Pappus's hexagon theorem

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Pappus's hexagon theorem: Points X, Y and Z are collinear on the Pappus line. The hexagon is AbCaBc. Pappus-proj-ev.svg
Pappus's hexagon theorem: Points X, Y and Z are collinear on the Pappus line. The hexagon is AbCaBc.
Pappus's theorem: affine form

A
b
[?]
a
B
,
B
c
[?]
b
C
=
A
c
[?]
a
C
{\displaystyle Ab\parallel aB,Bc\parallel bC\Rightarrow Ac\parallel aC} Pappus-aff-ev.svg
Pappus's theorem: affine form

In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that

Pappus of Alexandria Greek mathematician of Antiquity

Pappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection, and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than, that he had a son named Hermodorus, and was a teacher in Alexandria.

Contents

In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear. In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".

It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. [1] Projective planes in which the "theorem" is valid are called pappian planes.

Projective plane Geometric concept of a 2D space with a "point at infinity" adjoined

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect in one and only one point.

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring under the looser definition where noncommutative ring refers to rings which are not necessarily commutative.

If one restricts the projective plane such that the Pappus line is the line at infinity, one gets the affine version of Pappus's theorem shown in the second diagram.

If the Pappus line and the lines have a point in common, one gets the so called little version of Pappus's theorem [2] .

The dual of this incidence theorem states that given one set of concurrent lines , and another set of concurrent lines , then the lines defined by pairs of points resulting from pairs of intersections and and and are concurrent. (Concurrent means that the lines pass through one point.)

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.

In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects A and B. The "theorem" states that, whenever a set of objects satisfies the incidences, then the objects A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.

Concurrent lines

In geometry, three or more lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point.

Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines. Pascal's theorem is in turn a special case of the Cayley–Bacharach theorem.

In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B but not vice versa, or equivalently, if B is a generalization of A. A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed.

Pascals theorem theorem

In projective geometry, Pascal's theorem states that if six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal.

In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example:

The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of intersection of and . [3] This configuration is self dual. Since, in particular, the lines have the properties of the lines of the dual theorem, and collinearity of is equivalent to concurrence of , the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.

Proof: affine form

Pappus theorem: proof Pappus-proof-ev.svg
Pappus theorem: proof

The projective form of Pappus's theorem is proven, if the affine form of the statement can be proven (the extension of a pappian plane to a projective plane is unique !).

Because of the parallelity in an affine plane one has to distinct two cases: and . The key for a simple proof is the possibility for introducing a "suitable" coordinate system:

Case 1: The lines intersect at point .
In this case coordinates are introduced, such that (see diagram). have the coordinates .

From the parallelity of the lines one gets and the parallelity of the lines yields . Hence line has slope and is parallel line .

Case 2: (little theorem).
In this case the coordinates are chosen such that . From the parallelity of and one gets and , respectively, and at least the parallelity .

Proof with homogeneous coordinates

Choose homogeneous coordinates with

.

On the lines , given by , take the points to be

for some . The three lines are , so they pass through the same point if and only if . The condition for the three lines and with equations to pass through the same point is . So this last set of three lines is concurrent if all the other eight sets are because multiplication is commutative, so . Equivalently, are collinear.

The proof above also shows that for Pappus's theorem to hold for a projective space over a division ring it is both sufficient and necessary that the division ring is a (commutative) field. German mathematician Gerhard Hessenberg proved that Pappus's theorem implies Desargues's theorem. [4] [5] In general, Pappus's theorem holds for some projective plane if and only if it is a projective plane over a commutative field. The projective planes in which Pappus's theorem does not hold are Desarguesian projective planes over noncommutative division rings, and non-Desarguesian planes.

The proof is invalid if happen to be collinear. In that case an alternative proof can be provided, for example, using a different projective reference.

Dual theorem

Because of the principle of duality for projective planes the dual theorem of Pappus is true:

If 6 lines are chosen alternately from two pencils with centers , the lines

are concurrent, that means: they have a point in common.
The left diagram shows the projective version, the right one an affine version, where the points are points at infinity. If point is on the line than one gets the "dual little theorem" of Ppappus' theorem.

If in the affine version of the dual "little theorem" point is a point at infinity too, one gets Thomsen's theorem, a statement on 6 points on the sides of a triangle (see diagram). The Thomsen figure plays an essential role coordinatising an axiomatic defined projective plane [6] . The proof of the closure of Thomsen's figure is covered by the proof for the "little theorem", given above. But there exists a simple direct proof, too:

Because the statement of Thomsen's theorem (the closure of the figure) uses only the terms connect, intersect and parallel, the statement is affinely invariant, and one can introduce coordinates such that (see right diagram). The starting point of the sequence of chords is One easily verifies the coordinates of the points given in the diagram, which shows: the last point coincides with the first popint.

Other statements of the theorem

Triangles
X
c
C
{\displaystyle XcC}
and
B
b
Y
{\displaystyle BbY}
are perspective from
A
{\displaystyle A}
and
a
{\displaystyle a}
, and so, also from
Z
{\displaystyle Z}
. Pappus hexagon.svg
Triangles and are perspective from and , and so, also from .

In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements:

That is, if are lines, then Pappus's theorem states that must be a line. Also, note that the same matrix formulation applies to the dual form of the theorem when etc. are triples of concurrent lines. [8]

Origins

In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, and 143 of Book VII of Pappus's Collection. [10] These are Lemmas XII, XIII, XV, and XVII in the part of Book VII consisting of lemmas to the first of the three books of Euclid's Porisms.

The lemmas are proved in terms of what today is known as the cross ratio of four collinear points. Three earlier lemmas are used. The first of these, Lemma III, has the diagram below (which uses Pappus's lettering, with G for Γ, D for Δ, J for Θ, and L for Λ).

Pappus-collection-7-129.svg

Here three concurrent straight lines, AB, AG, and AD, are crossed by two lines, JB and JE, which concur at J. Also KL is drawn parallel to AZ. Then

KJ : JL :: (KJ : AG & AG : JL) :: (JD : GD & BG : JB).

These proportions might be written today as equations: [11]

KJ/JL = (KJ/AG)(AG/JL) = (JD/GD)(BG/JB).

The last compound ratio (namely JD : GD & BG : JB) is what is known today as the cross ratio of the collinear points J, G, D, and B in that order; it is denoted today by (J, G; D, B). So we have shown that this is independent of the choice of the particular straight line JD that crosses the three straight lines that concur at A. In particular

(J, G; D, B) = (J, Z; H, E).

It does not matter on which side of A the straight line JE falls. In particular, the situation may be as in the next diagram, which is the diagram for Lemma X.

Pappus-collection-7-136.svg

Just as before, we have (J, G; D, B) = (J, Z; H, E). Pappus does not explicitly prove this; but Lemma X is a converse, namely that if these two cross ratios are the same, and the straight lines BE and DH cross at A, then the points G, A, and Z must be collinear.

What we showed originally can be written as (J, ∞; K, L) = (J, G; D, B), with ∞ taking the place of the (nonexistent) intersection of JK and AG. Pappus shows this, in effect, in Lemma XI, whose diagram, however, has different lettering:

Pappus-collection-7-137.svg

What Pappus shows is DE.ZH : EZ.HD :: GB : BE, which we may write as

(D, Z; E, H) = (∞, B; E, G).

The diagram for Lemma XII is:

Pappus-collection-7-138.svg

The diagram for Lemma XIII is the same, but BA and DG, extended, meet at N. In any case, considering straight lines through G as cut by the three straight lines through A, (and accepting that equations of cross ratios remain valid after permutation of the entries,) we have by Lemma III or XI

(G, J; E, H) = (G, D; ∞ Z).

Considering straight lines through D as cut by the three straight lines through B, we have

(L, D; E, K) = (G, D; ∞ Z).

Thus (E, H; J, G) = (E, K; D, L), so by Lemma X, the points H, M, and K are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon ADEGBZ are collinear.

Lemmas XV and XVII are that, if the point M is determined as the intersection of HK and BG, then the points A, M, and D are collinear. That is, the points of intersection of the pairs of opposite sides of the hexagon BEKHZG are collinear.

Notes

  1. Coxeter, pp. 236–7
  2. Rolf Lingenberg: Grundlagen der Geometrie, BI-Taschenbuch, 1969, p. 93
  3. However, this does occur when and are in perspective, that is, and are concurrent.
  4. 1 2 Coxeter 1969 , p. 238
  5. According to ( Dembowski 1968 , pg. 159, footnote 1), Hessenberg's original proof Hessenberg (1905) is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by Cronheim 1953.
  6. W. Blaschke: Projektive Geometrie, Springer-Verlag, 2013, ISBN   3034869320, S. 190
  7. Coxeter, p. 231
  8. 1 2 Coxeter, p. 233
  9. Whicher, chapter 14
  10. Heath (Vol. II, p. 421) cites these propositions. The latter two can be understood as converses of the former two. Kline (p. 128) cites only Proposition 139. The numbering of the propositions is as assigned by Hultsch.
  11. A reason for using the notation above is that, for the ancient Greeks, a ratio is not a number or a geometrical object. We may think of ratio today as an equivalence class of pairs of geometrical objects. Also, equality for the Greeks is what we might today call congruence. In particular, distinct line segments may be equal. Ratios are not equal in this sense; but they may be the same.

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