In mathematics, Pappus's hexagon theorem (attributed to
Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
) states that
*given one set of
collinear
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 (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
points
and another set of collinear points
then the intersection points
of
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
pairs
and
and
and
are
collinear
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 (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
, lying on the ''Pappus line''. These three points are the points of intersection of the "opposite" sides of the hexagon
.
It holds in a
projective plane over any field, but fails for projective planes over any noncommutative
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
. Projective planes in which the "theorem" is valid are called pappian planes.
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.
The
dual of this
incidence theorem states that given one set of
concurrent lines
In geometry, lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. They are in contrast to parallel lines.
Examples
Triangles
In a triangle, four basic types of sets of concurrent lines ar ...
, 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.)
Pappus's theorem is a
special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case ...
of
Pascal's theorem for a conic—the
limiting case when the conic
degenerates
Degenerates is a musical group which originated in Grosse Pointe Park, Michigan in 1979, during the formative years of the Detroit hardcore scene. The group predated the Process of Elimination EP, which some reviewers view as the beginning of the ...
into 2 straight lines. Pascal's theorem is in turn a special case of the
Cayley–Bacharach theorem.
The
Pappus configuration
In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point.
History and construction
This configuration is named after Pappus of A ...
is the
configuration
Configuration or configurations may refer to:
Computing
* Computer configuration or system configuration
* Configuration file, a software file used to configure the initial settings for a computer program
* Configurator, also known as choice bo ...
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
. 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
In the mathematical field of graph theory, the Pappus graph is a bipartite 3- regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek ...
, a
bipartite
Bipartite may refer to:
* 2 (number)
* Bipartite (theology), a philosophical term describing the human duality of body and soul
* Bipartite graph, in mathematics, a graph in which the vertices are partitioned into two sets and every edge has an en ...
distance-regular graph with 18 vertices and 27 edges.
Proof: affine form
If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as 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.
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
A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand.
Pencils create marks by physical abrasion, leaving a tra ...
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 Pappus' theorem.
Pappus-dual-proj-ev.svg, dual theorem: projective form
Pappus-dual-aff-ev.svg, dual theorem: affine form
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. 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 point.
Thomsen-kl-d-pap-ev.svg, ''Thomsen figure'' (points of the triangle ) as dual theorem of the little theorem of Pappus ( is at infinity, too !).
Thomsen-beweis.svg, Thomsen figure: proof
Other statements of the theorem
In addition to the above characterizations of Pappus's theorem and its dual, the following are equivalent statements:
* If the six vertices of a hexagon lie alternately on two lines, then the three points of intersection of pairs of opposite sides are collinear.
* Arranged in a matrix of nine points (as in the figure and description above) and thought of as evaluating a
permanent, if the first two rows and the six "diagonal" triads are collinear, then the third row is collinear.
::
: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.
[Coxeter, p. 233]
* Given three distinct points on each of two distinct lines, pair each point on one of the lines with one from the other line, then the joins of points not paired will meet in (opposite) pairs at points along a line.
* If two triangles are
perspective in at least two different ways, then they are perspective in three ways.
* If
and
are concurrent and
and
are concurrent, then
and
are concurrent.
Origins
In its earliest known form, Pappus's Theorem is Propositions 138, 139, 141, and 143 of Book VII of
Pappus's ''Collection''. 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
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
'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).
:
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:
[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.'']
: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
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...
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.
:
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:
:
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:
:
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
References
*
*
*
*
*
*
*
*
*{{Citation , last1=Whicher , first1=Olive , title=Projective Geometry , publisher=Rudolph Steiner Press , year=1971 , isbn=0-85440-245-4
External links
Pappus's hexagon theoremat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Dual to Pappus's hexagon theoremat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Pappus’s Theorem: Nine proofs and three variations Theorems in projective geometry
Euclidean plane geometry
Articles containing proofs