Euclidean & Non-Euclidean Geometries Part 5: Axioms (Cont.)

Continued from Part 4.

I knock a glass candleholder off the shelf during the video, and the sound, while not very loud, might surprise you or your cat. I also knock something else off the ledge, but I don't remember what it was.

EUCLID'S POSTULATE III. For every point O and every point A not equal to O there exists a circle with center O and radius OA.

DEFINITION. The ray AB is the following set of points lying on the line AB: those points that belong to the segment AB and all points C such that B is between A and C. The ray AB is said to emanate from A and to be part of line AB.

DEFINITION. Rays AB and AC are opposite if they are distinct, if they emanate from the same point A, and if they are part of the same line AB = AC.

DEFINITION. An "angle with vertex A" is a point A together with two nonopposite rays AB and AC (called the sides of the angel) emanating from A.

DEFINITION. If two angles BAD and CAD have a common side AD and the other two sides AB and AC form opposite rays, the angles are supplements of each other, or supplementary angles.

DEFINITION. An angle BAD is a right angle if it has a supplementary angle to which it is congruent.

EUCLID'S POSTULATE IV. All right angles are congruent to each other.

DEFINITION. Two lines m and n are parallel if they do not intersect, i.e., if no point lies on both of them.

EUCLID'S POSTULATE V. (THE PARALLEL POSTULATE) For every line l (el) and for every point P that does not lie on l (el) there exists and unique line m through P that is parallel to l (el).

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Tags:

• Geometry
• logic
• axiom
• postulate
• definition

[Prepoceros]

I get the sense that your videos are making almost the same point as my "proving theories" video, but with more rigor and less hand-waving. =) Looking forward to the conclusion!

[DarwinsHamster]

I did post my first video in the series as a response to yours, and in the meantime have used your name in response to other comments and explicitly noted the connection to your video in the expanded description to Part 1 and a longer comment I made on my channel page yesterday in response to a comment there.

[Prepoceros]

Ah, so you did. I hadn't noticed the comment.

Watching is my pleasure. =)

[DarwinsHamster]

There's no way you could have noticed the comments unless you monitored them over time because I change the video descriptions over time and the comments of others followed your viewing of the videos. Thanks again.

[DarwinsHamster]

I will be doing some hand-waving, but hope to make the disconnect between "reality" and mathematics as dramatic as possible. I hope to be able to ask for your help after the last formal video if there are questions from viewers that I can't deal with. Thanks for watching.

[shaynexus]

Bravo. But nowhere have you made the point that all the lines are or are not on a single plane.

Dennis

[DarwinsHamster]

I certainly neglected it way, way too long, but I think I finally said something near the end. I am going to go back and put that clarificantion in the comments for Part 1, and maybe even make a Part 0 to explain your point and to clarify that this all isn't ultimately about geometry, but some other mysterious point I want to eventually make about proofs of theorems vs proofs of theories.

[edgeArchitect]

About parallels:

What if a point P is not fixed, but is at Infinity?

[animimm]

In fact the uniqueness of the circle is included in the definition of a circle, as the def says it is "the set of ALL points.." :)

Thnx for the videos :)