• 1-20/22 Introductory Classes Summary (+).
  What are different aspects of geometry? How is the study of geometry organized? Analytic(numbers) , Synthetic(axiomatic),  Transformations (functions) are three ways to organize information and the study of geometry. Also Projective and Differential geometry were mentioned as alternative focuses for studying geometry.
  
• The Course Description was discussed. Still to be covered is the course project on the assignments webpage.
  
• Additional Relevant Notes:
• Different types of geometry:
• Euclidean: Lengths are important
  
• Similarity: Shape is important
• Affine: Parallel lines are important.
• Projective: "Shadows" are important
• Differential: Curvature is important.
• Topological: General shape- especially holes and connectedness- is important.
• What is synthetic geometry? A geometry that focuses on connecting statements (theorems, constructions) to a foundation of "axioms" by using proofs.
• What is analytic geometry? A geometry that focuses on connecting statements (theorems, constructions) to a foundation of number based algebra.
• What is "structural geometry"? A geometry that focuses on connecting statements (theorems, constructions) to a foundation of structures (relations and operations) on sets by using proofs.
• transformations: tools that allow for changing figures:[the basis for studying different geometries in the Klein Erlangen Program. See Felix Klein - YouTube Felix Klein (25 April 1849 – 22 June 1925)
  
• Euclidean: translations, rotations, and reflections.
• Similarity: magnifications, dilations
• Affine: Preserve parallel lines
• Projective:  "linear projections"...line preserving
• Differential:   "smooth".
• Topology: continuous
  

• geometry as an empirical science
• geometry as a formal system of information
• geometry focused on special objects like triangles, or special qualities like convexity.
• A figure C is called convex if given any two points in the figure, A and B, the line segment AB is a subset of the figure.
• Examples. An elliptical region in the plane is convex. A lunar region in the plane is not convex. [Here is an interesting piece of geometry related to the area of lunes.]
• Geometry on the web and using GeoGebra

• Geometry has traditionally been interested in both results- like the Pythagorean Theorem- and foundations -  using  axioms to justify the result in some rigorous organization. We will be concerned with both results and foundations.
  
• We will attempt to build an understanding of both the synthetic and analytic approaches to geometry by using the tools of a structural approach.

•  In the distinction between synthetic and analytic geometry the key connecting concept is the use of measurements.
  
• Initially we will try to avoid when possible the use of measurement based concepts in the structures we use .

•   Consider the on-line version of Euclid's Geometry- especially Book 1:
  
  • The list (on-line) of initial definitions and postulates and the end results of Book I- The Pythagorean Theorem and its converse.
  • Consider Proposition 1 and note it has a statement that was not justified by the definitions and postulates! A more detailed discussion of this proposition is found below.
    
• Consider the Pythagorean Theorem Activity Sheet.
  
• Discuss: What would we need in geometry to make a proof of the PT based on this activity?
• Back to Euclid for some further examination of the postulates:
  
  • Looking at the definitions and postulates, a key feature of any geometric structure appears early- What defines a point and a line?
  • An alternative- minimal geometry:  points: A,B.  lines: {A,B}. That's all!
  • Postulate 1. To draw a straight line from any point to any point.
  • Current view:  Two distinct points determine a unique straight line.
  • Issue: What is a "straight" line?
    
  • Alternative geometry- spherical. Points: points on a unit sphere in coordinate space: { (x,y,z): x² +  y² +  z²=1}
    Straight lines:  Great Circles on the sphere. Determined by a plane in space through (0,0,0).
    Fact: Two points on the sphere that are not antipodal- (Opposite) determine a unique straight line. [Use the plane determined by the two points and (0,0,0).] For this to make this spherical geometry satisfy Post 1- redefine what "distinct" means: If the points on the unit  sphere are antipodal, then they are considered "equal" and thus not distinct.
  
  

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  • Postulate 2. To produce a finite straight line continuously in a straight line.
  • Current view: A straight line has no "endpoints".
  • Issue: What is the nature of a straight line not being "finite"?
    Alternative Geometry.  Points: all numbers in an open interval {x: 0<x<1}. Straight line: The interval (0,1) and all subintervals.
    A "finite" straight line: Any closed sub-interval of (0,1).
    Fact: Any closed sub-interval of (0,1) is properly contained in another closed subinterval of (0,1).
    Proof: Suppose 0<a,b<1. Then [a,b] is contained in the interval `[a/2, (b+1)/2]` which is contained in (0,1).
  • Postulate 5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
  • Current view: Under the given circumstances, the two straight lines are not parallel.
    Issue: How can you find the actual point where the lines meet?
    Here is the most distinctive of the postulates that appear in Euclid.
    The definition of parallel: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
  • Looking at "Common Notions" we see the geometry has language for equality and addition- a relation and an operation for the objects.
    
    
  • Look at how the definitions and postulates are used in Euclid:
  • Consider Euclid's Proposition 1 and  Proposition 2.
  • These propositions demonstrate that Euclid did not treat moving a line segment as an essential property worthy of being at the foundations as an axiom. However, this is a fundamental tool  for all of geometry.
  • Note that in the proof of proposition 1 the point of intersection of circles is presumed to exist without reference to any of the postulates. This presumptions was left implicit for hundreds of years, but were cleared up in the late 19th century (see the work of David Hilbert) when careful attention was given again to the axioms as a whole system.
    
    
  •  Notes on Existence: Existence is an often overlooked quality in a mathematical statement:
    For example it is common to state that `sqrt{2}` is an irrational number.
    The proof starts by assuming `sqrt{2}` is a rational number,  say `sqrt{2} = p/q` where `p` and `q` are natural numbers, so that ... `2 q^2 =  p^2`. From this the usual proof of the statement deduces a contradiction. However, the statement assumes that there is a number the square of which is equal to `2`.
    One can justify the existence of the square root of `2` in many ways. One way presumes that any line segment has a length measured by a real number. Then by the Pythagorean theorem, the diagonal of a unit square will have length `sqrt{2}`.
    
    
  • Alternative "rational coordinate" plane geometry.  [A geometry in which circles do not always intersect.]
  Points: ordered pairs of rational numbers.
  
  An example of a point not in the rational coordinate plane is the point `(sqrt{2},0)`. This point can be constructed in the ordinary plane with straight edge and compass using the circle with center `(0,0)` and radius determined by the points `(0,0)` and `(1,1)`. This circle will meet the X-coordinate axis at the point `(sqrt{2},0)`
  
  The circle with center `(0,0)` and radius `1` and the circle with center `(1,0)` and radius `1` meet in the ordinary plane at the points with coordinates `(1/2, sqrt{3}/2)` and `(1/2, -sqrt{3}/2)` . Since `sqrt{3}/2`  is not a rational number, this ordered pair does not correspond to a point in the rational coordinate plane, so the two circles do not have a point of intersection in the rational coordinate plane.
  

• Revisit Euclid's Proposition 1 and  Proposition 2.
• Proposition 2 demonstrates a concern of Euclid about establishing a "motion" or transformation for this geometry based on fundamental postulates.
• An example of the "web" of information formed in this "axiomatic" synthetic geometry.
  
• Not just demonstrations- but the first two propositions are dedicated to constructions! A focus on existence as well as conditions that guarantee other conditions.

• Note that M&I build an informal presentation of a geometry structure based on using the real numbers.
  
• To explore some of these issues, let's looked at  the Pythagorean Theorem and its proofs.
•   Consider Euclid's statement and proof of Proposition 47.
In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.

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Note that Euclid's treatment in its statement or its "proof" never refers the traditional equation, a²+b²=c².


• Measurement and the Pythagorean Theorem (PT)
  

a² + b² = c²

• Pythagorean Activity Sheet
  
• Virtual Manipulative for PT
• Discuss Pythagorean Theorem and proofs.
• Over 30 proofs of the Pythagorean theorem!
• Many Java Applets that visualize proofs of the Pythagorean Theorem

• In one alternative proof for this theorem illustrated in the GeoGebra sketch below, follow the movie to consider 4 congruent right triangles and the square on the side of the hypotenuse arranged inside of a square with side "b+c" and then moving through the movie the same 4 triangles and 2 squares arranged inside of a square with side "b+c"  . Can you explain how this sketch justifies the theorem?

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Another proof using "shearing" illustrated in the GeoGebra visual proof  by choosing "Shear Pythogoras"and the Java sketch below taken from a Geometers' Sketchpad example that can be connected to Euclid's proof.


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(Based on Euclid's Proof)  D. Bennett 10.9.9
1. Shear the squares on the legs by dragging point P, then point Q, to the line. Shearing does not affect a polygon's area.
2. Shear the square on the hypotenuse by dragging point R to fill the right angle.
3. The resulting shapes are congruent.
4. Therefore, the sum of the squares on the sides equals the square on the hypotenuse.

What kind of structures and assumptions were needed in the first proof with the 4 triangles and squares?

Here are some considerations related to those assumptions:
• What types of objects are being considered?
• points
• lines
  
• line segments
  
• angles
• triangles
• squares
• What types of relations are being considered?
• points on lines
• Lines intersecting at points
• Congruent line segments
• congruent angles
• congruent figures (triangles, squares)
• Decomposing and recomposing a figure.
  
• How could we justify identifying "equal" objects (congruent figures)?
• How do the objects fit together?
• How do movements effect the shapes of objects.

 1. Dissections: How are figures cut and pasted together? What can be achieved using dissections?
 2. Transformations: How are figures transformed? What transformations will leave the measurements of  "area", "lengths", and "angle measures" of figures invariant (unchanged)?

A look at the possibilities of dissections .


Puzzles and Polygons

• Tangrams. More
  
• Tangrams – Use all seven Chinese puzzle pieces to make shapes and solve problems.
  
  
  

• Cutting and reassembling polygons. More ....

• Dissections (like Tangrams) and equidecomposable polygons.
• Use tangram pieces to make a square.
• Note that in putting the pieces together to form any other shape, the area of that shape would be the same as the area of the square unless there is some overlap of the pieces in the shape.
• Question: Is this necessary condition of equal areas sufficient to say that two polygonal regions could be decomposed (cut and pasted) into smaller regions that would be congruent?

Comment: In a sense a positive (yes) response to this question means that one could create a set of smaller shapes with which one could make either of the two regions using precisely these smaller shapes. The answer to this question is yes (in fact this is a late 19th - early 20th century result), which is the basis for the remainder of this section
[The analogous problem in three dimensional geometry: volume equality of polyhedra is a necessary but not sufficient condition for a similar result. This was the third of the famous 23 "Hilbert problems" for the 20th century. This was first demonstrated by Dehn almost as soon as the problem was announced. Dehn used another invariant of polyhedra related to the lengths of the edges and the dihedral angles between the faces of the polyhedra.[ "Ueber den Rauminhalt," Math. Annalen, 55 (1902), 465-478].

See also A New Approach to Hilbert's Third Problem - University of ...by D Benko.

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First, consider some of the background results which were known to Euclid: (1) parallelograms results and (2) triangle results. The justifications for these results can be  reviewed briefly.

(1) a. Parallelograms between a pair of parallel lines and on the same line segment are equal (in the sense of being able to decompose one to reconstruct the other).  Proposition 35.
b. Parallelograms between a pair of parallel lines and on  congruent segments are equal (in the sense of being able to decompose one to reconstruct the other). Proposition 36.

(2) a. The line segment connecting the midpoints of  two sides of a triangle is parallel to the third side and is congruent to one half of the third side.
[The justification of this result is left as an exercise in traditional Euclidean Geometry.]
b. By rotating the small triangle created by connecting the midpoints of two sides of a triangle 180 degrees about one of the midpoints, we obtain a parallelogram. (This shows that the triangle's area is  the area of this parallelogram which can be computed by using the length of the base of the triangle and 1/2 of its altitude- which is the altitude of the parallelogram.)
Compare this with Euclid  Prop. 42 and Prop. 44.

(ii) The triangulation of any polygonal region in the plane is a key element in a proof of the equidecomposable polygon theorem.
The proof of this proposition examines a more careful characterization of the polygonal regions being considered. The key idea of the proof goes by induction on the number n = the number vertices = the number of sides in the polygon.  Proof of the triangulation proposition.

Another component of the proof of the equidecomposable polygon theorem is the ability to "add two parallelograms to form a single parallelogram which is scissors congruent to the two separate parallelograms". Here's how:
Intersect two pairs of parallel lines, l and l' with m and m'- one from each of the given parallelograms. Draw a diagonal HI in the resulting parallelogram.
Cut and translate one parallelogram so that it is scissors congruent to a parallelogram HIJK within the same parallel lines l and l' with one side being the diagonal.
Cut and translate the other parallelogram so that it is scissors congruent to a parallelogram HINO within the same parallel lines m and m' with one side being the diagonal and on the other side of the diagonal HI from the transformed first parallelogram. Now draw the parallel NO to the diagonal in the second transformed parallelogram HINO so that it intersects the parallels  l and l' from the first parallelogram at the points P and Q. This makes one larger parallelogram JKPQ which is scissors congruent to the original two parallelograms.
Compare this with  Euclid Proposition  45.

The film Equidecomposable Polygons also proves the result:
If two polygonal regions in the plane have the same area, then there is a decomposition of each into polygons so that these smaller polygons can be moved individually between the two polygons by translations or half turns (rotations by 180 degrees).
 

REVIEW of basic plane geometry concepts and definitions based on M&I.
M&I build their foundations for Euclidean geometry on a one to one correspondence between points on a line and real numbers and the ability to match angles with numbers between 0 and 180.

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Review materials defining  rays, segments, angles, triangles, and planes  in M&I.
These definitions present a model for geometry based on the real numbers. Points, rays, line segments, etc., are identified with real numbers and intervals of real numbers. Notice that angles in the plane are not oriented and  are always considered to have a measure between 0 and 180.

Review eight of the basic Euclidean constructions described in M&I section 1.2. Note that several of these constructions rely on some foundations that assert the existence of points of intersection of circles. Thus these constructions will not be guaranteed to be effective in a geometric structure where such points do not exist- such as the geometry of the rational number plane.

Some comments about Constructions: It is important to notice that constructions also require a justification (proof) that the construction has in fact been achieved.


In proving the constructions we use some basic euclidean results, such as the congruence of all corresponding sides in two triangles is sufficient to imply the triangles are congruent (SSS). [Other basic Euclidean results are SAS and ASA congruence conditions, as well as the result that corresponding parts of congruent triangles are congruent (CPCTC).]


For these particular constructions to be justified by the same arguments given by Euclid in a geometric structure, the structure will need both an equivalence relation called congruence for line segments, angles, triangles and propositions that connect congruence of triangles to sufficient conditions  like  SSS, SAS, and ASA.
 
The constructions play two important but different roles in (euclidean) geometry:
(i) Construction allow us to "move", deconstruct and reconstruct figures, while maintaining the magnitudes of the pieces and angles . Thus constructions provide the tools for transformations such as rotations and translations.
(ii) Constructions allow us to develop comparative measurements based on a "unit" segment and the "straight" angle.

It should also be noted that the three transformations (translation, rotation, and reflection) commonly used in geometry are connected to constructions as well. For example, to translate a figure by a vector it would be useful to know how to construct parallelograms.
 
 




Note on midpoints: With the construction of midpoints in Euclidean Geometry, we can show that a Euclidean line segment has an infinite (not finite) number of distinct points. Furthermore, if we think of approximating real number distance with points on a segment after establishing a unit length, then we can construct the position of a euclidean point as close as we want to the position where a real number might correspond to a point in that position.

• The construction of points that correspond to numbers on a line.

First construct `P_k` where `k` is an integer using circles.
We can construct points with fractions using powers of `2` for the denominator by bisection.
(However, with only bisection we could not construct a point for `1/3` although we could get very close to that point using a binary representation of that common fraction.) Using  midpoints of segments we can construct points that correspond to `n/(2^k)` for any `n` and `k`. Unfortunately real numbers like `sqrt{2}` and `pi`  cannot be expressed in this way. To obtain a correspondence of points with all real numbers using bisection we need a geometric property that will give a point to correspond to any convergent sequence of numbers of the form `n/(2^k)`.

The continuity axiom for a euclidean line:
• Any non empty family of nested segments will have at least one point in the intersection of the family.
This axiom allows us to make a 1 to 1 correspondence between any real number and a point on a euclidean line once points have been determined to correspond to 0 and 1.

I.e., given P₀ and P₁   for any real number x where a<x<b there is a point P_x where P_x is between P_a  and P_b if and only if  the point `P_{n/(2^k)}` corresponds to the number `n/(2^k) ` for any integer `n` and natural number `k` .

[Review the construction of a line though a given point parallel to a given line. See Euclid  I.31.]
We can also construct Pk when k is a rational number using the theory of similar triangles.

For example: We can bisect or trisect a line segment, giving us the ability to find points representing rational numbers with denominators involving powers or 2 and 3, such as `5/6, 7/18`, etc.
The figure below gives two ways to achieve these constructions. One can see how to generalize these to allow one to construct points to represent any rational number on the line so that the arithmetic of numbers is consistent with the arithmetic of geometry. [Adding segments and adding numbers, etc.]

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 Consider M&I's constructions of the same correspondence of integer and rational points. These also rely on the ability to construct parallel lines.

Tangents to circles.
In considering constructions of tangents to circles we use the characterization of a tangent line as making a right angle with a radius drawn at the point it has in common with the circle. ( Book III Prop. 16.) In our construction, not Euclid's (Book III Prop. 17), we also use the result that any angle inscribed in a semi-circle is a right angle. ( Book III Prop. 31.)
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Note on Three Historical Problems of Constructions.
(1) Trisection of an angle: Since it possible to bisect and trisect any line segment and bisect any angle, the issue is, is it possible to trisect any angle?
(2) Duplication of a cube: Since it is possible to construct a square with twice the area of a given square, the issue is, is it possible to construct a cube with twice the volume of a given cube?
(3) Squaring a circle: Since it is possible to construct a square the same area as any given polygonal region, the issue is, is it possible to construct a square with the same area as a given circle.

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Doing arithmetic with constructions in geometry. Note that the construction above allows one to construct a point Px' from a point Px as long as x is not 0 so that x' x = 1.[ Use the circle of radius 1 with center at P0 to construct the inverse point for Px.]


    The relation of the inversion transformation with respect to a circle and orthogonal circles.
Proposition: If C2 is orthogonal to  C1  (with center O) and A is a point on C2 then the ray OA will intersect C2 at the point A' where A and A' are inverses with respect to the circle C1. Click here for the proof.

Solution: First construct the inverse A'  of A with respect to C1 and then the tangent to C1 at B and the perpendicular bisector of AA' will meet at the center of the desired circle.

2. Construct a circle C2  through two points A and B inside a circle C1 so that C2 is orthogonal to C1.
Solution: This solution is demonstrated in the sketch below.