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Geometry


Foundations of Geometry


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Geometry

Geometry is the study of figures. The most common types of geometry are plane geometry, solid geometry and spherical geometry to study the properties of a figure about its shape, size or relative position in space. 

The Elements of Geometry [1]

There are three fundamental objects in describing elements of geometry. The first element is point which is a 0-dimensional object used to represent a unique location in space, or Euclidean space. Points are usually denoted by capital letters A, B, C. The second element is line or straight line which is a 1-dimensional object used to represent the linear dimension or length of a straight object with no breadth in space, or Euclidean space. Lines are usually denoted by small letters a, b, c. The third element is plane or flat plane surface which is a 2-dimensional object used to represent the plane area of a flat object with no thickness in space, or Euclidean space. Planes are usually denoted by Greek letters α, β, γ. In other words, points are the fundamental elements of linear geometry. points and straight lines are the fundamental elements of plane geometry. And points, staight lines and flat planes are the fundamental elements of space which form the geometry of space.

The Axioms of Geometry[1]

In order to make the system of geometry work, some mutual relations are developed based on a small set of axioms which are called the axioms of geometry. These fundamental facts of intuition, that forms the foundations of geometry, can be arranged in five groups according to Hilbert, D.

  1. Axioms of Connection

  2. Axioms of Order

  3. Axiom of Parallels

  4. Axioms of Congruence

  5. Axiom of Continuity

Axioms of Connection[1]

The axioms of connection describe and establish a connection between the fundamental concepts of those fundamental elements, point, straight line, and plane.

  1. Two distinct points A and B always completely define or determine a staight line a, i.e. AB=a or BA=a.

    In other words, a line a can be constructed by joining A and or with B. Geometrically, points A and B lie upon line a, or both points A and B are points of line a, or line a goes through points A and B.

  2. Any two distinct points of a straight line completely determine that line. if AB=a and AC=a, where B≠C, then  points A, B, and C are points of line a and implys BC=a by joining BC also.

  3. Three distinct points A and B not lie in the same straight line always completely define or determine a plane α, i.e. ABC=α.

  4. Any three distinct points, which do not lie in the same straight line, of a plane completely determine that plane.

  5. If two points A, and B of a straight line a lie in a plane α, then every point of line a lies in plane α.

    In other words, the line a lies in the plane α.

  6. If two planes α, and β have a point A in common, then two planes have at least a second point B in common.

  7. Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not liying in a plane.

Axioms of Order[1]

The axioms of order describe and establish an order of sequence of points upon a straight line, in a plane and in space. The order of points can usually be expressed in form of a point between the other two points. In stead of focusing on the whole straight line, a new object called segment, named AB or BA are defined to name a part of the straight line between two points, A and B. Only points lying between A and B within the segment AB are called points of the segment AB. The points A and B are called the extremities of the segment AB.

  1. If A, B, and C are points of a straight line and B lies between A and C, then B lies also between C and A.

  2. If A, C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D such that C lies between A and D. i.e of order A, B, C, D.

  3. Of any three points situated on a straight line, there is always one and only one which lies between the other two.

  4. Any four points A, B, C, D, of a straight line can always be so arranged that B shall lie between A and C, and also between A and D, and furthermore, so that C shall lie between A and D, and also between B and D. i.e of order A, B, C, D.

    Note: This axiom is proved to be redundant as shown is "On the Projective Axioms of Geometry" by Moore, E.H.[1].

  5. Let A, B, C be three points, not situated in the same straight line, lying in the plane ABC and let a be the straight line lying in the plane ABC without passing through any of the points A, B, C. If the straight line a passes through a point of the segment AB, then the straight line a will also pass through either a point of the segment BC or apoint of the segment AC.

Axiom of Parallels (Euclid's Axiom)[1]

The axiom of parallels describes the relationship between two straight line.

  • In a plane α, there can be drawn through any point A, lying outside fo a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel line to line a through the given point A.

Axioms of Congruence[1]

The axioms of congruence describes the relationship between two objects of geometry. In order to handle complex figure, a new object called angle, named ∠(h,k) or ∠(k,h) are defined to name the system of two half-rays lying in a plane α and emanating from an the intersecting point O.

  1. If A, B are two points on a straight line a, and if A' is a point upon the same or another straight line a;, then, upon a given side of A' on the straight line a', there is one and only one point B' so that the segment AB or BA is congruent to the segment A'B'. The relation is denoted by AB≡A'B'. Ane every sement is congruent to itself; that is, AB≡AB

  2. If a segment AB is congruent to the segment A'B' and also to the segment A"B", then the segment A'B' is congruent to the segment A"B"; that is if AB≡A'B' and AB≡A"B", then A'B'≡A"B".

  3. Let AB and BC be two segements of a straight line a which have no points in common aside from the point B, and, furthermore, let A'B' and B'C' be two segments of the same or of another straight line a' having, likewise, no point other than B' in common. Then, if AB≡A'B' and BC≡B'C' then AC≡A'C'.

  4. Let an angle (h,k) be given in the plane α and let a straight line a' be given in a plane a'. Suppose also that, in the plane α', a definite side of the straight line a' be assigned. If a half-ray h' of the straight line a' emanating from a point O' of this line, then in the plane α' there is one and only one half-ray k' such that the angle (h,k), or (k,h), is congruent to the angle (h',k') and at the same time all interior points of the angle (h',k') lie upon the given side of a'. That is ∠(h,k)≡∠(h',k') and every angle is congruent to itself, ∠(h,k)≡∠(h,k) and ∠(h,k)≡∠(k,h).

  5. If the angle (h,k) is congruent to the angle (h',k') and to the angle (h",k"), then the angle (h',k') is congruent to the angle (h",k"). That is if ∠(h,k)≡∠(h',k') and ∠(h,k)≡∠(h",k"), then ∠(h',k')≡∠(h",k").

  6. If, in the two triangles ABC and A'B'C', the congruences AB≡A'B', AC≡A'C', ∠BAC≡∠B'A'C' hold, then the congruences ∠ABC≡∠A'B'C' and ∠ACB≡∠A'C'B' also hold.

Axiom of Continuity (Archimedes's Axiom)[1]

The axioms of continuity describes the existence of any given point between two points in a straight line by dividing the straight line into segments continuously.

  • Let A1 be any point upon a straight line between the arbitrarly chosen points A and B. Take the points A1, A2, A3,... so that A1 lies between A and A2, A2 between A and A3, A3 between A and A4, etc. Moreover, let the segments  AA1 , A1A2 , A2A3 , A3A4 ,... be equal to one another. Then, among this series of points, there always exista a certain point An such that B lies between A and An.

Axiom of Completeness[1]: An additional axiom to approach geometry in a more theoretical point of view.

  • To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.


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References

  1. Hilbert, D. (translated by Townsend E.J.), 1902, The Foundations of Geometry, The Open Court Publishing Company, Chicago
  2. Moore, E.H., 1902, On the projective axioms of geometry, Transactions of the American Mathematical Society 3: 142-158
  3. Fitzpatrick R. (translated), Heiberg J.L. (Greek Text), Euclid (Author), 2008, Euclid's Elements of Geometry, (Richard Fitzpatrick)
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ID: 140600014 Last Updated: 2014/6/23 Revision: Ref:

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