Homework:Non-Euclidean Geometries, Math 3379

Homework:Non-Euclidean Geometries, Math 3379

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1.The Three Point Geometry:

Prove Theorem 2: There are exactly 3 distinct lines in this geometry.

Part A, Suppose there are fewer than 3 lines…

Part B, Suppose there are more than 3 lines…

2.Using the finite geometries as a “go by”, write a set of axioms that would do as the ones that produced the following model. Add an axiom that details the situation with respect to parallel lines.

This is an octahedron;“points” are interpreted as verticesand

“lines” are the edges.

3.An Incidence Geometry with 7 points.

Fano’s Geometry is an incidence geometry with exactly seven points.

Find Fano’s Geometry on the internet: give the reference here:

Copy out the whole geometry:

• Undefined terms
• Axioms

Copy out the model with the points lettered. Write an explanation of how the model matches each axiom exactly.

Write out a list of possible

• definitions – at least 3.

Find and learn the proofs for at least 2

• theorems. Include these in your homework. Legibly.

4.Find 10 same/different pairs for TCG and EG. An example is:

the same:they have the same undefined terms and axioms up to SAS

different:SAS isn’t an axiom in TCG and it IS in EG.

Find 9 more pairs.

You may NOT list axioms singly. Each similarity and difference must be

substantially different from the other ones on the list.

5.In Euclidean Geometry, circles can share no points (ie, they don’t intersect), intersect in one point (tangent, internally tangent) or intersect in two points.

How do TCG circles interact?

Clearly there can be “no intersection”.

What about tangency – are there both internally tangent and externally tangent? Look up internally tangent circles if you’ve never heard the term before.

Intersect in two points, clearly this exists.

Are there intersections of more than two points?

Write a 1 page illustrated essay about two TCG circles and how they interact.

6.Write an essay about Saccheri Quadrilaterals and why they are important.

Include some details comparing and contrasting them across all 3 of the “big geometries”: Euclidean, Spherical, and Hyperbolic.

Fill in the following chart

Geometry / Summit angles type
(acute, right, obtuse) / Summit to base comparison (<, >, =)
Euclidean
Spherical
Hyperbolic

A Saccheri Quadrilateral in Hyperbolic Geometry is illustrated below:

This is the only kind of quadrilateral that you can have in all 3 of the big geometries:

a Saccheri Quadrilateral.

7.Here is a list of statements that are equivalent to the Euclidean Parallel Postulate:

• The area of a triangle can be made arbitrarily large.
• The angle sum of all triangles is a constant.
• The angle sum of any triangle is 180.
• Rectangles exist.
• A circle can be passed through any 3 noncollinear points.
• Given an interior point of a angle, a line can be drawn through that point intersecting both sides of the angle.
• Two parallel lines are everywhere equidistant.
• The perpendicular distance from one of two parallel lines to the other is always bounded.

Consider each statement in turn and find a counterexample in Spherical or Hyperbolic Geometry.

Illustration:

• Two parallel lines are everywhere equidistant.

8.Parallel Lines in Hyperbolic Geometry

H is the line we’re interested in. Point D is our point NOT on H.

To which line is H “asymptotically parallel”?

Which lines are “divergently parallel” to H ?

9.Vertical angles are congruent…in Euclidean Geometry for sure, but what about the other 2 “big ones”?

Analyze the picture below for your consideration of Hyperbolic geometry:

Sketch in an illustration that is an example or is a counterexample in Spherical Geometry:

10.Triangles and Exterior Angles in Hypebolic Geometry

Yes, we still have triangles. No, the sum of the interior angles is not equal to 180; it is LESS THAN 180 as promised. The difference between 180 and the sum of the interior angles of a given Hyperbolic triangle is called the DEFECT of the triangle. In Spherical geometry the difference between the sum of the interior angles of a spherical triangle and 180 is called the EXCESS of the triangle. Note that the defect of the triangle above is 58.6. (The sum of the angles is 121.4) We use the lower case Greek letter delta for defect (  ).

Given that Exterior Angle Inequality Theorem in EG is true, is the exterior angle (LBM) in the picture above demonstrating that the Theorem is true in Hyperbolic Geometry?

Is it true in Spherical Geometry that an Exterior Angle is greater than either remote interior angle? How do you know?

11. Right triangles – we’ve got them in all 3 “Big geometries”.

Here’s a right triangle – the measures of the angles are shown. I built it using an H and the Hperpendicular bisector of that line (again: Hyperbolic tools on the left menu).

 = 54.3

Does the Pythagorean Theorem hold in Hyperbolic Geometry?

Does it hold in Spherical Geometry?

12.Distance

Measure the following distance in Euclidean Geometry:

AM

AN

BM

BN

You use these Euclidean measurements to calculate the Hyperbolic distance with a formula:

the Hdistance from A to B is

is called the “cross product”.

The absolute value of the natural log of the “cross product” is a very clever way to measure distances.

Let’s look at some consequences of this formula.

Here’s a picture with Hyperbolic distances on the left and Euclidean distances on the right. The hyperbolic distances were measured using the hyperbolic distance tool on the left and the Euclidean distances were measured using the Euclidean distance tool on the top menu. On the right, the calculation for Hyperbolic distance is shown. Remember that the calculation uses Euclidean distances. The Hyperbolic tools do the calculation automatically for you.

In Euclidean Geometry the distance between points on a segment is fixed and independent of location in the plane. In Hyperbolic Geometry, however, you have an interesting stretching of calculated distances that depends on whether the points are close to the center of the disc or close to the edge of the disc. Points can be the SAME Euclidean distance apart and have different Hyperbolic distances depending on their location in the disc. This is a function of the distance formula.

Here’s an illustration with two points that are .09 apart in Euclidean geometry and located in two different spots in the disc. Note that the Hyperbolic distances are different and the points are further apart out near the edge of the disc.

Euclidean distance .09

Hyperbolic distance .52

Euclidean distance .09

Hyperbolic distance.12

Notice that the Hyperbolic distance depends on WHERE you are in the disc. Points that are the same Euclidean distance apart in different locations on the disc are different Hyperbolic distances apart.

We will use the Unit Circle, and the x-axis as our Hline.

We will find the distance from D to E using the formula. I’ll do one as an example.

If D has the Cartesian coordinates (1/3, 0) and E has the Cartesian coordinates (3/5, 0), then they are 4/15 .267 apart in Euclidean Geometry.

The Hdistance from D to E is =

DA is 4/3.

EB is 2/5.

DB is 2/3.

EA is 8/5.

Putting these together in the cross product:

And the absolute value of ln (.5) is approximately .69, the Hdistance.

Now, I’m going to give you 5 points. Fill in the following chart with both the Euclidean Distances and the Hyperbolic Distances and draw a conclusion about Hyperbolic Distance and location in the disc.

Points / Distances / Euclidean d / Hyperbolic d
P0 / (0, 0) / NA
P1 / (1/3, 0) / P0 to P1
P2 / (3/5, 0) / P1 to P2
P3 / (7/9, 0) / P2 to P3
P4 / (15/17, 0) / P3 to P4
P5 / (31/33, 0) / P4 to P5

What do you observe about the distances as you move further from the center of the disc?

How is this like or different from Spherical geometry and Euclidean geometry?

Problem 13

Find the Hdistances

d1 from (0, 0) to ( ½, 0)

d2 from (1/2, 0) to (15/16, 0)

d3 from ( ½, 0) to (½,½)

What does this tell you about Hyperbolic distances?

Sketch the Unit disc in Euclidean Geometry, put these points in (as “to scale” as you can) and put both the Hyperbolic and Euclidean distances on your sketch in textboxes. By hand is just fine.