Euclidean GeometryEuclid’s Axioms
Before we can write any proofs, we need some common terminology that will make it easier to talk about geometric objects. These are not particularly exciting, but you should already know most of them:
A A point is a specific location in space. Points describe a position, but have no size or shape themselves.
In Mathigon, large, solid dots indicate interactive points you can move around, while smaller, outlined dots indicate fixed points which you can’t move.
A A line is a set of infinitely many points that extend forever in both directions. Lines are always straight and have no width.
Lines are labeled using lower-case letters like a or b. We can also refer to them using two points that lie on the line, for example
A A line segment is the part of a line that lies between two points, without extending to infinity. We can label line segments from point A to point B as
A A ray is the part of a line that starts at a point and extends infinitely in one direction. We can label rays starting at
When labelling rays, the arrow shows the direction where it extends to infinity, for example
A A circle is the set of all points in two dimensions, at a fixed distance (the radius) from a given point (the center). The radius of a circle is the distance (or a line segment) between its center and any point on its circumference.
Congruence
These two shapes basically look identical. They have the same size and shape, and we could turn and slide one of them to exactly match up with the other. In geometry, we say that the two shapes are In geometry, two figures are congruent if are identical in size, shape and measure. This means we could move, flip or rotate them to exactly fit on top of each other.
The symbol for congruence is
Here are a few different geometric objects – connect all pairs that are congruent to each other. Remember that more than two shapes might be congruent, and some shapes might not be congruent to any others:
Two line segments are congruent if they
Note the that “congruent” does not mean “equal”. For example, congruent lines and angles don’t have to point in the same direction. Still, congruence has many of the same properties of equality:
- Congruence is symmetric: if
X ≅ Y then alsoY ≅ X . - Congruence is reflexive: any shape is congruent to itself. For example,
A ≅ A . - Congruence is transitive: if
X ≅ Y andY ≅ Z then alsoX ≅ Z .
Parallel and Perpendicular
Two straight lines that never intersect are called Two or more lines are parallel if they never intersect. They have the same slope and the distance between them is always constant.
A good example of parallel lines in real life are railroad tracks. But note that more than two lines can be parallel to each other!
In diagrams, we denote parallel lines by adding one or more small arrows. In this example,
The opposite of parallel is two lines meeting at a 90° angle (right angle). These lines are called Two lines are perpendicular (sometimes called normal or orthogonal) if they intersect at a right angle.
In this example, we would write a
Euclid’s Axioms
Greek mathematicians realised that to write formal proofs, you need some sort of starting point: simple, intuitive statements, that everyone agrees are true. These are called An axiom, sometimes called postulate, is a mathematical statement that is regarded as “self-evident” and accepted without proof. It should be so simple that it is obviously and unquestionably true. Axioms form the foundation of mathematics and can be used to prove other, more complex results.
A key part of mathematics is combining different axioms to prove more complex results, using the rules of logic.
The Greek mathematician Euclid of Alexandria (Εὐκλείδης, around 300 BCE) was a Greek mathematician and is often called the father of geometry. His book The Elements first introduced Euclidean geometry, defines its five axioms, and contains many important proofs in geometry and number theory – including that there are infinitely many prime numbers. It is one of the most influential books ever published, and was used as textbook in mathematics until the 19th century. Euclid taught mathematics in Alexandria, but not much else is known about his life.
First Axiom
You can join any two points using exactly one straight line segment.
Second Axiom
You can extend any line segment to an infinitely long line.
Third Axiom
Given a point P and a distance r, you can draw a circle with centre P and radius r.
Fourth Axiom
Any two right angles are congruent.
Fifth Axiom Two or more lines are parallel if they never intersect. They have the same slope and the distance between them is always constant.
Given a line L and a point P not on L, there is exactly one line through P that is
Each of these axioms looks pretty obvious and self-evident, but together they form the foundation of geometry, and can be used to deduce almost everything else. According to none less than Sir Isaac Newton (1642 – 1726) was an English physicist, mathematician, and astronomer, and one of the most influential scientists of all time. He was a professor at Cambridge University, and president of the Royal Society in London. In his book Principia Mathematica, Newton formulated the laws of motion and gravity, which laid the foundations for classical physics and dominated our view of the universe for the next three centuries. Among many other things, Newton was one of the inventors of calculus, built the first reflecting telescope, calculated the speed of sound, studied the motion of fluids, and developed a theory of colour based on how prisms split sunlight into a rainbow-coloured spectrum.
Euclid published the five axioms in a book “Elements”. It is the first example in history of a systematic approach to mathematics, and was used as mathematics textbook for thousands of years.
One of the people who studied Euclid’s work was the American President Thomas Jefferson (1743 – 1826), was one of the Founding fathers of the United States, their third president, and the principal author of the Declaration of Independence. In addition to his work in politics, he studied mathematics, horticulture, mechanics, and worked as an architect. After retiring from public office, Jefferson founded the University of Virginia.
This is just one example where Euclid’s ideas in mathematics have inspired completely different subjects.