$$$$

5  Modern Axioms

Euclid’s postulates were chosen with care to be both self-evident and useful. But they are by no means the only possible axiom set one could choose to base Euclidean geometry off of. Just like it is possible for other statements to be equivalent to Postulate 5, it’s also possible for another set of axioms to be Equivalent to Euclid’s:

Definition 5.1 (Equivalent Axiom Systems) Two axiom systems $\mathcal{A}$ and $\mathcal{B}$ are equivalent if you can both use the axioms of $\mathcal{A}$ to prove the axioms of $\mathcal{B}$, and vice versa.

In modern math, when defining something axiomatically we often prefer to choose axioms whose meaning is clear. Can we formulate a collection of axioms equivalent to Euclid’s, that capture the essence of the geometry of the plane?

5.0.1 Axioms 1,2 & 3: Space is Complete & Infinite

The first three axioms of Euclid focus on the ability to draw lines (between any points, and of any length) and circles (of any radius). All of these together work to capture the property that space doesn’t have any holes, and goes on forever.

Definition 5.2 A space $X$ is complete if it does not have any holes, gaps or boundaries. Intuitively, a space is complete if you can continue walking straight in any direction, for as long as you like.

It is easiest to explain this notion by giving non-examples. The unit disk $D=\{(x,y)\in {\mathbb{R}}^2\mid x^2+y^2\le 1\}$ is not complete because if you start at the center you only have to walk one unit before you have to stop: you’ve reached the edge of space!

The punctured plane (all of ${\mathbb{R}}^2$, except the origin has been removed) is also not complete: any line segment passing through the origin in ${\mathbb{R}}^2$ cannot exist in this space, if you were to try and walk along it you would have to stop when you hit the missing point!

But being complete does not imply that space is infinite: indeed, the surface of the earth is complete, but finite in size! Anyone who starts walking in any direction on the earth’s surface can continue walking forever without falling off the world, they’ll just come back to their old location over and over.

The other property that us moderns would see as implicitly underlying the first three axioms of Euclid is the infinitude of space.

Definition 5.3 A space $X$ is infinite if there are pairs of points arbitrarily far apart from one another.

The way to check if a space is infinite is to ask, “*for every natural number $N$, can I find a pair of points farther apart than $N$?” From this reasoning, we can see that the real line $\mathbb{R}$ is infinite, as we can look at the points $0$ and $N+1$: they’re at distance $N+1$ apart, which is greater than $N$. The same argument applies to the plane or 3-dimensional space, or any ${\mathbb{R}}^m$.

But this fails for the sphere: while it is complete its finite in size the farthest two points can possibly be from one another is if they are opposites on the sphere (like the north and south pole). And these points are only distance $\pi$ apart, so there are no points on the unit sphere at distance greater than $4$.

5.0.2 Axiom 4: Space is Homogeneous and Isotropic

Euclids fourth postulate is short and intuitive: all right angles are equal. But it’s actually doing a lot of work! To see this, we must unpack what Euclid meant. Two angles are equal (in their measure) if they are congruent: that is, if there is a rigid motion of space that carries one to the other. Thus, Euclid here is claiming that you can always translate and rotate space so that any right angle is carried to any other.

Us moderns would naturally separate this into two actions: you can translate space to carry any point to any other, and then you can separately rotate space about any point, carrying any direction to any other. These properties are called homogenity and isotropy respectively.

Definition 5.4 (Homogeneous Space) A space is homogeneous if for every pair of points in the space, there is a rigid motion taking one to the other.

Definition 5.5 (Isotropic Space) A space is isotropic if for any point $p$ and any two directions leaving $p$, there is a rotation of the space taking one direction to the other.

A space that is not homogenous. Space looks different near points on the hill to points away from the hill. This space is also not isotropic (on the sides of the hill there’s no symmetry between up and down), even though it does have rotational symmetry around the very apex of the hill.

We will be able to make these notions much more precise shortly, when we come back and redevelop geometry from calculus. But even before having everything rigorously defined, its useful to see these properties in action.

Example 5.1 Isotropy implies homogenity.

Proof. Let $p$ and $q$ be distinct points of a space $X$, and draw the line segment between them. Say this line segment is of length $L$, and mark the point $m$ which is of length $L/2$ along it: the midpoint. Since $X$ is isotropic there are rotations about $m$ of any angle we wish.

Rotate about $m$ by 180 degrees: this exchanges the points $p$ and $q$. Thus there is a motion of $X$ taking $p$ to $q$, so $X$ is homogeneous as claimed.

In two dimensions, it turns out that homogenity also implies isotropy: if a space looks the same at every point then it also looks the same in every direction. But this is false in higher dimensions! Indeed, some of my favorite spaces are three dimensional worlds which are homogeneous but not isotropic.

A quick peek at a geometry called ${\mathrm{SL}}_2$, which is homogenous but not isotropic. You can tell, because the center of your field of view looks very different than the view off to the side.

5.0.3 Axiom 5: Space is Flat

The fifth axiom, and all of its equivalents, capture something about space above and beyond the fact that it is infinite in extent and looks the same at every point.

By the list of equivalents to postulate 5, this additional bit of information has a lot of effects on the space: it determines how lines, circles, and triangles behave and it forces the Pythagorean theorem to be true!

It is difficult for us to give a full definition of “flatness” here, but we will in due time. Indeed - much of this course’s purpose is to specifically get acquainted with this notion. For now, we’ll make do with the following intuition: the plane is flat, and any surface you can make by bending the plane without stretching is also flat. Thus, the surface of a cylinder is flat, as you can roll up a sheet of paper without stretching it, as is the surface of a cone.

Definition 5.6 (Modern Axioms for Euclidean Geometry) The Euclidean plane is

• Complete
• Infinite
• Homogeneous
• Isotropic
• Flat

There are spaces which are not flat - the surface of a sphere, for one. Our definition of flatness (and the lack thereof - curvature) will require mathematics beyond the Greeks, and we will return in detail once we have construction our geometric foundations from calculus.