Geometry Extended

Our purpose in compiling and presenting this simple geometry is to provide an example of how higher dimensional geometry can successfully be integrated into secondary school level mathematics courses, without sacrificing mathematical rigor. Towards that end, our geometry is a highly derivative work, drawing many postulates, theorems, and sometimes even proofs from Houghton-Mifflin's outstanding School Mathematics Geometry . This book is not mathematically naive, but is nevertheless acessible to the beginner, and so provided a comfortable starting point for our work on this project.

Our geometry adheres as much as conveniently possible to the traditional form for geometries, deriving theorems about sets of undefined primitives from unproven postulates. In addition, in order to keep this geometry as user-friendly as possible, it is synthetic and Euclidean. The only unfamiliar element of it should be the fact that it applies to sets of points in four dimensions.

Included is a list of all postulates and theorems.

It is necessary in any logical system to start with some undefined terms, sometimes called primitves. All other terms may be defined in terms of these few primitives. While presenting the undefined terms below, we will offer some intuitive examples of physical representations of them, but these intuitive explanations are not to be confused with rigorous definitions. They are simply to provide a more gut-level understanding of what the theorems and postulates to follow are saying.

Our first undefined term is point. Euclid described the point as "that which has no part." Euclid meant that the point has no extension, in neither breadth, nor width, nor height. It is usually represented in geometric figures as a tiny dot, but keep in mind that this tiny dot is just a convenient representation of a point; if it truly had no width or breadth or depth, it would be impossible to see. The abstract points which concern geometry are without extension at all. The point is an abstraction of location; it corresponds perfectly to a precise place.

The next undefined term is line. A line, as will be made clearer later on, is in fact a set of points, but it is nonetheless not definable entirely in terms of points. When we talk about lines, we are talking about straight lines, such as the edge of a ruler. The line only extends in one dimension, and extends infinitely in both directions along that dimension. A line is also thought of as infinitely dense with points, such that if you grab it at any place along the infinite line, you are guaranteed to touch it at a point-- another way of saying this is that there are no holes in the line.

Next we have the plane. A plane, like a line, is a set of points, but one which cannot be defined solely in terms of points, and so is treated as an undefined term. The plane has extension in width and breadth, but not depth. It is analogous to the line, in that as the line is straight, the plane is flat; as the line is infintely dense with points, so is the plane; and as the line extends infinitely in one dimension, the plane extends infinitely in two dimensions.

Finally, we will also be using the term space, or sometimes hyperplane to denote a set of points which extends in three mutually perpendicular directions. A space is the logical extension of the progression from point (0 dimensions) to line (1 dimension) to plane (2 dimensions). Since our geometry is treating points in four dimensions, however, the notion of space as used in it may be quite different from the ordinary understanding of a space being the set of all points. In fact, there are an infinite number of spaces in our geometry, and we will often make reference to two spaces intersecting, for example. In ordinary experience, one is unlikely to encounter spaces intersecting, but for the purposes of our geometry, they will.

2.1 Initial Postulates

Before continuing, we must make some unproven assumptions on which to base our proofs in this chapter.

Another way of putting this is to say that two points determine a line. When talking about lines, it is often handy to use this postulate to only refer to two points when speaking about the line; since the two points determine a line, there is no possibility of confusion as to which line one is referring to. When we write, "A<->B", we mean, "the line determined by points A and B," which is the same as "B<->A."

Notice that we aren't using any units to measure distance in. Geometry is just an abstraction; if one wanted to apply it to real world problems, one would have to be concerned with units of measurement, but for our abstraction generic real numbers will suffice. When referring to the distance between points A and B, we will write, "AB," which is equivalent to "BA."

• there is a one to one correspondence between the set of points on the line and the set of real numbers, and
• the distance between two points equals the absolute value of the difference of the corresponding numbers

Such a correspondence is called a coordinate system. Given a coordinate system for any line, the number corresponding to a given point is called the coordinate of the point.

The Ruler Postulate tells us that these coordinate systems exist, but doesn't give us any way of quickly creating one. For that purpose, we will use the Ruler Postulate to prove our first theorem, the Ruler Placement Theorem.

2.2 Betweenness

Betweenness is an excellent example of how intuitive concepts can be codified mathematically. Nearly everyone has an intuitive understanding of what it means to say that an object is between two other objects. This intuitive understanding is not mathematically rigorous enough for our purposes, however. Thus the following definition:

B is between A and C if

1. A, B, and C are distinct points on the same line, and
2. AB + BC = AC.

The definition uses distance, and applying the concept of a coordinate system we have the following theorem.

• AB = |y-x|
• BC = |z-y|
• AC = |z-x|.

• AB = y-x
• BC = z-y
• AC = z-x

Since AB + BC = AC, A-B-C by the definition of between.

Let the respective coordinates of A, B and C be x, y and z. Either x < y or y < x. If x < y, then there are three possibilites:

1. z < x < y,
2. x < z < y,
3. x < y < z.

1. y < x < z,
2. y < z < x,
3. z < y < x.

If A-B-C, then by definition of between, AB + BC = AC, and by the Distance postulate, AB, BC and AC are all positive numbers. This means that AC > BC and AC > AB. If A-C-B, similarly, AC + BC = AB, so that AB > AC and AB > CB. Similarly, if B-A-C, then BA + AC = BC, so that BC > AB and BC > AC.

Now, it is clear that at most one of the possible cases can be true: if A-B-C, (meaning that AC > AB and AC > BC) then it is impossible for C to be between A and B, as AB would be simultaneously less than and greater than AC. It is also impossible for A to be between B and C, because BC would be both less than and greater than AC. Similar logic applies to the other two cases, so that at most one point can be between the other two points.

2.3 Segments, Rays and Midpoints

Obviously, one rarely encounters lines of infinite length in problems involving geometric truths. It is much more likely that one is concerned with problems concerning finite, straight lengths. For this reason, geometry provides an abstractions of a portion of a straight line. These abstractions are called segments and rays.

Given any two points A and B. The set containing A, B, and all points between A and B, is a segment, and is denoted by A--B or B--A. The points A and B are the endpoints of the segment. The segment is sometimes said to join its endpoints. The length of a segment is simply the distance between its endpoints, and as such is written as AB.

The ray A->B is the union of

1. the segment joining A and B, and
2. the set of all points C such that A-B-C.

Let us first examine the case where x < y. Now, either y < z or y > z. Suppose for the time being that y > z. We do not know whether x < z < y or z < x < y, but it is immaterial. If either one is true, then we have violated Thm 2.3, because we are given that A-B-C, whereas the inequalities above would assert respectively that A-C-B and C-A-B. Therefore, if x < y, x < y < z.

The other case, where x > y, works out similarly.

By the lemma, either x < y < z, or z < x < y. Let us first examine the case where x < y < z. Let P be any point on B->C, and let p denote its coordinate. We wish to show that p >= y.

By the definition of ray, either P is B, B-P-C, or B-C-P. In the first case, p=y and p >=y. In the second case, by the lemma, either y < b < z or z < b < y. Since we are examining the case where y < z, we know that b > y. If, on the other hand, B-C-P, the lemma tells us that either y < z < p or p < z < y, and again we know that y < z < p because we are examining the case where y < z. So indeed, for any point P on ray B->C, its coordinate is greater than or equal to the coordinate of B.

What of the opposite ray? Again, let P be any point on B->A, and let p be P's coordinate. Either P is B, or A-P-B, or P-A-B. We wish to show that in all three cases, p <= y. If P is B, we have the case where p = y. If A-P-B, the lemma indicates that either x < p < y, or y < p < x, and just as above we know that x < p < y because we are examining the case where x < y < z. So we know that p < y. Again, if P-A-B, by the lemma, either p < x < y or y < x < p, and we know that p < x < y because we know that x < y. So again, p < y.

To sum up: we first observed that either x < y < z or z < y < x. We decided to examine the first case, and have proven that the demonstrandum holds for it, because for any point on B->C, its coordinate is greater than or equal to B's coordinate, and for any point on B->A (which is, by definition of opposite ray, B->C's opposite ray), its coordinate is less than or equal to B's. The same logic would apply for the case where z < y < x, and it would be unnecessarily repetitive to include the same arguments again here.

The next theorem pertains to the midpoints of segments. A point B is called a midpoint of segment A--C if A-B-C and AB = BC.

We first designate a coordinate system for A<->C such that the coordinate of A is zero and the coordinate of C is positive. By the Point-Plotting Theorem, there is exactly one point B of A->C such that AB =1/2*AC. Since 0 < 1/2*AC < AC, B is between A amd C by Thm 2.2, so AB + BC = AC. Since AB = 1/2*AC, we can see by sybtraction that BC = 1/2*AC as well. Since A-B-C and AB = BC, B, is a midpoint of A--C.

For any other point D between A and C, AD is the coordinate of D and AD != 1/2*AC. We know that AD + DC = AC. Then AD != DC, because if AD = DC, then AD + AD = AC, so AD = 1/2*AC, which is impossible. So, D is not a midpoint of A--C, and B is the only midpoint of A--C.

Before proceeding, a simple definition: Hyperspace is the set of all points.

1. Every plane contains at least three noncollinear points.
2. Every space contains at least four noncoplanar points.
3. Hyperspace contains at least five noncospatial points.

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   Postulate 5 If a plane contains two points of a line, then the plane contains the whole line.

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   The next postulate is analogous to Postulate 5.

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   Postulate 6 If a space contains three noncollinear points of a plane, then the space contains the whole plane.

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   Theorem 3.2 If a line intersects a plane not containing it, then the intersection contains exactly one point.
   P R O O F The proof follows directly from postulate 5; if the intersection of the line and the plane contained more than one point, the plane would contain the line, which violates the given. Therefore, no more than one point can be in the intersection. Since it is given that the line intersects the plane, the intersection can contain no more than one point.

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   3.2 Sets Determining a Plane

   Postulate 7 Any three points lie in at least one plane, and any three noncollinear points lie in exactly one plane.
   Thm 3.4 Let l be a line and P be a point not on l. Then there is exactly one plane containing l and P.
   P R O O F By the Ruler Postulate, l contains at least two different points, which we call A and B. The points, A, B and P are noncollinear. By Postulate 7, there is exactly one plane containing A, B and P, which we will call E. Postulate 5 insures that E contains l, because it contains A and B. No other plane can contain l and P because no other plane contains A, B and P.

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   Thm 3.5 Let l1 and l2 be any two intersecting lines. Then there is exactly one plane containing them.
   P R O O F Let P be a point on l2, but not on l1, and let Q be the point o fintersection of l1 and l2. By Thm 3.4, there is a plane E containing P and l1. Since P and Q are in E and on l2, Postulate 5 insures that E contains l2. Thus the plane E contians both l1 and l2. No other plane can contain l2 and l1 because no other plane can contain l1 and P.

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   Postulate 7 If two distinct planes intersect, then their intersection is a line.

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   3.3 Sets Determining a Space

   This section is developed in parallel with 3.2. The theorems and postulates in it are the higher dimensional analogues of the facts about planes in 3.2

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   Postulate 8 Any four points lie in at least one space, and any four noncoplanar points lie in exactly one space.

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   Thm 3.6 Let E be a plane and P a point not on it. Then there is exactly one space containing E and P.
   P R O O F By Postulate 4, E contains at least three noncollinear points. Let these three points be called A, B and C. The set of points {A,B,C,P} is noncoplanar because A, B and C lie in exactly one plane, E, by postulate six, and so, by postulate 6x, the four points lie in exactly one space. This space contains E by Postulate 6 because it contains three points of E. No other space can contain E and P because no other space can contain A, B, C and P.

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   Thm 3.7 Let E and F be two intersecting planes. Then there is exactly one space containing them.
   P R O O F Let l be the line of intersection of E and F. Let P be a point on E that is not on F, and let A and B be distinct points on l. By Thm 3.5, there exists a space S containing F and P. It also contains E by postulate 6, because it contains three noncollinear points of E. No other space can contain E and F because no other space can contain F and P.

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   Postulate 9 If two distinct spaces intersect, then their intersection is a plane.

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   3.4 Convexity and Separation

   In this section we examine mathematically the intuitively obvious ideas of convexity and separation. First a definition: a set of points is convex if for any two points P and Q of the set, the entire segment P--Q is in the set. Using this definition, we will state the following important postulate:

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   Postulate 10: The Plane Separation Postulate Given any line and any plane containing it. The points of the plane that do not lie on the line form two disjoint sets such that
   1. each of the sets is convex and non-empty and
   2. if P is any point of one set and Q is any point in the other set, then the segment P--Q intersects the line.

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   Given a line l and a plane E containing l, each of the two sets established by postulate 8 is called a half-plane with edge l. N.B: neither half-plane contains its edge. We say that l separates E into two half-planes, called opposite half-planes.

   The next

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   Thm 3.8 Given any plane E, any line l in E, and any two points A and B in E but not on l. If no point of l is between A and B, then A and B lie in the same half-plane with edge l.
   P R O O F Suppose that A and B were in opposite half-planes with edge l, then by the Plane Separation Postulate, A--B would intersect l, and hence there would be a point of l between A and B. But this contradicts our hypothesis. Therefore A and B are in the same half-plane with edge l.

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   Thm 3.9 Given any plane E, any line l in E, and any two points A and B that are in E but not on l. If some point of l is between A and B, then A and B are in opposite half-planes with edge l.
   P R O O F Suppose that A and B were in the same half-plane with edge l. Then that half-plane, which is convex, would contain A--B, and thus no point of l would be between A and B. But this would contradict the given data. Therefore, A and B are not in the same half-plane with edge l.

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   3.5 More on Separation

   Postulate 11: The Space Separation Postulate Given any plane and any space containing it. The points of space that do not lie on the plane E form two disjoint sets such that
   1. each of the sets is convex and non-empty and
   2. if P is any point in one set and Q is any point in the other setm then the segment P--Q intersects the plane.

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   Each of the sets described in the Space Separation Postulate is called a half-space with face E. A plane is said to separate a space into two half-spaces that are called opposite half-spaces.

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   Postulate 12: The Hyperspace Separation Postulate The points of hyperspace that do not lie in a given space form two disjoint sets such that
   1. each of the sets is convex and non-empty, and
   2. if P is any point in one set and Q is any point in the other set, then the segment P--Q intersects the space.

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   Similarly, these two disjoint sets are called opposite half-hyperspaces with boundary E.

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   Thm 3.10 Given any space S and any plane E in S and any two points A and B that are in S but not on E. If no point of E is between A and B, then A and B are in the same half-space with face E.
   P R O O F Suppose that A and B were in opposite half-spaces with face E. Then, by the Space Separation Postulate, A--B would intersect E. Then, however, a point of E would be between A and B. This would violate the hypothesis. Thus, A and B must be in opposite half-spaces with face E.

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   Thm 3.11 Given any space S and any plane E in S, and any two ponts A and B that arein S but not on E. If some point of E is between A and B, then A and B are in opposite half-spaces with face E. The proof is analogous to that of Thm 3.9

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   Thm 3.12 If a ray contains a point of a half-plane, and if the endpoint of the ray is on the edge of the half-plane, then the interior of the ray lies in the half-plane. P R O O F Let H be a half-plane with edge l. Let A->B be a ray with its endpoint on l and B in H. We wish to show that the interior of A->B lies in half-plane H.

   By the Plane Separation Postulate, H is convex, so A--B is contained in (H U l) (by def. of convexity). For any C such that A-B-C, B--C is contained in H, also by definition of convexity. A->B's interior consists of the union of the interior of A--B with all such C's, so A->B's interior is containted in half-plane H.

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   The final two theorems are offered without proof, as their proof runs along lines quite similar to that of thm 3.12.

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   Thm 3.13 If a ray contains a point of a half-space, and if the endpoint of the ray is on the face of the half-space, then the interior of the ray lies in the half-space.

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   Thm 3.14 If a ray contains a point of a half-hperspace, and if the endpoint of the ray is on the face of the half-hyperspace, then theinterior of the ray lies in the half-hyperspace.

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