Euclid described the world using an abstract model that transitions from 3D objects to dimensionless points.
Dimensions:
Solid: \(3D\) (has shape, size, and position).
Surface: \(2D\) (boundaries of solids).
Line: \(1D\) (boundaries of surfaces).
Point: \(0D\) (ends of lines; has no part).
Surface: \(2D\) (boundaries of solids).
Line: \(1D\) (boundaries of surfaces).
Point: \(0D\) (ends of lines; has no part).
Undefined Terms:
In modern geometry, point, line, and plane are considered undefined terms because they are the simplest concepts used to define everything else.
Parallel Lines: Lines in the same plane that never intersect and maintain a constant distance.
Perpendicular Lines: Lines that intersect at a \(90°\) angle.
Circle: Defined by a center and a radius.
Definitions:
Line Segment: A "terminated line" with two definite endpoints.Parallel Lines: Lines in the same plane that never intersect and maintain a constant distance.
Perpendicular Lines: Lines that intersect at a \(90°\) angle.
Circle: Defined by a center and a radius.
Euclid's Axioms:
Axioms are self-evident truths applicable to all fields of mathematics, not just geometry.1. Transitivity: Things equal to the same thing are equal to each other.
2. Addition: If equals are added to equals, the wholes are equal.
3. Subtraction: If equals are subtracted from equals, the remainders are equal.
4. Coincidence: Things that coincide with one another (overlap perfectly) are equal.
5. The Whole: The whole is always greater than the part.
6. Doubles: Things which are double of the same things are equal.
7. Halves: Things which are halves of the same things are equal.
Euclid's Five Postulates:
Postulates are assumptions specific to the field of geometry.
Postulate 1: A straight line can be drawn from any one point to any other point. (Axiom: This line is unique).
Postulate 2: A terminated line (segment) can be produced indefinitely in a straight line.
Postulate 3: A circle can be drawn with any center and any radius.
Postulate 4: All right angles are equal to one another.
Postulate 5 (Parallel Postulate): If a line falling on two straight lines makes interior angles on the same side totaling less than 180°, the two lines will eventually meet on that side.
Postulate 2: A terminated line (segment) can be produced indefinitely in a straight line.
Postulate 3: A circle can be drawn with any center and any radius.
Postulate 4: All right angles are equal to one another.
Postulate 5 (Parallel Postulate): If a line falling on two straight lines makes interior angles on the same side totaling less than 180°, the two lines will eventually meet on that side.
Equivalent Versions of the Fifth Postulate:
Because the Fifth Postulate is complex, simpler versions were developed:
- Playfair’s Axiom: For every line \(l\) and point \(P\) not on \(l\), there exists a unique line \(m\) passing through \(P\) that is parallel to \(l\).
- Intersection Version: Two distinct intersecting lines cannot be parallel to the same line.
Important!
Theorems (Propositions): Statements that are proven using axioms, postulates, and previously proven theorems.
Non-Euclidean Geometry: While Euclidean geometry deals with flat surfaces, non-Euclidean geometry (like Spherical geometry) deals with curved surfaces where these postulates (especially the fifth) do not apply.
Non-Euclidean Geometry: While Euclidean geometry deals with flat surfaces, non-Euclidean geometry (like Spherical geometry) deals with curved surfaces where these postulates (especially the fifth) do not apply.