In projective geometry, Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if six arbitrary points are chosen on a conic (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon. The theorem is valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.

Contents

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• 1 Euclidean variants
• 2 Related results
• 3 Hexagrammum Mysticum
• 4 Proofs
• 5 Proof using cubic curves
• 6 Proof using Bézout's theorem
• 7 A Property of Pascal's Hexagon
• 8 Degenerations of Pascals's theorem
• 9 See also
• 10 Notes
• 11 References
• 12 External links

Euclidean variants[edit]

The most natural setting for Pascal's theorem is in a projective plane since all lines meet and no exceptions need be made for parallel lines. However, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel, the theorem remains valid in the Euclidean plane.

If exactly one pair of opposite sides of the hexagon are parallel, then the conclusion of the theorem is that the "Pascal line" determined by the two points of intersection is parallel to the parallel sides of the hexagon. If two pairs of opposite sides are parallel, then all three pairs of opposite sides form pairs of parallel lines and there is no Pascal line in the Euclidean plane (in this case, the line at infinity of the extended Euclidean plane is the Pascal line of the hexagon).

Related results[edit]

The intersections of the extended opposite sides of hexagon ABCDEF (right) lie on the Pascal line MNP (left).

This theorem is a generalization of Pappus's (hexagon) theorem – Pappus's theorem is the special case of a degenerate conic of two lines. Pascal's theorem is the polar reciprocal and projective dual ofBrianchon's theorem. It was formulated by Blaise Pascal in a note written in 1639 when he was 16 years old and published the following year as a broadside titled "Essay povr les coniqves. Par B. P.".^[1]

A degenerate case of Pascal's Theorem (four points) is interesting; given points ABCD on a conic Γ, the intersection of alternate sides, AB ∩ CD, BC ∩ DA, together with the intersection of tangents at opposite vertices (A, C) and (B, D) are collinear in four points; the tangents being degenerate 'sides', taken at two possible positions on the 'hexagon' and the corresponding Pascal Line sharing either degenerate intersection. This can be proven independently using a property of pole-polar. If the conic is a circle, then another degenerate case tells us that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the Gergonne triangle, are collinear.

Six is the minimum number of points on a conic about which special statements can be made, as five points determine a conic.

The converse is the Braikenridge–Maclaurin theorem, named for 18th century British mathematiciansWilliam Braikenridge and Colin Maclaurin (Mills 1984), which states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line, then the six vertices of the hexagon lie on a conic; the conic may be degenerate, as in Pappus's theorem.^[2] The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a synthetic construction of the conic defined by five points, by varying the sixth point.

The theorem was generalized by Möbius in 1847, as follows: suppose a polygon with 4n + 2 sides is inscribed in a conic section, and opposite pairs of sides are extended until they meet in 2n + 1 points. Then if 2n of those points lie on a common line, the last point will be on that line, too.

Hexagrammum Mysticum[edit]

If six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This configuration of 60 lines is called the Hexagrammum Mysticum.^[3]

As Thomas Kirkman proved in 1849, these 60 lines can be associated with 60 points in such a way that each point is on three lines and each line contains three points. The 60 points formed in this way are now known as the Kirkman points.^[4] The pascal lines also pass, three at a time, through 20 Steiner points. There are 20 Cayley lines which consist of a Steiner point and three Kirkman points. The Steiner points also lie, four at a time, on 15 Plücker lines. Furthermore, the 20 Cayley lines pass four at a time through 15 points known as the Salmon points.^[5]

Jump in!

Jump in!

I want to begin with a confession.  I love projective geometry!  I've studied it for over 30 years and I still can't get enough.  I'm convinced that this modern geometry (discovered in the Renaissance and re-discovered in the Romantic era),  has much to offer we humans as we evolve to higher levels of understanding of ourselves and the world we live in.  I'll begin by introducing the subject in a way which may not seem like mathematics (no equations, no variables, no algebra) to some readers.   Once I've established the key ideas I'll turn to some themes which hopefully will help the reader understand my enthusiasm for the subject, by connecting it to larger issues in science, society, and human development.

There are in fact some very good web sites devoted to projective geometry and its potential significance for the human future.  For example, Nick Thomas's projective geometry site is one such. It gives an overview of projective geometry and how it has begun to be applied to scientific research, using abundant illustrations and non-technical language.

This blog will represent my particular perspective on projective geometry. For example, one of my special interests is creating interactive software for all kinds of geometry.  I'd like to use this blog to make available interactive software which I've written over the years for exploring themes in projective geometry.  I'd also like to present in understandable form some ideas which form part of my Ph. D. thesis (TU-Berlin, 2011).

For this beginning post, I'd like to close with a couple of examples which give a flavor of the kind of phenomena one meets in projective geometry.

One of the fundamental theorems of projective geometry is Desargues Theorem, which concerns the relationship of two triangles.  It states that if the joining lines of corresponding vertices of the two triangles meet in a point, then the intersection points of corresponding sides (considered as infinite lines!) lie on a line.  And vice-versa!  This interactive applet allows you to play around with this theorem.   Pay especial attention to what happens as pairs of lines become parallel.  In projective geometry such pairs still have an intersection point, allowing the fluid motion to continue undisturbed.


A second famous theorem of projective geometry is Pascal's Theorem. It begins with 6 points A, B, C, D, E, and F on a conic section.  Consider the six (infinite!) joining lines of adjacent points  AB, BC, etc.  These six lines are arranged in pairs of opposite lines, for example, AB and DE,  BC and EF, and CD and FA.  Then the theorem asserts that the intersection points of these three pairs of lines lie on a line.  This interactive application allows you to explore this configuration. 

Note: in this figure point B has a distinguished role: it cannot be moved by the user.  In fact, the five other points determine a conic section, and B is constructed from these five points using Pascal's Theorem.   Also, by moving the other points one obtains a wide variety of conic sections, including ellipses and hyperbolas, but also parabolas, even a pair of straight lines can be obtained.

Before proceeding:  please play with these apps!  If they don't work, let me know (cgunn3@gmail.com).  Hands-on experience is invaluable in developing a relationship to this geometry. 

With a little experience, I think you'll agree that both of these theorems are "different" from the geometry you learned in school. In fact, they illustrate a fundamental quality of projective geometry: the geometric phenomena are much more dynamic and flexible than in ordinary "school" geometry.   We can simply note how many different configurations one can arrive at by moving one or the other of the special points of the configurations. Later perhaps we can consider why that is. 

This quality of projective geometry is related to its genesis in the birth of perspective painting in 15th century Italy.  The human being at this time learned to see the world in a new way, and projective geometry in this sense is the mathematics of this seeing.  "School" geometry,  more accurately known as euclidean geometry for its great expositor Euclid, can be thought of as the mathematics of touch.   Many of the paradoxes and peculiarities of projective geometry can be grasped in terms of this tension between these two fundamental human senses.  And the relative "strangeness" of projective geometry can be understood as an expression of its relative youth in comparison to euclidean geometry, rather than any intrinsic deficiency. 

Perfect partnerships

The previous post led to the creation of projective geometry by extending "normal" geometry by anideal plane filled with ideal points and lines.  In this post I want to explore the consequences of this extension in one particular direction, revealing a startling symmetry permeating projective geometry.

SIMPLE CHAOS THEORY

Simple Chaos Theory

Can chaos be explained in a very fundamental way, without resorting to Hamiltonians and phase space, to give an intuitive feel for what is going on? This is attempted here.

Chaos theory is to be found in many places from the giant red spot on Jupiter to dripping taps, and in the biological realm in heart fibrillation and brain seizures. Feigenbaum discovered a way of describing it, although he was not the first to discover chaos, it being known to Einstein, and even before him in the 19th Century from the study of dynamical systems where phase-space orbitals could cease to be well defined. It was largely ignored until the meteorologist Lorentz found that his simple model of the atmosphere did not give repeatable results. The advent of the PC with sufficient power to implement chaotic systems finally opened up the subject to wide research and application, although we might recall that Feigenbaum used a simple calculator to make his initial discovery! The actual existence of chaos as a fundamental fact rather than a mere appearance arising from inadequate precision in the calculations interested the engineer writing this. In other words he was sceptical: was it just 'hype'? What is actually happening is not easy to grasp from the advanced maths used. Below we show the classic figure for the equationy=rx(1-x) when handled recursively i.e. the calculated value of y is re-inserted as x in the equation, and so on. The value of r is increased from 1 to 4 along the x-axis.

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, in a given dimension, and that geometric transformations are permitted that move the extra points (called "points at infinity") to traditional points, and vice versa.

Properties meaningful in projective geometry are respected by this new idea of transformation, which is more radical in its effects than expressible by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometric language is adequate to the novel situation? It is not possible to talk about angles in projective geometry as it is inEuclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly inperspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See projective plane for the basics of projective geometry in two dimensions.

While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century. A huge body of research made it the most representative field of geometry of that time. This was the theory of complex projective space, since the coordinates used (homogeneous coordinates) were complex numbers. Several major strands of more abstract mathematics (including invariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programme leading to the study of theclassical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of synthetic geometry. Another field that emerged from axiomatic studies of projective geometry is finite geometry.

The field of projective geometry is itself now divided into many research subfields, two examples of which are projective algebraic geometry (the study of projective varieties) andprojective differential geometry (the study of differential invariants of the projective transformations).