Euclidean or plane geometry begins with notions of points and lines, and the notion that a point lies on a line. Think of lines as sets, and a point as an element belonging to a set. Points and lines satisfy certain axioms. In Euclidean Geometry (or plane geometry), the axioms are based on Euclid’s original axioms. From these axioms, we can use the rules of logic to derive theorems (or propositions) that can be regarded as truthful statements that apply to the plane. Here a plane is a model, or a mini-universe where those axioms and theorems hold.

For example, consider the theorem: The sum of angles in a triangle is $180$ degrees. The various terms in this theorem (angle, triangle etc.) are constructs in the plane that we wish to study. The theorem itself is a property that will hold in our mini-universe. The proof should proceed from the axioms, use the definitions of the various constructs, and follow the rules of logic.

Even though the theorem is true, it does not imply that the sum all triangles is $180$ degrees. For example, consider the surface of the earth. Draw a triangle with a right angle at the North Pole. Suppose the two sides of this angle go down to the equator, and the third side of the triangle is the equator. The sum of angles of this triangle—made on the surface of the earth—is $270$ degrees!

In fact, in this non-euclidean geometry, the sum of angles in a triangle is always greater that $180$ degrees.

Can you find a surface where a sum of angles in a triangle is always less than $180$ degrees?

## Friday, November 15, 2002

## Friday, November 08, 2002

### Experience Mathematics #19 -- Euclid's axioms

Just like elements and sets,

We can think of a line as a set of points. These satisfy certain

However, several gaps were found in Euclid’s axioms. For example, consider Euclid’s proof that the base angles of an isosceles triangle are equal. Suppose we have an isosceles triangle $ABC$, where the side $AB$ is equal to the side $AC$. Drop a perpendicular $AD$ from a vertex to the side $BC$. There is nothing in Euclid’s axioms that says that the point $D$ is between the points $B$ and $C$. Nevertheless, Euclid proves that the triangles $ABD$ and $ACD$ are congruent. From this it is easy to see that the base angles of an isosceles triangle are equal.

The great mathematician Hilbert completed Euclid’s work by listing a few more axioms. These included the

To return to Euclid’s proof, some steps need to be added to show that $D$ is between $B$ and $C$.

But that is not all. We could consider a geometry where given a line $l$ and a point $P$ not on the line, there are no lines parallel to $l$ containing the point $P$. Such a

**Points**and**Lines**are undefined notions.We can think of a line as a set of points. These satisfy certain

**axioms**, such as: Given a line $l$ and a point $P$ not on the line, there is only one line that is parallel to $l$ containing the point $P$. Axioms are considered to be self-evident truths.However, several gaps were found in Euclid’s axioms. For example, consider Euclid’s proof that the base angles of an isosceles triangle are equal. Suppose we have an isosceles triangle $ABC$, where the side $AB$ is equal to the side $AC$. Drop a perpendicular $AD$ from a vertex to the side $BC$. There is nothing in Euclid’s axioms that says that the point $D$ is between the points $B$ and $C$. Nevertheless, Euclid proves that the triangles $ABD$ and $ACD$ are congruent. From this it is easy to see that the base angles of an isosceles triangle are equal.

The great mathematician Hilbert completed Euclid’s work by listing a few more axioms. These included the

**betweenness axioms**. For example, given three points $A, B and C$, one of the axioms said either $B$ is between $A$ and $C$, or $C$ is between $A$ and $B$ or $A$ is between $C$ and $B$.To return to Euclid’s proof, some steps need to be added to show that $D$ is between $B$ and $C$.

But that is not all. We could consider a geometry where given a line $l$ and a point $P$ not on the line, there are no lines parallel to $l$ containing the point $P$. Such a

**non-euclidean geometry**exists on the surface of the Earth. So one of Euclid’s axioms cannot be considered to be a self-evident truth after all.

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Gaurav Bhatnagar
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