Conventional Euclidean Rules

Euclidean Geometry

A geometry is a particular system of mathematical laws that govern and regulate a dimension. We live in the Euclidean dimension.

Euclidean geometry is the name for the geometry that follows all of the usual rules that apply to our world on Earth, such as:

• The shortest line between two points is a straight one.
• The length to which any straight line segment can be extended is limitless.
• A circle can be drawn with any centerpoint coupled with any radius.
• Right angles are all equal to one another.
• If two straight lines cross another straight line, and the interior angles formed by their crossing are each less than 90 degrees, then on that smaller side of the third line, the other two lines will eventually meet each other.

(simplified)

These five rules are known as Euclid's postulates. They are the most core, the most fundamental rules of Euclidean geometry and are the basis for most other rules.

• Predictably enough, that last one, known as the Parallel Postulate, is the source of much frustration, since it is not a simple and concise statement. Simplicity is especially preferred in math. Mathemeticians tried for centuries to prove the results that the other postulates produce without using the Parallel Postulate, or to prove it as a theorem from the other four, but they failed. However, there are several non-Euclidean geometries made with only the first four postulates and not the fifth.
  Another way to word this fifth postulate: For any given line A and separate point B, there is exactly one line that goes through/uses point B but does NOT cross line A. (This statement is, essentially, equivalent to Euclid's fifth postulate. It is named the Playfair postulate.)

The term "non-Euclidean geometry" means any geometry other than ours; i.e. that breaks at least one of these postulates. However, sometimes scientists use the term to mean hyperbolic geometry (which is a specific type of non-Euclidean geometry), as opposed to any unusual geometry.

Some other rules that you may be more familiar with include:

• Parallel lines are a constant distance from each other.
• The angles of a triangle always add up to 180 degrees.
• All right angles are congruent.

The core basis for Euclidean versus non-Euclidean geometry is parallel lines. That is the main difference between each geometry: the rules that parallel lines follow.

History (WIP)

The man known as Euclid was a Greek mathematician, around 300 BCE. He was most famous for publishing Elements, a treatise regarding geometry and detailing these five postulates. Euclidean geometry is named as such because Euclid is the person who defined it as it is understood today.

Euclid began Elements with a strict set of few mathematical assumptions and attempted to use them to prove all other results in the treatise. These assumptions (listed as definitions, common notions, and postulates) were the foundations for defining Euclidean geometry. Almost as soon as he published his work, widespread discussion arose as others tried to disprove his theories or prove him correct. However, this debate led to the accidental discovery of the possibility of laws of geometry that fit together, but did not follow conventional rules of the known universe at the time. This became known as non-Euclidean geometry, although the idea wasn't really accepted as probable or real until the early nineteenth century.

Footnote: A Fun Fact About Spheres

Realistically, spheres do not exist. After all, for a sphere to be a true sphere, every single one of its infinite points must be at an exactly equal distance from its center, by definition. Similarly precise rules govern other basic polygons like triangles and squares. The material (e.g. plastic) of real spheres must occupy all of their points, otherwise the sphere is either not a sphere or nonexistent and nothing more than a specified section of space.

Atoms never stop moving and there are a finite number of subatomic particles in anything, so one cannot construct such material to fit that criteria. Therefore, despite there being innumerable imperfect and irregular spheres in the universe, you will never come across a technically, mathematically accurate sphere. Thus why topology is a largely theoretical field (but it has indeed been useful in the real world).

Also, I would say circle, but circles already can't exist in a 3D world, since circles are 1D; and I would say disc, but discs are 2D.

• Discs are the reigon in a plane whose boundary is a circle. In simplified terms, it is a filled-in circle, or your classic polka dot.

I SWEAR I meant for this to be a footnote. The words and questions just kept coming. I hope it contributes to clarity instead of befuddlement, tautology, convolution, etc.