John, Adam, and Steve's excellent adventure: The Hexagrammum Mysticum

The John here is John Conway, who is already famous. The Adam is Adam Marcus, who will be famous. The Steve is me, who will never be famous. The adventure is our research into Pascal's Hexagramum Mysticum.

Fun, fun, fun. But then, Geometry is fun.

Note: most pictures in this document were generated in Mathematica and finished in Illustrator. The Mysticum and Salmon cube pictures were generated in Geometer's Sketchpad and finished in Illustrator. Some of these pictures are excerpted from pictures in the forthcoming The Triangle Book by J.H. Conway and S. Sigur.

This is an expanded version of the original web pages. It includes new material on the S5, S6, and S7 symmetries of the Hexagrammum Mysticum. This material is more difficult but is well worth the effort to understand. It makes the properties of the whole figure easy to understand, probably for the first time. If permutation symmetries are not to your taste, then you can skip these and focus on the famous points and lines.

Pascal Lines

A long time ago and when he was very young Pascal showed that, if six points are chosen on a conic, the three pairs of opposite edges meet in collinear points. This line has been called the Pascal line ever since.

Opposite edges are extended, meeting in three points on one Pascal line. In the picture one of the edge meets is off the page.

But the hexagon vertices can be connected in a different order, producing a different hexagon. Each hexagon can be considered a permutation of its six vertices, which gives the order vertices are connected. Each Pascal line can be indexed by this permutation. The group structure of these permutations give relations among corresponding Pascal lines.

This picture shows hexagons made from permutations of the same six points. Note that in the third and fourth images different permutations give the same Pascal line.The red number tells the order in which the points are connected.

Equivalent permutations

Early on it was noticed that there are many hexagons that can be created from the same points. Most but not all have a different Pascal line. In fact here are 12 equivalent permutations for each hexagon, six because cyclic permutations give the same hexagon doubled by the fact that forwards and backwards permutations give the same hexagon. Hence there are 6!/12 = 60 different hexagons and 60 different Pascal lines.

The Pascal structure

The six points are joined by lines, called edges (there are 6 choose 2 = 15 of these), which meet at edge meets (45 of these). The edge meets form, 3 at a time, 60 Pascal lines. The Pascal lines form, 3 at a time, 20 Steiner points, which, 4 at a time, lie on 15 Plucker lines.

There is also a reciprocal structure built from points, called Kirkman points, which is maximally related to the Pascal configurations.

There is a dual structure built by dualizing the Pascal lines in the conic. These points are called Brianchon points and this configuration is as elaborate as the Pascal one but only incident with a small part of it.

This web-page is devoted to all of these.

The 15 Pascal edges.

Brianchon and duality

A Brianchon point is the dual in the conic of a Pascal line. Pascal lines, from an inscribed hexagon, with their incident points are mirrored by an equivalent Structure of Brianchon points, from a circumscribed hexagon, and their incident lines. With a small exception the Brianchon and Pascal configurations are independent. The whole of the Pascal structure was named by Pascal the Hexagrammum Mysticum. John Conway has named its dual as the the Hexagrammum Reciprocum. The following picture shows the relation of the two. Note that the two structures are independent of eachother with the sole exception that each Steiner point is on the dual in the conic of another Steiner point. This picture gives too much information at first read, but should be consulted frequently.

Figure: Incidence relations in the Mysticum and Reciprocum. Each vertical or horizonal line indicates that the indicated point and line are incident. Capital P and L designate Points and Lines. The red and blue numbers tell how many of each object there are. The black numbers tell how many of one are incident with the other. For example there are 4 Pascal lines going through each edge-meet and that there are 5 edge-meets on each Pascal line. This picture was assembled by John Conway from the information in the appendix of Salmon's Conics.

Disjoint hexagons, Kirkman points, reciprocal structure, and Desargues configurations:

Hexagons are disjoint if they share no side in common. Three hexagons are disjoint to a given hexagon. For example 135264, 351426, and 136425 are disjoint to 123456. The Pascal lines determined by disjoint hexagons concur at a Kirkman point. Hence each Kirkman point can be indexed by the given hexagon.

Kirkman points thus correspond 1-1 to Pascal lines. Just as 3 Pascal lines determine a Kirkman point, so 3 Kirkman points lie on a Pascal line.The three Kirkmen points on a Pascal line are the ones whose permutations are disjoint to that of the Pascal line. This is an example of what Hesse called "a certain reciprocity."

Figure: The Pascal lines from 3 hexagons disjoint to 123456 concur at Kirkman point 123456.

The disjointness relation forms a closed group of 10 permutations that give a Desargues configuration of 10 Pascal lines and 10 Kirkman point, 3 points per line and 3 lines per point.

Figure: 10 Pascal lines and 10 Kirkman points form a Desargues configuration of a particular color, red in this case.

Veronese discovered that the Pascal lines divide into 6 groups of 10 lines, each forming a Desargues configuation with their 10 corresponding Kirkman points. We give each of the 6 Desargues configurations a color. John Conway has discovered that this color symmetry is the secret to understanding the Hexagrammum Mysticum, the full figure of 60 Pascal lines, 60 Kirkman points, 20 Steiner points, 20 Cayley lines, 15 Plucker lines, and 15 Salmon points.

Indexing a Desargues configuration, Disjointness, and S₅.

Each of the 10 points of a Desargues configuration can be indexed by P_ij ( = P_ji) where i and j are distinct digits from 1 to 5. Any permutation of these digits yeilds a symmetry of the configuration, so these permutaions form S₅. Similarly L_ij indexes the 10 lines so that the reciprocity between the Kirkment and Pascal lines implies the symmetry of the whole thing is S₅ x C₂. With this notation P_ij lies on L_kl iff ij is disjoint from kl, meaning that no digits are in common. For a given kl, there are three choices of ij that are disjoint so that three points will lie on the given line. The following picture shows a possible indexing for a Desargues configuration.

This is the first example of a valuable lesson: Disjointness in the group implies geometric incidence.

The Geometry of a Desargues Configuration

Each point contains three incident lines and has six neighbors on these lines. The three points that are not neighbors are colinear and form the line with the same index as the point. Of the six neighbors, it is always possible to form exactly two triangles from these points using lines of the configuration. These two triangles are in perspective at the point with the line as perspectrix.

In the above picture triangles P₁₃P₁₄P₁₅ (edges L₂₃L₂₄L₂₅) and P₂₃P₂₄P₂₅ (edges L₁₃L₁₄L₁₅) are perspective at P₁₂ (containing L₃₄, L₃₅, and L₄₅) with perspectrix L₁₂ (containing P₃₄, P₃₅, and P₄₅).

Each point can be the perspector so there are 10 versions in a Desargues configuration.

Disjointness closes

The three points disjoint from a given point are collinear and are on the perspectrix for that point. The disjointness relation between points form the Peterson graph below.

These points and this graph are symmetric under S₅. The complete Desargues configuration includes two copies of the Petersen graph, one for the points and one for the lines. (I picture one above the other). This would be symmetric under S₅ x C₂.

The Color Symmetry

Each Pascal line can be indexed as a permutation of the six vertices of the Pascal hexagon. There are three disjoint permutaions, described above.These permutations also form a Petersen graph, indicating that the Pascal lines and Kirkman points from these hexagons form a Desargues configuration as Veronese discovered.

These 10 permutations form one of the 6 Desargues configurations discovered by Veronese. We assign a color to each. Each Pascal line and each Kirkman point thus has a particular color, indicating which of the 6 Desargues configurations they reside.

There were 6 hexagon vertices and are 6 colors. Any part of the Hexagrammum Mysticum can be described in terms of either description. Color will turn out to be a powerful way to determine what points are on which lines.

The color classifications are greatly aided by this chart compiled by Adam Marcus, a graduate student at the Georgia Institute of Technology. In his chart hexagon permutations are letters and colors are numbers.

The Color indexing of a Pascal line or a Kirkman point

We choose the colors Red, Yellow, Green, Cyan, Blue, Magenta and use their first letters as idices. So to give color indices to a Pascal line or a Kirkman point we first begin with the color of the Desargus group in which it resides, R, say. Now 5 colors are left. We know that each member of a Desargues configuration is indexed by two entries of 5, so each Pascal line or Kirkman point has a color index such as R (YG), R indicating the color of the Desargues configuration and YG indexing the particular member.

The colors left over, CBM in this case, turn out to be the color index of the Steiner point incident with the Pascal line, or, reciprocally, the Cayley line incident with the Kirkman point. Details on this below.

The Hexagramum Mysticum

This picture shows all 60 Pascal lines and most of the 60 Kirkman points. Each point and line is colored by the Desargues configuration in which it resides. The colors are Red, Yellow (Brown), Green, Cyan, Blue, Magenta. We can see much of the color structure in this picture. Three lines of the same color meet at a Kirkman point. Three Pascal lines of different colors meet at a Steiner point. Four of different color meet at an edge-meet.

The Hexagrammum showing all 60 Pascal lines and all 60 Kirkman points. Initially this looks like a gibberish of lines, but each Pascal line and Kirkman point has a color.  By focusing on the colors we make sense of this Hexagrammum Mysticum. The dotted lines are the 15 Pascal edges.

How to look at this picture. Choose a color, blue say. There are 3 blue Pascal lines through each blue Kirkman point and 3 Kirkman points (possibly offscreen) on each line. The colors interact. Pascal lines of three different colors concur at a Steiner point so try to find some of them. Pascal lines of four different colors concur at an edge intersection. These last are the easiest to see. Harder to see are Cayley lines, each through a Kirkman point of a different color.

Another Mysticum based on the Jerabek hyperbola for of triangle geometry. Here H and K are the orthocenter and the symmedian point of ABC.

Steiner points

Three Pascal lines, each of a different color, concur at a Steiner point.
The hexagons, considered as a permutation of the six vertices, that correspond to these three Pascal lines are those whose squares are dquivalent permutations. For example the hexagons described by permutations (123456), (325416),  and (521436) square to (135)(246). The Pascal lines described by these hexagons will concur at a Steiner point, which can thus be indexed by (135)(246).

Figure: Three Pascal lines from hexagons whose permutations have the same square concur at a Steiner point. The squared permutation indexes the Steiner point.

There is 1 Steiner point per Pascal line and 3 Pascal lines, each of a different color, per Steiner point. The Steiner point is traditionally indexed in color space by the color of the 3 lines not through it.

Each Steiner point is the harmonic conjugate of one other in the conic. They are indexed as (135)(246) and (135)(264). The conjugate is made from the 3 colors not used by the first Steiner point although these web pages are not consistent in this as the next pictures illustrates. The various color relationships are shown in this chart.

Here is the Mysticum including its Steiner points. The colors here are red, yellow (brown substituted), green, cyan, blue, magenta. RMY means that the Steiner point has Pascal lines of colors red, magenta, and yellow (brown) going through it. It is more common and slightly better to use the three colors not used to index the point.

Plucker lines

There are 4 Steiner points on a Plucker line and 3 Plucker lines per Steiner point. There are 15 Plucker lines.

Figure: Three Plucker lines go through a Steiner point and there are 4 Steiner points on a Plucker line (shown for only 1 of the Plucker lines).

Plucker lines result from Steiner points, which result from Pascal lines. They are 3rd order constructions from a given hexagon. They are in fact indexed by the cube of the permutation of the original hexagon. The cube of (123456) is (14)(25)(36). There are 4 hexagons with this permutation as cube, so there are 4 Steiner points on each Plucker line. There are 3 Plucker lines through each Steiner point so that there are 15 Plucker lines total. The following picture shows all 15 Plucker lines, indexed by two colors, with the Steiner points, indexed by 3 colors.

A Plucker line is indexed by 2 colors, leaving 4 colors, which can be grouped into 4 groups of 3. A group of three represents a Steiner point.

This Figure shows all 15 Plucker lines for the given conic and 16 of the 20 Steiner points. Each Plucker line is indexed by two colors (from Red, Yellow, Green, Cyan, Blue, Magenta). The 4 Steiner points on each Plucker line are indexed by three colors from the 4 colors not indexing the Plucker line.

There is additional structure in the Steiner points and Plucker lines. The 10 Steiner points that contain a given color lie four at a time on 5 Plucker lines with each Plucker going through 2 points. The 10 Steiner point that do not contain a given color combine with the other 10 Plucker lines to form a Desargues configuration. There are 42 such Desargues configurations.

This figure shows the 10 Plucker lines that come from cubing the 10 permutations of a Desargues configuration. The 10 points are the Steiner mates of those that come from the same Desargues configuration.

This figure shows the other 10 Steiner points, which lie, 4 at a time, on the remaining 5 Plucker lines, which meet the Steiner points two at a time. Each Steiner point in this picture is determined by the square of one of the 10 Desargues permutations.

Permutations

I use six different groups of Permutations in creating these figures. Each is a set of permutations that is intended to operate on a given hexagon, generating new hexagons with the stated properties.

The 10 permutations in the Desargues configuration. Applied to a hexagon, this will generate a set of 10 hexagons whose Pascal lines and Kirkman points form a Desargues configuration.

The three permutations disjoint from a given permutation.Applied to a given permutations they will generate the permutation of the Kirkman points for the given Pascal line, or the three Pascal lines for the given Kirkman point.

The 3 permutations that square to the same permutation. Used to generate the three Pascal lines through the given Steiner point or the three Kirman points on a given Cayley line as well as the three Plucker line through a given Seiner point or the 3 Salmon points on a Cayley line.

The 4 permutations that cube to a given permutation.Used to find the 4 Steiner points on a Plucker line, or the 4 Cayley lines through a Salmon point.

The permutation that gives the "mate" of the given one.

The six permutations that represent the six colors. Applied to a given permutation, this will generate 6 permutations one of each color. To generate the Hexagamum Mysticum apply this to any hexagon. Apply the Desargues' permutations above to each of the 6, this will generate 60 non-equivalent hexagons.

"A Certain Reciprocity:" Cayley Lines

Just as three Pascal lines, each of a different color, concur at a Seiner point, three Kirkman points, each of a different color, determine a Cayley line. There are 60 Pascal lines and 60 Kirkman points. There are 20 Steiner points and so 20 Cayley lines. This relationship has often been referred to as "dual" but this word is inadequate to the behavior of these points and lines. Hesse termed it "a certain reciprocity."

Each Cayley line is described by a permutation of the form (135)(246) or by 3 colors. The three hexagons that square to this determine the three Kirkman points on the Cayley line. There is one Steiner point on each Cayley line, which is the one with different colors.

Figure: "A certain reciprocity." Three Pascal lines go through a Steiner points and three Kirkman points lie on a Cayley line.

Salmon points

Just as Pascal lines determine Steiner points which determine Plucker lines, there is the reciprocal structure where Kirkmen points determine Cayley lines which determine Salmon points. There are four Cayley lines through a Salmon point and three Salmon points on a Cayley line. Salmon points are determined by 2 colors, so that there are 6 choose 2 = 15 of them.

This figure shows three black Kirkman points, which determine a Cayley line. Four Cayley lines concur at a Salmon point. There are three Salmon points on a Cayley line (shown for one of the lines). This has a "certain reciprocity" to the fact that there are four Steiner points on a Plucker line and three plucker lines through a Salmon point.

The projective cube "around" a Salmon point

Figure: The central Salmon point is surrounded by 4 Kirkman (blue) and 4 Steiner points (red), connected by Cayley lines. The 12 edges of the cube are Pascal lines (blue and continued in dotted red) which meet, 4 at a ime, in the meeting points of two edges. The Salmon point is described by the permutation (14)(25)(36) and the three meet points are the edges 14, 25, 36. The notations for each point are their color designations, with numbers in place of colors.

The next pictue illustrates the color relations of the points. The Kirkman points and Pascal lines are colored, 4 different ones; the Steiner points and Cayley lines have 3 colors. The Salmon points and the edge meets share the same two colors.

This projective cube was discovered by Adam Marcus, currently a graduate student at the Georgia Institute of Technology.

Created by Mathematica  (August 9, 2005)