扩展：介绍一些非欧氏几何

What is Geometry? That could, quite possibly, be a question worthwhile of a year-long course. Therefore, I will only write an article giving an introduction to certain geometries.

Essentially, almost all of the geometries are derived by using or changing Euclid's 5 postulates, so it might be useful listing those.They are:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. Given a point $P$ and a line $l$, there is exactly one parallel line of $l$ that goes through $P$.

Euclidean Geometry-this is the geometry we learn in grade school, derived from Euclid's 5 postulates. Essentially, this is the geometry that the 5th postulate satisfies. This is the "normal" geometry. Within this geometry, triangle angle sums add up to 180 degrees, and quadrilateral angle sum add to 360 degrees. There is exactly one parallel line, so there isn't too much weird stuff like in Hyperbolic Geometry and Elliptic Geometry. Euclidean Geometry is often represented by the Cartesian plane, where each point has an $x$ and $y$ coordinate. Euclidean Geometry was derived by, well, Euclid.

Hyperbolic Geometry. This is an interesting geometry, mostly because they changed the final postulate. Now, there is an infinite number of parallel lines that go through P to line $l$. Within this geometry, triangle angles sum to less than 180. The triangles in Hyperbolic Geometry are known as either singly-asymptotic, doubly-asymptotic, or triply-asymptotic. Even though the triangle sides seem to meet at infinity (parallel), you can still calculate the area of a triangle within this geometry. If you want to know how (kind of hard to explain with words/equations), go google it! (Warning: there may be many, many greek letters.) An interesting thing to note is that all triply-asymptotic triangles are congruent.

Elliptic Geometry. Sometimes known as Riemannian geometry, this is almost the opposite of Hyperbolic Geometry. Instead of an infinite number of parallels, there are none. Any two lines will intersect. This results in a triangle with angle sums of greater than 180 degrees, in fact, you may see a triangle that sums to 270 degrees (3 right angles). The "model" for this plane would be if you took a sphere's surface and took lines as "great circles". Interesting properties is that all lines have length of $\pi$ and the area of an elliptic plane is $2\pi$. The formula for a spherical triangle in this geometry is the sum of the angles subtracting $\pi$. This was proved by Albert Girard, with many, many, many greek variables (trust me, I had to take a lecture on this proof on my birthday).

Affine Geometry. This is quite possible my favorite type of geometry. There are 3 axioms of Euclidean geometry :

1. Given any two points, there exists a line.

2. Playfair's Axiom (Essentially Euclid's 5th postulate)

3. There exists 3 non-collinear points.

Affine Geometry is a generalization of Euclidean Geometry. There are two types of Affine Geometry; a plane with an infinite number of points (Euclidean Plane), and a finite affine plane. A finite affine plane has a finite number of points. The number of points within a plane is $n^{2}$, and the number of lines is $n^2+n$. The smallest plane has 4 points, and 6 lines. Every point within a finite affine plane has $n+1$ lines going through it, and every line has $n$ points on it.

Projective Geometry. As George Thomas, founder of MathPath and USA/Canada MathCamp, wisely said, "Let's go home to mommy. Projective Geometry." This Geometry is the mother of all geometries. All geometries are contained within this. The axioms within this geometry are:

1. There is exactly one line going through points A and B.

2. Any two lines intersect, whether at an ideal point (will be explained later) or a finite point.

3. There exists 3 non-collinear points.

Projective Geometry is essentially Affine Geometry with an addition: The introduction of ideal points and ideal lines. Ideal points are points at infinity where parallel lines meet. It is quite hard to wrap your mind around, but essentially they are points beyond the finite plane. The ideal line is the line that connects all ideal points.

I recommend going to MathPath or MathCamp, as George Thomas teaches week-long classes of all of these geometries. I also recommend reading H.S.M Coxeter's book on the Real Projective Plane.

I wrote an article that covered this within "What is Geometry?", but the article should be here.

What is Geometry? That could, quite possible, be a question worthwhile of a year-long course. Therefore, I will only write an article giving an introduction to certain geometries.

Essentially, almost all of the geometries are derived by using or changing Euclid's $5$ postulates, so it might be useful listing those.

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. Given a point $P$ and a line $\ell$, there is exactly one parallel line of $\ell$ that goes through $P$.

Euclidean Geometry. This geometry is the one we learn in grade school, derived from Euclid's $5$ postulates. Essentially, this is the geometry that the $5$th postulate satisfies. This is the "normal" geometry. Within this geometry, triangle angle sums add up to $180^\circ$, and quadrilateral angle sum add to $360^\circ$. There is exactly one parallel line, so there isn't too much weird stuff like in Hyperbolic Geometry and Elliptic Geometry. Euclidean Geometry is often represented by the Cartesian plane, where each point has an $x$ and $y$ coordinate. Euclidean Geometry was derived by, well, Euclid.

Hyperbolic Geometry. This is an interesting geometry, mostly because they changed the final postulate. Now, there is an infinite number of parallel lines that go through $P$ to line $\ell$. Within this geometry, triangle angles sum to less than $180^\circ$. The triangles in Hyperbolic Geometry are known as either singly-asymptotic, doubly-asymptotic, or triply-asymptotic. Even though the triangle sides seem to meet at infinity (parallel), you can still calculate the area of a triangle within this geometry. If you want to know how (kind of hard to explain with words/equations), go google it! (Warning: there may be many, many greek letters.) An interesting thing to note is that all triply-asymptotic triangles are congruent.

Elliptic Geometry. Sometimes known as Riemannian geometry, this is almost the opposite of Hyperbolic Geometry. Instead of an infinite number of parallels, there are none. Any two lines will intersect. This results in a triangle with angle sums of greater than $180^\circ$, in fact, you may see a triangle that sums to $270^\circ$ ($3$ right angles). The "model" for this plane would be if you took a sphere's surface and took lines as "great circles". Interesting properties is that all lines have length of $\pi$ and the area of an elliptic plane is $2\pi$. The formula for a spherical triangle in this geometry is the sum of the angles subtracting $\pi$. This was proved by Albert Girard, with many, many, many greek variables (trust me, I had to take a lecture on this proof on my birthday).

Affine Geometry. This is quite possible my favorite type of geometry. There are $3$ axioms of Euclidean geometry.

1. Given any two points, there exists a line.

2. Playfair's Axiom (Essentially Euclid's $5$th postulate)

3. There exists $3$ non-collinear points.

Affine Geometry is a generalization of Euclidean Geometry. There are two types of Affine Geometry; a plane with an infinite number of points (Euclidean Plane), and a finite affine plane. A finite affine plane has a finite number of points. The number of points within a plane is $n^2$, and the number of lines is $n^2+n$. The smallest plane has $4$ points, and $6$ lines. Every point within a finite affine plane has $n+1$ lines going through it, and every line has $n$ points on it.

Projective Geometry. As George Thomas, founder of MathPath and USA/Canada MathCamp, wisely said, "Let's go home to mommy. Projective Geometry." This Geometry is the mother of all geometries. All geometries are contained within this. The axioms within this geometry are:

1. There is exactly one line going through points $A$ and $B$.

2. Any two lines intersect, whether at an ideal point (will be explained later) or a finite point.

3. There exists $3$ non-collinear points.

Projective Geometry is essentially Affine Geometry with an addition: The introduction of ideal points and ideal lines. Ideal points are points at infinity where parallel lines meet. It is quite hard to wrap your mind around, but essentially they are points beyond the finite plane. The ideal line is the line that connects all ideal points.

I recommend going to MathPath or MathCamp, as George Thomas teaches week-long classes of all of these geometries. I also recommend reading H.S.M Coxeter's book on the Real Projective Plane.

The geometry most of us learn in school is Euclidean geometry. Euclidean geometry is based off Euclid's findings and postulates of his book "Elements." However, what if we did not use his logical findings? What if we looked at geometry differently? Let's look at Euclid's fifth postulate, the Parallel Postulate. It says that within a two-dimensional plane, for any given line $\ell$ and a point A, which is not on $\ell$, there is exactly one line through A that does not intersect $\ell$.

But, what if this postulate is looked at differently?

That is what hyperbolic and elliptic geometry says.

In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting $\ell$, while in elliptic geometry, any line through A intersects $\ell$.