered cave drawings believed to have been created during the Paleolithic age. Many cave paintings are of animals but there is also a large body of geometric art (particularly in Spain). Visit the site linked below to find more about prehistoric art and geometric drawings:

Test Question #1: What is Lascaux, where is it, and how old is it thought to be?

Visit the site below, and read a timeline of mathematical history. This website was created by students at the University of the Virgin Islands, as a class project!

That first line drawn by a man or woman perhaps 100,000 years ago has brought us over the centuries to the creation of art, graphic design, architecture, science, and mathematics. Geometry is perhaps the most visual form of mathematics, but all fields of mathematics, from arithmetic through calculus use the symbols, lines, graphs, shapes and numerals that create meaning out of dots, lines, and curves. Human beings have used sticks, quill pens, chalk, pencils, printing presses, typewriters, and now computers to create, make sense of and record scientific and mathematical ideas.

The history of mathematics is a long and interesting tale, with mathematical heroes from all parts of the world. The Egyptians were great mathematicians, and you will learn more about their work designing and building the pyramids, in Chapter 4 Solids. The Greeks are among the most famous mathematicians, and Euclid is given credit for being the creator of Geometry. The word geometry itself (as with many mathematical terms) comes from the Greek language. Geometry derives from two Greek words: geo "earth" and metri "measure". Geometry was originally used to measure fields and and distances on the earth, and still is used today in surveying and navigation.

Visit the website below to learn more about the history of mathematics:

Test Question #2: What is the name of the longest-used math textbook in history, what was its main topic, where, when and by whom was it written?

Another website, with much interesting information about the history of mathematics, and particularly Greek mathematicians:

Geometry is one of the most visual branches of mathematics. The word geometry comes from two Greek words: geo (earth) and metron (measure). One of the many applications of geometry is measuring the earth, and things on the earth.The basic elements of geometry are points and lines. Using these basic figures, the rest of geometry was created. Learn more about these basic elements by visiting the website linked below:

Test Question #3: If the vertex of an angle is point N, and two points (one on each side of the angle) are points P and Q, what is the name of the angle?

Using these basic elements, mathematicians create graphs to analyze and record information. An example of a line graph is shown below. This is a graph of the equation y = x + 1

There are other types of graphs. Besides purely mathematical graphs, there are applications in statistics, insurance, finance, science, and practically every field of business from real estate to the stock market.If you look, you can find many examples of graphs in the newspaper and on the internet, showing scientific, financial and mathematical data. Visit the site below for some more examples of statistics and graphs:

Test Question #4: How do US high school students perform in math and science compare to the international average?

Many of the shapes we see around us have mathematical names. Those made of straight lines are called polygons. A polygon is a straight-sided closed geometric figure. The sides may not be curved. The quilt below is made up of different types of polygons:

To learn more about polygons, click on the link below:

Test Question # 5: Which one or ones of the following words apply to the figure below; list ALL that apply: polygon, pentagon, hexagon, octagon, regular, equilateral, equiangular

There are many different polygons, named by the number of their sides. You will find much more information about other polygons at the following website:

If you look around you, you will find polygons in many places, from fabric patterns to buildings. The triangle is a particularly useful polygon, because of its strength. Visit the website below to find out more about the strength of triangles, and their usefulness in engineering:

What concepts of math and physics are commonly used by civil engineers? Dr. Math answers this question on the following web page:

How would you span a freeway? A canyon? A river? Or an ocean waterway? Go to the website below and learn about the four major types of bridges and then test your knowledge by matching the right bridge to the right location:

Besides their strength, triangles have other applications. To discover applications of triangles and gother polygons to music, visit the following website about Fractals and hear "Fractal Music, The Sound of Chaos":

When an object is the same on one side as one the other, we say that it is symmetrical. The human body is roughly symmetrical, and so is the crab below.

If you draw a vertical line down the middle, the left side of each would be a "mirror image" of the right side. Actually, this is only one kind of symmetry: we call this "reflection symmetry". There are other kinds of symmetry, which you will discover by clicking on the link below.

Test Question #6: How many lines of reflection symmetry a regular hexagon would have? Is there only one type of line or two types? If two, describe the two types of lines of symmetry. Does it have rotation symmetry? If so, how many degrees?

You will learn more about symmetry at the website below:

This beautiful geometric patterns can be constructed using a compass and a straight-edge, or using a computer.This drawing was done by a student like yourself. One of the projects for this chapter is to create a geometric design, using constructions. You can learn how to create geometric designs like this by going to the webpage linked below:

Test Question #7: What are the two main tools used to do geometric constructions?

You may wonder why mathematicians do constructions. Dr. Math, at the Math Forum, answers this question at the following web page:

A polygon is made up of sides and angles. There are two kinds of angles, the exterior angles and the interior angles. You will learn much more about the angles of polygons by going to the web page linked below:

Test question #8: What is the measure of each exterior angle, and each interior angle, of a regular hexagon?

You can find examples of polygons all around you. Often, you will see them in tile patterns and in buildings. Some polygons will fit together better than others, depending on their angles.

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Tessellations

When polygons do fit together perfectly, we call the pattern a tessellation. Many fabric designs are tessellations, and we also see tessellations in ornamental tile work. Examples of tessellations can be seen at the following web pages about a Spanish palace called The Alhambra:

A Dutch artist named M.C. Escher visited the Alhambra and was fascinated by the geometric patterns he saw there. He created many beautiful tessellations. You can see some of his work at the following website:

Not all polygons will fit together to form a tessellation. You can find out which ones will, and which will not, but visiting the following website:

Test Question #9: Will regular octagons tessellate? Explain why or why not, using mathematics.

Tessellations are easy to construct, and fun to make. Click on the link below to learn how to create a tessellation like the one above, created by a mathematics student.

Test Question #10: To create a tessellation like the one above, you would use translation (sliding). What transformation would you use to create a tessellation based on triangles, and what transformation would you use to create a tessellation based on hexagons?

1) Construct each of the assigned regular polygons: equilateral triangle, square, pentagon, hexagon, octagon on one or more blank sheets of white paper. Use a compass and straightedge or The Geometer's Sketchpad computer software. If drawn by hand, your construction lines should be very light, and your final lines should be clean and dark.

2) Design and draw a colorful geometric graphic following the examples on the web page. Your design should have either reflection symmetry or rotation symmetry. You may use dilations, similar and congruent figures. Write at least 5 sentences explaining the symmetry and the types of figures used, using correct mathematical terminology. Use colored pens or colored pencils. If drawn by hand, your construction lines should be very light, and your final lines should be clean and dark.

3) Design and draw a colorful tessellation based on a polygon. Use a compass and ruler, graph paper, and/or computer software such as The Geometer's Sketchpad, other painting software, or HyperCard. If drawn by hand, your construction lines should be very light, and your final lines should be clean and dark.