March 25, 2014 - The word geometry is derived from the Greek “geometria” (to measure the earth). In its earliest applications, this branch of mathematics concerned itself with land measurement.
Ancient Egyptians were one of the earliest practitioners of geometry. Most of the farming in Egypt occurred along the banks of the Nile River. Plots of land, carefully laid out in rectangular plots, would have all of their boundaries obliterated by the annual flooding of the river. Techniques were needed to restore those boundaries so that farmers could resume their work without engaging in lengthy disputes with neighbours. Of course, each farmer knew the length and width of his plot, but the problem was to get each section reconstructed exactly. One technique for making exact 90 degree corners for each plot involved the use of a length of rope divided into 12 even sections by a series of knots.
By holding two ends of the rope together, and pulling out sections of the rope to make a triangle with sides of lengths 3, 4 and 5, the 90-degree corner would be formed at the intersection of the sides 3 and 4. The length and width could then be measured off from that corner and a perfect rectangle would be created.
Much was learned about lengths and angles, areas and volumes, through experience. One of the earliest known texts to document this accumulated knowledge was Book 2 of the Egyptian Rhind Papyrus (approx. 1650 B.C.).
Solutions were given that showed how to calculate the volumes of cylindrical and rectangular granaries, how to calculate the areas of various figures (including circles), and how to determine the slope of the sides of pyramids.
The Greeks, however, were noted for establishing rules of logical reasoning for geometry. In the sixth century B.C., Thales of Miletus (624 – 546 BC) is acknowledged as the first to do so. Recognized as being the world’s first “true mathematician”, he studied the computations used by the Egyptians and by using deductive methods of reasoning, he was able to prove which methods were correct and which were not. Thales understood similar triangles and right angles, and used this knowledge in practical ways.
In the diagram at left, the two red lines are parallel to each other.
Angle ADE = Angle ABC
Thus, the three angles inside triangle ADE are equal to the three angles inside triangle ABC. We say that the two triangles are similar (think of scale models; the little triangle is a small replica of the big one).
Thales’ Theorem states that these ratios are equal:
(which can also be stated as AE:AC=AD:AB=DE:BC)
It is said he used the fact that “the ratio of corresponding sides of similar triangles are equal” to determine the heights of the pyramids. The same method is still taught to Boy Scouts today to calculate the heights of trees or the distance across rivers.
Consider the picture at the right. The small line labelled as “A” could be a boy scout (or Thales) standing erect. The line labelled as “B” is his shadow. The tall line “D” could be a tree, or the height of the Pyramid of Cheops in Egypt, and “C” is the length of its shadow. “A” and “D” are parallel to each other, so we have the same situation of similar triangles that was described in the previous box. This makes the following equal ratio: A:D = B:C
Since the height “A” of the person and the lengths of the two shadows “B” and “C” can be measured, the height “D” of the large object can be calculated.
Pythagoras (570 – 495 BC), who likely studied under Thales, is well-known to most students of mathematics because a famous geometrical relationship is named after him. The Pythagorean Theorem states that “the square on the hypoteneuse of a right-angled triangle is equal to the sum of the squares on the other two sides”.
This picture illustrates the rule by using a 3-4-5 right-angled triangle. The longest side of the triangle is 5 units long (this is the hypoteneuse). The square that is 5 by 5 has an area of 5² = 5X5 = 25 square units. The squares on the other two sides have areas of 3² = 3X3 = 9 sq. units and 4² = 4X4 = 16 sq. units.
The square of the hypoteneuse (25) is equal to the sum of the squares on the other two sides (i.e. 9 + 16). 5² = 3² + 4² for this right-angled triangle. Pythagoras also showed that if any triangle has this relationship c² = a² + b² among its three sides a, b and c then it must have a 90 degree angle in it. This proved that the rope technique used by the Egyptians truly formed right angles.
Many other cultures, such as Mesopotamian, Babylonian, Indian and Chinese, had been making practical uses of geometric techniques for centuries. One of the oldest and most famous Chinese mathematical texts, the Zhou Bi Suan Jing (dating from the period 500 – 200 BC), contained a visual proof of the Pythagorean relationship for the 3-4-5 right triangle.
I’ve added red letters to help explain this visual proof. Each square has an area of 1 sq. unit. The rectangle ABCD contains 12 of these small squares, so it has an area of 12 sq. units. The triangle ABC has a 90 degree angle at B, and we can clearly see that it has a vertical side 3 units long and a horizontal side that is 4 units long. Is side AC = 5 units?
Well, triangle ABC is half of rectangle ABCD, so it must have an area of 6 sq. units. There are three other triangles which, by a similar argument, also have areas of 6 sq. units. The 4 triangles together have an area of 4X6 = 24 sq. units. Include the centre square: the total area of the large square is 24 + 1 = 25 sq. units. Because the large figure is a square with an area of 25 sq. units, each of its four sides must have a length of 5 units (the area of a square is found by multiplying any of its sides by itself: 25 = 5X5). Side AC is one of those sides. Thus, AC = 5.
This visual proof shows that a right-angled triangle, with sides of length 3 and 4 at the 90 degree angle, has a hypoteneuse of length 5.
At left, the diagram begins with a small right-angled triangle with two sides of length 1. Call the length of the hypoteneuse the letter c.
c² = 1² + 1²
c² = 1 + 1
c² = 2
Thus, c = the square root of 2 = √2
Build a right-angled triangle on top of the first triangle so that the outer side is still of length 1. Call the new hypoteneuse length d.
d² = 1² + (√2)²
d² = 1 + 2
d² = 3
d = √3
If we continue to add new right-angled triangles to the diagram in the same fashion (each new outer side is still 1 unit long), we build a beautiful spiral. Similar spirals exist in nature, in creatures such as the Nautilus, inside the human ear, and in many plant forms.
Plato (427 – 347 BC) was a Greek philosopher, not a mathematician, who emphasized the importance of accuracy and proof. This philosophy inspired another Greek by the name of Euclid (323 – 283 BC) to document, organize and extend the work of other outstanding geometers such as Pythagoras and Hippocrates in a work called “The Elements”. In this book, geometry was developed in a logical, step-by-step method, proceeding from a series of ‘self-evident truths’ called axioms and extending through a logical sequence of theorems which were carefully proved.
Many of today’s grandparents will remember when the study of Euclidean Geometry was a part of the mathematics curriculum in the 1950s and 1960s before the ‘new math’ was introduced. Euclid’s approach dominated the teaching of geometry over the centuries, and he is credited as being the most widely read author in the history of mankind.
One of the axioms in “The Elements” can be simply stated as “parallel lines never meet” (the exact statement is “For any line L and any point P, there is exactly one line through P not meeting L”). No proof was offered for this seemingly self-evident statement known as the Parallel Postulate.
In my next article (“Non-Euclidean Geometry”), we’ll see how a few 19th century mathematicians challenged this statement and developed entirely different, yet completely valid, alternative geometries.
The Parallel Postulate is one of the axioms that Euclid considered fundamentally true, with no proof required. Many theorems can be proved by making use of parallel lines, such as “the sum of the three interior angles in any triangle is 180 degrees” (the Triangle Postulate), so their truth depends entirely on the truth of the Parallel Postulate. Next month we’ll see why the challenge of this axiom led to new developments in geometry.
A straight line may be considered to be an angle of 180 degrees
(think of it as a “flat” angle). 180 degree angles are called straight angles.
When a straight line (called a transversal) crosses two parallel lines (labelled as line 1 and line 2), the alternate angles (which make a Z-pattern) are equal in size.
Angle C = Angle F
Angle D = Angle E
These last two facts can be used to prove that the sum of the internal angles of a triangle is : 180°
In the next diagram, the two red lines are parallel to each other. A triangle with the vertices (i.e. the ‘corners’) A, B and C is drawn between them so that side BC sits on one line and point A lies on the other line.
Side AB is a transversal that meets two parallel lines, so the yellow alternate angles labelled y are equal.
Side AC is another transversal meeting two parallel lines, so the green alternate angles labelled x are equal.
Because the top red line is a straight line, the multi-coloured angle that curves around point A is equal to 180 degrees in size. That is, y + z + x = 180 degrees.
But the three interior angles of triangle ABC are y (at vertex B), x (at vertex C), and z (at vertex A). Since x + y + z = 180 degrees for the straight line, then x + y + z = 180 degrees for the triangle.
This result (that the sum of the three interior angles in any triangle is 180 degrees) is one of many dozens of proofs that are accomplished by logical reasoning, not by trial and error measurement.
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For two thousand years Euclidean Geometry was recognized as being the only geometry. Next month we’ll learn about Non-Euclidean Geometries and how our perception of the universe was changed as a result.
Marvellous Mathematics - Introduction
# 2 Non-Euclidean Geometry - Issue 24 - May 1, 2014
# 3 Rational Numbers – Fractions, Decimals and Calculators - Issue 25 - June 1, 2014
This article is the first of a series of mathematics articles published by CHASA.
CHASA has received many communications from concerned parents about the difficulties their children are having with the math curriculum in their schools as well as their own frustration in trying to understand the concepts - so that they can help their children.
The intent of these articles is to not only help explain specific areas of history, concepts and topics in mathematics, but to also show the beauty and majesty of the subject.
Dave Didur is a retired secondary school mathematics teacher with a B. Sc. degree from the University of Toronto majoring in Mathematics and Physics. He was Head of Mathematics for over twenty years, as well as the Computer Co-ordinator and consultant for the Board of Education for the City of Hamilton. He served with the Ontario Ministry of Education for three years as an Education Officer.
A stamp commemorating Euclid, issued by the Maldive Islands on January 10, 1988
Parents will find educational expectations for geometry at different grade levels, printable worksheets, games, videos and other materials in the following list of resources. More advanced materials for older students are also included.
WikiHow: How to Prove the Pythagorean Theorem
A comprehensive list of geometry topics can be found at http://www.ck12.org/geometry/
U. of Georgia Dept. of Mathematics: More Advanced Proofs of the Pythagorean Theorem
Alexander Bogomolny: 101 Proofs of the Pythagorean Theorem (advanced)
Alfred Posamentier: The Pythagorean Theorem – The Story of its Beauty and Power
Clark University, MA Dept. of Math & Computer Science: An Introduction to Euclid’s “Elements”
About.com: An Introduction to Geometry (Terminology) - This page contains many links dealing with geometry concepts.
Math is Fun: Plane Geometry
Math is Fun: Solid Geometry
KidsMathGamesOnline.com: Geometry Games for Elementary School Students
HoodaMath.com: Geometry Games (for various levels)
Math-Play.com: Geometry Math Games for Elementary School Students
Fact Monster: Baseball Geometry Game (Terminology & Calculations)
NeoK12.com: Geometry Lessons for Grades 4 - 12 (Videos)
Khan Academy: 113 Videos on Topics in Geometry
MathPlayground.com: Geometry Board Explorations of Perimeter & Area – This site contains a comprehensive set of video lessons for geometry topics. Other math topics can be found at http://www.mathplayground.com/mathvideos.html
National Council of Teachers of Mathematics: What Students are Expected to Know in Geometry (Pre-Kindergarten to Grade 12)
Common Core State Standards in Mathematics: Look up expectations by grade or topic
EdHelper.com: Printable Geometry Worksheets – organized by grade level
Math-Drills.com: Printable Geometry Worksheets – organized by topic
“The laws of nature are but the mathematical thoughts of God.”
Marvellous Mathematics – Introduction to the mathematics series of articles