yes, the red lines are parallel

Main ideas / concepts

• Math can change meaning in a broader “context.” Statements that are “wrong” under standard Euclidean geometry can become “correct” in other geometric systems.

• The parallel-lines meme is comparing Euclidean vs. non-Euclidean (specifically finite affine) geometry.

  • Euclidean definition of parallel lines: lines in the same plane that never meet.
  • In the meme, green lines appear to intersect if you use ordinary Euclidean intuition. The claim, however, is that in another geometry they can be considered parallel because they do not share a point in that geometry.

• Background: a “math meme war.”

  • The video describes meme variations and replies (including references to “midwit” and “extremes”) that argue about whether the green lines should count as parallel depending on which geometric context you assume.
  • The creator’s point: confusion comes from how the diagram is drawn and how people import Euclidean intuition into a non-Euclidean setting.

Method / argument presented (finite geometry → affine geometry → proof)

1) Shift from Euclidean to finite geometry

• The video starts from Euclid’s Elements:
  • Euclidean parallelism is governed by Euclid’s parallel postulate (a long axiom).
• It notes that it’s possible to build consistent geometries where the parallel postulate is negated, yielding useful alternative geometries.

2) Check finite “plane” geometries

Projective geometry (finite plane geometry)

• Projective geometry is described using three axioms (as given in a cited textbook).
• Key point: Axiom 3 forbids parallel lines because it states that any two distinct lines intersect.
• Therefore, projective geometry does not match the meme’s claim about green lines being parallel.

Affine geometry (finite plane geometry)

• Affine geometry does allow parallel lines.
• An “affine transformation” is explained:
  • Does not preserve angles or distances
  • Does preserve collinearity and parallelism
• Hence, affine geometry provides the right framework for the meme’s interpretation.

3) Axioms of finite affine geometry (affine planes)

The video presents affine-plane axioms (numbered):

• Affine Axiom 1: For every two distinct points, there is exactly one line containing both.

• Affine Axiom 2 (Playfair’s axiom / parallel postulate form): Given a line (L) and a point (P) not on (L), there exists exactly one line (L’) through (P) that shares no points with (L). (Equivalently: it is the unique “parallel line” through (P).)

• Affine Axiom 3 (non-degenerate / not trivial): There exists a set of four points with no three collinear, ensuring the geometry isn’t just one line.

4) “Order” and the smallest example: affine plane of order 2

• The order (n) of an affine plane is the number of points on each line.
• From the axioms, the smallest affine plane is order 2:
  • Each line contains exactly 2 points.
• A concrete picture can be misleading if drawn with Euclidean-looking lines/intersections.
• Crucial correction: in the affine plane, “parallel” means the lines do not share a point—and in order 2 there may be no available point for them to “intersect” in the Euclidean sense.

5) Main proof: affine plane of order (n) has exactly (n^2) points

The video proves a general statement:

• Claim: A finite affine plane of order (n) contains exactly (n^2) points.

Given setup: From Axiom 3, choose four points with no three collinear, so you can pick three non-collinear points. Let:

• Two of these define a line (L)
• Another pair define a line (L’)
• Let (Q) be a point where the relevant lines “meet” (i.e., lies on both in the sense used by the argument)

Line sizes: In order-(n) affine geometry, each line has (n) points.

Construction / counting argument (lower bound idea)

1. Consider the points on (L’) that are not on (L). There are (n-1) such points.
2. By Axiom 2, through each such point there exists a unique line parallel to (L).
3. These parallel lines contain (n) points each.
4. The structure created this way yields the plane has at least (n^2) points.

Finishing (show it’s not just a lower bound)

• Take an arbitrary point (P) not on (L).
• There is a unique line through (P) parallel to (L) (Axiom 2).
• The video argues that (P) must lie in the previously constructed set of points (i.e., it must be on one of the counted parallel lines).
• Therefore the earlier count is actually the complete point set, proving the total is exactly (n^2).

6) Conclusion tying back to the meme

• In an affine plane (including order 2):
  • The “green lines” are parallel precisely because they do not share a point.
  • In order 2, the geometry is so small that the drawing’s “intersection” (as seen in Euclidean ink) cannot correspond to an actual common point in the affine-plane structure.

Key lessons

• Visual diagrams can mislead if you assume Euclidean meaning of “crossing” and “intersection” when the underlying structure is different.
• Parallelism is definition-relative: here, “parallel” means no common point, not necessarily “doesn’t look like it crosses” in a Euclidean sketch.
• Finite affine planes provide a concrete setting where the meme’s claim is mathematically consistent.

Speakers / sources featured

• Speaker(s) / narrator: An unnamed speaker (the video’s presenter; no personal name given in the subtitles).
• Sources mentioned:
  • Euclid — The Elements (Book I; definitions and axioms including the parallel postulate).
  • O’Hara and Ward — textbook cited for projective geometry axioms.
  • Joy Morris — Combinatorics (proof referenced; link mentioned).
  • Caliba fan — credited with the original “bell curve meme” related to the concept.
  • Cougall Blitzka — mentioned as replying to the meme (“midw” style comment).